Numerical Determination of Exotic Option Price

Numerical Determination of

Exotic Option Prices

Diplom Thesissupervised by the

Swiss Banking Institute (ISB)at the University of Zurich

Prof. Dr. Thorsten Hens

Author: Esad CekicStudent ID: 03-714-276

Address: Bulachhof 3/15, 8057 ZurichE-Mail: [email protected]

Abstract

In this diplom thesis different numerical approaches are examinedin terms of pricing the exotic options. It will be tested, which of thethree numerical approaches (the trees, Monte Carlo simulation andfinite difference methods) approximates the exotic option price themost accurately. The focus will lie on the down-and-out put barrieroption. Since the barrier option is very difficult to hedge, the invest-ment banks shift the barrier when they price the exotic. It will beexamined to what extent the barrier is shifted. Stated differently,in terms of the down-and-out put, it will be shown by how much thedown-and-out put is too expensive. It will be examined why and whena client will consider buying such an exotic. More generally, it willbe shown that the investor can benefit if the derivatives (plain vanillaoptions) are introduced into the portfolio. Simultaneously, it will beshown how the investment bank hedges such an exotic position, if atall, and additionally, it will be shown why an asset manager should/ should not use such a product for hedging a portfolio. Finally, itwill be examined if the structured products containing a plain vanillaoption or a barrier option respectively, are fairly priced.

1

Contents

1 Introduction 4

2 Theory 52.1 Different types of the Exotics . . . . . . . . . . . . . . . . . . 6

2.1.1 Binary Options . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Barrier Options . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Look back Options . . . . . . . . . . . . . . . . . . . . 92.1.4 Asian Options . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Different Numerical Approaches . . . . . . . . . . . . . . . . . 102.2.1 The Trees . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . 132.2.3 Finite Difference Method . . . . . . . . . . . . . . . . . 14

3 Empirical Part 153.1 Empirical Part 1: Application of the three numerical approaches 15

3.1.1 Plain vanilla option, European Call . . . . . . . . . . . 163.1.2 Convergence-Plain Vanilla Option . . . . . . . . . . . . 193.1.3 Barrier Option, Down-and-Out Put . . . . . . . . . . . 21

3.2 Empirical Part 2: How much do the IB shift the barrier? . . . 253.2.1 Brief review of the markets . . . . . . . . . . . . . . . . 253.2.2 Offer prices from the seven IBs . . . . . . . . . . . . . 263.2.3 The BS price overprice the market offer price strongly . 293.2.4 How do the IBs price a Barrier Option . . . . . . . . . 303.2.5 What does it mean when the Barrier is shifted 2.5-3%? 333.2.6 Barrier Option valuation, revisited . . . . . . . . . . . 353.2.7 Trinomial Tree, revisited . . . . . . . . . . . . . . . . . 35

4 Application Part 364.1 Application Part 1: Bonus Certificate . . . . . . . . . . . . . . 37

4.1.1 From client’s point of view . . . . . . . . . . . . . . . . 384.1.2 From IB’s point of view . . . . . . . . . . . . . . . . . 434.1.3 From Asset Manager’s point of view . . . . . . . . . . . 46

4.2 Application Part 2: Are SP fairly priced? . . . . . . . . . . . . 504.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 504.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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4.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Conclusion 58

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1 Introduction

In recent decades, the exotic options (exotics) have become constantly moreimportant in the financial sector and beyond it. Every bank uses/issuesthem actively whenever structuring is requested. For a bank, they are moreprofitable than the plain vanilla options (options) due to their special payoffwhich is slightly different compared to the options.

The Black and Scholes[12] Formula1 (BS) was a significant research break-through in the field of pricing the options, or generally spoken, in the fieldof pricing the derivatives. Since the exotics are slightly modified options interms of their payoff (in the case of the barrier options), it is important tounderstand how the options are priced before the pricing of an exotic is con-sidered. For the path-depended options, an analytical closed pricing formuladoes not exist. Barrier options are path-dependent but the path-dependenceis (very) weak, so that an analytical closed pricing formula exists. In thisdiplom thesis, the price obtained by this closed analytical formula is con-sidered correct and will be regarded as ”benchmark”, especially in the partwhere the numerical approaches are tested (Empirical Part 1).

This diplom thesis is divided in three parts: the theoretical part, the em-pirical part and the application part, where the derived results are applied.

In the first subsection of the theoretical part, different types of the exoticsare presented. In the second subsection, the different numerical approachesadequate for pricing the derivatives are reviewed.

The empirical part itself is divided in two parts. In the empirical part 1the different numerical approaches are applied for pricing a derivative andthe convergence of the corresponding method to the BS price is shown. In afirst step a plain vanilla call is priced and then a down-and-out put is priced.In the empirical part 2 it will be shown to what extent the investment banks(IB) shift the barrier when they price such an exotic. It will be shown laterwhy such a barrier bending is needed in the first place.

In the application part, the results obtained are applied. Since a privateclient does not have the opportunity to buy a barrier option exclusively, onestructured product will be presented where a barrier option is a component ofthat product. The screening for the Swiss Market will be conducted in terms

1Robert C. Merton has to be mentioned at this place since he contributed very muchfor the success of the Black and Scholes formula and financial engineering in general withhis paper[21] published parallel to the one of Black and Scholes. For the rest of the diplomthesis the BS abbreviation will be used though.

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of which stock qualifies as a ”good” candidate for the underlying. From theclient’s point of view, it will be shown when a structured product with sucha payoff structure seems to be attractive. From the IB’s point of view, it willbe shown how such a position can be approached in terms of hedging. Fromthe asset manager’s point of view, it will be shown why a barrier option couldbe used to hedge a portfolio, and why such a barrier option should not beused for this purpose. A more general study regarding the investor’s pointof view and the inclusion of the derivatives into the portfolio, will be made.In the application part 2, it will be shown if the obtained results by Wilkens,Erner and Roder[5] and Stoimenov and Wilkens[4] hold as well for the Swissmarket. Both found that the structured products are not fairly priced andthat the seller of the product is favored. Furthermore, they claim that the”fairness” decreases as the complexity and therefore the transparency of theStructured Product increases.

In the conclusion the major findings will be reviewed and the short sum-mary of the whole thesis will be given.

2 Theory

The analytical pricing formula developed by Black and Scholes might belooked at as a pricing formula for a portfolio consisting of two options: anasset-or-nothing-option and cash-or-nothing-option. Therefore the call op-tion priced by the BS formula, for instance, consists of two exotics.

Here, the analytical BS formula is briefly reviewed

c = S0N(d1) − Ke−rtN(d2)

p = Ke−rtN(−d2) − S0N(−d1)

where c and p denote the European Call and Put option respectively. S0 isthe initial asset price, K is the strike, r is the risk free rate, N(d1) is thecumulative normal distribution function and d2 = d1 − σ√

T, where σ is the

volatility and T is the maturity of the option.As already mentioned in the introduction, the exotics exhibit a very sim-

ilar structure to the plain vanilla options, but they also embody at least onefeature, which distinguish them significantly from the corresponding plainvanilla option. For instance for a barrier option, down-and-out put, the ex-otic is kind of a part of the corresponding plain vanilla option according to

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the in-out parity. (Opv = Oin+Oout, where pv stands for plain vanilla). Sincethe exotic exists only if a certain barrier is not touched (in the case of down-and-out put), the price of the exotic is less than the price of the correspondingplain vanilla option. Moreover, the exotics are nonstandard derivatives, thustraded over-the-counter. These two properties make the exotics (at leastin this case) compared to the options, less valuable (cheaper) and thereforemore attractive as a building component of a derivative product, which couldbe looked at as a portfolio. In the following subsection different types of theexotics are presented. Clearly, there are more exotics traded in the marketsthan will be presented below. As already said, the exotics are nonstandardderivatives and therefore the creativeness and innovation ability of the mar-keters on the one hand and the needs of the clients on the other hand, makesure that new types of the exotics always get created.

2.1 Different types of the Exotics

In this part different exotics are presented2.

2.1.1 Binary Options

ι) Asset-Or-Nothing-Option[26]

The exotic pays only if the derivative closes in the money (ITM) and nothingelse. The respective payoff at the expiry is for Call:

c =

{

ST , if ST > K

0, otherwise

Put:

p =

{

ST , if ST < K

0, otherwise

ιι) Cash-Or-Nothing-Option[27]

The exotic pays a fix amount M only if the exotic closes ITM. The respectivepayoff at the expiry is for

2The following homepage gives a nice overview over the exotics considered in thissubsection http://www.sitmo.com/

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Call:

c =

{

M, if ST > K

0, otherwise

Put:

p =

{

M, if ST < K

0, otherwise

2.1.2 Barrier Options

There are three different characteristics, which need focus when a barrieroption[28] is considered. The option is either call or a put, either up or downand either in or out. The characteristic up (down) indicates that the barrieris above (below) the initial stock price and the characteristic in (out) indi-cates that option starts (ceases) to exist when the certain barrier is breached.Further down the exotic type with the respective payoff formula is presented.

ι) Down-and-in Call

max(ST − K, 0) if min0≤t≤T

(St) ≤ B

ιι) Down-and-out Call

max(ST − K, 0) if min0≤t≤T

(St) ≥ B

ιιι) Up-and-in Call

max(ST − K, 0) if min0≤t≤T

(St) ≥ B

ιν) Up-and-out Call

max(ST − K, 0) if min0≤t≤T

(St) ≤ B

ν) Down-and-in Put

max(K − ST , 0) if min0≤t≤T

(St) ≤ B

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νι) Down-and-out Put

max(K − ST , 0) if min0≤t≤T

(St) ≥ B

νιι) Up-and-in Put

max(K − ST , 0) if min0≤t≤T

(St) ≥ B

νιιι) Up-and-out Put

max(K − ST , 0) if min0≤t≤T

(St) ≤ B

Similarly to the put-call-parity, there exists a parity relation for barrier op-tions, which is very important and useful for their pricing.

Opv = Oin + Oout

That means, e.g, once a down-and-out put price is obtained, the down-and-input price can be deduced. It makes sense to divide the barrier options intwo groups: the intrinsic and non-intrinsic barrier options. The non-intrinsicoptions are those, which do not have any intrinsic value when the barrier isbreached. For instance the down-and-in call is a non-intrinsic option sincethe barrier is set at a level below the strike price when the option is issued. Sowhen the option starts to exist, there is no intrinsic value in it. Contrastingly,the intrinsic options, they do have an intrinsic value when the barrier isbreached. So, for instance, the up-and-out call is an intrinsic option sincethe barrier is set at a level above the strike price when the option is issued.That indicates that the option loses all the intrinsic value once the barrieris touched. In the table below the intrinsic and non-intrinsic barrier optionstypes are summarized.

non-intrinsic intrinsic

down-and-out call up-and-out calldown-and-in call up-and-in callup-and-out put down-and-out putup-and-in put down-and-in put

Additionally, there are several variations of the barrier option type.

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1. Regarding the monitoring of the barrier option: the barrier could bemonitored on a daily basis, e.g the closing level, or on a continuousbasis. If the barrier is monitored once a day, the barrier option is moreexpensive.

2. Regarding the number of the barriers: barrier option exists with morethan just one barrier, e.g a double knock out barrier option is a bar-rier option, which knocks out if either of the barriers is breached andtherefore cheaper than the barrier option with only one barrier.

3. Regarding the possibility of an early exercise: a barrier option mighthave the feature of an early exercise, which gives the option holder theright to exercise the option at any point before the maturity. It seemsobvious that this feature allows the option to be almost as expensiveas the plain vanilla option, since the option holder is going to exercisethe option when the underlying seems to break through the barrier.

4. Regarding the rebates offer: If the barrier is breached for instance forthe down-and-out put barrier option, the option holder might still get acertain amount, which is fixed prior to the issue. This feature obviouslyagain makes the barrier option more expensive.

2.1.3 Look back Options

For the look back options the distinction between a fix and floating strikehas to be done.

ι) Fixed Strike[29]

c = max( max0≤t≤T

(St) − K, 0)

p = max(K − min0≤t≤T

(St), 0)

ιι) Floating Strike[30]

c = max(ST − min0≤t≤T

(St), 0)

p = max( max0≤t≤T

(St) − ST , 0)

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2.1.4 Asian Options

Equally also for the Asian Options, the distinction between a fix and floatingstrike has to be done. However in this case rather the average is takeninto account so that the impact from the big movements of the under-lying is reduced. The respective payoff functions are given below, whereAT = St,1+St,2+...+St,n−1+ST

n.

ι) Fixed Strike[31]

c = max(AT − K, 0)

p = max(K − AT , 0)

ιι) Floating Strike[32]

c = max(ST − AT , 0)

p = max(AT − ST , 0)

2.2 Different Numerical Approaches

In this subsection the different numerical approaches adequate for pricing aderivative are reviewed. The trees, the Monte Carlo simulation (MCS) andfinite difference method (FDM) are reviewed.

2.2.1 The Trees

ι) Binomial Trees

The binomial tree is the one of the most applied and studied approachesfor the pricing of the derivatives. This numerical approach is based on Cox,Ross and Rubinstein[13]. They assume that the price of the underlying eithermoves up (Su = S0 ∗ u) or down (Sd = S0 ∗ d) after each time step. The

size of an up and down movement is u = eσ√

∆t and d = 1

urespectively.

The corresponding (risk neutral) probabilities for up (down) movements are

pu = er∆t−du−d

(pd = 1 − pu). Therefore the tree is recombinant and symmetric

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as shown in the graph below.3

As the time step number increases, the time step size converges to zero andthe option price converges to BS option price.

ιι ) Trinomial Trees

Trinomial trees are based on Boyle[14]. The stock either moves up, down ordoes not change after each time step. Hence, the stock can take three differentstates: Su, Sd and Sm after each move and the corresponding probabilitiesare then pu, pd and pm. Here in this case, u = eσ

√3∆t, d = 1

u,m = 0 and the

corresponding probabilities are

pu =er∆t/2 − e−σ

√∆t/2

eσ√

∆t/2 − e−σ√

∆t/2

pd =eσ√

∆t/2 − er∆t/2

eσ√

∆t/2 − e−σ√

∆t/2

pm = 1 − pu − pd.

When the terms of higher order ∆t are ignored, then the probability can berewritten as

pu =

√

∆t

12σ2(r − σ2/2) +

1

6

The graph below shows such a trinomial tree for a stock with six time stepsand S0 = K = 50, where S0 is the initial stock price and K is the strikeprice.

3The graph has been taken from http://www.global-derivatives.com

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

Elapsed time

Ass

et P

rice

The convergence for the trinomial tree to the BS price is much faster thenfor the binomial tree. This will be proved in the empirical part 1.

ιιι ) Trinomial Trees (Ritchken and Kamrad)

It has already been said that plain vanilla options priced by the trinomial treeconverge much faster to the BS price than the ones priced by the binomialtree. When the barrier options are considered this convergence slows down.Since the focus in this diplom thesis lies on the barrier options, the trinomialtrees modification given by Ritchken and Kamrad [16] should be taken intoaccount, which again speeds up the convergence process. The up, downand middle movements are multiplied by a certain factor u = eλσ

√∆t, d =

e−λσ√

∆t, and m = 0 and the corresponding probabilities are then

pu = 1

2λ2 + µ√

∆t2λσ

, pd = 1

2λ2 − µ√

∆t2λσ

, pm = 1 − 1

λ2

where µ = r − 0.5σ2. If λ = 1 then the modified trinomial tree becomesclassical binomial tree and when λ =

√3 the modified trinomial tree becomes

the classical trinomial tree introduced by Boyle. λ indicates the spacingbetween the underlying movements from one time period to an other. In thegraph below Kamrad and Ritchken show in their paper how the prices of thetwo models compared to the BS model deviate.

They compare the accuracy of the two models for pricing the at-the-money (ATM) option, which is as well priced here in this diplom thesis. Thesize of the error of the trinomial tree model is in almost all cases smaller thanthe error obtained by the binomial tree model even though the binomial treemodel has twice as much time steps. The trinomial model is slightly moretime consuming and with the assumption about the number of the time

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steps the two researches wanted to account for that. This empirical studypresented below is done for the plain vanilla option.

2.2.2 Monte Carlo Simulation

The asset price process can be simulated by SDE equation, as shown in JohnC. Hull’s book [1]

∆S = µS∆t + σS∆W

or in the risk neutral world, where the expected return is replaced by riskfree interest rate

∆S = rS∆t + σS∆W

where ∆W = ǫ√

∆t, which indicates the changes of the Brownian Motion.The more time steps are integrated in the model, respectively, the smallerthe time step size is in the model, the closer the model comes to a real assetprice movements. See the graphs below for three different time step numbers,10, 30 and 365 respectively.

10 time steps

49.9

50

50.1

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50.6 30 time steps

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50.8 365 time steps

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50

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The left hand side of the equation above represents the asset changes, whichare realized after each time step. The right hand side of the same equationdefines the asset price moving process. So the first part of the right hand sideof the equation determines the drift (graph below on the left). The secondpart of the right hand side of the equation, which involves the BrownianMotion, determines the noise (graph in the middle). Both together determinehow the asset price process evolves over time (graph on the right). In thisexample the following input data has been chosen: r = 0.1, S = K = 50, σ =0.4, T = 1, timesteps = 365, ∆t = 1

365= 0.00274.

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49.4

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So far, only one asset path has been generated. Since the value of a callat the maturity is c = max(ST − K, 0), a few more asset paths need to begenerated so that the estimated call price gets close to the market call price.In the graph below 10 asset paths are generated. Later, in the empirical partvastly more asset paths are generated and compared with the BS price.

30

35

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45

50

55

60

65

70

2.2.3 Finite Difference Method

Finite Difference Method (FDM) is the last numerical pricing method, whichis going to be presented here.

The starting equation for the FDM is the Partial Differential Equation(PDE) of the BS-Formula:

∂f

∂t+ rSt

∂f

∂S+

1

2σ2S2

t

∂2f

∂S2= rf

S(t) is the underlying asset price, f(S, t) is the unknown function of thederivative depending on both the asset price and the time. The solution off(S, t) has to satisfy the PDE for every value S and t. The range for S andt needs to be defined. For that reason, the time and asset price partition aswell as the maximum asset price level need to be determined.

In the graphs below the time and asset partition are shown for values 1,10 and 100. In the graph on the left (partition corresponds to one asset pricechange) and on the right (partition corresponds to one hundred asset pricechanges), a random path is shown so that the partitioning process is madeclear.

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0

0.5

1 020

4060

80100

0

10

20

30

40

50

Asset price

Approximated call value surface

Time to maturity

Opt

ion

valu

e

0

0.5

1 020

4060

80100

0

10

20

30

40

50

Asset price

Approximated call value surface

Time to maturity

Opt

ion

valu

e

The FDM can be divided in two parts: the implicit difference method (IDM)and explicit difference method (EDM). The point (i, j) represents a point onthe grid, where i stands for ∆t and j for ∆S. It follows then that fi,j =f(i∆t, j∆S). For IDM, fi,j+1, fi,j, fi,j−1 lead to fi+1,j, where for the EDMfi,j leads to fi+1,j+1, fi+1,j, fi+1,j−1, which exactly matches the set up of thetrinomial tree. IDM is very robust but the calculation is time consuming.EDM simplifies the method, assuming that the first and second derivativesof f respective to S are the same on the grid (i,j) and (i+1,j).

3 Empirical Part

In the theory part the different exotic options and the different numerical ap-proaches have been presented. Here, in this part, the numerical approachesare applied and they will show the convergence to the BS price, which isassumed, at least here, to be correct. It will be proven that the trinomialtree converges much faster to the BS price when the plain vanilla option isconsidered. When a barrier option is considered, here the down-and-out put,it will be shown that the convergence for the trinomial tree model is betteragain, but that the convergence slows down and some modification speedsup the process again. In a first step the numerical approaches are applied toprice a plain vanilla option and then applied to price a barrier option. Forthe plain vanilla option only the call is considered, whereas for the barrieroption, the down-and-out put is considered.

3.1 Empirical Part 1: Application of the three numer-ical approaches

For the modeled option prices the following parameters have been used: S0 =50, K = 50, r = 0.1, σ = 0.4 and T = 1. Stated in words: the option is ATM,

15

time to maturity is one year, risk free rate is 10% and the volatility is 40%.As for all the graphs presented in this diplom thesis, I have used matlab4,excel, bloomberg and fincad.

3.1.1 Plain vanilla option, European Call

ι ) Binomial Tree

In the graphs below the convergence of the option price to the BS price isshown as a function of the time partition. The more time steps are included,the closer the option price is to the BS price. The graph on the left showsthe deviation from the BS price in absolute values, where the graph on theright shows the deviation in percentage of the BS price, which is assumedto be the correct price. When the time partition is more then 15, then thedeviation from BS price is smaller then 2%.

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-0.5

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-8%

-3%

2%

7%

12%

17%

ιι ) Trinomial Tree

Trinomial trees, additionally to the binomial tree, introduce a price state afterevery time step, which does not change. As already mentioned, the trinomialtree option pricing is equivalent to the explicit finite difference method shownin the previous section. Taking this additional feature, it is to be expectedthat the convergence to the BS price is much faster. The deviation from theBS price, which is less than 2% is already achieved after the time partitionhas been divided in 6 time steps. Compared to the binomial model, thedeviation less than 2% is achieved after the time partition is set to 15. But,it has to been said that the trinomial tree is slightly more time consumingto be set up and calculated.

4In the course[17] taught by Prof. Markus Leippold the application published on his un-official page has been used. The matlab application can be downloaded from the followingpage:http://leippold.googlepages.com. For few of my calculations I used this application.Rosli[20] gives in his semester thesis a good introduction to that application.

16

Here again, the left graph shows the deviation in absolute values andthe graph on the right the deviation in percent. Here the graph is moreinteresting compared to the previous one, since the trinomial tree approachseems to underestimate the call for every single time partition and thereforeconverges to the BS price from below.

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ιιι ) Monte Carlo Simulation

In this simulation below, I assumed 365 asset price changes per year sincethe year has 365 days. For this simulation, the random variable has beengenerated and then implemented in the asset price process equation. Sincethe value of the call option is f = (ST − K, 0), the value of the optionis dependent on the number of the generated asset price paths. The moreasset price paths are generated, ”the less noise” the option price exhibits.The graphs below show how the option value price changes dependent ondifferent number (500, 1500, 3000 and 5000) of asset price paths.

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9

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11

250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

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11

250 750 1250 1750 2250 2750 3250 3750 4250 4750

The simulated price after 1000 paths deviates less then 5% from the BSprice, which here is assumed to be correct. After 3000 generated asset pathsthe deviation from the BS price is less than 2%.

8.5

9

9.5

10

10.5

11

250 750 1250 1750 2250 2750 3250 3750 4250 4750

-1

-0.8

-0.6

-0.4

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0

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0.4

0.6

8.5

9

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10

10.5

11

250 750 1250 1750 2250 2750 3250 3750 4250 4750

0%

2%

4%

6%

8%

10%

17

ιιι )FDM

The last method considered here is the FDM.The graphs presented below are divided in three steps. In FDM, the time

and asset price have to be partitioned.In the first case, the time partition is varied and the asset price partition

is kept constant at 100. The maximal asset price is set at 100. The graphbelow shows that the option price evolves in the same manner as the optionprice when a trinomial tree is considered.

7.5

8

8.5

9

9.5

10

10.5

1 2 3 4 5 6 7 8 9

10

20

30

40

50

75

100

0

0.2

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0.6

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1

1.2

1.4

1.6

7.5

8

8.5

9

9.5

10

10.5

1 2 3 4 5 6 7 8 9 10 20 30 40 50 75 100

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

Once the time partition has been varied and the asset price partition hasbeen kept constant, here in the second case, the reverse case can be shown.That means that the asset price partition is varied and the time partition iskept constant at 100. The graphs below show that the option price evolvesin the same manner as when the binomial tree is considered.

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 20 30 40 50 75 100

0

0.5

1

1.5

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4

4.5

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5

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15

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1 2 3 4 5 6 7 8 9 10 20 30 40 50 75 100

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

In the third and last case, both the asset price and the time partition isvaried and this results then in a mixture model between the binomial andtrinomial tree.

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 20 30 40 50 75 100

0

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1

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5

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1 2 3 4 5 6 7 8 9 10 20 30 40 50 75 100

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%

18

3.1.2 Convergence-Plain Vanilla Option

In this subsection the convergence and the convergence speed is shown.

ι ) Binomial Tree vs. Trinomial Tree

Here the binomial and trinomial deviation are examined. In the graph belowon the left, the deviation to the BS price are shown. The green columnrepresent the binomial model, the blue column the trinomial model. It isclear that the deviation with the binomial model starts from a much higherlevel and for every single time partition stays higher.

The graph on the right, represent the deviation reduction in percentagewhen more time steps are included. Equally, for instance, is the reduction ofthe deviation with binomial model lower when a second time step is included.Therefore, for example, if two time steps are included instead of one, thebinomial price deviation reduces by 45% compared to the 47% reduction bythe trinomial model. So, the binomial model starts from a higher level andreduces the deviation with a lower rate.

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

18.00%

1 3 5 7 9

15

25

35

45

60

80

100

140

180

250

350

450

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

2 4 6 810203040507090120160200300400500

ιι ) MCS

Here the MCS is examined. In the left graph, the deviation from the BSprice already encountered in the previous section is presented again. Thedeviation after 250 asset paths is quite small. However, because a randomvariable is included here in this model, the price of the option rises and fallsthrough the whole graph. It seems that this up and down occurs all the time,but becomes smaller and smaller the more asset paths are included in themodel. On the right graph, the deviation reduction in percent is presented.How should the graph on the right be interpreted? The closer the modeledprice is to the BS price, the bigger is the relative deviation when a relativebig price change occurs when one more asset price path is included in pricingof the option. Thus, i.e when 3755 asset price paths are included the option

19

deviation from the BS price is slightly above 1%. When one more path isincluded, the price deviates from the BS price more than 3%, which resultsin an increase for more than 300%. Therefore the changes above 100% havebeen truncated.

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

250

500

750

1000

1250

1500

1750

2000

2250

2500

2750

3000

3250

3500

3750

4000

4250

4500

4750

5000 0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

251 751 1251 1751 2251 2751 3251 3751 4251 4751

Generally speaking, since a random variable is included in this model and nodeviation reduction pattern exists, it is difficult to make a clear statementabout the convergence and convergence speed. It can been said that the moreasset price paths are included, the less ”noise” the option price exhibits as afunction of the asset paths.

ιιι )FDM

Here as well, the FDM is examined. On the left graph below, all the threedeviations are shown. The green one shows the deviation when the assetpartition is set equal to 100 and the time partition is varied. The blue oneshows the deviations when the time partition is set equal to 100 and the assetprice partition is varied. And the purple one shows the deviation when thetime and asset price partition are changed at the same time. It is obviousthat the one with the time variation has the lowest deviation from the BSprice. (That is actually the explicit difference method, or the trinomial tree.)

The graph on the right shows the deviation reductions for all of them.Here as well, the same ones seem to be better than when the asset price isvaried. When both are varied, the result obtained is not better that the one,where only the time is varied, which is intuitively clear.

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

1 2 3 4 5 6 7 8 9 10 20 30 40 50 75 100

0%

50%

100%

150%

200%

250%

300%

350%

3 4 5 6 7 8 9 10 20 30 40 50 75 100

20

Generally, it can be said, that the best result is obtained when the assetpartition is set high and the time is varied. Again, this method is EDM andis equal to the trinomial tree and will not be considered for the rest of thediplom thesis.

3.1.3 Barrier Option, Down-and-Out Put

In this part the numerical approaches, except from the FDM, will be testedfor a barrier option. Since there are 8 different types of the barrier option,one is going to be picked out. I chose one with an intrinsic value when thebarrier is hit, since they exhibit discontinuous payoff and those are mostlyused as a component of a structured derivative product. Because of the bar-rier option parity (in-out), the down-and-in put can easily been derived sothat 2 out of 8 are covered here in this diplom thesis.

A down-and-out put means that the option only exists if the underlyingdoes not breach the barrier until the maturity, or stated differently, the op-tion ceases to exist if the barrier gets touch until the maturity. In the graphbelow it becomes clear, that the option price increases as the underlying de-creases, analogues to the plain vanilla put option, but as the underlying getscloser to the barrier the option value decreases. When the barrier is touchedthe option value becomes 0 and the option ceases to exist.

21

30 40 50 60 70 800

5

10

15

Asset price

Val

ue d

own−

and−

out p

ut

t =0

t =0.5

t =1

Value down−and−out put2.1534

Current asset price = 50

Strike price = 50

Barrier = 30

Black Scholes value function

Current Black Scholes value = 5.4011

Pricing the barrier option with a tree causes some difficulties. In the fol-lowing example below, it will become clear what the problem is. The graphon the left shows a binomial tree with only one time step and a barrier, whichdoes not get breached. In this case, the option price of the model equals theplain vanilla option price, valuated with the same method. The graph on theright shows a case with 5 time steps, where the barrier gets breached afterthe 4th time step. Since the model is discrete, it means that the asset pricejumps from one node to the other, and breaching the barrier seems to causea problem. This can be solved by placing the barrier on the node or refiningthe tree as suggested by adaptive mesh model when the underlying is tradingclose to the barrier.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 130

35

40

45

50

55

60

65

70

75

Elapsed time

Ass

et P

rice

Down−and−out put

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

30

40

50

60

70

80

90

100

110

120

Elapsed time

Ass

et P

rice

Down−and−out put

22

Below, it will be shown how the option price evolves when the time par-tition is varied. As in the case of the plain vanilla option, the binomial andtrinomial trees as well as MCS is considered. The trinomial tree will addi-tionally get some modification suggested by Ritchken and Kamrad. FDMwill not be tested.

As for the plain vanilla option, the parameters are the following: S0 =50, K = 50, r = 0.1, σ = 0.4 and T = 1. Stated in words: the option is ATM,time to maturity is one year, risk free rate is 10% and the volatility is 40%.Additionally, the barrier is set at 30 or in relative terms 60%.

ι ) Binomial Tree

Binomial tree does not seem to be an adequate method for pricing the barrieroption. The deviation seems to be pretty persistent so that even after a bignumber of time steps, a descent result is not obtained (even after 250 timesteps the deviation is bigger than 10%). The graph on the left again showsthe deviation from the BS price in absolute values, whereas the graph on theright shows it in percentage. For the first 10 time steps, the deviation fromthe BS price has not been reported, since the BS price for the barrier optionis small and the deviation relatively huge.

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9

10

15

20

25

30

35

40

45

50

60

70

80

90

100

120

140

160

180

200

250

300

350

400

450

500

-1

0

1

2

3

4

5

6

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9

10

15

20

25

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35

40

45

50

60

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80

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100

120

140

160

180

200

250

300

350

400

450

500

0%

5%

10%

15%

20%

25%

30%

35%

ιι )Trinomial Tree

It has been said in the theory part, that the trinomial tree is not an adequatemethod neither for pricing the barrier option since the convergence slowsdown dramatically when the barrier option is valuated. In the graphs below,obviously the trinomial tree does a better job compared to the binomialtree, but still, even after the time has been partitioned in 50 time steps thedeviation is slightly above 10%.

23

0

0.5

1

1.5

2

2.5

3

3.5

4

1 2 3 4 5 6 7 8 9 10 15 20 25 30 40 50

0

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

3

3.5

4

2 3 4 5 6 7 8 9 10 15 20 25 30 40 50

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

In the theory, Kamrad and Ritchken came up with an additional parameterλ, which speeds up the convergence. They mention that the λ should bebetween 1 and

√3. In the graphs below the λ (=1.15) has been chosen so

that the average deviation from the BS price is minimized.It is obvious that with λ = 1.15 the convergence speeds up again. So,

already after the time partition is bigger than 15 a deviation smaller than10% is guaranteed.

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10 15 20 25 30 40 50

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

1.5

2

2.5

3

3.5

4

2 3 4 5 6 7 8 9 10 15 20 25 30 40 50

0%

10%

20%

30%

40%

50%

60%

70%

ιιι ) Monte Carlo Simulation

The MCS again, seems to do a very good job. The deviation from the BSprice is throughout the entire graph below 10%. After 5000 generated assetprice paths, the deviation less than 3% can be achieved.

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

250 750 1250 1750 2250 2750 3250 3750 4250 4750

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

250 750 1250 1750 2250 2750 3250 3750 4250 4750

-1%

1%

3%

5%

7%

For the barrier option, the convergence and convergence speed will not bereported. It seems to be obvious that the trinomial tree here as well performsbetter than the binomial tree, which produces quite huge deviations. Ifthe modification suggested by Ritchken and Kamrad is introduced, even abetter result is obtained. The barrier option valuation with MCS should bepreferred for pricing the barrier option. In the industry, among others, thisis actually the way in which a barrier option is priced.

24

3.2 Empirical Part 2: How much do the IB shift thebarrier?

In the previous part, it has been shown how the barrier option price valuatedwith numerical approaches converges to the BS option price. But is the BSprice correct? The biggest drawback of the BS model nevertheless is thatconstant volatility is assumed. The volatility observed in the markets is any-thing but constant, which causes the closed analytical formula valuation todeviate strongly from the market prices.

In this part, it will be shown that the BS price will overprice the marketprice strongly due to assumption of the constant volatility. Moreover, it willbe shown how the IBs value the barrier options and to what extent they shiftthe barrier when they price the barrier options. Later, the consequences ofthis barrier shifting are examined.

The following markets have been chosen as the underlyings for the optionpricing: Swiss Market Index (SMI), UK Stock Market (FTSE), Standard &Poors 500 (SPX) and DJ Euro Stoxx 50 (SX5E). The general market viewand the corresponding implied and realized volatility will here be briefly re-viewed.

3.2.1 Brief review of the markets

For the market analysis, the time period of the last 8 years has been chosen.This period contains both big stock market crashes, the Internet bubble andthe Real Estate bubble in the US (but also in the UK). It is worth mentioningthat the markets after 8 years have almost not changed at all compared tothe initial value from the beginning of 2000. Even a couple of indices arebelow the initial level.

0

20

40

60

80

100

120

140

01.01.2000

01.05.2000

01.09.2000

01.01.2001

01.05.2001

01.09.2001

01.01.2002

01.05.2002

01.09.2002

01.01.2003

01.05.2003

01.09.2003

01.01.2004

01.05.2004

01.09.2004

01.01.2005

01.05.2005

01.09.2005

01.01.2006

01.05.2006

01.09.2006

01.01.2007

01.05.2007

01.09.2007

01.01.2008

01.05.2008

SMI Index UKX Index SPX Index SX5E Index

25

Additionally to the general market view, the volatility will be reviewed here.The volatility serves as a risk measure and is from crucial importance forpricing the derivatives. In the graphs below, the implied and the realizedvolatility are reported. The implied volatility stems from an average of 3calls, which are the closest to the ATM in terms of the moneyness. Thematurity is 12 months for all of them. The realized volatility is calculated on260 days basis, that means excluding the weekends. The difference betweenthe implied and the realized volatility is reported in form of the green columngraph and indicates that the implied volatility is pretty much always higherthan the realized. Selling the volatility seems to be a very attractive tradingstrategy (e.g going short a variance swap). The high implied volatility (rel-ative to the realized) indicates as well that the options are more expensivethan they should be, respectively, that the investors are too risk averse andbuy too much protection. Regarding the implied volatility, it can generallybe said that the volatility in 2007 increased a lot and recently came downagain dramatically.

SMI

0

5

10

15

20

25

30

22.09.2004

22.11.2004

22.01.2005

22.03.2005

22.05.2005

22.07.2005

22.09.2005

22.11.2005

22.01.2006

22.03.2006

22.05.2006

22.07.2006

22.09.2006

22.11.2006

22.01.2007

22.03.2007

22.05.2007

22.07.2007

22.09.2007

22.11.2007

22.01.2008

22.03.2008

-4

-2

0

2

4

6

8

imp.vol-realised VOLATILITY_260D 12MO_CALL_IMP_VOL

FTSE

0

5

10

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20

25

30

35

22.09.2004

22.11.2004

22.01.2005

22.03.2005

22.05.2005

22.07.2005

22.09.2005

22.11.2005

22.01.2006

22.03.2006

22.05.2006

22.07.2006

22.09.2006

22.11.2006

22.01.2007

22.03.2007

22.05.2007

22.07.2007

22.09.2007

22.11.2007

22.01.2008

22.03.2008

-4

-2

0

2

4

6

8

10

12

imp.vol-realised VOLATILITY_260D 12MO_CALL_IMP_VOL

SPX

0

5

10

15

20

25

30

22.09.2004

22.11.2004

22.01.2005

22.03.2005

22.05.2005

22.07.2005

22.09.2005

22.11.2005

22.01.2006

22.03.2006

22.05.2006

22.07.2006

22.09.2006

22.11.2006

22.01.2007

22.03.2007

22.05.2007

22.07.2007

22.09.2007

22.11.2007

22.01.2008

22.03.2008

-2

0

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4

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8

10

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imp.vol-realised VOLATILITY_260D 12MO_CALL_IMP_VOL

SX5E

0

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30

22.09.2004

22.11.2004

22.01.2005

22.03.2005

22.05.2005

22.07.2005

22.09.2005

22.11.2005

22.01.2006

22.03.2006

22.05.2006

22.07.2006

22.09.2006

22.11.2006

22.01.2007

22.03.2007

22.05.2007

22.07.2007

22.09.2007

22.11.2007

22.01.2008

22.03.2008

-4

-2

0

2

4

6

8

10

12

imp.vol-realised VOLATILITY_260D 12MO_CALL_IMP_VOL

3.2.2 Offer prices from the seven IBs

The barrier option is a derivative, which belongs to the class of the exotics.The exotics, generally, are traded over the counter and therefore their priceis very difficult to obtain. For the purpose of comparing the BS model pricewith the market price, I have contacted few IBs and asked them for a pricing.

26

The prices received are indicative and not tradable.Seven companies provided prices for a down-and-out put option with the

following characteristics: the option is ATM, barrier is set at 70%, time tomaturity is 1 year and the underlying is one of the following: Swiss MarketIndex (SMI), Standard & Poors 500 (SPX), DJ Euro Stoxx 50 (SX5E) andFTSE (UKX). I thank the companies, which provided the prices and theparameter values included in the pricing. The companies are: JPMorgan5,Goldman Sachs6, UBS7, CS8, Deutsche Bank9, Dresdner Kleinwort10 and Ex-ane11.

The prices are offer prices, since the mid price would not contain anybarrier shifting and would therefore not really be interesting here. Addition-ally to the barrier shifting, the offer price includes a certain amount of hedgecosts, which is minor (approximately at about 10 bps) and can be neglected.

As already shown in the theory part, there are 8 different barrier options.They can be divided in intrinsic and non-intrinsic barrier options. Only forthe intrinsic barrier options a barrier shifting is needed. Why? The intrinsicbarrier option has a certain value when the barrier is breached, that meansthat the option loses all the intrinsic value at some point, which causes thepayoff to be discontinuous. The discontinuity is actually the reason why abarrier bending is needed. This situation is especially observable, when theunderlying trades close to the barrier and the time to maturity is a small.The barrier shifting is made so that the trader can react if the market all of asudden breaks down and a sell off takes place. That means that down-and-output is too expensive (the barrier is shifted to the left) and the down-and-input is too cheap ceteris paribus. This is already a known fact and it willbe discussed among others in the paper written by Schmock, Shreve andWystup[10].

Several telephone calls with the brokers made it clear, that the barrierin this market circumstances is shifted between 2.5-3%, depending on thevolatility skew level when the pricing is done. That means in this case here,that the shifting will be around the same for each index but different among

5www.jpmorgan.com6www.goldmansachs.com7www.ubs.com8www.credit-suisse.com9www.db.com

10www.dresdnerkleinwort.com11www.exane.com

27

indices.The IBs, which provided the prices also provided the ATM volatility and

the reference price of the underlying. For the sake of fairness the IBs willremain anonymous.

In the graphs below the prices provided by the IBs are shown. The offerprice is shown on the x axis and expressed in percentage of the spot priceand on the y axis the ATM volatilities are shown.

SMI

A

B

C

E

F

G

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

SX5E

G

F

E

D

C B

A

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.2 2.4 2.6 2.8 3 3.2 3.4

SPX

G

F

E

D

C

B

A

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.2 2.4 2.6 2.8 3 3.2

FTSE

G

FE

D

C

B

A

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

When the above presented graphs are observed, it can not be concluded thatthe higher ATM volatility causes the offer price to be higher. Actually, forthe down-and-out put the volatility has two effects. On the one hand, thehigher implied volatility increases the probability that the option ends ITMat expiry. On the other hand, the higher implied volatility increases the prob-ability that the barrier will be breached and the option loses all the value.That means that the down-and-out put might be very insensitive to the ab-solute value of the implied volatility since the two effects might cancel out.What can be concluded is the aggressiveness of the pricing of each company.So for instance is the company F persistently the most aggressive one andcompany G the least aggressive one. In the graph for the SMI index, thecompany D could not provide a price for that underlying because they donot have a book for that underlying.

In a first step, the prices will be calculated with the BS analytical for-mula using the same input as the IBs provided additionally to the offer price.

28

Yet two parameters need to be estimated, risk free rate (r) and the dividendyield (q), which together are the forward basis (r − q). This two parameterscould not be obtained by the IBs, since they were included in the pricing by”default” according to the brokers. For the risk free rate, the 1y Libor in thecorresponding currency has been used and for the dividend yield I used theestimate reported on Bloomberg.

3.2.3 The BS price overprice the market offer price strongly

As already mentioned above, the biggest drawback of the BS pricing model isthe assumption about the volatility. The volatility is assumed to be constant,whereas the volatility is observed to be non constant in the market. Oftenthe financial instruments (especially the single stocks and the currencies)exhibit a smile, the index a skew. In the graphs reported below additionallyto the offer prices from the IBs, the BS mid price is reported. Analyzingthe BS price of the barrier option it becomes clear that the higher volatilitycauses the BS price to be be lower, or stated differently, the effect that thebarrier will be breached overweights, at least here, the effect that the optionwill end ITM. Graph below indicates that the BS price overprice the barrieroptions strictly and the overpricing is up to 100% compared to the marketprices. Here, it could be argued that the offer price received by the IB iscompared to the mid price obtained by the BS model, thus not really ”fair”.Even if the BS offer price would have been plotted here, the effect would bemarginal. The prices would shift parallel to the left and the situation wouldnot have changed.

29

SMI

BS-G

BS-F

BS-E

BS-C

BS-B

BS-AA

B

C

E

F

G

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.5 3 3.5 4 4.5 5 5.5

SX5E

BS-G

BS-F

BS-E

BS-D

BS-CBS-B

BS-A

G

F

E

D

C B

A

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.5 3 3.5 4 4.5 5

SPX

BS-G

BS-F

BS-E

BS-D

BS-C

BS-B

BS-A

G

F

E

D

C

B

A

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.5 3 3.5 4 4.5 5 5.5

FTSE

BS-G

BS-FBS-E

BS-D

BS-C

BS-B

BS-A

G

FE

D

C

B

A

20

20.5

21

21.5

22

22.5

23

23.5

24

2 2.5 3 3.5 4 4.5 5

So, taking the barrier shifting into account for the BS model and choosing abarrier between 67-67.5% would not change a lot. The BS price would comecloser to the offer market price but the deviation would still be huge.

3.2.4 How do the IBs price a Barrier Option

Why does the BS price deviate by such a big amount from the market price?The answer is the volatility. Because the BS model assumes a constantvolatility, the skewness is not taken into account so that the ATM volatilityis taken for the whole ”moneyness” (St/K) of the option. And this is exactlythe point, which makes the BS model strictly overprice the option. Throughthe volatility surface, the BS model underestimates on average the volatility,which then consequently results in a higher price.

30

The graph above shows the volatility as the function of the stock price andthe time. The time axis can be neglected, at least then when the time tomaturity is big.

What is the driver of the down-and-out put option? Down-and-out putis very sensitive to the skew. The drivers of the down-and-out put pricecan be divided in two components: a long Gamma around the strike andshort Gamma around the barrier. That means, that the slope of the Deltais positive around the strike (falling underlying will increase the price of thedown-and-out put) and negative around the barrier (falling underlying willdecrease the price of the down-and-out put). Because the barrier option cancease to exist at any point during the life time when the barrier is breached,the Gamma at the barrier is of American type, whereas the Gamma at thestrike is of European type. The same holds for the Vega. So being long Vegaat the strike means, that increasing volatility will increase the price of theoption ceteris paribus. Being short Vega at the barrier means, the higherthe volatility the more we would get for selling the option. Both togetherimply then that the steeper the skew (low volatility ATM and high volatil-ity OTM), the cheaper the barrier option. So the Gamma and Vega of theAmerican type at the barrier will dominate, thus determine the price. TheGreeks will be further examined and portrayed in the application part 1.

In the graph below the volatility skew is presented for all four under-lyings, where the volatility is the function of the underlying. The graphexhibits, that the volatility increases as the underlying moves from ITM toOTM and that volatility decreases as the underlying gets deeper ITM. Stateddifferently, the volatility tends to be higher on the downside. Observing theimplied volatility for a single stock, the implied volatility increases even ifthe underlying moves from OTM to ITM, resulting in a so called volatilitysmile.

31

So far, it has been shown that the BS price overestimates the market pricefor the barrier options. Additionally, the forward basis (r − q) needs to beestimated since every bank has an own system for estimating these two pa-rameters. But estimation of these two parameters does not make the BSprice to deviate strongly. It is the volatility, which is from crucial impor-tance for an accurate pricing. On the one hand the IBs have again their ownsystems for estimating the volatilities and on the other hand their marketview sometimes influences the pricing. So e.g can an IB make a bet, whichcan not be modeled or the IB might prefer to close a open position, whichthey have on their books. Assuming that these kind of influences do notoccur, only the volatility impact will be from interest.

How do the IBs price the barrier options? Contrary to the BS model,they take into account that the volatility is not constant. So for pricing thebarrier options they include the whole volatility surface, the so called localvolatility model, into their pricing model. Dupire [7] and [8] shows in histwo papers how the pricing and hedging can be done when the volatilityis not constant. More from a point of view of a practitioner, Overhaus etal.[9] show in their paper the link between the market and the correspondingvolatility and how such a pricing will be done in the industry. Why doesthe whole volatility surface needs to be taken into account when a barrieroption is priced? One reason is because the option might terminate at anytime before the maturity. That means that the ”maturity” of the barrieroption can not be predicted for sure. For the down-and-out put option, theoption ceases to exist as soon as the barrier is breached. If the barrier is notbreached through the whole life time of the option, then the maturity equalsthe predicted maturity. For a plain vanilla option this is not the case. The

32

maturity is known so that only the volatility with a certain time to maturityneeds to be accounted for in that pricing. The second and the much moreimportant reason is that the volatility is stochastic itself. That means thatthe stock price follows a stochastic process, and the volatility, which is partof this stochastic process is again a stochastic process for itself!

As above mentioned, an intrinsic barrier is very difficult to hedge. This com-pels the IB to shift the barrier to the left when they price the down-and-output option. That means that the mid price will be at the true barrier (70%)and the offer price at a barrier further to the left. The most brokers, whoprovided the prices mentioned that the barrier shifting is depended on thevolatility level and volatility skew. That means that the higher the currentvolatility and the skew are, the higher the shift will be. At the moment thebarrier is shifted between 2.5-3%. But what does that mean in terms of theoffer price compared to the mid price? How much more expensive will thedown-and-out put be?

3.2.5 What does it mean when the Barrier is shifted 2.5-3%?

In graphs below it is shown how the down-and-out put price will change asa function of the barrier shifting. The used model here is the local volatilitymodel, which takes the whole volatility surface as a pricing parameter. Thegraph on the left shows how the price evolves from a starting point 0, whichrepresents the true barrier of 70%. The barrier shifting up to 4.5% is thenshown with a barrier shifting step of 50 bps. The barrier seems to influencethe option price positively, as expected, that means, that the more the barrieris shifted to the left the higher the price of the down-and-out put will be.

33

The graph on the left shows how the price of the barrier option changes inpercent, where again the true barrier of 70% represents the starting point.The result is very surprising! The barrier shifting of 2.5-3% results in anmore expensive down-and-out put option offer compared to the mid. Theagio is up to more than 22%!! From the both graphs, it is obvious that theSMI has the smallest volatility and therefore the highest prices on one hand,and on the other hand, the same underlying will be affected the least froma barrier shifting. The FTSE has the highest volatility and therefore thesmallest prices, but will be affected by the barrier shifting the most. Here,it can be assumed, that the volatility skew (110/90) has the same slope forall of them.

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

SMI UKX SPX SX5E

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

SMI UKX SPX SX5E

The finding above was quite impressive. In the graphs below it is examinedhow the barrier option prices evolve for barriers ranging from 95% to 10%.The graph on the left shows, pretty intuitively, that the lower the barrieris the higher the option price is. The barrier option converges to the BSprice as the barrier converges to 0. The lower the volatility the higher theconvergence to the BS price since a lower volatility makes the barrier optionmore expensive c.p. and therefore closer to the BS price. Here, again, theskew of all them is around the same. In the graph on the right it is shownhow the price changes evolve compared to the previous barrier. So e.g theprice change in percent is the highest when the barrier is moved from 65% to60%. This means that shifting the barrier in this area results in the highestincrease of the price.

34

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SMI FTSE SPX SX5E

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

SMI FTSE SPX SX5E

3.2.6 Barrier Option valuation, revisited

So far, it has been shown that the BS model strictly overprice the barrieroption. It has only been shown how big the overpricing is only for a certainbarrier level (70%). In this part, and as the end of this section, the exami-nation is done for a barrier option with a range of different barriers and thencompared to the corresponding BS price.

In the graph below such an examination is done for the SMI index. Thebarrier option price valued with the BS model does not overestimate stronglythe barrier option price valued with the local volatility model for both, thevary high barriers and very low barriers. The overpricing is the biggest inbetween, being more concrete, at the barrier at around 70%.

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

0 0.2 0.4 0.6 0.8 1 1.2

BS Barrier Option (LN) Local volatility model Difference

3.2.7 Trinomial Tree, revisited

In the empirical part 1, it has been shown that the trinomial tree gives quitea good approximation for the barrier option, especially when the Kamrad

35

and Ritchken [16] modification is included in the model. In the empiricalpart 2, it has been shown that the volatility is not constant and that the BSmodel strictly overprice the barrier option. Therefore, for an accurate pricingrather the whole volatility surface should be considered than only one certainvolatility point.

Derman, Kani and Chriss[6] show how a trinomial tree can be constructedwith relaxing the assumption about constant volatility. They chose the trino-mial tree because of the additional parameters, which they use for the statespace, just to mention one of the most important. This subsection is notmeant to go into detail, rather to provide a connection between the empiri-cal part 1 and 2.

The graph below shows the comparison between two trinomial trees, theone on the left assuming constant volatility and the one on the right relaxingthis assumption.

4 Application Part

In the empirical part it has been shown that due to hedging difficulties theintrinsic barrier option needs some barrier adjustment when such an exoticoption is priced. In the application part 1, a structured product (SP) con-taining such a barrier option as component is developed and priced. Sincea private client can not buy solely the down-and-out put option (of coursethe very big private clients are excluded) a SP, the bonus certificate, is heretherefore introduced. After the introduction of the product, the view of theclient, the IB and the asset manager is examined.

In the application part 2, the focus will be on the pricing of the structuredproducts. More concretely, the question if the SP are fairly priced for theSwiss market is answered. Similar empirical investigation to the one given byWilkens, Erner and Roder[5] and Stoimenov and Wilkens[4] for the German

36

market is going to be done for the Swiss market, but some modifications areallowed.

4.1 Application Part 1: Bonus Certificate

Bonus Certificate is a SP with the following payoff:

Bonus Ceritficate

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140 160

price underlying

p&l

barrier not breached barrier breached

The components creating this payoff are:

- long the underlying- long the down-and-out put

The specialty of this product is, that the long down-and-out put is fi-nanced by the withheld dividend. Therefore, the underlyings with a highdividend seem to qualify as ”good” candidates for such a product, becauseeither more of the down-and-out put can be bought or the barrier can beshifted further to the left. Another important factor is, as shown above, thevolatility or rather the volatility skew. Since the screening for the volatilityskew is very difficult and time consuming, here it has been assumed that thehigher volatility causes the volatility skew to steepen, thus makes the barrieroption more attractive. This assumption is quite realistic, since no memberof the SMI index has a persistently high volatility level. So for instance, in

37

volatile markets, the investors tend to buy protection (put), therefore theprice of the option will increase, hence the implied volatility will do. Addi-tionally, to enhance their returns, the investor will prefer to sell OTM calls,which will further cause the volatility skew to steepen. In the graph belowthese two important parameters, dividend and volatility, are plotted for allof the SMI Index members. The stocks on the top-right have both, the highvolatility and the high estimated yield. It is not surprising, that these stocksare very often encountered in this product type.

SMI Members

ZURN VX Equity

UBSN VX Equity

SYST VX Equity

SYNN VX Equity

SCMN VX Equity

RUKN VX Equity

SLHN VX Equity

UHR VX Equity

ROG VX Equity

NOVN VX Equity

NOBN VX Equity

NESN VX Equity

HOLN VX Equity

CSGN VX Equity

CLN VX Equity

CFR VX Equity

BALN VX Equity

BAER VX Equity

ADEN VX Equity

ABBN VX Equity

0

5

10

15

20

25

30

35

40

45

50

0 1 2 3 4 5 6 7

dividend yield indication

hist_call_imp_vol

4.1.1 From client’s point of view

The payoff structure of the bonus certificate is very popular in the SP world.But, why should a client buy such a product? One benefit for the clientbuying this product structure, equivalent to buying the underlying and thebarrier option, are the transaction costs. For buying a ”package”, only onetransaction is needed, whereas constructing such a structure on its own needstwo transactions. Additional benefit is the access to the exotic markets. Aprivate client does not have the access to the OTC markets, thus can not buya down-and-out put option. And the last benefit mentioned in this contextis the denomination: the client can enter such a market with a quite a smallamount of money (in Switzerland already with 1000 CHF). A better reasonto buy a down-and-out put is the possibility to trade the skew. It has beenmentioned above, that the down-and-out put is very sensitive to the skew.Stated differently, buying a down-and-out put means buying the skew. Soif the skew flattens, the barrier option would become less expensive ceterisparibus. Thus, the down-and-out put could be used as a vehicle to trade the

38

skew. In the graph[15] below it is shown, how the prices will change whenthe skew reduces. Four down-and-out puts with different barriers and oneplain vanilla put are plotted. It is clear, that the skew reduction will havethe biggest impact on the down-and-out put with the smallest barrier (80%).The skew reduction of 4% will there almost halve the initial price.

The above mentioned arguments give some reason why an investor should buysuch a structure. There are more important reasons why an investor shouldnot invest in such a product. Bradbury[22] shows in her diplom thesis thatthe investors systematically underestimate the risks involved in a down-and-out barrier products, which explains why this product structure has gainedsuch a popularity in the recent years. That means that the investors do notreally realize what kind of risk they go into when they buy such a product.Rieger[23] shows how a SP with the same payoff could be created. Thisproduct would be much easier to hedge, this means that the IBs would nothave to shift the barrier when pricing such a product, which then wouldnot have any overpricing as shown above. But, such a product does notexist! Hens and Rieger[3] show in their paper that the most structuredproducts are not optimal for a perfectly rational investor. Additionally tothe mis-estimation, they report the heterogeneous beliefs. The heterogeneousbelief indicates that the market will behave differently than the probabilitydistribution p forecasts (often a sign of overconfidence). More, they show thatsuch a barrier product is not optimal for any decision model and only themis-estimation (the investor tend to underestimate the probability that thebarrier is breached) can explain their popularity. The heterogeneous beliefscan explain it partly, if even.

To buy such a structure does not seem to be very attractive from the

39

client’s point of view. But, there is very high demand for products with sucha structure.

Here, as the last point of this subsection, it is shown that adding thederivatives to a portfolio can result in a higher utility function. For thisstudy, the S&P 500 and BXM[25] indices have been used. The latter hasbeen launched by the options exchange CBOE and is basically an overlaystrategy on the S&P 500 index, or stated differently, a covered call strategy.That means, that the BXM Index is selling monthly slightly OTM calls on theS&P 500. Therefore the BXM Index is going to outperform in sideways andbearish markets, whereas S&P 500 is going to outperform in strong bullishmarkets. Both are presented below.

Peformance of SPX and BXM

0

100

200

300

400

500

600

700

Jan 90

Jan 91

Jan 92

Jan 93

Jan 94

Jan 95

Jan 96

Jan 97

Jan 98

Jan 99

Jan 00

Jan 01

Jan 02

Jan 03

Jan 04

Jan 05

Jan 06

Jan 07

Jan 08

spx index bxm index

A good starting point for the explanation is the normal distribution function,which is presented below (X ∼ (0, 1)). It is a very common assumption inthe financial industry to assume that the prices are log normally distributedand the returns therefore normally distributed. A more realistic way tomodel the returns would be to allow the existence of the fat tails. Thefat tails imply that the more extreme events, positive and negative, occurmore often than predicted by the normal distribution, which increases thevariance, therefore the volatility. And this is actually what is happening withthe overlay strategy. The form of the distribution will slightly be amendedand shifted to the right. The fat tails will be reduced and the distribution willbecome skinnier. In terms of the mean and variance, the mean will increaseand the variance will decrease. In the world of the CAPM, the investors willchoose the asset with the lowest volatility and highest expected return. So,in the world of CAPM, the investor will prefer the BXM index over SPXindex! The above given benefits of an overlay strategy will be proven below.

40

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Here, the distribution of the monthly returns of both indices from 1990 tilltoday are shown. Both graphs exhibit what has been stated above. The SPXindex has much stronger tails and fatter shoulders. The BXM on the otherside has much skinnier shoulders and flatter tails. The expected return of theSPX (BXM) is 0.66% (0.86%) and the standard deviation is 3.95% (2.70%).So the BXM has higher expected return and lower variance. Its distributionis slightly shifted to the right and the variance is slightly reduced as shown inthe graphs below. The graph on the right shows both distribution together(normalized).

SPX

0

10

20

30

40

50

60

70

-0.158

-0.131

-0.105

-0.079

-0.052

-0.026

0.000

0.027

0.053

0.079

0.106

0

0.2

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0.6

0.8

1

1.2

BXM

0

10

20

30

40

50

60

70

-0.126

-0.106

-0.086

-0.066

-0.046

-0.026

-0.006

0.013

0.033

0.053

0.073

0

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0.8

1

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-70

-50

-30

-10

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50

In terms of the utility function, two different models, expected utility theory(EUT) and prospect theory (PT) are qualitatively presented. It is going tobe shown that for both the overlay strategy increases the expected utility.

First, the EUT is examined. In the graphs below the utility function isplotted. The utility of money is concave and the marginal utility is decreasingas the money increases (lim

k→0f ′(k) → ∞), lim

k→∞f ′(k) → 0). The graph on the

left shows the case, where only the variance has been reduced ceteris paribus.Due to lower variance, the dispersion is reduced and therefore the expected

41

utility increased. In the graph on the right, the case is shown where theexpected return has been increased ceteris paribus. Due to higher expectedreturn, again a higher expected utility can be obtained. Both combined resultin a even better case for the investor, which an overlay strategy can provide.

Variance Reduction Exp. Return Increase

Last, the PT is examined. There, the gains are assumed to be concave andthe losses convex. More, rather a reference point and not the final wealthis relevant, which lies where the both axes cross. So, the gains will be tothe right of the reference point and the losses to the left. In the graphbelow the corresponding utility function has been plotted. We see, that theinvestor prefers ofter a smaller gain than once a high gain. In contrast, dueto convexity in the losses, the investor prefers a high loss once rather thanofter few smaller losses. And this is actually what the BXM index offers.The graph on the right shows the difference of the frequencies of occurrencesof a certain return. So for instance, the overlay strategy offers both, a higherfrequency of relatively small returns and lower frequency of relatively lowreturns.

42

BXM-SPX

-15

-10

-5

0

5

10

15

20

25

30

35

-0.158

-0.131

-0.105

-0.079

-0.052

-0.026

0.000

0.027

0.053

0.079

0.106

In both, EUT and PT respectively, the overlay strategy seems to be a strictlydominate strategy. The overlay has been done here with plain vanilla option.But how does the situation change when the barrier option is sold instead ofthe plain vanilla option? Let assume that the up-and-out call is sold. Thesituation does not change very much. The premium will be indeed smallerbut the upside would not been capped. Further study is not done here.

4.1.2 From IB’s point of view

It has been shown above that from the client’s point of view, an investmentin a bonus certificate does not really seem to make sense. From the IB’s pointof view, the whole situation looks completely differently. It has already beenmentioned above, that such a product is very difficult to hedge and that abarrier shifting is needed. That means, that such a product will consequentlybe more expensive. Additionally to the value after the barrier bending, letscall it ”fair value”, there need to be be added up the IB margin (CVA:Client Value Added), the private bank margin (UF: Upfront) and the hedgecosts(HC).

0 Price=Offer

CVAUF Hedge CostFair

Value

So from the IB’s point view, the sales and the trading are involved in theissue process of the product and both divisions make money. So for instance,if the market is breaking down and the product is coming close to the barrier,the IB often calls up the client and tries to unwind the position. Therefore

43

such a barrier shifting would not be of need in the first place, if the clientand the IB could agree in the beginning to unwind the position in such acase.

At this point it has to be said that the market (primary) is quite com-petitive when it comes to pricing a product. The private bank starts acompetition and the IB with the best offer gets the deal. (For big banks,such as UBS & Credit Suisse in Switzerland, their own IB is preferred if theprice is not very far away from the best offer.) Such a case is vary similarto the Bertrand Monopoly, where already two competitors are sufficient tomake the market look like perfect. So when an IB loses a competition, theyalways ask for a feedback so that they know how far away they are fromthe best offer, so that they can adjust their pricing parameters and price thenext competition more aggressively.

In the secondary market, the situation looks differently for the IB, whichnow acts as a monopoly. The private bank, the investment bank and hedgecost margin will be amortized over short period of time and the ”fair value”of the product will be moved in between the bid and ask, which means thatIB will make money on the gap spread.

Bid Ask

Fair

Value

At the maturity of the product, the IB often places the ”fair value” at thebid price so that the product price equals the ”fair value” in case that theproduct is bought back by the IB. So, it is to expect that the price of the SPwill deviate from the fair value the most at the issue date.

At the end of this subsection it is shown how the IB could hedge sucha position. It will be shown, that such a position is difficult to hedge andthat the barrier shifting is ”justified”. One way, the most obvious one, ofhedging the position is to try to buy the down-and-out put, since the IB isshort when such a SP is issued. An other way to hedge such a position wouldbe to create a portfolio consisting of the following instruments

• long put

• short American digital option

• short down-and-in call

44

or more formally, pdao = p − cdai − Digitalamerican. So if the barrier is nothit, then the down-and-out put is equal to the long put and the hedge worksperfectly, since neither of the two exotic options kick in. If the barrier ishit, the down-and-out put loses all the intrinsic value at that point. TheAmerican digital option kicks in and has the same value as the barrier optionsat that point and accounts for that loss in intrinsic value. The down-and-in call kicks in and has the same time value as the long put. The wholeportfolio is then sold. But the problem of hedging is not really solved, sincethe American digital option again exhibits discontinuity (This will be shownbelow in detail). What additionally needs to be solved is the down-and-in calloption, which is classified as the non-intrinsic barrier option, which knocksin without being ITM. The hedge hear is again not very straight forward:Long put and short the underlying is the portfolio needed in this case. Soif the barrier is not breached the down-and-in call does not knock in, theput expires without any value and the short underlying corresponds to thedown-and-out call, obtained by the in-out parity. This on the other handmeans, that the down-and-out call is hedged by the buying the underlyingand charging S0 − K when down-and-out call is sold. The graphs are takenfrom an internal paper written by Allen, Einchcomb and Granger[15]. Theyclarify the hedge mentioned above. On the left, the hedge for a down-and-out call is presented, and on the right the hedge for a down-and-in call ispresented. It has to be said at this point that the hedge for the down-and-outcall demonstrated so far is too trivial. A more realistic hedge is presentedfurther below.

A more appropriate way to hedge the down-and-out call (the IB is short) isdone by a risk reversal, long (OTM) call and short (OTM) put. The barriervalue equals the ATM of the underlying. So if the underlying never breachesthe barrier, the long call accounts for the down-and-out call and the putexpires worthless. When the barrier is hit, the risk reversal will be sold. Theshort put is needed to cancel out the long theta by being long the call. So,

45

the call will account for the intrinsic value and the put for the time value.This shows again that the volatility skew is very important pricing driver fora barrier option.

Now, back to the American Digital, which again has an discontinuouspayoff. The American digital option can be hedged by the increasingly tightcall spread, that means a call (ITM) is bought and a call (OTM) is sold.Comparing the American digital to the European one, the first will be twicethe second, assuming zero drift and zero skew. The intuition is the following:for the European digital only the maturity is relevant for the payoff, whereasfor the American digital the whole lifetime of the exotic. So once the barrieris touched, there is an even chance that the underlying will be above or belowthis barrier, or stated differently the European digital will end up not belowthe barrier with a probability of p = 1

2. So neglecting the theta, the down-

and-out put will be priced as a portfolio, with being long put and short twicethe European Digital.

The description above shows that the barrier options (intrinsic ones) aredifficult to hedge and the barrier shifting seems to provide an appropriatesolution from the perspective of the IB.

4.1.3 From Asset Manager’s point of view

Approaching the SP from the CAPM, Capital Asset Pricing Model, or stateddifferently, assigning them a place in that framework, is not possible. In theframework of Two-Fund-Separation, there is no space for SP, since it is onlyoptimal to hold the market portfolio (or a friction of it) and the risk freerate.

So what is the reason for the asset manager to include such a structurein the portfolio or using the down-and-out put as a portfolio hedge?

In Switzerland the overlay strategy is very popular in terms of generatingadditional returns. That means that the asset manager tends to buy theasset and sell an OTM call on that asset, which is very often a single stock.On the other hand, the asset manager hedges the portfolio by being long theput. From the supply and demand perspective, their is an excess supply onthe call side and an excess demand on the put side. The excess demand onthe put side makes the put more expensive. It has been said several times,that the barrier down-and-out put is much cheaper than the correspondingplain vanilla put in terms of costs. That means that a down-and-out putcan be used as a ”hedge” by an asset manager, but the market view has to

46

match with the downside protection of the barrier option! The down-and-output should not be used only because a cheaper ”hedge” is obtained, becausethe ”hedge” disappears once certain threshold is touched. So far, the barrieroption as a hedge has been considered. Now the question regarding whetherit makes sense to include a SP with such a structure is briefly mentioned.The barrier option is taxed in different way than the plain vanilla put since itdoes not really provide a hedge as the latter. Therefore from the perspectiveof taxes such a product might be preferred by the client ant therefore as wellby the asset manager. Wild [24] discusses several SPs and correspondingtaxes implication in his diplom thesis.

Regarding the SP with such a structure, there are several hidden risks,which need to be studied thoroughly. David Ruppert mentions in his booka negative correlation between the prices and the volatility, which meansthat the falling prices and increasing volatility move along with quite strongempirical evidence. The graphs below shows the regression of the SPX andthe corresponding volatility index, the VIX. The correlation between themis −0.06.

That means that the volatility will increase as the market falls. The graphbelow12 shows a Bonus Certificate. On the left the volatility is assumed to beconstant, and on the right the volatility increases as the market falls down.The graph on the right assumes a minimum volatility of 10%, which is persis-tent even if the market is extreme bullish. With this assumption it is avoidedthat the volatility falls to an implausible level, which is certainly above thelevel assumed here. Both graphs show the scenarios at different time to ma-turities. It becomes clear that the smaller the time to maturity is, the closerthe payoff is to the final payoff, which is labeled with the green line. So,concluding the stated above, it can be said, that the barrier option loses theprotection when it is needed the most, that means, when the volatility is

12The graphs have been generated with the application Fincad. The explanation re-garding the application can be found on the web page: www.fincad.com

47

very high.

4000

5000

6000

7000

8000

9000

10000

11000

12000

50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 110.0% 120.0% 130.0% 140.0% 150.0%

Underlying Level

Portfolio

24.04.2008 24.06.2008 24.08.2008 24.10.2008 24.12.2008 24.02.2009 24.04.2009

4000

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50.0% 60.0% 70.0% 80.0% 90.0% 100.0% 110.0% 120.0% 130.0% 140.0% 150.0%

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24.04.2008 24.06.2008 24.08.2008 24.10.2008 24.12.2008 24.02.2009 24.04.2009

An other point which the asset manager should be aware of are the Greeksand their dynamics when a barrier option, or a SP containing such an exotic,is considered. The Greeks represent the sensitivities of the derivative and arefrom crucial importance for the investment bankers as well as for the assetmanagers when they manage their positions. So for instance just to mentionthe most used and applied:

• Θ = ∂f∂t

• ∆ = ∂f∂S

• Γ = ∂∆

∂S

• V ega = ∂f∂σ

Generally speaking, the asset manager manages a portfolio with severalpositions. The Delta of each position within the portfolio shows the marketexposure and the mark-to-market risk of each single position. Special atten-tion should be given to the Delta exposure, which shows how the derivativechanges when the underlying changes. In terms of the exposure, it showshow much of the underlying needs to be bought so that the a Delta neutralposition is obtained. An other Greek, which is here examined is the Vega.It shows how the underlying will change when the volatility changes. First,let here assume that a Bonus Certificate, and therefore a down-and-out put

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(long), is a part of such a managed portfolio.The below shown example is a down-and-out put option on UBS, ATM

(25), barrier is set to be 20, time to maturity is 89 days, the barrier is ob-served on a continuous basis.

All the graphs show the scenarios very close to the maturity. The graphsbelow from left to the right are Delta, Gamma and Vega (in percent).

Lets start the explanation when the underlying is ATM and moves to-wards the barrier. Delta is negative, the usual situation encountered as forthe plain vanilla put. But as the underlying moves towards the barrier, theDelta switches from negative to positive. That means that an increasingprice would increase the value of the down-and-out put, which is now differ-ent to the plain vanilla put. When the underlying is very close to the barrierand the time to maturity is very small, then the value of the Delta literatelyexplodes and becomes very big. Since Gamma is the second derivative, thesituation is very similar to the one described above for Delta. The scenarioindicates that such a situation within the portfolio will imply that the Deltaexposure of the whole portfolio will be very misleading.

The situation for the volatility change is again different to the one of theplain vanilla option. Again the situation is considered when the underlyingmoves from ATM to the barrier. The Vega is positive, the same as for theplain vanilla put. Higher volatility makes the option more expensive. Buthere again, the sign switches at some point. As the underlying moves to thebarrier, the volatility increases since the negative correlation between themexists. When the underlying is close to the barrier, the high volatility in-creases the probability that the barrier is hit and the option knocked out.Therefore the Vega is the lowest the closer the underlying to the barrier andthe time to maturity very small.

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4.2 Application Part 2: Are SP fairly priced?

4.2.1 Introduction

In this part it is going to be examined if the conclusions, at least some ofthem, made by Wilkens, Erner and Roder[5] and Stoimenov and Wilkens[4]hold for the Swiss market. That means that the most products will containSwiss single stock underlyings, but not exclusively.

[5] examined the German Market for the reverse convertible (RC) anddiscount certificate (DC) for one month in 2001. The three researchers foundout a strong pricing bias favoring the product seller for both, the RC andDC.

[4] examined the German market for equity-linked structured product(EL), a more general study. EL can be divided in two groups, the group con-taining the plain vanilla option component (classic, corridor, guarantee, turboproducts) and the group containing the exotic option component (knock-in,knock-out products). In their paper the two researchers postulate three hy-pothesis, where only the first two are mentioned here:

(H1) In the primary market, equity-linked structured products arepriced, on average, above their theoretical values.

(H2) The overpricing is highera) for products with stock underlyings than for those with indexunderlyings andb) for more complex products, compared to ”classic” instruments.

Speaking of the complex products, the products having the barrier option asa component are meant.

In their summary the two researchers conclude that products with anembedded exotic option are subject to even higher premiums, compared tothe common ”classic” products. They as well mention that this overpricingis due to higher hedging costs, which is as shown above somehow justified.In this part it will be shown if the above proven facts for the German markethold as well for the Swiss market. Due to limited capacity, time restrictionand the subject of this diplom thesis, the analysis will be concentrated onthe RC, DC, barrier reverse convertible (BRC) and bonus certificate (BC).

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4.2.2 Methods

The pricing for the SP containing the plain vanilla option (RC, DC) willbe done by a pricing function available on bloomberg, which uses the BSmodel. The tool on bloomberg delivers automatically all the input param-eters needed, such as volatility, estimated dividend and risk free rate. Asalready mentioned above, the BS model provides a very accurate price forthe plain vanilla option and is widely applied in the industry. All the param-eters included in the pricing will be the closing level on the date when theproduct has been issued.

For the latter two SPs containing the barrier option, a local volatilitymodel will be applied for their pricing. There, the problem is that the tool,which applies the local volatility model can not construct the volatility sur-face from the past, so that the product containing the barrier option has beenpriced a couple of days after it has been issued. The issue date time framewill be the whole year 2008 so far, where the products issued recently (inJune 2008) will be preferred (especially for the BC and BRC). The productsare picked randomly from the two, [33] and [34], online platforms. Thirtyproducts will be priced, twenty of them containing the plain vanilla option(RC and DC, each ten) and ten of them containing the barrier options (BCand BRC, each five).

What can be expected regarding the deviation from the market price tothe fair value? It is to expect that RC and DC will slightly, if even, overshootthe fair price ((H1) will be rejected). (H1) postulated above will be rejectedin terms of, that the overshooting, if anything, will be very small so that sucha product can be obtained cheaper than buying the two structures separately.Especially, the strong bias favoring the seller found by Wilkens, Erner andRoder[5] will be rejected. The product containing a plain vanilla option isquite transparent. The price of the plain vanilla option component can notdeviate strongly from the plain vanilla option traded on the exchange. Sostated differently, if the SPs containing the plain vanilla options are overpriced, then it could be argued that the plain vanilla options traded on theexchanges are overpriced as well.

The products containing the barrier option are expected to have a strongerovershooting ((H2) will be confirmed). Even though the two researchers usea wrong model (not the local volatility model) to price a barrier option, theirhypothesis will be confirmed due to barrier shifting, which makes the barrieroption more expensive. The barrier shifting is of course the major driver of

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the higher premium. An other important driver, which has to be mentionedat this place is the liquidity. On the one hand, the barrier options are tradedOTC, therefore not very liquid, and on the other hand the liquidity of the un-derlying is very important. Several brokers mentioned as well in this context,that the quotient of the notional amount and the liquidity could representthe hedging speed. That means, in an extreme case, a very high notionalamount and low liquidity of the underlying would result in a extraordinarypremium, due the fact that the notional amount to be hedged is high andthe hedging due the liquidity restrictions difficult.

Both studies have been done in 2003 and 2005 respectively. In does notseem to be very far away, but it is. The market became more and morecompetitive in the recent years so that too high premiums would not havebeen sustainable. Basic economic theory predicts that such an ”arbitrage”would have been removed by the market participant and this is especiallywhat is expected to have happened. Both studies examine the primary andthe secondary market. Here the focus will be on the primary market, sincethe biggest deviation occurs at the issue date.

In both papers, [5] and [4], the researchers try to duplicate the pay off ofthe SP. In one of the papers[5], they use for their duplication portfolio theoptions traded on the Eurex (European Exchange), where their try to mini-mize the differences in both, the strike and the time to maturity. Since thebiggest deviation occurs at the issue date, the focus will lie on the issue date.In the other paper[4], they duplicate the SP by calculating the theoreticalvalue of the SP. For the SP with the barrier option as a component, they usethe formulas given by Rubinstein and Reiner[18], which does not use the localvolatility model. And this is actually what has been shown above, wrong!The closed analytical formula is not taking the whole volatility surface intoaccount, thus overpricing the market price of the exotic option significantly.Hence, the SP will be overpriced as well.

Before the results are presented the products priced are briefly reviewed.They are all together somehow connected. To see the link, a good startingpoint is the Put-Call parity, which is

Ke−rT + c = p + S0.

With some basic algebra the equation below can be rewritten into

S0 − c = Ke−rT − p

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where the left hand side is the DC (long the underlying, short the call) andthe right hand side the RC (long the zero bond, short the put). Since theleft hand side is equal to the right hand side, both products have exactly thesame pay off. The taxation is different and is reviewed in detail by Wild [24]in his Diplomarbeit. When the plain vanilla put on the right hand side isreplaced by the in-out parity (ppv = pdao + pdai) and after one rearrangementthe BRC is created. (For the sake of completeness, the left hand side is thebarrier discount certificate, which will not be examined here).

S0 − c + pdao = Ke−rT − pdai

For the BC, the call needs to be moved to the other side so that followingequation evolves:

BC = S0 + pdao = c + Ke−rT − pdai = c + BRC

4.2.3 Results

Here, the results obtained will be presented. First, the results for products,which have a plain vanilla option as a component will be presented, thatmeans DC and RC. Later, the results for products, which contain a barrieroption as a component will be presented, that means BRC and BC.

a) Discount Certificate and Reverse Convertible Note

The DC is a certificate, where a stock is offered at a certain discount. Thediscount is possible because a call option is sold and the resulting cash flowcan be used to buy the underlying. Normally, if the stock closes above thestrike then a cash settlement takes place, otherwise the certificate holder getsthe underlying stock at a predefined discount. The IBs price such a structurewith a call spread, that means a call is sold, normally ATM or slightly OTM,and a call is bought being deeply ITM (Delta close to one). The latter will beachieved by being long a LEPO (Low Exercise Price Option). Such a optionhas the same payoff as a forward or the stock itself, since the volatility andtime value (Vega and theta are close to zero) do not influence the price ofthat option.

The RC is a note, where a zero coupon bond is bough and put, normallyATM, is sold. The reverse convertible has the same payoff as the DC, butit should be mentioned again, that the both are taxed in a different way.Since the option value is positively influenced by the volatility, especially in

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the high volatility regimes, high coupons, respectively high discounts can beachieved for both products.

As already mentioned, both structures have been priced on bloombergsince they both contain plain vanilla options as a building component. Forthe DC, a two leg - 2 plain vanilla options - have been priced13. For theRC, there is a bloomberg14 function, which allows a direct pricing of such astructure.

The results of this subsection can be summarized as follows: Both struc-tures are quite fairly priced. The overshooting is as expected very low. Fromthe investors’s perspective, both products can be obtained cheaper than buy-ing corresponding components separately. Although the price at the issue ishigher than the fair value, the overshooting is smaller than 1% (on average)for single stocks. Therefore, the founding of both, [5] and [4], will be rejectedfor the SP with plain vanilla options as a component of that product.

The brokers as well mentioned here, that for these products the marketis very competitive and that on average one Vega for single stocks and 0.5Vega for indices can be charged when options are priced. Stated differently,there will be a bid-ask spread on the volatility when such a product is is-sued. Therefore, it can be argued that the plain vanilla options are as welloverpriced since there as well a spread is charged.

Below the results for the DC are presented. In the first table the discounthas been deducted from the issue price. This price has then been comparedto the fair value price. The normal case is that the fair value price lies underthe offer price and therefore the discount offered is smaller than it could havebeen offered. But, there are products for which this fair value price actuallylies above the price offered by the IBs, which indicates that the SPs (A andE) are actually under priced! This discount which could have been offered isthen shown in the last column of the table. The one product for which thedeviation is huge is a DC traded in a different currency than the underlyingitself. These kind of products are called Quanto. That means that for sucha product additionally a Swap needs to be entered. But the deviation is stillto big and a wrong data input might be suspected.

13For purposes of recalculation: the bloomberg function is OVME14For purposes of recalculation: the bloomberg function is OVSN

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name discount(d) price(p) (p) − (d) pricefv(pfv) discountfv(dfv)A 14.12 100 85.88 86.21 13.79B 13.71 100 86.29 86.10 13.90C 12.20 100 87.80 87.41 12.59D 16.78 100 83.22 82.49 17.51E 11.27 100 88.73 89.29 10.71F 10.46 100 89.54 85.42 14.58G 13.00 100 87.00 86.21 13.79H 9.31 100 90.69 90.32 9.68I 7.38 100 92.62 90.85 9.15J 22.80 100 77.20 77.09 22.91

In the table below additionally to the offer price and the fair value price, thefair value price with the volatility spread is shown. There one Vega percentpoint spread has been added. That means, that the volatility is reduced by1% since a call is a sold, therefore the discount obtained is reduced. Thelast two column indicate the deviation from the fair value price (deviation1) and the deviation from the fair value price including the volatility spread(deviation 2). The average deviation 1 is 0.76% points and average deviation2 is 0.41% points. Excluding the outlier F the deviation 1 is 0.39% pointsand deviation 2 is 0.02% points. When the volatility spread has been addedalready 4 products seem to be under priced, which indicates that even lessthan 1% point is charged.

name (p) − (d) pricefv pricefvinc.vol.spread deviation1 deviation2A 85.88 86.21 86.60 -0.33 -0.72B 86.29 86.10 86.48 0.19 - 0.19C 87.80 87.41 87.78 0.39 0.02D 83.22 82.49 82.78 0.73 0.35E 88.73 89.29 89.68 -0.56 -0.95F 89.54 85.42 85.64 4.12 3.90G 87.00 86.21 86.62 0.79 0.38H 90.69 90.32 90.72 0.37 -0.02I 92.62 90.85 91.13 1.77 1.49J 77.20 77.09 77.39 0.11 -0.19

Here below the results for the RC are presented. In the second and thirdcolumn the coupon and the issue price are presented respectively. The fourthcolumn indicates the fair value price. It is clear that here again the fair value

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is very close to the issue price. The fair value price lower than the issue priceindicates that the client is paying more than he should be doing. But hereagain there are fair values which are above the issue price (B, F, G, I andJ). This ”issue” will be examined more in detail in the discussion part. Inthe fifth column the fair value is indicated when volatility spread is added.Here again, because the put is sold, the volatility will be reduced by 1%point. Therefore the premium collected by the sold put will be smaller andthe price higher for such a structure. Here again the situation reported belowindicates that 8 out of 10 SPs are under priced when the volatility spread of1% is added, there it follows that even less than 1% point is charged.

name rate price pricefv pricefvinc.vol.spread deviation1 deviation2A 14.50 100 99.17 99.52 0.83 0.48B 9.76 100 100.47 100.78 -0.47 -0.78C 15.00 100 99.80 100.14 -0.14 0.20D 10.50 100 99.40 99.79 0.60 0.21E 11.50 100 99.72 100.05 0.28 -0.05F 14.00 100 101.65 102.31 -1.65 -2.31G 13.00 100 100.54 100.94 -0.54 -0.94H 9.30 100 99.92 100.27 0.08 -0.27I 13.00 100 100.52 100.92 -0.52 -0.92J 13.125 100 100.83 101.22 -0.83 -1.22

For both structures the critical input parameter was again the volatility. Forthe calculations done here the implied volatility15 calculated from a weightedaverage of the volatilities of the three call options closest to the ATM hasbeen used.

b) Bonus Certificate and Barrier Reverse Convertible Note

Contrary to the first part of this subsection, the barrier options get involvedwhen these two products are considered. For a BC, the investor buys again aLEPO and a down-and-out put option, which barrier is usually set between60-80%. That means, that the two options cancel out in the range betweenthe strike of the call and the barrier of the put and anywhere else the calldetermines the payoff of the SP. A BRC is slightly modified compared tothe classic RC. Here, the clients is not selling a plain vanilla put, rather a

15This volatility can be obtained on bloomberg via the function ”hist call imp vol”available for those securities.

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down-and-in put.The barrier option in both SPs is of intrinsic type. Thus, these options

will be difficult to hedge and therefore the barrier will be shifted. Moreover,it has been shown above that barrier option in general should not be pricedby the BS model. And this is actually what the [4] applied when they calcu-lated the fair values of the SPs.

In the calculations conducted here, the local volatility model has beenapplied. The pricing has been conducted by a pricing tool of an IB, whichwill stay anonymous.

The results are here again as expected. The deviation of the theoreticalfair value from the market price is smaller for the SPs priced by the localvolatility model. But still, the deviation from the theoretical fair value cannot be neglected. The most important driver of the overshooting is the bar-rier shifting, which as shown above, causes the barrier option to be up to20% more expensive. Thus, if the barrier option is 10% of the ”portfolio”,the deviation will be at least 2%. Additionally to that, even if the barriershifting is included in the model, the theoretical fair value price deviate fromthe market price. Non of the SPs containing the barrier option had a fairvalue above the market value. H(2) will be confirmed, even though, the tworesearchers did not use the appropriate model.

4.2.4 Discussion

In the section above where the results have been presented, it has becomeclear that the SPs containing a plain vanilla option are pretty fairly priced.There were all in all 6 products which were even under priced; somethingwhich is almost too good to be true. When such a product is issued, thereare a lot of people involved in the process such as sales, trading, legal andcompliance just to mention the most important. And all of them need tobe paid! However, the result still showed that the SPs containing the plainvanilla option are competitively priced. The fair value price of the SPs shoulddefinitely be below the market price. So, what could be the error?

All the products priced in this diplom thesis have been priced using theclosing level of all the input parameters. Some of the SPs presented abovehave been priced during a trading day.

The markets in the first half of the year 2008 have significantly decreased.The daily changes in prices and volatilities were relatively extreme. So if acertain security has dropped dramatically, the implied volatility has increased

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dramatically due to the negative correlation. And for both SPs containingthe plain vanilla option (DC and RC), this effect has a positive impact ontheir price since they both sell an option. Therefore, a SP priced at the endof the day will have either a higher coupon or a higher discount, which inturn allows us to argue that the extremely volatile markets can partly explainthis under priced SPs which were priced during a trading day.

In the first half of the year 2008, the financial sector continued the verynegative trend which was initiated by the subprime crises in 2007. Mostof the financial firm stocks fell to a relatively very low level due to hugewrite downs and decreasing client confidence. This on the other hand, hada negative impact on the credit default swaps (CDS) which are positivelycorrelated with the volatility. CDS are of crucial importance when bondsare issued, which is the case when an option needs to be hedged. Since theoption can be hedged by a portfolio consisting of stocks and bonds, the highCDS indicate again that the price offered can not be far away from the fairvalue.

5 Conclusion

In the empirical part 1, it has been shown that the price obtained by thetrinomial tree method converges much faster to the BS price than the priceobtained by the binomial tree method when the plain vanilla option is priced.MCS provides an accurate price as well. When the barrier option is priced,the trinomial tree method should be used again, but a modification suggestedby Kamrad and Ritchken[16] needs to be included. Here again, the MCS pro-vides a good result too.

In the empirical part 2, it has been shown that a barrier option shouldnot be priced with the BS model, rather the local volatility model shouldbe applied. There, a barrier shifting is needed due to hedging difficultiesand discontinuity. That means that the down-and-out put is too expensivebecause the barrier is shifted to the left. It has been shown that a barriershifting from 2.5-3% (which is, by the way, a very realistic case) makes theexotic option up to 20% more expensive. The major price driver of the bar-rier option is the skew and not the absolute value of the implied volatility.The steeper the skew, the cheaper the down-and-out put option and viceversa.

In the application part 1, a SP (Bonus Certificate) containing such a

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barrier option has been examined thoroughly. The client’s, IB’s and assetmanager’s point of view have been reported. From the client’s point of view,it has been shown that including the derivatives in the portfolio can increasethe investor’s utility (the overlay strategy). This has been shown with theplain vanilla options, but a similar result can be expected with the barrier op-tions. From the asset manager’s point of view, especially the Greeks (Delta,Gamma and Vega) and their dynamics, should be reviewed briefly. On theone hand, the Delta, and therefore the Gamma, could become very high whenthe underlying approaches the barrier close to maturity. On the other hand,the sign of the both switches at some point when the underlying approachesthe barrier.

In the application part 2, it has been shown that SPs with a plain vanillaoption as a component, such as DC and RC, are not overpriced relative totheir theoretical fair value. The SPs with a barrier option as building compo-nent of that SP are strictly overshooting the theoretical fair value, even thenwhen the local volatility model has been applied and the barrier hedging hasbeen accounted for.

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