Computational Finance Review

Computational FinanceReview

Truong Ngo

Contents

1 Risk Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Binomial Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 From binomial trees to Black/Scholes . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Greeks and hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Calibration with market data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Random number generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Simulation of SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8 Exotic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9 Finite difference methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1 — Risk Neutral Pricing

Question 1.1 Explain why numerical methods are required for the valuation of financialderivatives. �

Many problems in financial mathematics do not have a closed form solution. Even if one is avail-able, it often require the evaluation of computationally laborious functions (e.g., Black/Scholesformula). Therefore, numerical methods are needed for the valuation of financial derivatives.

Question 1.2 What is the principle behind risk-neutral valuation? What are the advantagesof the concept in practice? �

Principle of risk-neutral pricingThe price of an option or other derivative when expressed in terms of the price of the underlyingstock is independent of risk preferences. Options therefore have the same value in a risk-neutralworld as they do in the real world.

This simplifies the analysis because:• Risk-free rate can be used for discounting• “True” probabilities or expected returns do not have to be estimated.

Question 1.3 What is the meaning of “risk-neutral probabilities”?How are they derived? �

Risk-neutral probabilities is an artificial probability measure of future outcomes adjusted for risk.Under this measure, the expected return on the stock corresponds to the return on an investmentat the risk-free rate.

p =erT −du−d

(1.1)

Derivation• 1st step: Construct portfolio with stock and derivative that yields the same return in both

states (∆ is the fraction of the stock):

∆.uS0−V u = ∆.dS0−V d ⇒ ∆ =V u−V d

uS0−dS0

6 Risk Neutral Pricing

• 2nd step: The return of this portfolio must correspond to an investment at the risk-freerate; therefore the current value V0 of the derivative is given by:

∆.S0−V0 = (∆.uS0−V u)e−rT

V0 = ∆.S0− (∆.uS0−V u)e−rT = [pV u +(1− p)V d ]e−rT

with

p =e−rT −d

u−d

Question 1.4 What does arbitrage-free mean? Which conditions must be satisfied in abinomial tree? �

Arbitrage-free means the market prices do not allow for profitable arbitrage.

In binomial model, the arbitrage-free condition must be satisfied, as the possibility of a risklessprofit could lead to contradictory results from the model. This implies:

d ≤ erT ≤ u (1.2)

Question 1.5 Describe the binomial model of Cox/Ross/Rubinstein (CRR). Which parame-ters are relevant for the valuation of call and put options? �

The Cox-Ross-Rubinstein binomial model is a discrete-time numerical method we use to pricecontingent claim financial derivatives such as European options, American options, and exoticoptions with nonstandard structures. Binomial model option pricing generates a pricing tree inwhich every node represents the price of the underlying financial instrument at a given point intime. In CRR model:• Investors do not have to agree on µ

• Option price depends only on u and d, not µ .

u = eσ√

T ; d = e−σ√

T

Question 1.6 What is the numerical advantage of recombining tree? What condition mustbe satisfied to achieve the recombination in the binomial method? �

If tree is non-recombining, the number of nodes doubles with each step and it will exhaust thecomputational power of most modern workstation even before 20 steps. Recombining tree, onthe other hand, has only n+1 different final states (although there are 2n different paths).

In order for tree to recombine, the price of the underlying assets after an up then down movementmust be the same as that after a down then up.

Question 1.7 What determines the accuracy of the binomial method? What can be saidabout the convergence? �

The accuracy of the binomial method depends on the number of time intervals. Increasing thenumber of time intervals would increase the method’s accuracy because the model would thenbe a better approximation of the time continuous model.

7

The binomial method approximation converges to the BS value as n→ ∞. The convergencepattern is not monotonic, but oscillatory. The option price calculated with binomial methodfluctuates around the value according to the BS model.

2 — Binomial Method

Question 2.1 What are the characteristics of European and American call and put options?How can the optimal exercise strategies be characterized? �

If a stock pays no dividend:• American Call is best exercised at time T• Optimal exercise time of American Put can before T. Indeed, at any given time during its

life, a put option should always be exercised early if it is sufficiently deep in the money.For example, suppose that the strike price is $10 and the stock price is virtually zero. Byexercising immediately, we make an immediate gain of $10. If we wait, the gain might beless than $10, but cannot be more than $10, since negative stock prices are impossible.

In case of dividend payments, American Call should be exercised if the dividend is sufficientlylarge and the dividend date is close to the expiry date of the option, i.e., T − tn is small.

Dn > K(1− e−r(T−tn)) (2.1)

Question 2.2 Describe why the early exercise of American call options on a no-dividendpaying stock is never profitable. �

We have the lower bound of European call option:

c≥ St −Ke−r(T−t)

Because the owner of an American call has all the exercise opportunities open to the owner ofthe corresponding European call, we must have C ≥ c. Hence,

C ≥ St −Ke−r(T−t)

Given r > 0, it follows that C > St−K when t < T . This means that C is always greater than theoption’s intrinsic value prior to maturity. If it were optimal to exercise at a particular time priorto maturity, C would equal the option’s intrinsic value at that time. Thus it can never be optimalto exercise early.

10 Binomial Method

Question 2.3 How can American options be priced with the binomial method? �

For American option, the procedure is to work back through the tree from the end to the beginning,testing at each node to see whether early exercise is optimal. The value of the option at the finalnodes is the same as for the European option. At earlier nodes the value of the option is thegreater of

1. The value given by the risk neutral valuation2. The payoff from early exercise

Question 2.4 Describe the algorithm of the binomial method. What is different for Europeanand American options? �

• Let Si j be the stock price and Vi j be the option value in state j of stage i( j = 0, · · · , i; i =0, · · · ,n)• In the final stage n (corresponding to time T = n∆t) by definition

Vn j =

{(Sn j−K)+ if call(K−Sn j)

+ if put(2.2)

• The values Vi j in the previous stages i < n depend on the option style (European vsAmerican)

– For European options:

Vi j = e−r∆t .(pVi+1, j+1 +(1− p)Vi+1, j) (2.3)

– For American options check for early exercise:

Vi j =

{max{(Si j−K)+,e−r∆t .(pVi+1, j+1 +(1− p)Vi+1, j)} if callmax{(K−Si j)

+,e−r∆t .(pVi+1, j+1 +(1− p)Vi+1, j)} if put(2.4)

Question 2.5 How can options on stocks with dividends be priced? What cases must bedistinguished and what are the corresponding approaches? Which assumptions are required?�

Assumption: dividend is predictable (this is realistic for only short-life options)

Case 1: Continuously paid dividend• In case of no dividend, stock price is assumed to grow on average by riskless rate r. With

dividend, this growth becomes r−δ since the stock price is continuously reduced at therate δ

• The probabilities of the states now becomes:

p =e(r−δ )∆t −d

u−d(2.5)

• The discount factor in the tree is still e−∆t.r

Case 2: Dividend as percentage• A dividend yield of β makes the stock prices fall by βS• If K dividends β1, . . . ,βK are paid during the lifetime of the option, the stock price at the

expiry date of the option is

Sn j = S∗n j(1−β1)× . . .× (1−βK) = S0u jdn− jK

∏k=1

(1−βK) (2.6)

• The effect of dividends is absorbed in the tree construction; it does not affect the probabili-ties of the states and the algorithm for the option valuation remains the same

11

Case 3: Constant dividend• Stock prices decrease by the amount of dividend paid.• The tree is no longer recombining and there are 2(k+1) nodes rather than k+2 in stage

k+1 after the dividend payment.• The same arguments may be applied in the case that dividend is an arbitrary function of

the stock price (not just a percentage as in case 2)

Question 2.6 How can the recombining property be preserved in the binomial method in thepresence of dividends? �

In the case of a constant dividend D, the recombining property can be preserved if the stock priceS is decomposed in• An uncertain component with value (τ = dividend date)

S∗ ={

S if i.∆t > τ

S−De−r(τ−i.∆t) if i.∆t ≤ τ

i.e., S∗ changes as if there were no dividends• A certain component (that represents the value of the known future dividend during lifetime

of option)The tree is constructed for the uncertain component with the same values for u, d, p and∆t.

Question 2.7 How can the numerical accuracy of the binomial method be increased? De-scribe an approach which does not require an increase in the step size. �

The accuracy of the binomial method can generally be improved by increasing the number ofsteps.

Another approach for American options is to implement control variate technique, in whichthe option price is adjusted by a correction term.• The error is estimated from the difference of an evaluation of the same tree for a European

option and the corresponding price according to the Black/Scholes model• Let VA be the price for the American option, VE the price for the European option and VBS

the Black/Scholes price, then the corrected price for the American option becomes

VA +VBS−VE

Question 2.8 What is the meaning of the “put/call parity”. �

Theorem 2.0.1 — Put-Call parity. A relationship between the price, c, of a European calloption on a stock and the price, p, of a European put option on a stock. It shows that the valueof a European call with a certain exercise price and exercise date can be deduced from thevalue of a European put with the same exercise price and exercise date, and vice versa.

c+Ke−rT = p+S0 (2.7)

For a dividend-paying stock, the put-call parity relationship is

c+D+Ke−rT = p+S0 (2.8)

3 — From binomial trees to Black/Scholes

Question 3.1 What is the relation between the CRR and the Black/Scholes (BS) model? �

Both BS model and CRR model are underpinned by similar assumptions. CRR model providesa discrete time approximation to the continuous process underlying the BS model. In fact, forEuropean options without dividends, the binomial model value converges on the BS formulavalue as the number of time steps increases.

Question 3.2 What are the assumptions in the original derivation Black and Scholes? Whyare they required? �

The original derivation of Black/Scholes requires the following assumptions:• The stock price follows a geometric Brownian motion with constant drift µ and constant

volatility σ .• The short selling of securities with full use of proceeds is permitted.• There are no transactions costs or taxes. All securities are perfectly divisible. There are no

liquidity restrictions. The Black-Scholes valuation relies on the perfect replicability of theoption.• There are no dividends during the life of the derivative.• There are no riskless arbitrage opportunities.• Security trading is continuous.• The risk-free rate of interest, r, is constant and the same for all maturities.

The necessity of the assumptions:• Option pricing models that account for the stochastic property of volatilities can be highly

complex, discouraging many market participants from using them.• The problem that arises from using stochastic interest is similar to that for volatilities.

However, this assumption can be relaxed provided that the stock price distribution atmaturity of the option is still lognormal.• The Black-Scholes valuation relies on the perfect replicability of the option. In order to

achieve an exact replication, it requires continuous adjustments. Since securities cannotbe traded around the clock and not all underlyings are perfectly liquid, trades are onlypossible at certain times and in limited volumes.• Costs associated with each transaction can have a big impact on trading. An adjustment of

14 From binomial trees to Black/Scholes

the hedge position several times a day is practically impossible because of these marketfrictions.• The hedge portfolio of a put option is created by entering a short position in the underlying

and investing the proceeds in a bond. The constraints on short selling may cause someadditional costs, and often also complicates systematic short selling activities.

Question 3.3 Describe the derivation of the BS equation (in principle). What quantitiesappear in the solution? What changes in the case of American options? �

The derivation of BS equation:• Apply Ito’s Lemma to the stock price process:

dS = µSdt +σSdW

This provides the process V(S,t) for the price of a derivative:

dV =

(δVδ t

+µSδVδS

+12

σ2S2 δ 2V

δS2

)dt +σS

δVδS

dW

• Construct a portfolio consisting of the derivative and a hedge position of ∆ shares:

Π :=V −∆.S

The value of this portfolio follows the process

dΠ = dV −∆.dS

This portfolio can be made riskless if ∆ = δV/δS; the change in value dΠ during the timedt is then purely deterministic:

dΠ =

(δVδ t

+12

σ2S2 δ 2V

δS2

)dt (3.1)

• Investment of the amount Π at the riskless rate over the time dt yields:

dΠ = rΠdt =(

rV − rSδVδS

)dt (3.2)

• By comparison of (3.1) and (3.2) and the exclusion of arbitrage opportunities we obtainthe BS equation:

δVδ t

+12

σ2S2 δ 2V

δS2 + rSδVδS− rV = 0 (3.3)

For an American option, the BS equation would become an inequality of the type “≤”

Solution:C(S, t) = S.Φ(d1)−K.e−r(T−t).Φ(d2)

P(S, t) = K.e−r(T−t).Φ(−d2)−S.Φ(−d1)

}S > 0, 0≤ t < T

d1,2 =ln(S/K)+(r±σ2/2)(T − t)

σ√

T − t

15

Question 3.4 What is the meaning of Ito’s lemma? �

Ito’s lemma is a way of calculating the stochastic process followed by a function of a variablefrom the stochastic process followed by the variable itself.

Definition 3.0.1 — Ito’s Lemma. Suppose that x follows the Ito process. A function G of xand t follows the process:

dG =

(δGδx

a+δGδ t

+12

δ 2Gδx2 b2

)dt +

δGδx

bdz (3.4)

It plays a very important part in the pricing of derivatives and is central to the Black-Scholesmodel because derivatives can be viewed as a function of their underlying.

Question 3.5 Given the BS formula for a European call option, how can the price of a putwith same strike and time to expiry be determined? �

Knowing the formula for a European call option, we can derive the formula for a put with samestrike and time to expiry using put/call parity: S+P = Ke−r(T−t)+C

Question 3.6 How can the c.d.f. of the standard normal distribution be calculated (approxi-mately)? �

The c.d.f. of the standard normal distribution can be calculated approximately with the imple-mentation of the error function:

Φ(X) =1+ erf(x/

√2)

2(3.5)

Error function is defined as:

erf(x) :=2√π

∫ x

0exp(−t2)dt (3.6)

Question 3.7 How (and under which conditions) can the BS formula be applied to optionson dividend paying stocks? Describe the value function of a European option in the presenceof dividends. �

The stock price can be decomposed into two parts: a riskless component that corresponds to theknown dividends during the life of the option and a risky component. By the time the optionmatures, the dividends will have been paid and the riskless component will no longer exist.Therefore, BS formula is correct if S0 is equal to the risky component of the stock price and σ isthe volatility of the process followed by the risky component.BS formulas can be used provided that the stock price is reduced by the present value of alldividends during the life of the option, the discounting being done from the ex-dividend dates asthe risk-free rate. Instead of the current stock price S, the adjusted stock price S∗ is inserted inthe BS formula.

S∗ = S−∑PV (dividend) (3.7)

Question 3.8 Describe the shape of the value functions for European and American call andput options on no-dividend paying stocks. How do they depend on the time to exercise? �

16 From binomial trees to Black/Scholes

For call option, as the time to maturity is reduced, the option price curve moves closer towardsthe intrinsic value. In contrast to the graph for a call option, the price curves for put optionintersect. The out-of-the-money put options have a higher price if the remaining time is longer.From a certain degree of “moneyness”onwards, this relationship reverses and options that maturesooner are more expensive than those that mature later.

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

S

C

Call

time = 1time = 0.8time = 0.6time = 0.4time = 0.2time = 0.0

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90

100

S

P

Put

time = 1time = 0.8time = 0.6time = 0.4time = 0.2time = 0.0

4 — Greeks and hedging

Question 4.1 What does “delta hedging” mean? �

Delta hedge is a hedging strategy that aims to reduce (hedge) the risk associated with pricemovements in the underlying asset by offsetting long and short positions.

Question 4.2 How is it applied? What problems arise in practice? What determines theaccuracy of a hedge strategy? �

The formula for the change in the portfolio value in the derivation of the BS equation:

dΠ =

δVδ t

+µS(

δVδS−∆

)︸ ︷︷ ︸

∗

+12

σ2S2 δ 2V

δS2

dt +(

δVδS−∆

)︸ ︷︷ ︸

∗

σSdW

To eliminate risk, i.e., the term (*), we set the corresponding hedge ratio ∆ = δV/δS.

Problem in practice: The delta of an option does not remain constant, the trader’s positionremains delta hedged for only a relatively short period of time. However, in reality, trading isonly possible at discrete points in time, the portfolio cannot be rebalanced continuously. Thus,there will be some error in the risk elimination.

Question 4.3 Assume that option on a stock is replicated by a portfolio of the underlyingand an investment at the riskless rate. What is the impact of the drift of the stock price processon the terminal value of the portfolio? What is the impact on its composition? Describereasons for the replication error. �

Question 4.4 What are the Greeks (in the context of financial derivatives)? How do theydepend on the price of the underlying, strike and time to expiry? �

The ’greeks’ are the sensitivities of derivatives prices to underlyings, variables and parame-ters. They can be calculated by differentiating option values with respect to variables and/orparameters, either analytically, if you have a closed-form formula, or numerically.

18 Greeks and hedging

• Delta: ∆ = δV/δSδC/δS > 0 or δP/δS < 0, for t < T• Gamma: Γ = δ 2V/δS2

Positive for both calls and puts due to convexity• Vega(Kappa): κ = δV/δσ

Positive for calls and puts since higher volatility of underlying increases probability thatoption gets into money• Theta: θ = δV/δ t

Always negative for calls since prices fall monotonously with shorter time to expiry; forputs positive or negative• Rho: ρ = δV/δ r

Positive for calls since present value of strike price K falls (negative for puts for the samereason)

Question 4.5 How do the Greeks affect the prices of European call and put options? �

Question 4.6 How do the values of the Greeks change over time? �

• Delta:

Question 4.7 Explain the behavior of the Delta of a European option when the expirationdate is approached. �

Analysis shows that:

limt→T−

δC(S, t)δS

=

1 if S(T )> E12 if S(T ) = E0 if S(T )< E

(4.1)

limt→T−

δP(S, t)δS

=

0 if S(T )> E−1

2 if S(T ) = E−1 if S(T )< E

(4.2)

The delta always finishes at 1 for options that expire in-the-money and 0 for options that expireout-of-the-money.

An explanation can be that for t ≈ T there is little time left for the asset value to change – ifit is currently in/out-of-the-money then it will probably remain in/out-of-the-money. In otherwords, the call option and the asset are highly correlated – they share the same risk. Since theportfolio is designed to replicate the risk in the option, it follows that it will hold approximately1 unit of asset, so ∆i ≈ 1. Conversely, if the call option is out-of-the-money close to expiry thenthe payoff is very likely to be zero whatever happens to the asset – there is no risk, so we shouldnot be holding any asset.

Question 4.8 How can the Greeks be calculated for American options or other more generalcases when no closed formula for the option value is known? What distinguishes the differentcases? �

Greeks can be calculate from the binomial tree.• Delta: at time ∆t, there are two estimates for the option value, namely for the stock prices

S0u and S0d:

∆≈ ∆V∆S

=V u−V d

uS0−dS0

19

• Gamma: At time 2∆t there are two estimates for ∆:

S =12(u2S0 +S0)→ ∆ = (V uu−V ud)/(u2S0−S0)

S =12(S0 +d2S0)→ ∆ = (V ud−V dd)/(S0−d2S0)

The difference between the upper and lower value of S is:

h =12(u2S0−d2S0)

Approximate Gamma as the change in ∆, divided by h.• Theta: observe that Sud = S0 for d = 1/u

θ ≈ V ud−V0

2∆t

• Vega: obtained from the difference quotient

κ ≈ V0(S,T,K,R,σ +h)−V0(S,T,K,r,σ)

h

i.e. from evaluating two binomial trees with different volatilities.• Rho: same as vega, apply binomial method for 2 different r

Question 4.9 Why is the knowledge of the Greeks required in practice? �

A financial institution that sells an option to a client in the over-the-counter markets is faced withthe problem of managing its risk. If the option happens to be the same as one that is traded onan exchange, the financial institution can neutralize its exposure by buying on the exchange thesame option as it has sold. But when the option has been tailored to the needs of a client anddoes not correspond to the standardized products traded by exchanges, hedging the exposure isfar more difficult.

An alternative approach to this problem is to calculate the Greeks. Each Greek letter measuresa different dimension to the risk in an option position and the aim of a trader is to manage theGreeds so that all risks are acceptable.

5 — Calibration with market data

Question 5.1 What must be observed when the volatility is estimated from historical data? �

Using historical prices, calculate the changes in the logarithms of the prices:

yi = lnS(ti)− lnS(ti−1), i = 1, . . . ,n, y =1n

n

∑i=1

yi

The sample mean and standard deviation of the n changes in the logarithms of the stock pricesare:

an :=1n

n

∑i=1

yi→ estimator for (µ−σ2/2)∆t

bn :=

√1

n−1

n

∑i=1

(yi−an)2 → estimator for√

σ2∆t

An estimator for the historical volatility is given by

σhist :=bn√∆t

(5.1)

We have

an ∼ N((µ−σ

2/2)∆t,σ2∆t/n

)For small ∆t the expectation and variance of an are also small, in practice one works with theapproximation

σhist ≈

√√√√ 1∆t

(1

n−1

n

∑i=1

y2i

)(5.2)

22 Calibration with market data

ProblemsIf the accuracy of the estimation should be improved, one may

1. either keep ∆t fix and go further back in time to increase n or2. keep the sample period n∆t fix and decrease ∆t

But:1. Older data are considered less relevant for today’s prices (sometimes handled by assigning

a lower weight to them)2. For very short time intervals ∆t the price volatility may be influenced by variations in the

bid-ask spreadIn practice time is measured in trading (not calendar) days

Question 5.2 What does “implied volatility” mean and how is it calculated? Is the solutionunique? �

The one parameter in the BS pricing formulas that cannot be directly observed is the volatilityof the stock price. In practice, traders usually work with implied volatilities, which are impliedby option prices observed in the market. It gives the market price of the option when used inconjunction with the BS option pricing formula.

To calculate implied volatility, we determine the root of the function:

f (σimpl) :=C(σimpl)−C0

Depending on the properties of f , there can be no root, exactly one root or more than oneroot.

Question 5.3 Describe some procedures for root finding and their pros and cons. �

Some of the methods for root finding are bisection, Newton’s and secant method.

Bisection method• Steps

1. Start with a and b (a < b), such that f (a). f (b)≤ 02. Set m := (a+b)/2 and calculate f (m)3. If f (a). f (m)< 0, then set b = m, otherwise set a = m4. If b− a < ε (tolerance value), then stop. (a+ b)/2 is approximation for root x∗.

Otherwise continue with step 2.• Pros and Cons

– The method is easy to understand and remember– It always works.– But it is also slow. The computation time depends on the choice of the initial interval[a,b]

Newton’s method• Idea: Given some point xn with function value f (xn), construct the next value xn+1 such

that it is closer to the root• Steps:

1. Linearize f at xn, i.e., approximate f by its tangent in the point (xn, f (x,)) with slopef ′(xn)

t(xn +h) := f (xn)+ f ′(xn).h

23

2. For h : x− xn define the linear function

t(x) = f (xn)+ f ′(xn)(x− xn)

3. Choose xn+1 := x as its single root:

xn+1 := x = xn−f (xn)

f ′(xn)

4. Possible stopping criteria: | f (xn)|< ε1 or |xn+1− xn|< ε2• Pros and Cons

– In general, Newton’s method has quadratic convergence: the error is squared at eachiteration (the number of accurate digits doubles)

– But it requires calculating the derivative which may not be easy in many cases.– If the initial value is too far from the true zero, Newton’s method may fail to converge.– If the function is not continuously differentiable in a neighborhood of the root, it is

possible that Newton’s method will always diverge or fail.– Newton’s method will fail in cases where the derivative is zero– In some cases the iterates converge, but do not converge quadratically.

Secant method• Secant method approximates derivative by difference quotient:

f ′(xn)≈f (xn)− f (xn−1)

xn− xn−1

⇒ xn+1 = xn−xn− xn−1

f (xn)− f (xn−1)f (xn)

• Advantages:– Knowledge of derivative not required– Often faster since only f (xn) must be calculated and not f ′(xn)

• Disadvantages:– If f ′(x) is zero at the root, expressions of the type 0/0 occur– The increase in accuracy near the root may be wiped out by rounding errors.

• If prerequisites are given, Newton’s method is preferable.

Question 5.4 What does the term “volatility smile” describe? How can this effect beexplained for currency and stock options? �

Volatility smile is the term for implied volatility curve.• For equity markets, the curve when plotted against strike price is usually downward

sloping. The distribution for stocks typically has a heavier left tail and less heavy right tail.The heavier left tail should lead to high prices, and therefore high implied volatilities, forout-of-the-money (low-strike-price) puts. Similarly the less heavy right tail should leadto low prices, and therefore low volatilities for out-of-the-money (high-strike-price) calls.This results in a curve where volatility is a decreasing function of strike.• In currency market, symmetric jump risk leads to implied distribution with fat tails, since

both currencies can crash. This leads to higher prices (and implied volatilities) for OTMputs and calls.

6 — Random number generation

Question 6.1 For which types of financial derivatives is the Monte Carlo method appropriate?�

Monte Carlo simulation is often used for pricing European-style derivative securities withcomplicated payoffs, especially those dependent upon several Brownian motions.

• path-dependent payoffs• a payoff depending on a larger number of variables• stochastic volatility

Question 6.2 What do you understand by “pseudo random number”? �

Pseudo random number is a term for numbers that appear to be random but is not. Unlike a“truly”random number, which is generated by a natural random process such as radioactivedecay, pseudo random numbers are obtained from deterministic algorithms (and thus repro-ducible).

Question 6.3 What are criteria for good random number generators and what is their moti-vation? How can they be tested? �

Criteria for good random number generators are:

• Uniform distribution: The empirical distribution of the generated numbers must be equiva-lent to the U(0,1)–distribution.• Serial independence: No autocorrelation between successive random numbers becomes

apparent.• Density: Random numbers can produce only a finite number of different values by

construction; this results in “gaps”between the realizations although the criteria of uniformdistribution is not violated.• Efficiency: The underlying algorithm should be quick and require only little memory.• Reproducibility: The same initial state should always generate an identical sequence (e.g.,

important for program tests).

Question 6.4 Describe a linear congruential generator. Which theoretical properties shouldbe observed? �

26 Random number generation

An LCG generates pseudo random numbers by starting with a value called the seed, andrepeatedly applying a given recurrence relation to it to create a sequence of such numbers.

xi+1 = (axi + c) mod m (6.1)

• a > 1 is called multiplier, c≥ 0: increment and m: modulus (m > a,m > c); m is usually aprime number when c = 0• By the division ui = xi/m, all values can be standardized to the U(0,1)-distribution

The period p (i.e., the smallest number for which the sequence repeat itself) does not alwaysmeet the largest possible value m−1. Based on number theory, combinations of the parameterscan be found that guarantee a full period (i.e., p = m−1)

Question 6.5 What does “lattice structure”mean in the context of random number generators?How can it affect the results of the MC method? �

The pairs, triples, etc. from most congruential pseudo-random number generators are know to lieon a lattice. The idea behind the generated random number is that the numbers should be reallyrandom. That means that the numbers should appear to be distributionally independent of eachother; that is, the serial correlations should be small. How bad the structure of a sequence is (thatis, how much this situation causes the output to appear nonrandom) depends on the structure ofthe lattice.

Consider k-tuples of consecutive realizations:

(ui,ui+1, . . . ,ui+k−1) ∈ [0,1]k

If they are good uniform deviates, the random vectors generated in this way should be uniformlydistributed over [0,1]k.

The tuples (ui, . . . ,ui+k−1) are located on max. (k!.m)1/k parallel hyperplanes in the k-dimension unit cube [0,1]k

Question 6.6 How does a Fibonacci generator work? �

The original Fibonacci motivates trying the formula

Ni+1 := (Ni +Ni−1) mod M

But it won’t lead to true random numbers since the cases xi−1 < xi+1 < xi and xi < xi+1 < xi−1can never occur. Lagged Fibonacci generators use larger gaps ν and µ (ν > µ > 1) that mayalso be variable:

xi+1 = (xi−ν ⊕ xi−µ) mod m, m = 2M;ν ,µ ∈ N

where ⊕ can be any of the binary operations +,−,× or XOR

Question 6.7 What are the advantages of other methods than LCG for uniformly distributedvariates? �

LCGs are very efficient and work well for many applications. However they have some draw-backs:• The maximum period is M− 1, which is much too small for 32-bit generators where

M ≤ 232≈ 109. Meanwhile, Fibonacci generator has the maximum period (2ν−1)2M−1;Mersenne-Twister generator has an extremely long period of 219937−1(≈ 4.3.106001).• d-tuples of such numbers show a regular lattice structure when plotted in d-dimensions.• These generators have been proven to have correlations between numbers that are 2n apart

in the sequence. This can be a major problem for applications using a regular grid orlattice.

27

Question 6.8 How can uniformly distributed random numbers be transformed into standardnormally distributed ones? Describe some methods with their pros and cons. How are theylinked up? �

Application of Central Limit Theorem

Theorem 6.0.2 — CLT. Let Xi(i= 1, . . . ,n) be i.i.d. random numbers with expectation E(Xi)=µ and variance Var(Xi) = σ2 < ∞, then the sequence Zn of the standardized sums

Zn =∑

nj=1(X j−µ)

σ√

n

converges asymptotically for n→ ∞ against the standard normal distribution

It holds for any U(0,1)-distr. random number Ui : E(Ui) = 0.5 and Var(Ui) =112 . For sufficiently

large n we have

Zn =∑

nj=1U j−n/2√

n/12

a standard normally distributed random number. Although the CLT is valid only for n≥ 30 inthe strict sense, one obtains a good approximation already for n = 12 (note than only values in[−6;6] will be generated):

Z =12

∑i=1

Ui−6

Inversion methodAssume that for a random variable X , its cdf F(X) is known. General procedure following theinversion method:• Generate a U(0,1)-distributed random number u• Solve F(x) = u for x, then the variable x is a random number according to the distribution

F .Pros and cons• Preserves monotonicity and correlation but• requires a closed form expression for F(x) (does not exist for normal distribution, possible

remedy is to use approximations)• often very slow

Box-Muller methodAlgorithm:• generate u1 ∼U(0,1) and u2 ∼U(0,1)• Calculate z1 = r cosϕ,z2 = r sinϕ with r :=

√−2lnu1,ϕ := 2πu2

Pros and cons:• The computation of the trigonometric functions is time-consuming

Polar methodAlgorithm:• Generate U(0,1)-distributed random numbers u1, u2• Set v1 := 2u1−1, v2 := 2u2−1, s = v2

1 + v22

• If s > 1, go to 1

28 Random number generation

• Set

z1 :=

√−2lns

sv1; z2 :=

√−2lns

sv2

Notes:• The uniformly distributed random numbers are transformed to the interval [−1,+1]; this

is repeated if they are outside the unit circle• The algorithm needs 4/π iterations on average(more efficient than Box-Muller method

although 1−π/4≈ 21 of the random number pairs are rejected)

Question 6.9 How can random numbers for a normal distribution with arbitrary parametersbe derived in the uni- and multivariate case? �

UnivariateAny standard normally distributed random variable can be transformed into a normally distributedone:

zµ,σ = µ +σz

Multivariate• Calculate the elements ci j of the Cholesky decomposition C such that:Σ =C.C′

• Generate independent N(0,1)-distributed random numbers for example with Marsaglia’spolar method• Calculate for i = 1, . . . ,n the components Xi = µi +∑

ij=1 ci jZ j

• X = (X1, . . . ,Xn) is the desired random vector for N(µ,Σ)

Question 6.10 What would you observe when you must calculate the Delta of an optionwith the MC method (if δV/δS is approximated by the difference quotient)? �

In order to calculate Delta with Monte Carlo method:• Compute an N(0,1) sample ξi

• Calculate underlying price with S(0) = S0

Si = S0e(r−12 σ2)T+σ

√T ξi

• Calculate underlying price with S(0) = S0 +h

Si = (S0 +h)e(r−12 σ2)T+σ

√T ξi

• Calculate the payoff ∧(Si) and ∧(Shi )

• Calculate ∆ of each path

∆i = e−rT (∧(Si)−∧(Shi ))/h

• Finally:

aM =1M

M

∑i=1

∆i

b2M =

1M−1

M

∑i=1

(∆i−aM)2

7 — Simulation of SDEs

Question 7.1 What determines the accuracy of the MC method? What is the computationaleffort for its improvement without further measures? �

Monte Carlo method has two accuracy issues:• The discretization error: financial models usually define processes in continuous time.

However, the sampling step in MC method requires a discretization w.r.t. time, resultingin errors.• The statistical error of MC method.

The computational effort for improvement:• Reduce ∆t• Increase the number of realizations

Question 7.2 What do you understand by a variance reduction method? �

Every output random variable from the simulation is associated with a variance which limitsthe precision of the simulation results. Variance reduction schemes are aimed at improving theaccuracy of MC methods (in the sense of narrower confidence intervals) with no or only modestincrease in the numerical effort instead of increasing the sample size.

The idea is to replace X by a random variable Z that has the same expectation but a lowervariance.

Question 7.3 Which methods do you know? How do they work? What are their prerequi-sites? �

Control Variates• Use the random variable

Zθ := X +θ [E(Y )−Y ],θ ∈ R

(with θ > 0, if cov(X ,Y )> 0 or θ < 0, if cov(X ,Y )< 0)• Obviously, we still have E(Zθ ) = E(X)• The variance is now

var(Zθ ) = var(X−θY ) = var(X)−2θcov(X ,Y )+θ2var(Y )

30 Simulation of SDEs

• There is a value θmin that minimizes the variance:

θmin =cov(X ,Y )

var(Y )(7.1)

• To reduce the variance, θ must be between 0 aand 2θmin

Antithetic Variables• For each original path, this method create a “partner” path, which looks like a mirror

image of the original. For example, if a random variable satisfies X ∼ N(0,1), then also−X ∼ N(0,1).• Use in the i-th iteration zi := [ f (xi)+ f (−xi)]/2 and calculate from the zi’s the correspond-

ing estimator• This approximation in many cases is more accurate than the original one.

var(

f (ui)+ f (1−ui)

2

)=

12[var( f (ui))+ cov( f (ui), f (1−ui))]

If f is monotonic, cov( f (ui), f (1−ui)) will be negative thus:

var(

f (ui)+ f (1−ui)

2

)≤ 1

2var( f (U))

Question 7.4 How can the accuracy/convergence of the control variate method be improved?What must be observed? �

Solve for θ that gives the minimum variance:

θmin =cov(X ,Y )

var(Y )

Usually cov(X ,Y ) is unknown, an estimator for it can be calculated from the random numbersgenerated during MC-simulation.

Question 7.5 Name some applications from derivative pricing where the methods are useful.�

These methods can be used for the Monte Carlo estimation of path-dependent exotic options.The classic example for control variates method is the pricing of an arithmetic average price

Asian option. An analytical formula exists for the geometric average price Asian option thus wemay use this option as a control variate when valuing the arithmetic version.

Question 7.6 What are the properties of a Wiener process? �

A Wiener process (or Brownian motion; notation Wt or W ) is a time-continuous process with theproperties• W0 = 0• Wt −Ws ∼ N(0, t− s) for all 0≤ s≤ t, this implies: E(Wt −Ws) = 0 and var(Wt −Ws) =

E[(Wt −Ws)2] = t− s

• All increments ∆Wt :=Wt+∆t−Wt on nonoverlapping time intervals are independent: Thatis, the displacements Wt1−Wt1 and Wt4−Wt3 are independent for all 0≤ t1 < t2 ≤ t3 < t4• Wt depends continuously on t

31

Question 7.7 Which methods for the solution of SDEs do you know? How do they workand what distinguishes them? �

Stochastic differential equation:

dXt = a(Xt , t)dt +b(Xt , t)dWt (7.2)

Its solution is a stochastic process Xt that satisfies:

Xt = X0 +∫ t

0a(Xs,s)ds+

∫ t

0b(Xs,s)dWs (7.3)

Two of the numerical methods for SDE are Euler-Maruyama and Milstein method.

Euler-MaruyamaThis is an extension of the classical Euler method. We approximate the original SDE as follows:∫ t j+1

t j

a(Xs,s)ds≈ a(y j, t j)∆t j and∫ t j+1

t j

b(Xs,s)dWs ≈ b(y j, t j)∆Wj

where:

∆t j = t j+1− t j =∫ t j+1

t j

ds; ∆Wj =Wj+1−Wj =∫ t j+1

t j

dWs

Then we have:

y j+1 = y j +a(y j, t j)∆t j +b(y j, t j)∆Wj (7.4)

The Euler-Maruyama scheme is• weakly convergent with order 1• strongly convergent with order 0.5

Milstein methodIt has the form:

y j+1 = y j +a(y j)h+b(y j)∆Wj +12

b′(y j)b(y j)(∆W 2j −h) (7.5)

For non-autonomous SDEs, it has a corresponding structure:

y j+1 = y j +a(y j, t j)h+b(y j, t j)∆Wj +12

b′(y j, t j)b(y j, t j)(∆W 2j −h) (7.6)

The high order of the Milstein scheme comes from the fact that in Euler-Maruyama method, aterm of magnitude h is ignored.

The Milstein scheme is:• weakly convergent with order 1• strongly convergent with order 1

Question 7.8 How is the absolute error of the approximation defined? �

We assume that step sizes ∆t1, ∆t2,. . . are identical. The approximation w.r.t. time induces anerror. Since the accuracy of the approximation depends on h, we denote the solution of the(approximated) SDE at time T by yh

m. We simulate the SDE for various step sizes h and comparethe results with the analytical solution XT for a case where the latter is known. Then the absoluteerror is calculated:

ε(h) := E(|XT − yhm|) (7.7)

It is desirable that the error will vanish for h→ 0, i.e., that we have a convergence against thetrue solution XT .

32 Simulation of SDEs

Question 7.9 What is the motivation for higher order schemes? �

If we decrease the step size by a factor k and the error of the approximation decreases by thefactor kγ , then the order is γ . Formally, the order is γ if a constant C exists such that:

ε(h)≤C.hγ (7.8)

Certainly, we favor a scheme with high order since it will not require much more computationalefforts to reduce the error.

Question 7.10 What distinguishes strong and weak convergence? �

Strong convergenceA numerical scheme is strongly convergent with order γ if

E(|XT − yhm|)≤CT .hγ (7.9)

where the constant C depends on T and the considered SDE.

Weak convergenceIn many applications, we do not require that y j.∆t ≈ xt j holds also for intermediate points att j < T . Instead we want to calculate the expectation of a function g:

E[g(yhm)]≈ E[g(XT )]

(e.g., the payoff function of an option). This is a weaker condition than yT ≈ XT . A numericalscheme is weakly convergent with order β if

|E(g(XT ))−E(g(yhm))| ≤C.hβ (7.10)

for every polynomial g; the constant C depends on g, T and the considered SDE

Question 7.11 When is strong and when is weak convergence appropriate? What does thismean for the choice of a particular approximation scheme? �

Weak convergence is useful• if the distribution at T only is required• for pricing European path-independent options

Strong convergence is a pathwise property and• should be used if whole paths play a role• is appropriate for path-dependent options

8 — Exotic options

Question 8.1 What characterizes exotic options? How can they be distinguished? �

Exotic options are options whose payoff is different from European or American options. Theycan be distinguished by:• the nature of its path dependency - the way in which the payoff depends upon the asset

path S(t) for 0≤ t ≤ T• single- vs. multi-asset• possibility of early exercise, but with restrictions

Question 8.2 Which types of exotic options do you know? �

Barrier options• Payoff > 0 if price of underlying S crosses (or does not cross) a pre-defined level.

– Down-and-out call: Payoff = 0 if price S in [0,T ] crosses barrier B < S0; otherwisepayoff like European call.

– Down-and-in call: Payoff = 0 unless price S in [0,T ] crosses barrier B < S0; other-wise payoff like European call.

– Up-and-out call: Payoff = 0 if price S in [0,T ] crosses barrier B > S0; otherwisepayoff like European call.

– Up-and-in call: Payoff = 0 unless price S in [0,T ] crosses barrier B > S0; otherwisepayoff like European call.

• Analogously for puts• Options have lower value than European options, because the payoff opportunities are

more limited compared to Europeans.• An analytical formula for the option value can be obtained by solving the Black_Scholes

PDE with appropriate final time and boundary conditions.• Variations:

– Double barrier options impose upper and lower bounds on the asset price, and payoffmay knock in (or out) if either barrier is (or both barriers are) crossed.

– Partial barrier options have barriers that apply for a limited time interval.– Parisian options have barriers that must remain crossed for some pre-specified amount

of time.

34 Exotic options

Lookback option• Payoff depends either on the minimum or maximum value attained by the asset.

Smax := max[0,T ]

S(t) and Smin := min[0,T ]

S(t)

• There are two broad categories, fixed and floating strikes:– Fixed strike lookback call: Payoff in T is max(Smax−K,0)– Fixed strike lookback put: Payoff in T is max(K−Smin,0)– Floating strike lookback call: Payoff in T is S(T )−Smin– Floating strike lookback put: Payoff in T is Smax−S(T )

• Lookback options have higher value than the corresponding Europeans. The fixed strikelookbacks differ from European options in that the final asset value is replaced by the’best’ asset price. In the case of floating strike, the strike price becomes the extremelyfavorable minimum price for a call and maximum price for a put.

Bermudan options• Are between European and American options as the exercise is only possible at a limited

set of points in time

Canary options• Are between European and Bermudan: the holder may exercise at some predefined points

in time but not before a certain period has passed.

Shout options• Allows the holder to “shout” once between times 0 and T (possibly more often in more

complex cases):{max(S(T )−K,S(τ)−K) if holder shouted at time τ

max(S(T )−K,0) otherwise(8.1)

Question 8.3 What types of Asian options exist and how can they be priced efficiently? �

• The payoff of Asian options depends on average prices:– Average price Asian call:

max(

1T

∫ T

0S(τ)dτ−E,0

)– Average price Asian put:

max(

E− 1T

∫ T

0S(τ)dτ,0

)– Average strike Asian call:

max(

S(T )− 1T

∫ T

0S(τ)dτ,0

)– Average strike Asian put:

max(

1T

∫ T

0S(τ)dτ−S(T ),0

)

35

• In practice the continuous average cannot be calculated from discrete observations. There-fore it is approximated by the arithmetic or geometric average:

1n

n

∑i=1

S(ti)→ arithmetic

(n

∏i=1

S(ti)

)1/n

→ geometric

• Due to path dependency, Asian options are more difficult to price than Barrier or Lookbackoptions. Exact solutions in the BS framework can only be found in some cases, e.g., thegeometric average price Asian call/put. In general, Monte Carlo methods are appropriate.

Question 8.4 How can Bermudan, Canary and shout options be priced? �

• Bermudan and Canary options can be priced using binomial method (analogous to Ameri-can options but check for early exercise only at the pre-described dates)• Shout option: if shout happened at time τ , the payoff may be written

max(S(T )−S(τ),0)+S(τ)−K (8.2)

Once τ and S(τ) are known, the first term corresponds to the payoff of a European optionand may be calculated with the BS formula for K = S(τ) and t = τ .Again, the Binomial method can be applied, but instead of checking for early exerciseas for American options, the value from shouting at a certain node is compared with thevalue of the continuation strategy:

V in = max

[value (8.2) from shouting at (ti,Si

n),e−rδ t (pV i+1

n+1 +(1− p)V i+1n)]

(8.3)

Question 8.5 What impact does the convexity of an option have on the valuation of afinancial derivative? Where does it result from? �

If the payoff is convex then

E[P(ST )]≥ P(E[ST ])

Using a Taylor series approximation around the mean of S:

E[ f (S)] = f (E[S])+12

f ′′(E[S])E[ε2]

This shows that the convexity f ′′(E[S]) adds an inherent value to an option at t = 0. Any intuitionwe may get from linear contracts (forwards and futures) might not be helpful with non-linearinstruments such as options.

Question 8.6 What is the meaning of Jensen’s inequality? �

Definition 8.0.2 — Jensen’s inequality. states that if f (·) is a convex function and x is arandom variable then

E[ f (x)]≥ f (E[x]) (8.4)

Jensen’s inequality justifies why non-linear instruments, options, have inherent value.

9 — Finite difference methods

Question 9.1 Describe the principle behind finite difference methods. �

Finite difference methods involve approximating the differential operator by replacing thederivatives in the equation using differential quotients. The time and state space are discretizedat a finite number of points and approximations of the solution are computed at the space or timepoints.

Question 9.2 What is understood by explicit and implicit methods? �

Explicit Method (FTCS)The temperature at time v+1 depends explicitly on the temperature at time n. The explicit finitedifference discretization of heat equation is:

yv+1i − yv

i

∆τ−

yvi+1−2yv

i + yvi−1

(∆x)2 = 0 (9.1)

Rewrite (9.1):

yv+1i = λyv

i+1 +(1−2λ )yvi +λyv

i−1 (9.2)

where λ := ∆τ/(∆x)2 is called mesh ratio. Since we know yvi+1, yv

i , and yvi−1, we can compute

yv+1i .

Implicit Method (BTCS)yv

i − yv−1i

∆τ−

yvi+1−2yv

i + yvi−1

(∆x)2 = 0 (9.3)

In implicit method, we no longer have an explicit relationship for yv+1i−1 , yv+1

i and yv+1i+1 . Instead,

we have to solve a linear system of equations.

yvi = yv−1

i +λ (yvi+1−2yv

i + yvi−1) (9.4)

38 Finite difference methods

Question 9.3 What distinguishes the FTCS and the BTCS method and why? What deter-mines their convergence? �

FTCS vs BTCS• FTCS involves approximating the time derivative δ/δ t by the scaled forward difference in

time.• BTCS replaces the forward difference in time in FTCS by a backward difference.

Figure 9.1: FTCS and BTCS

ConvergenceLax equivalence theorem states that a method converges if and only if its local accuracy tends tozero as ∆x→ 0, ∆τ → 0 and a stability criterion is satisfied.

Question 9.4 What is understood by stability and how does it relate to the underlying gridin the corresponding methods? �

Definition 9.0.3 A finite difference method generating approximations yvi is stable (in the

sense of von Neumann) if, ignoring initial and boundary conditions, under the substitutionyv

i = ξ veiβ i∆x, it follows that |ξ | ≤ 1 for all β∆x ∈ [−π,π].

FCTS is only useful for λ ≤ 12 . If we consider refining the gird, that is reducing ∆τ and ∆x to get

more accuracy, then this condition must be respected. BTCS is unconditionally stable for allλ > 0. A large value of λ does not give rise to any instabilities.

Question 9.5 Describe the Crank/Nicolson method and their relation to the former two. �

FTCS and BTCS are both of local accuracy O(k)+O(h2). The O(k) accuracy in time arises fromthe use of first order forward or backward differencing in time. The Crank-Nicolson method usesa clever trick to achieve second order in time without the need to deal with more than two timelevels.

Crank-Nicolson was obtained from taking the average of the FTCS (for v) and BTCS (forv+1) equations.

yv+1i − yv

i

∆τ−

yv+1i+1 −2yv+1

i + yv+1i−1 + yv

i+1−2yvi + yv

i−1

2(∆x)2 = 0 (9.5)

Its local accuracy is the average of the corresponding local accuracies.

Question 9.6 Compare the three methods. �

• Explicit method– easy to implement– stability problem (stable only for λ ≤ 0.5)– approximation error of order O(∆τ)+O((∆x)2)

• Implicit method

39

– more difficult to implement– Unconditionally stable (for λ > 0)– approximation error of order O(∆τ)+O((∆x)2)

• Crank-Nicolson method– More difficult to implement– Unconditionally stable (for λ > 0)– approximation error of order O((∆τ)2)+O((∆x)2)

Question 9.7 How can these methods be applied to the solution of the BS equation? �

Question 9.8 How do finite differences relate to the binomial method? �

Similarities• Both are based on discretization of the time and state space,• advance in the time direction and• are designed to be more accurate when the discretization is refined

Differences• The binomial method works backward in time; it starts with option values at t = T and

finishes with a single value for t = 0 and S = S0• Finite difference methods produce option values at all grid points; in particular at t = 0

option values are available for all initial asset prices S ∈ 0,∆x,2∆x, . . . ,L