PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS

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1 PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS RENÉ CARMONA AND VALDO DURRLEMAN ABSTRACT This paper provides with approximate formulas that generalize Black-Scholes formula in all dimensions Pricing and hedging of multivariate contingent claims are achieved by computing lower and upper bounds These bounds are given in closed form in the same spirit as the classical one-dimensional Black-Scholes formula Lower bounds perform remarkably well Like in the onedimensional case, Greeks are also available in closed form We discuss an extension to basket options with barrier INTRODUCTION This paper provides with approximate formulas that generalize Black-Scholes formula in all dimensions The classical Black-Scholes formula gives in closed form the of a call option on a single stock whose dynamics is a geometric Brownian motion Its use has spread to fixed income markets to caps and floors in Libor models or swaptions in Swap models when volatilities are deterministic Many options however have multivariate payoffs Although the mathematical theory does not present any particular difficulties, actual computations of s cannot be done in closed form any more Financial practitioners have to resort to numerical integration, simulations or approximations In high dimensions, numerical integration and simulation methods may be too slow for practical purposes Many areas of computational finance require robust and accurate algorithms to these options In this paper we give approximate formulas that are fast, easy to implement and yet very accurate These formulas are based on rigorous lower and upper bounds These bounds are derived under two assumptions First, we restrict ourselves to a special class of multivariate payoffs Throughout payoffs are of the European type options can only be exercised at maturity and when exercised these options pay a linear combination of asset s This wide class includes basket options ie, options on a basket of stocks, spread options ie, options on the difference between two stocks or indices and more generally rainbow options but also discrete-time average Asian options and also combination of those like Asian spread options ie, options on the difference between time averages of two stocks or indices Second, we work in the so-called multidimensional Black-Scholes model In this model, assets follow a multidimensional geometric Brownian motion dynamics In other words, all volatilities are constants As usual, to extend the results for deterministic time dependent volatilities one just has to replace volatilities by their root-mean-square over the option life Date: November 8, 3

2 RENÉ CARMONA AND VALDO DURRLEMAN To continue the discussion, let us fix some notations In a multidimensional Black-Scholes model with n stocks S,, S n, risk neutral dynamics are given by ds i t S i t = rdt + ij db j t, with some initial values S,, S n B,, B n are independent standard Brownian motions Correlations among different stocks are captured through the matrix ij Given a vector of weights w i i=,,n, we are interested, for instance, in valuing the following basket option struck at K whose payoff at maturity T is + w i S i T K Risk neutral valuation gives the at time as the following expectation + p = e rt E w i S i T K i= i= Deriving formulas in closed form for such options with multivariate payoffs has already been tackled in the financial literature For example, Jarrow and Rudd in [] provide a general method based on Edgeworth sometimes also called Charlier expansions Their idea is to replace the integration over the multidimensional log-normal distribution by an integration over another distribution with the same moments of low order so that this last integration can be done in closed form In the case where the new distribution is Gaussian, this approximation is often called the Bachelier approximation since it gives back formulas alike those derived by Bachelier Another take on this problem introduced in [4] is to replace arithmetic averages by their corresponding geometric averages The latter have the nice property of being log-normally distributed; they therefore lead to formula alike the Black-Scholes formula See, for example, [3] pp 8-5 for a presentation of these results This method assumes however that the weights w i i=,,n are all positive Our method does not require this assumption and will prove to be more accurate There are two difficulties in computing : the lack of tractability of the multivariate log-normal distribution on the one hand and the non linearity of the function x x + on the other Whereas [] and [4] circumvent the first difficulty, our approach relies on finding optimal one-dimensional approximations thanks to properties of the function x x + In one dimension, computations can then be carried out explicitly Approximations in closed form are given in Proposition 4 and 6 below Various sensitivities, the so-called Greeks, are given in Proposition 9,, and Section 3 shows actual numerical results as well as an extension to multivariate barrier options j= APPROXIMATE LOWER AND UPPER BOUNDS As we have just explained, our goal is to compute EX + where X is the random variable X = ε i x i e G i VarGi

3 PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS 3 G i,,n is a mean zero Gaussian vector of size n + and covariance matrix Σ ε i = ± and x i > for all i =,, n In view of, this is just ε i = sgnw i and x i = w i S i Note that entries of Σ are dimensionless, that is, they are volatilities squared time to maturity Without loss of generality, we suppose that not all of the ε i have the same sign If this were the case, computing EX + would not present any difficulty Note also that Σ is symmetric positive semi-definite but not necessarily definite Before we explain our approximation method we need the following definition and proposition Definition For every i, j, k =,, n, we let and Σ k ij = Σ ij Σ ik Σ kj + Σ kk i = Σ ii k i = Σ k ii Proposition For every k =,, n, let G k i,,n be a mean zero Gaussian vector with covariance Σ k Then, + EX + = E ε i x i e Gk i VarGk i Proof This is an easy consequence of Girsanov s transform Indeed, + EX + = E e G k VarG k ε i x i e G i G k VarG i VarG k + = E Q k ε i x i e G i G k VarG i VarG k, where probability measure Q k is defined by its Radon-Nikodým derivative dq k dp = eg k VarG k Under Q k, G i G k i n is again a Gaussian vector Its covariance matrix is Σ k Without loss of generality, we will also assume that for every k =,, n, Σ k Indeed if such were the case, Proposition above would give us the without any further computation Two optimization problems The following proposition will provide us with bounds Proposition For any X L, sup EXY = EX + = inf EZ Y X=Z Z,Z,Z Proof On the left-hand side, letting Y, EXY = EX + Y EX Y EX + and taking Y = X shows that the supremum is actually attained On the right-hand side, it is well known that if X = Z Z with both Z and Z non negative, then Z X + These two optimization problems are dual of each other in the sense of linear programming

4 4 RENÉ CARMONA AND VALDO DURRLEMAN Derivation of the lower bound Our lower bound is obtained by restricting the set over which the supremum in is computed We choose Y of the form u G d where u R n+ and d R are arbitrary Let us let p = sup E X u G d u,d The next two propositions give further information on p First, we need the following definition Definition Let D to be the n + n + diagonal matrix whose ith diagonal element is / i if i and otherwise Let C to be such that C = DΣD C is also a positive semi-definite matrix and we denote by C a square root of it ie, C = C C T Proposition 3 p = sup d R sup u Σu= ε i x i Φ d + Σu i = sup d R sup v = ε i x i Φ d + i Cv i Here and throughout the paper, we use the notation ϕx and Φx for the density and the cumulative distribution function of the standard Gaussian distribution, ie, Proof ϕx = π e x / and Φx = π x p = sup d R u R sup E E X u G u G d n+ = sup d R sup u R n+ = sup d R sup u Σu= = sup d R sup u Σu= ε i x i E e u / du e CovG i,u G u G CovG i,u G u Σu u Σu u G d ε i x i E e CovG i,u Gu G CovG i,u G u G d ε i x i Φ d + Σu i By defining D to be the n + n + diagonal matrix whose ith diagonal element is i, we easily check that Σ = D C C T D Therefore by taking v = D C T u, we have the second equality of the proposition To actually compute this supremum, it is interesting to look at the Lagrangian L: Lv, d = ε i x i Φ d + i Cv i µ v denotes the usual inner product ofr n+

5 Proposition 4 PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS 5 p = ε i x i Φ d + i Cv i where d and v satisfy the following first order conditions ε i x i i Cij ϕ d + i 3 Cv i µvj = for j =,, n 4 5 ε i x i ϕ d + i Cv i = v = Note that expression for p is as close to the classical Black-Scholes formula as one could hope To conclude this subsection, we give a necessary condition for d to be finite It is interesting when it comes to numerical computations but it also ensures us that lower bounds are not trivial We need to make a non-degeneracy assumption Recall that the matrix C was introduced in Definition Through its definition, C may have columns and rows of zeros We are now assuming that the square matrix C obtained by removing these rows and columns is non-singular C is well defined because we assumed that none of the Σ k and therefore Σ were actually the zero matrix Condition Proposition 5 Under Condition, det C p > E X +, or equivalently d < + Proof Assume for instance that E X We want to show that p > E X Let us let f v d = n ε ix i Φ d + i Cv i First note that for any v, lim d + f v d = E X We are going to show the claim by showing that there exists a unit vector v such that f vd < when d is near + Under Condition, Range C = n ir and we can pick a unit vector v such that i > ε i Cv i > For such a v, write f v as f vd = ϕd x i e d i Cv i i Cv i x i e d i Cv i i Cv i i:ε i =+ i:ε i = By denoting = min i:εi =+ i Cv i and = max i:εi = i Cv i, we get the following bound, valid for d : f vd ϕd x i e d x i e d i:ε i =+ i:ε i = Without loss of generality, we can assume that and are not simultaneously zero and the above upper bound is strictly negative for d large enough The case where E X is treated analogously by showing that f v > around

6 6 RENÉ CARMONA AND VALDO DURRLEMAN 3 Derivation of the upper bound Our upper bound is obtained by restricting the set over which the infimum in is computed For every k =,, n, let E k = i : i k = Let us also let x k = i/ E k ε i x i and ε k = sgn i/ E k ε i x i Without loss of generality, we can assume x k > Then, choose reals λ k i i E k such that i E k λ k i = ε k and rewrite X as X = i E k ε i x i e G i VarG i λ k i x k e G k VarG k = i E k ε i x i e G i VarG i λ k i x k e G k VarG k + ε i x i e G i VarGi λ k i x k e G k VarG k i E k The family of random variables Z that we choose consists of those of the form ε i x i e G i VarGi λ k i x k e G k VarG + k i E k where k =,, n, i E k λ k i = ε k and λ k i ε i > for all i E k Because all the ε i do not have the same sign, the set of such λ k is nonempty for each k Proposition 6 6 p = min k n where d k is given by the following first order conditions ε i εi x i i k ln λ k i x ε ii k = ε j ε j x j k j k ln λ k j x k λ k i = ε k i E k λ k i ε i > for i E k ε i x i Φ d k + ε i i k ε j k j = d k for i, j E k Again, note that expression for p is as close to the classical Black-Scholes formula as one could hope Proof p = min k n = min k n inf P i E k λ k i = ε k P inf i E k λ k i = ε k E ε i x i e G i VarGi λ k i x k e G k VarG + k i E k εi εi x i ε i x i Φ k ln i E k i λ k i x + ε i k i k λ k εi εi x i i x k Φ ln λ k i x ε i k i k k i

7 Forming the Lagrangian L k PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS 7 L k λ k = i E k ε i x i Φ we find the first order conditions L k λ k i εi k i λ k εi i x k Φ i k = x k Φ εi x i ln λ k i x + ε i k i k εi k i εi x i ln λ k i x ε i k i k εi x i ln λ k i x ε i k i µ =, k from which we deduce that the arguments of Φ must all equal each other µ λ k i + ε k, i Ek 4 Cases of equality It is easily seen that when n =, lower and upper bounds both reduce to the Black-Scholes formula and therefore give the true value Let us stress that n = not only contains the classical call and put options but also the exchange option of Margrabe The following proposition gives other cases where the lower and upper bounds are in fact equal to the true value Proposition 7 If for all i, j =,, n, then Σ ij = ε i ε j i j, p = p Proof Exactly as in Proposition, note that for any k, p = sup d R sup ε i x i Φ d + Σ k u i u Σ k u= Therefore, for any k, 7 sup u Σ k u= ε i x i Φ d k + Σ k u i p p ε i x i Φ d k + ε i i k Let us choose k such that k = min i n i Note that under the hypothesis, Σ k ij = ε i i ε k k ε j j ε k k Notice further that since all the ε i do not have the same sign, we can define the following vector u: u i = sgnε i i ε k k n j= ε j j ε k k One trivially checks that u Σ k u = and that Σ k u i = ε i i ε k k Because of the way we chose k, ε i i ε k k = ε i ε i i ε k k = ε i k i This proves that the inequalities in 7 are in fact equalities

8 8 RENÉ CARMONA AND VALDO DURRLEMAN 5 Bound on the gap Although an estimate of the gap is readily available as soon as lower and upper bounds are computed, it is interesting to have an a priori bound on the gap p p Proposition 8 Proof p p min k n min k n p p j= l= π min x i i k k n ε i x i Φ d k + ε i i k max k n By Cauchy-Schwarz inequality, Σ k u i = Σ k lj u jδ il It follows that p p min k n which is the desired upper bound on the gap 6 Computation of the Greeks ε i x i Φ d k + Σ k u i ε i x i Φ d k + ε i i k Φ d k + Σ k u i j= l= Σ k lj u j u l j= l= x i ϕ k i Σ k u i ϕ min k n Σ k lj δ ilδ ij = i k x i i k, Lower bound To compute partial derivatives with respect to the coefficients of C ie, the various correlation parameters, we again need to make a non-degeneracy assumption Assume Condition holds true Then, C is also non-singular and we define C to be the n + n + matrix obtained with the entries of C and putting back the rows and columns of zeros that we first removed from C Proposition 9 Under Condition, i = p = ε i Φ d + i Cv i x i V ega i = p T = εi x i Cv i ϕ d + i Cv i T i χ ij = p ρ ij = Θ = p T δij = if i = j and otherwise = T ε k x k i C kj v j + j C ki k= k= v i ϕ d + k Cv k ε k x k k Cv k ϕ d + k Cv k

9 PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS 9 Proof First order derivatives are easily computable thanks to the following observation dp = p + p d + v p v = p + µ v dx i x i d x i x i x i x i because p satisfies the first order conditions 3-5 at d, v = p x i Second order derivatives are more difficult to obtain since the previous trick is no longer possible There exist however simple and natural approximations that satisfy the multidimensional Black- Scholes equation Proposition Let ϕ d + i Cv i ϕ d + j Cv j Γ ij = ε i ε j n k= ε kx k k Cv k ϕ d + k, Cv k then Θ + T j= Σ ij x i x j Γ ij = The unusual term /T comes from our convention on Σ Proof It suffices to show that: ε k x k k ϕ k Cv k + k= where we used the short-hand notation ϕ k = ϕ j= ε i ε j Σ ij x i x j ϕ i ϕ j = Again because of 3 and 5, we have: This completes the proof µ = = µ j= ε i ε j Σ ij x i x j ϕ i ϕ j = d + k Cv k Simply note that because of 3, j= k= ε i ε j i j Cik Ckj x i x j ϕ i ϕ j ε j j x j ϕ j Cv j j= ε i i x i ϕ i Cv i Upper bound We now turn ourselves to the case of upper bounds The function min is only almost everywhere differentiable Therefore the next two propositions have to be understood in an almost sure sense k denotes the value for which the minimum is achieved in 6

10 RENÉ CARMONA AND VALDO DURRLEMAN Proposition with the convention / = i = p = ε i Φ d k + ε i i k x i V ega i = p i ρ ik k T = xi ϕ d k + ε i k i i k i T χ ij = p ρ ik i k = δ jk x i ϕ d k + ε i i k ρ ij Θ = p T = T l= i k x l l k ϕ Proof We proceed in the same way as for the lower bound Proposition Let then Proof Straightforward d k + ε l k l ϕ d k +ε i Γ i k ij = δ x i i k ij if i E k for all j if i / E k, Θ + T j= Σ k ij x i x j Γ ij = 3 NUMERICAL EXAMPLES AND EXTENSIONS 3 Basket options As a first example, we shall consider the case of a basket option For simplicity, let us suppose that there are n stocks whose initial values are all $ and whose volatilities are also all the same, equal to Correlation between any two distinct stocks is ρ This amounts to the following: ρ ρ /n ε =, x = /n K ρ and Σ = T ρ ρ ρ The option has maturity year We present the results in Figure when n = 5 and for different volatilities = %, %, 3% and different correlation parameters ρ = 3%, 5%, 7% We plot lower and upper bounds against For the sake of comparison we also plot results of brute force Monte Carlo simulations with, simulated paths Agreement of the lower bound with Monte Carlo results is excellent, Monte Carlo results being sometimes slightly below the lower bound Obviously, upper bounds are really not as good as lower bounds

11 PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS In view of these plots one can make two comments The gap between lower and upper bound tends to decrease as correlation increases, which is in total agreement with Proposition 7 On the other hand the gap increases with the volatility, this, in turn, could be suspected from Proposition 8 ρ = 3% = % ρ = 3% = % ρ = 3% = 3% ρ = 5% = % ρ = 5% = % ρ = 5% = 3% ρ = 7% = % ρ = 7% = % ρ = 7% = 3% FIGURE Lower and upper bound on the for a basket option on 5 stocks each one having a weight of /5 as a function of K + denote Monte Carlo results 3 Discrete-time average Asian options In the case of Asian option, we compare the lower bound with another often used approximative lower bound for Asian option This lower bound is obtained by replacing an arithmetic average by a geometric one see, for example, [4] Results are reported in Figure Again, we take an option with year to expiry and an initial value for the stock of $ Averaging is performed over 5 equally spaced dates Results are given for different stock volatilities = %, %, 3% The lower bound is uniformly better than the geometric average approximation

12 RENÉ CARMONA AND VALDO DURRLEMAN = % = % = 3% FIGURE Lower and upper bound on the of an Asian option The dotted line represents the geometric average approximation 33 Basket options with barrier In this subsection, we show how to extend the previous results to the case of a basket option with a down-and-out barrier condition on the first stock of the basket More specifically, the option payoff is + w i S i T K inft T S t H i= With the notation used so far, the option is 3 E + ε i x i e G i i, inf θ x e G θ θ H where Gθ; θ is a n + -dimensional Brownian motion starting from with covariance Σ We propose to approximate the option and its replicating strategy with an optimal lower bound p = sup E ε i x i e G i i d,u inf θ x e G θ θ H;u G d Using Girsanov s theorem, this rewrites p = sup d,u ε i x i P inf G θ + Σ i / H θ ln ; u G d Σu i θ x Let us define a new standard Brownian motion W θ; θ independent of G θ; θ by u Gθ = u Σu Σu W θ + Σu G θ 3 Assuming, without loss of generality, that ε = +, > and H < x

13 PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS 3 We choose to normalize u by setting Σu = Letting λ i = Σ i / and Y = d Σu i u Σu W, we get p = max sup sup d R Σu = sup sup d R Σu = H ε i x i P inf G θ + λ i θ ln θ ε i x i P inf G θ + λ i θ ln θ x H ; G Y ; x ; G Y To compute these probabilities we use the following result see, for example, [3] pp 47 Lemma Let B be standard Brownian motion and Xθ = Bθ + λθ, then for y, Φ y+λ e λy Φ y+λ Φ x+λ P inf Xθ y; X x = θ +e λy Φ x+λ+y if y x otherwise, Φ λy x+λ e Φ x+λ+y if y x P inf Xθ y; X x = θ Φ y+λ e λy Φ y+λ otherwise Therefore, we have, by first conditioning on Y : 4 H P inf G θ + λ i θ ln ; G Y = θ x λi lnh/x H λ i E Φ x H λ i + Y + lnh/x Φ x λi lnh/x Φ H λ i x Φ λi + lnh/x Φ λi + lnh/x Y Φ lnh/x Y +λi = d Σu i + λ i lnh/x Φ u Σu Φ d Σu i, d Σu i + λ i lnh/x, u Σu u Σu u Σu H λ i + Φ d Σu i lnh/x, d Σu i + λ i lnh/x, x u Σu u Σu u Σu 4 Φ denotes the cumulative distribution function of the standard bivariate Gaussian distribution, ie, Φ x, y, ρ = p Z x π ρ Z y exp u ρuv + v dudv ρ

14 4 RENÉ CARMONA AND VALDO DURRLEMAN Similarly, H P inf G θ + λ i θ ln ; G Y = θ x λi lnh/x H λ i Φ λi + lnh/x Φ d Σu i λ i + lnh/x Φ u Σu x d Σu i +Φ, d Σu i λ i + lnh/x, u Σu u Σu u Σu H λ i d Σu i lnh/x Φ, d Σu i λ i + lnh/x, x u Σu u Σu u Σu Table below gives s for such options for various parameters Framework and notation are the same as section 3 There are n stocks whose initial values are $ and whose volatilities are all equal to Correlation between any two distinct stocks is ρ and options are at-the-money, ie, K = ρ H/x n = n = n = TABLE Lower bounds for a down-and-out call option on a basket of n stocks 4 CONCLUSION This paper showed how to efficiently compute approximate s and hedges of options on any linear combination of assets Our general method allowed us to treat all these options in a common

15 PRICING AND HEDGING MULTIVARIATE CONTINGENT CLAIMS 5 framework Lower bounds prove to be extremely accurate This methodology was applied to the pricing of basket, discrete-time average Asian options and basket options with barrier As an important by-product of this method, first and second order sensitivities are given in closed form at no extra cost This is a clear advantage over Monte Carlo methods Indeed, first order derivatives are also easily computable along with the as explained, for example, in [] In the one-dimensional case, there is only one second order derivative Gamma and it can be computed by imposing that it satisfy Black-Scholes equation In dimension n, this PDE involves all n second order partial derivatives, and it seems we need to compute n of them, if we want to use the same trick Acknowledgments The authors thank Robert Vanderbei for helping them out with AMPL and LOQO REFERENCES [] P Glasserman Monte Carlo Methods in Financial Engineering Springer-Verlag, New York, 3 [] R Jarrow and A Rudd Approximate option valuation for arbitrary stochastic processes J Finan Econ, : , 98 [3] M Musiela and M Rutkowski Martingale Methods in Financial Modelling Springer Verlag, 997 [4] T Vorst Price and hedge ratios of average exchange options Internat Rev Finan Anal, :79 93, 99 DEPARTMENT OF OPERATIONS RESEARCH AND FINANCIAL ENGINGEERING, PRINCETON UNIVERSITY, PRINCE- TON, NJ 8544, ALSO WITH THE BENDHEIM CENTER FOR FINANCE AND THE APPLIED AND COMPUTATIONAL MATHEMATICS PROGRAM address: rcarmona@princetonedu DEPARTMENT OF OPERATIONS RESEARCH AND FINANCIAL ENGINGEERING, PRINCETON UNIVERSITY, PRINCE- TON, NJ address: vdurrlem@princetonedu