Pricing of basket options

Project Abstract

Basket options are financial derivatives that are based on the performance of a portfolio of underlying assets rather than just a single asset. These options are widely used by investors and financial institutions for hedging purposes and to gain exposure to a diversified set of assets. Pricing basket options accurately is crucial for managing risk and maximizing returns in a portfolio. The pricing of basket options is a complex task due to the multiple underlying assets involved and their correlations. Various pricing models have been developed to estimate the value of basket options, including the Black-Scholes model, Monte Carlo simulations, and copula-based approaches. Each of these models has its strengths and weaknesses, making it important for investors to carefully choose the most suitable model for their specific needs. One of the key challenges in pricing basket options is determining the correlations between the underlying assets. Inaccurate correlation estimates can lead to mispricing of options and potentially significant losses for investors. Therefore, robust methods for estimating correlations, such as historical data analysis, implied correlations from market prices, or copula models, are essential for accurate pricing. Another important factor in pricing basket options is the choice of the option pricing model. While the Black-Scholes model is commonly used for pricing European basket options, it has limitations when dealing with more complex options or non-normal distributions of asset returns. Monte Carlo simulations provide a more flexible approach that can handle a wide range of option types and underlying asset distributions. Additionally, copula models have gained popularity in pricing basket options due to their ability to capture complex dependence structures between assets. Copulas can model the marginal distributions of individual assets separately from their joint distribution, allowing for more accurate pricing of basket options. Overall, pricing basket options requires a combination of accurate correlation estimation, appropriate choice of pricing model, and consideration of the unique characteristics of the underlying assets. By utilizing advanced pricing models and robust estimation techniques, investors can better manage risk and make informed decisions when trading basket options in financial markets.

Project Overview

<p> </p><p>General Introduction<br>In this chapter we give some denitions in probability theory needed for our<br>thesis and provide some introduction to the work.<br>1.1 Preliminaries:<br>We begin by introducing a number of probabilistic concepts.<br>1.1.1 -algebra<br>Let<br>be a non-empty set and B a non-empty collection of subset of<br>, B is<br>called a -algebra if the following properties hold:<br>i<br>2 B<br>ii A 2 ) A0 2<br>iii fAj : j 2 Jg B )<br>S<br>j2J<br>Aj 2 B for any nite or innite countable<br>subset J of N<br>7<br>1.1.2 Probability Space<br>1. Let<br>be a nonempty set and B a – algebra of subsets of<br>. Then<br>the pair (<br>; B) is called a measurable space and a member of B is<br>called a measurable set.<br>2. Let (<br>; B) be a measurable space and : B ô€€€! R be a real valued<br>map on . Then is called a probability Measure if the following<br>properties hold:<br>i (A) 0 8A 2<br>ii (<br>) = 1<br>iii For fAngn2N , with Aj Ak = ; for j 6= k<br>(<br>S<br>n2N<br>An) =<br>P<br>n2N<br>(An) i.e is -additive(or countably additive).<br>3. If (<br>; ) is a measurable Space and is a probability measure on<br>(<br>; ),then the triple (<br>; ; ) is called Probability Space.<br>1.1.3 Borel -algebra<br>If is a collection of subsets of<br>,then the smallest -algebra of subsets of</p><p>which contains , denoted by () is called the -algebra generated by<br>.<br>Let X be a nonempty set and a topology on X, i.e is the collection<br>of all open subsets of X.Then ( ) ia called the Borel -algebra of the<br>topological space (X; ).<br>8<br>1.1.4 A random variable:<br>Let (<br>; ; ) be an arbitrary probability space,B(Rd) be the Borel -algebra<br>of Rd and (Rd;B(Rd)) the d-dimension Borel measurable space. Then, a<br>measurable map X :<br>ô€€€! Rd is called a random vector. In the case<br>d=1, X is called a random variable.<br>1.1.5 Probability distribution<br>Let (<br>; ; ) be a probability space, (Rd; (Rd)) be the d-dimensional Borel<br>measurable space, and X :<br>ô€€€! Rd a random vector. Then the map<br>X : (Rd) ô€€€! [0; 1] dened by X(A) = (Xô€€€1(A)), A 2 (Rd) is called<br>the probability distribution of X:<br>1.1.6 Normal distribution<br>A standard univariate normal distribution(i.e of mean zero and variance 1)<br>has density (x) = p1<br>2<br>eô€€€x2<br>2 ,ô€€€1 &lt; x &lt; 1 and cumulative distribution<br>function<br>(x) =<br>R x<br>ô€€€1<br>p1<br>2<br>eô€€€u2<br>2 du<br>In general a normal distribution with mean and variance 2; &gt; 0<br>has density ;(x) = 1</p><p>p<br>2<br>eô€€€(xô€€€)2<br>22 and cumulative distribution function<br>;(x) = (xô€€€<br>)<br>The notation X N(; 2) means the random variable X is normally<br>distributed with mean and variance 2.<br>If Y N(0; 1) (i.e Y has the standard normal distribution, then +Y<br>N(; 2). Thus given a method for generating the samples Y1; Y2; from<br>the standard normal distribution, we can generate samples X1;X2; from<br>N(; 2). It therefore suces to consider methods for sampling from N(0,1).<br>[2]<br>9<br>1.1.7 A d-dimensional Normal distribution<br>This is characterised by a d-vector and a dd covariance matrix ; and is<br>abbreviated as N(; ). If is positive denite (i.e xTx &gt; 0,8x 6= 0 2 Rd),<br>then the normal distribution N(; ) has density<br>;(x) =<br>1<br>(2)<br>d<br>2 jj<br>1<br>2<br>exp(ô€€€<br>1<br>2<br>(x ô€€€ )Tô€€€1(x ô€€€ ));<br>x 2 Rd with jj the determinant of :<br>The standard d-dimensional normal distribution N(0; Id); with Id the dd<br>identity matrix, is the special case<br>1<br>(2)<br>d<br>2<br>exp(ô€€€1=2 xT x):<br>If X N(; ) (i.e the random vector X has a multivariate normal<br>distribution)then its ith component Xi has distribution N(i;2i<br>) with 2<br>i =<br>ii.The ith and jth component have covariances cov(Xi;Xj) = E[(Xi ô€€€<br>i)(Xj ô€€€ j )] = ij which justies calling the covariance matrix. The<br>correlation between Xi and Xj is given by ij = ij<br>ij<br>.<br>If a d d symmetric matrix is positive semi-denite but not positive<br>denite then the rank of is less than d, fells to be invertible, and there<br>is no normal density with covariance matrix . In this case we can dene<br>the normal distribution N(; ) as the distribution of X = + AZ with<br>Y N(0; Id) for any d d matrix A: AAT = . The resulting distribution<br>is independent of which A is chosen. The random vector X does not have a<br>density in Rd, but if has rank then one can nd k component of X with<br>multivariate normal density in Rk.<br>Any linear transformation of a normal vector is again normal, X<br>N(; ) ) AX N(A;AAT ) for any d-vector and dd matrix and<br>any d k matrix A, for any k.[2]<br>10<br>1.1.8 Log-normal Distribution<br>In simple terms: A random variable X is said to have a lognormal distri-<br>bution if its logarithm has a normal distribution. I.e ln[X] N(; ). An<br>important property of this distribution is that it does not take values less<br>than 0.<br>A lognormal distribution is very much what the name suggest “lognor-<br>mal”. Imagine that you have a function that is the exponent of some input<br>variable X. The input variable itself is a normal distribution function . e.g.<br>y = k:eX<br>Now, if we take a natural log of this function gives a normal distribution.<br>1.1.9 Mathematical Expectation<br>Let (<br>,,) be a probability space. If X 2 L1(<br>; ; ), then<br>E(X) =<br>Z</p><p>X(!)d(!)<br>is called the mathematical expectation or expected value or mean of X:<br>The map X 7ô€€€! E(X) ,X 2 L1(<br>; ; ) has the following properties:<br>i E is linear: E(X + Y ) = E(X) + E(Y ), for all X; Y 2 L1(<br>; ;<br>and, ; 2 R<br>ii Markov’s inequality holds, i.e let X 2 L1(<br>; ; ) be R-valued. Then<br>(f! 2<br>: jX(!)j g) E(jXj)<br>) = kXk1<br>, where &gt; 0.<br>iii E is positivity preserving i.e if X is real-valued and lies in L1(<br>; ; )<br>and X 0, then E(X) 0.<br>11<br>iv Chebychev’s inequality holds: Let X 2 L1(<br>; ; ) be a R-valued ran-<br>dom variable with mean E(X) = and variance 2X<br>.Then for &gt; 0<br>(f! 2<br>: jX(!) ô€€€ j g) 2X<br>2 ).<br>v Jensen’s inequality holds i.e, if X is real-valued and lies in L1(<br>; ; ).<br>: R ô€€€! R is convex and (X) 2 L1(<br>; ; ), then E((X))<br>(E(X)).<br>1.1.10 Variance and covariance of random variables:<br>Let (<br>; ; ) be a probability space and X an R-valued random variable on<br>,<br>such that X 2 L2(<br>; ; ). Then X is automatically in L1(<br>; ; ) (because<br>in general if p q, then Lq(<br>; ; ) Lp(<br>; ; ) for all p 2 [1;1) [ f1g:)<br>The variance of X is dened as<br>V ar(X) = E((X ô€€€ E(X))2):<br>The number X =<br>p<br>V ar(X) is called the standard deviation/error of<br>X. Now let X, Y 2 L2(<br>; ; ). Then the covariance of X and Y is given<br>by:<br>Cov(X; Y ) = E((X ô€€€ E(X))(Y ô€€€ E(Y )))<br>And the correlation is given by:<br>corr(X; Y ) = (X; Y ) =<br>Cov(X)Cov(Y ) p<br>V ar(X)V ar(Y )<br>Two random variables X,Y are called uncorrelated if cov(X; Y ) = 0.<br>1.1.11 Characteristic function<br>Let (<br>; ; ) be a probability space and X 2 L0(<br>;R). Dene the C-valued<br>function on R by:<br>(t) = E(eitx) = E(costx) + iE(sintx):<br>12<br>Then is called the characteristic function of X.<br>Note: 0(0)<br>i = E(X)<br>1.1.12 Stochastic process<br>A stochastic process X indexed by a set J is a family X = fX(t) : t 2 Jg of<br>members of L(<br>;Rd).The value of X(t) at ! 2<br>is written as X(t,!).<br>1.1.13 Sample Paths<br>If X is a stochastic process and w 2<br>then the map t 7ô€€€! X(t;w) 2 Rd is<br>called a sample path or trajectory of X.<br>1.1.14 Brownian Motion<br>Let Z = fZ(t) 2 L(<br>;Rd) : t 2 4g ,where 4 R+ = [0;1] be an Rd<br>Stochastic process on<br>with the following properties:<br>i Z(0) = 0, almost surely.<br>ii Z(t) ô€€€ Z(s) is an N(0,(t-s)I) random vector for all t s 0,where I<br>is the d d identity matrix.<br>iii Z has stochastically independent increments i.e for 0 &lt; t1 &lt; t2 &lt; &lt;<br>tn, the random vectors Z(t1);Z(t2) ô€€€ Z(t1); ;Z(tn) ô€€€ Z(tnô€€€1) are<br>stochastically independent.<br>iv Z has continuous sample paths t 7ô€€€! Z(t;w) for xed w 2</p><p>Then Z is called the standard d-dimensional Brownian Motion or d-dimensional<br>Weiner process.<br>13<br>For a d-dimensional Brownian motion Z(t) = (Z1(t); ;Zd(t)) we have<br>the following:<br>i E(Zj(t)) = 0, j = 1; 2; ; d<br>ii E(Zj(t)2) = t, j = 1; 2; ; d<br>iii E(Zj(t)Zk(s)) = jkt^k = jkminft; sg, for t; s 2 4, j; k = 1; 2; ; d<br>1.1.15 Filtration:<br>Let (<br>; ; ) be a probability space and consider F() = ft : t 2 g a<br>family of -subalgebras of with the following properties:<br>i For each t 2 , t contains all the -null members of .<br>ii s t whenever t s, s; t 2<br>Then F() is called a ltration of and (<br>; ; F(); ) is called a ltered<br>probability space or stochastic basis.<br>We interpret t as the information available at time t and F() describe the<br>ow of information.<br>1.1.16 Adaptedness<br>A Stochastic process X = fX(t) 2 L(<br>;Rn) : t 2 Tg is said to be adapted<br>to the ltration F() = ft : t 2 Tg if X(t) is measurable with respect to<br>t for each t 2 T. It is plain that every stochastic process is adapted to its<br>natural ltration.<br>1.1.17 Conditional expectation<br>Let (<br>; ; ) be a probability space, X a real random variable in L1(<br>; ; )<br>and a -subalgebra of . Then the conditional expectation of X given<br>14<br>written E(X j ) is dened as any random variable Y such that:<br>(i) Y is measurable with respect to i.e. for any A 2 (R), the set Y ô€€€1(A) 2<br>.<br>(ii)<br>R<br>B X(!)d(!) =<br>R<br>B Y (!)d(!) for arbitrary B 2 :<br>A random variable Y which satises (i) and (ii) is called a version of E(X j<br>):<br>1.1.18 Martingale<br>Let X = fX(t) 2 L1(<br>; ; ) : t 2 g be a real-valued stochastic process on<br>a ltered probability space (<br>; ; F(); ). Then X is a<br>1. submartingale, if E(X(t)=s) X(s) a.s whenever t s<br>2. Supermartingale, if E(X(t)=s) X(s) a.s whenever t s<br>3. Martingale, if X is both a submartingale and supermartingale i.e E(X(t)=s) =<br>X(s) a.s whenever t s<br>1.1.19 Quadratic variation<br>Let X be a stochastic process on a ltered probability space (<br>; ; F(); ):<br>Then the quadratic variation of X on [0; t], t &gt; 0, is the stochastic process<br>hXi dened by<br>hXi(t) = limjPj!0<br>nXô€€€1<br>j=0<br>jX(tj+1) ô€€€ X(tj)j2<br>where P = ft; t1; ; tng is any partition of [0; t] i.e. 0 = t1 &lt; t2 &lt; &lt; tn = t<br>and jPj = max0jnô€€€1jtj+1 ô€€€ tj j<br>Note: If X is a dierentiable stochastic process, then hXi=0.<br>15<br>1.1.20 Stochastic dierential equations<br>These are dierential equations in which one or more terms is a stochastic<br>process, resulting in a solution which is itself a stochastic process. SDE are<br>used to model diverse phenomena such as uctuating stock prices or physical<br>system subject to thermal uctuations. They are of the form<br>dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)<br>with initial condition X(t) = x, where W denotes a Wiener process (stan-<br>dard Brownian motion). These are equations of the form<br>dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)<br>with initial condition X(t) = x<br>1.1.21 Ito formula and lemma<br>Let (<br>; ; F(); ) be a ltered probability space, X an adapted stochastic<br>process on (<br>; ; F(); ) with quadratic variation hXi and U 2 C1;2([0; 1]<br>R): Then<br>U(t;X(t)) = U(s;X(s)) +<br>Z t<br>s<br>@U<br>@t<br>(;X( ))ds +<br>Z t<br>s<br>@U<br>@x<br>(;X( ))dX( )<br>+<br>1<br>2<br>Z t<br>s<br>@2U<br>@x2 (;X( ))dhXi( )<br>which may be written as<br>dU(t; x) =<br>@U<br>@t<br>(t;X(t))dt +<br>@U<br>@x<br>(t;X(t))dX(t) +<br>1<br>2<br>@2U<br>@x2 (t;X(t))dhXi(t)<br>The equation above is referred to as the Ito formula. If X satsies the<br>stochastic dierential equation (SDE)<br>dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)<br>X(t) = x;<br>16<br>then<br>dU(t;X(t)) = gu(t;X(t))dt + fu(t;X(t))dW(t)<br>U(t;X(t)) = U(t; x)<br>where<br>gu(t; x) =<br>@U<br>@t<br>(t; x) + g(t; x)<br>@U<br>@x<br>(t; x) +<br>1<br>2<br>(f(t; x))2 @2U<br>@x2 (t; x);<br>fu(t; x) = f(t; x)<br>@U<br>@x<br>(t; x)<br>We obtain a particular case of the Ito formula called the Ito lemma, if we<br>take X = Z, by setting g 0 and f 1 on T R. Then<br>dU(t;Z(t)) = [<br>@U<br>@t<br>(t;Z(t)) +<br>1<br>2<br>@2U<br>@x2 (t;Z(t))]dt +<br>@U<br>@x<br>(t;Z(t))dZ(t)<br>The equation above is referred to as the Ito lemma.<br>Table 1.1: Ito Multiplication Table<br>x dt dZ(t)<br>dt 0 0<br>dZ(t) 0 dt<br>1.1.22 Gamma distribution<br>The probability density function gô€€€ of a gamma distributed variable is given<br>by gô€€€(x; ; ) =<br>e<br>ô€€€x<br>( x<br>)ô€€€1<br>ô€€€() ,x , ; 0<br>The corresponding cumulative distribution function Gô€€€ is dene as:<br>Gô€€€(x; ; ) =<br>Z x<br>0<br>gô€€€(u; x; )du<br>=<br>R x<br>0 uô€€€1eudu<br>ô€€€()<br>=<br>(; x<br>)<br>ô€€€()<br>;<br>17<br>where<br>ô€€€(z) =<br>Z 1<br>0<br>tzô€€€1eô€€€tdt<br>The ith moment of the gamma distribution is given by:<br>E[Y i] =<br>iô€€€(i + )<br>ô€€€()<br>The ith moment of the inverse gamma distribution can be obtained for<br>ô€€€ &lt; i 0 for i ô€€€ the moments are 1<br>If Y is reciprocally gamma distributed then:<br>E[Y i] =<br>1<br>i( ô€€€ 1)( ô€€€ 2) ( ô€€€ i)<br>:<br>Let gR be the inverse gamma probability distribution function. Then<br>gR(x; ; ) =<br>gô€€€( 1<br>x ;;)<br>x2 , x 0; ; &gt; 0<br>1.1.23 Risk-neutral Probabilities<br>These are probabilities for future outcomes adjusted for risk, which are then<br>used to compute expected asset values. The benet of this risk-neutral pric-<br>ing approach is that once the risk-neutral probabilities are calculated, they<br>can be used to price every asset based on its expected payo. These theo-<br>retical risk-neutral probabilities dier from actual real world probabilities;<br>if the latter were used, expected values of each security would need to be<br>adjusted for its individual risk prole. A key assumption in computing risk-<br>neutral probabilities is the absence of arbitrage. The concept of risk-neutral<br>probabilities is widely used in pricing derivatives.<br>1.2 Overview<br>A nancial derivative is a contract whose price is dependent upon or derived<br>from one or more underlying assets. The underlying assets could be stocks,<br>18<br>commodities, currencies e.t.c. An option is a nancial derivative that gives<br>the holder the right but not the obligation to buy or sell an underlying asset<br>at a certain date and price. Options were rst traded on the Chicago Board<br>Options Exchange on April 26th, 1973. Basket option is a type of derivative<br>security where the underlying asset is a group of commodities, securities or<br>currencies. Since the early 1990s, basket options have been used as a tool<br>for reducing risks (Hedging).<br>The pricing and hedging of basket options is dicult, due to the number<br>of state variables. The usual methods employed in pricing options are not<br>used to price Basket options, like Black and Scholes(1973) model. A sin-<br>gle underlying asset is assumed to follow a geometric Brownian motion and<br>therefore log-normally distributed, the problem arises from the fact that sum<br>of correlated log-normally distributed random variables is not log-normal,<br>thereby making it dicult to price the basket options and have a closed form<br>pricing formula and hedging ratios. Some Practitioners sometimes take the<br>basket itself also as a log-normal distribution. However, it leads to an incon-<br>sistency in the basic assumption “The distribution of a weighted average of<br>correlated log-normals is anything but log-normal. Another diculty that<br>prevents the price of basket options from being exactly known the correla-<br>tion structure involved in the basket. Correlation is observed to be volatile<br>over time as is the volatility. A lot of research have been done to overcome<br>this diculty. Several methods has been proposed, comprising numerical<br>methods and analytical approximation.<br>Instead of buying an option on each underlying asset, one may buy a<br>single option on all the underlying assets “Basket options” as this will be<br>cheaper, since there is only one option to monitor and exercise.<br>In the second chapter we give a literature review in pricing of basket<br>options, highlighting some of the important contributions.<br>In the third chapter, we discuss nancial derivatives and basket options,<br>so as to have a clear idea of the nancial market. We provide the: Denition<br>19<br>of option, types of option, some examples of nancial derivatives, traders,<br>and some examples of basket options.<br>In the fourth chapter, we discuss the pricing of basket options and the<br>methods used in the pricing, which is the main work of this thesis. The seller<br>of a nancial derivative, in particular options, requires a compensation for<br>the risk he is bearing, by selling the option to the buyer. The buyer must<br>pay a certain amount called a premium, in order to get the right to buy or<br>sell the underlying asset and that is what is referred to as the price of the<br>option. Several factors aect the pricing of basket option which include the<br>initial prices, volatilities of the underlying asset, correlation e.t.c.<br>Various methods have been used in pricing of basket options, which<br>include Monte-Carlo simulation (by assuming that the assets follow corre-<br>lated geometric Brownian motion processes) rst suggested by Boyle(1977).<br>Monte-Carlo methods are suitable numerical methods used in pricing op-<br>tions that do not have an analytical closed form solution, especially basket<br>options, Cox and Ross (1976) noted that if a riskless hedge can be formed,<br>the option value is the risk-neutral and discounted expectation of its pay-<br>o, that is the price can be represented by an integral, therefore making<br>it possible to estimate the price of the option by Monte Carlo methods,<br>which is done by simulating many independent paths of the underlying as-<br>sets and taking the discounted mean of the generated pay-o’s. We also have<br>Tree based method (in the case of few state variables), analytical approx-<br>imations such as Taylor approximation, Reciprocal gamma approximation,<br>Log-normal approximation e.t.c.<br>In the last chapter, we give some applications of log-normal approxima-<br>tion by considering foreign exchange basket options, and give details on how<br>they are priced.</p> <br><p></p>