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1Pricing of Basket Options
Radhika NangiaMATH 608D87053120
"The four most dangerous words in investing are: this time its
different."- Sir John Templeton
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21 Introduction
In the fast growing financial markets, it has become imperative
to get accurate prices under uncertainties.Options are the
essential tools in hedging and risk management. Hence, numerical
methods are employedto price accurately and prevent any arbitrage
opportunities.An option on a stock A is a financial device that we
can buy at time t0 for a price V that allows us toeither buy or
sell the stock A at a fixed price K at later times. By saying the
price is fixed, we meanthat the price is already known at time
t0.Here we will consider the so-called European Call Option which
allows us to buy one unit of the stockat a fixed time t1 t0. A
Basket Option is one which is based on a basket of stocks, for
example, theindex option. An option is only exercised if it is
in-the-money i.e. it is in profit. The payoff for theEuropean call
option is written in the following manner
u(S, T ) = max(ST K, 0)where ST is the stock price at time T ,
and K is the exercise price of the option.For pricing simple
options on one underlying, the Black and Scholes model leads to a
closed formsolution since the stock price at a fixed time follows a
lognormal distribution. However, using thefamous Black and Scholes
model for a collection of underlying stocks, does not provide us
with a closedform solution for the price of a basket option. The
difficulty stems primarily from the lack of availabilityof the
distribution of a weighted sum of non-independent lognormals, a
feature that has hamperedclosed-form basket option pricing
characterization. Indeed, the value of a portfolio is the
weightedaverage of the underlying stocks at the exercise
date.Hence, we need numerical methods to price the equations. There
are two ways to go about it, usingMonte-Carlo methods which is a
much easier implementation or using Finite Differences which
becomesmuch more complicated for this higher degree simulation. We
will price the S& P 100 Index Optionwhich is popularly known by
its ticker symbol XEO, which as its name suggests has 100
underlyingstocks.
1.1 Some Terminology
This section introduces us to some of the "jargon" in finance.
It is necessary to clarify some of the termsbefore giving a
mathematical description.Financial Derivative A financial
derivative in simple terms is an instrument which is derived
fromvalues of some underlying assets/ variables.Portfolio- A
portfolio is the collection of all shares, options and other
derivatives owned by a investor.Arbitrage- Arbitrage indicates that
it is not possible in a financial market to make risk-free
profitslarger than just placing money in the bank.
There are several different types of portfolios. One of them is
the arbitrage portfolio where sharesand options are combined in
such a way that risk is eliminated and the portfolio makes money
withprobability 1 in time. This is an instantaneously risk-free
money making price process.Risk-free interest rate- The risk-free
interest rate, r(t), is the growth of money, M in time or theprofit
one makes from the interest rate when placing money in the bank
M(t) = K exp(t
0
r(t)dt)
Volatility- It is related to the standard deviation of the stock
price of a share. It is an indication forthe random behaviour of
the market.
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1.1 Some Terminology 3
Return- It is the ratio of money gained or lost (whether
realized or unrealized) on an investmentrelative to the amount of
money invested.Basket Option- A basket option is a financial
derivative, more specifically an exotic option, whoseunderlying is
a (weighted) sum or average of different assets that have been
grouped together in abasket. For example an index options, where a
number of stocks have been grouped together in anindex and the
option is based on the price of the index.Call Option- Call options
are a type of security that give the owner the right to buy N
number ofshares of a stock or an index at a certain price by a
certain date. That "certain price" is called thestrike price, and
that "certain date" is called the expiration date.European Option-
This option that can only be exercised at the end of its life, at
its maturity.Why buy Options?Options are used for several purposes.
The two most important are speculating and hedging.
1. Speculation is betting on the movement of a security. If the
holder buys a call, he expects thatthe stock price will increase.
The exercise price will be K. If the stock price S is greater than
K,the call will be exercised and the net profit of the option will
be where S K C, where C is thecost of the option. C is an important
parameter in the Black-Scholes analysis. Speculation withoptions
involves a greater risk than speculation with assets: the profit
and losses are multipliedwhen using options.
2. The second purpose to use options is hedging. It can be used
like an insurance policy; just asone would insure their house or
car, options can be used to insure ones investments against
adownturn. Suppose one wanted to take advantage of technology
stocks and their upside, but alsowanted to limit any losses. By
using options, you would be able to restrict your downside
whileenjoying the full upside in a cost-effective way. So, options
to reduce the risk of a portfolio.
An image showing the volatility in stocks which provide profit
making opportunities.
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42 The Black-Scholes Model
2.1 Assumptions
The assumptions to derive the Black-Scholes differential
equation are as follows:
Geometric Brownian Motion- The stock price follows a geometric
Brownian motion withconstant drift and volatility. Infinite
Divisibility-Stock can be sold in arbitrary non-integer amounts.
Infinite Credit- Market participants can borrow or lend arbitrary
amounts of money at arisk-free interest rate. No Transaction Costs-
The money paid by the buyer is exactly equal to the money
receivedby the seller. No Dividends- There are no dividends during
the lifetime of the derivative. No Arbitrage- There are no
risk-less arbitrage opportunities. Short Selling- Market
participants can borrow arbitrary amounts of stock, for an
arbitrary
amount of time, with no interest. Constant Risk-Free Rate- The
risk-free rate of interest, r(t) is a constant and the same forall
maturities i.e. r(t) = r. Infinite Liquidity- Market participants
can buy or sell a unit of stock at any time. Infinite Depth- The
sale of a unit of stock does not affect sale price of other units
of stock. Discrete Time- The time variable t increases in discrete
steps of size dt. No Storage Costs- Market participants can hold
onto arbitrary amounts of stock, for anarbitrary amount of time, at
no cost.
2.2 Stochastic Model
There are two sources of return in a price process modelled as a
stochastic process, namely, a deterministicand a stochastic
contribution. If is the average rate of growth of the asset price,
also known as the"drift", the deterministic contribution in time dt
is found to be dt. The other contribution relates tothe random
change in the asset prices. With the volatility related to the
standard deviation of thereturns and dX a sample from a normal
distribution, the contribution is assumed to be dX. Theresulting
equation reads
dS
S= dt+ dX (1)
where dSS is the return. This is the stochastic differential
equation. The normal distribution used is aWeiner process with the
following properties : E(dX) = 0 and E(dX)2 = dt. Also
E(dS) = E(Sdt+ SdX) = E(Sdt) + E(SdX) = Sdt; E(dX) = 0
V ar(dS) = E(dS2) (E(dS))2 = E((Sdt+ SdX)2) (Sdt)2 =
(2S2dX2)since E(S2dXdt) = 0. The standard deviation is the
square-root of the variance, so is proportionaltoV ar(dS)/S.
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2.3 Parabolic Equation 5
2.3 Parabolic Equation
The Black Scholes Model is a parabolic differential equation
which is given as follows:
u
t+ 12
2S22u
S2+ rS u
S ru = 0 (2)
where, r is the riskfree interest rate and is the volatility of
the underlying asset. The key financialinsight behind the equation
is that one can perfectly hedge the option by buying and selling
theunderlying asset in just the right way and consequently
eliminate risk. This hedge, in turn, impliesthat there is only one
right price for the option, as returned by the Black Scholes
formula.
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63 Problem Formulation
3.1 Monte Carlo
The Monte Carlo Approach is a relatively simple technique to
implement this otherwise very complicatedhigher dimensional problem
in numerical methods. The other advantages are that it does not
requireboundary conditions and other parameters unlike the other
equations. For pricing a basket option weare interested in the
following stochastic differential equation
dSi = rSi +dj=1
ijSidWj , i = 1, 2, , d (3)
where ij is the covariance and dWj is a Weiner process and can
be expressed as dWj =dtgj , with gj
being independent and normally distributed random numbers.Now to
get a form we will use for Monte Carlo, I will use Itos Calculus
covered in the last day of class:TheoremIf Xt solves an s.d.e. dXt
= udt + dWt & g : R R, g C2 and Y := g(Xt) solves dYt =g(Xt)dXt
+ 12g(Xt)(dXt)2 We will assume that Wj are independent B.Ms
dSiSi
= rdt+dj=1
ijdWj
d(logSi) =dSiSi 12S2i
(dS2i )
From the problem we have
dSi = rSidt+dj=1
ijSidWj
(dSi)2 =dj=1
2ijS2i dt
as Wi and Wj are independent. So,
d(logSi) =rSidt+
dj=1 ijSidWj
Si 12S2i
dj=1
2ijS2i dt
logSt logS0 = (r dj=1
2ij)dt+dj=1
ijdWj
St = S0 exp ((r dj=1
2ij)t+dj=1
ijWj)
This is modelled as
St = S0 exp((r dj=1
2ij)(T/N) +dj=1
ij
T
Ngj)
where T is the expiry of the option and N the number of
intervals at which I evaluate for each iterationand gj the
uncorrelated normal r.v.In a lot of places this is modelled
differently by taking correlated B.Ms and separating them down
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3.2 Finite Difference 7
to uncorrelated set of equations. This would need us to evaluate
the Cholesky decomposition. TheCholesky factorization says that
every symmetric positive definite matrix A has a unique
factorizationA = LL where L is a lower triangular matrix and L is
its conjugate transpose. In my case, since Idid not have a perfect
correlation matrix, so the lower triangular matrix turned out to be
zero. Hence Idid it like mentioned above.
Finally, after running the simulation multiple number of times,
the payoff is computed as
max(di=1
iSi K, 0)
3.2 Finite Difference
I will briefly write about this method to give an idea of how
complicated it gets when solving the sameequation as a parabolic
differential equation for n underlying assets. The equation is
given as
u
t+ 12
ni,j=1
2i 2j ijSiSj
2u
SiSj+ r
ni=1
Siu
Si ru = 0 (4)
where ij is the correlation between two assets. Correlation is
the extent to which assets perform inrelation to one another. For
instance, its widely considered good practice to reduce volatility
in yourportfolio by investing in a variety of assets whose values
rise and fall independently of one another, i.e.negatively
correlated with each other.
3.2.1 Principal Component Analysis
Solving the differential equation in its original form is a near
impossible task in a short time, so weneed to adopt some
short-cuts. First, we need to find the assets most responsible for
determining priceof the option, which is done by PCA. Principal
component analysis (PCA) is a mathematical procedurethat identifies
the direction of maximum variance or the principal component.
I will briefly illustrate the method for the problem in hand: We
first need to find the eigenvectors of thecovariance matrix of
stocks and denote it by Q. Then, we use the transformation by using
PCA and get
x = QS (5)
This transformation leads to a very cumbersome equation, hence
we make another transformation. Thistransformation to the principal
components can be written as
x = Q ln(S) + b (6)
where = T t and bi = dj=1 qij(r 2j2 ). By applying change of
variables to the Black ScholesEquation we get,
u
= 12
di=1
i2u
x2i ru (7)
where (x, ) Rd (0, T ) and i is the eigenvalue number i of the
covariance matrix.The payoff for the basket option is,
u(x, 0) = max(di=1
i exp(dj=1
qijxj), 0) (8)
where x Rd
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3.2 Finite Difference 8
3.2.2 Asymptotic Expansion
Now the equation (8) may be truncated to any number of
dimensions. But to account for all dimensionswithout solving the
original higher dimensional problem, each of the non-principal
dimensions areapproximated by a linear asymptotic expansion,
similar to taylor series expansion. This asymptoticexpansion is
given by
u = u(1) +dj=2
ju
j|=(1) +O(|| (1)||2)
where u(1) is the solution in the principal axis, is a parameter
vector of eigenvalues and (1) is aparameter vector when truncating
the equation to dimension 1. Using a finite difference approach,
theterms in the sum maybe approximated by:
u
j|=(1) =
u(1,j) u(1)j
+O(2j )
where u(1,j) corresponds to the solution of the 2D problem on
the plane spanned by the principal axisand variable j.
The following is the figure I got from running the finite
difference method with controlled inputs.
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94 Implementation
4.1 Monte-Carlo
For this implementation some of the concepts in class were very
vital. I will describe stepwise how Iimplemented this and some of
the theory behind it.
1. Calculate the covariance matrix from the daily returns of
each stock.
cov(i, j) =nd=1(Ri Ri)(Rj Rj)
n 1for stock i, j and where n is the number of time periods.
2. Correction for invalid correlation matrix. Many times the
data that one gets does not complywith the requirement of symmetry
and positive semi-definiteness. The invalid correlation matrixhas
negative eigenvalues. I used the method in Jckel [3] called
Spectral Decomposition.The method is as follows:
(a) Calculating the eigenvalues i and the right hand side
eigenvectors si of C, a real andsymmetric matrix.
C.S = S. with = diag(i)
(b) Setting all negative i to 0.
: i =
i : i 00 : i < 0(c) Muliplying the column vectors si by the
square roots of their associated corrected eigenvalues
i and arrange them as columns of B.
T : ti = [m
s2imm]1
B = S
B =TB =
TS
(d) Finally normalizing the row vectors B to unit length gives
us B.(e) The acceptable correlation matrix is given by
C = BB
3. Generate uncorrelated uniformly distributed random numbers.
As all pseudo-random generatorsare flawed, I use two of them
here:
(a) Mersenne Twister: Period of this sequence is a Mersenne
number i.e a prime number thatcan be written as 2n 1
(b) Congruential Generator
4. Convert uniformly distributed random numbers to normally
distributed random numbers byZiggurat method (inbuilt in
Matlab)
5. Simulate the Brownian motion by a random walk and calculate
payoffs.
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4.2 Finite Differences 10
4.2 Finite Differences
As discussed above the d dimensional problem is reduced to one
1D problem which is spanned by theprincipal axis and a 2D problem
in the plane spanned by the principal axis and one other
dimensioncorresponding to the variable j giving the solution
u(1,j). So we have to repeat the second step for theevery dimension
we have. In my case, we have to solve one 1D problem and 99 2D
problems.The scheme used is BDF2 method to solve the problem as it
is unconditionally stable and second orderaccurate in both space
and time. Most of the error introduced in this calculation is
through dimensionreduction approximation.
4.2.1 Scheme
The BDF-2 is a two step method, and hence we require another
scheme to do the first step. I usedBackward Euler for the first
time step.The Scheme is given by the following equations
uni =23tP (u
ni ) +
43u
n1i
13u
n2i (9)
where the spatial difference operator P in the 1D case is given
by
P (uni ) =12uni1 2uni + uni+1
x2 runi (10)
and in the 2D case it is given by
P (uni ) =121
uni1 2uni + uni+1x2 +
12d
uniNp 2uni + uni+Npx2d
runi (11)
The logarithmic domain must be specified. It was arbitrarily
chosen as x1 [0, 2x01] for the principalaxis and xd [x0d 3, x0d +
3] for the other axes. Here x0 corresponds to the transformation of
S0, thecurrent price of underlying assets.
The boundary conditions that were used were:
u = 0
u =di=1
ien
j=1 qij(xjbj) Ker
2u
x2d= 0
at x1 = 0at x1 = 2x01at xd = x0d 3, x0d + 3
(12)
with the linear boundary conditions discretized as
uni = 2uni+Np uni+2Np ,uni = 2uniNp uni2Np ,
i = 1, 2, , Npi = 1 + (Nd 1)Np, , NdNp
(13)
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11
5 Results
The following two figures illustrate how previously uncorrelated
sequences of random numbers undergometamorphosis after Cholesky
decomposition.
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12
The following figure illustrates the correlation matrix
The following is the 3 simulated paths of the price process for
the XEO option.
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13
I drew the following figures taking a basket of only two
underlying assets, Apple Inc and Google. Thisfollowing figure shows
the price process followed by the two assets in 1 iteration.
This figure shows the price process followed by the basket of
two options:
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5.1 Greeks 14
5.1 Greeks
The Greeks are quantities representing the sensitivity of the
price of derivatives such as options to achange in underlying
parameters on which the value of an instrument or portfolio of
financial instrumentsis dependent. The name is used because the
most common of these sensitivities are often denotedby Greek
letters. Collectively these have also been called the risk
sensitivities,risk measures or hedgeparameters. The Greeks are
vital tools in risk management. Each Greek measures the sensitivity
of thevalue of a portfolio to a small change in a given underlying
parameter, so that component risks maybe treated in isolation, and
the portfolio rebalanced accordingly to achieve a desired exposure;
see forexample delta hedging.
1. Delta -, measures the rate of change of option value with
respect to changes in the underlyingassets price. Delta is the
first derivative of the value V of the option with respect to the
underlyinginstruments price S. Using finite differences it can be
approximated as
= vS0
v(S0 + S0) v(S0)S0The following figure illustrates the Delta
calculated for two stocks:
The implication of Delta is that if for example, the delta of a
portfolio of options in XYZ (expressedas shares of the underlying)
is +2.75, the trader would be able to delta-hedge the portfolio
byselling short 2.75 shares of the underlying. This portfolio will
then retain its total value regardlessof which direction the price
of XYZ moves.
2. Gamma- , measures the rate of change in the delta with
respect to changes in the underlyingprice. Gamma is the second
derivative of the value function with respect to the underlying
price.Using finite differences it can be approximated as
= 2v
S20 v(S0 + S0) 2v(S0) + v(S0 + S0)S20
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5.1 Greeks 15
The following figure illustrates the Gamma calculated for two
stocks:
The values I worked with are :
Risk free rate r Expiry Time T Strike Price K0.16% 1/12 yr
805
The results were as follows:
Method Basket size & contents Value of option No of
iterations Time TakenMonte Carlo XEO option : 100 stocks 28.681807
5 104 3549.1475 secsMonte Carlo 2 basket option: AAPL & GOOG
254.0287 106 1564.1293 secs
Finite Difference 2 basket option: AAPL & GOOG 251.21133 301
1146.297641 secs
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16
The graph between value of option and value of the underlying
portfolio : XEO option
6 Conclusion
Monte Carlo is heavily used in the Financial Markets to deal
with higher dimensional data, infact thestatistics say that Monte
Carlo is used 60% of the time, finite difference methods 30% of the
time, andother methods the rest of the time. Finite difference
methods are preferred for data that has smallerdimensions 7 ,
usually Forex options (FX) because they have a faster convergence
rate , but MonteCarlo needs averagely about 107 iterations. It is
however preferred for its ease of use when dealingwith higher
dimensional data. There are many modifications that could have been
done here in thisproject if I had time. For e.g. instead of pseudo
random numbers one can use low discrepancy numbers,Sobol numbers to
perform this simulation. Also, I could have used variance reduction
techniques to getbetter results.
References
1. Bjrk, T. Arbitrage Theory in Continuous Time. Oxford
University Press, 2009.2. Brandimarte, P. Numerical Methods in
Finance and Economics. John Wiley & Sons,Ltd, 2002.3. Jckel, P.
Monte Carlo Methods in Finance. John Wiley & Sons,Ltd, 2002.4.
Reisinger, C., and Wittum, G. Efficient hierarchical approximation
of high-dimensional option
pricing problems. SIAM Journal on Scientific Computing 29, 1
(2007), 440458.
Copyright (c) 2013, All rights reserved by Radhika Nangia.
Duplication is strictly forbidden.