Bounds on some contingent claims with non-convex payoff based on multiple assets

Bounds on some contingent claims with non-convex payoff based on multiple assets Dimitris Bertsimas Xuan Vinh Doan Karthik Natarajan August 007 Abstract We propose a copositive relaxation framework to calculate both upper and lower bounds for prices of some European options with non-convex payoffs when first and second moments of underlying assets are known. Computational results shows that these upper and lower bounds are reasonably good for call options on the minimum of multiple assets and put options on the maximum of multiple assets. 1 Introduction Option valuation is important for a wide variety of hedging and investment purposes. Black and Scholes [3] derive a pricing formula for a European call option on a single asset with no-arbitrage arguments and the lognormal distribution assumption of the underlying asset price. Merton [9] provide bounds on option prices with no assumption on the distribution of the asset price. Given the mean and variance of the asset price, Lo [7] obtains an upper bound for the European option price based on this single asset. his result is generalized in Bertsimas and Popescu [1]. In the case of options written on multiple underlying assets, Boyle and Lim [4] provides upper bounds for European call options on the maximum of several assets. Zuluaga and Peña [13] obtain these bounds using moment duality and conic programming. Boeing Professor of Operations Research, Sloan School of Management, co-director of the Operations Research Center, Massachusetts Institute of echnology, E40-147, Cambridge, MA 0139-4307, dbertsim@mit.edu. Operations Research Center, Massachusetts Institute of echnology, Cambridge, MA 0139-4307, vanxuan@mit.edu. Department of Mathematics, National University of Singapore, Singapore 117543, matkbn@nus.edu.sg. 1

Contributions and Paper Outline he options considered in these papers have convex payoff functions. Given first and second moments of underlying asset prices, a simple tight lower bound can be calculated using Jensen s inequality. In this paper, we consider a class of European options with non-convex payoff, the call option written on the minimum of several assets. Similarly, put options on the maximum of several assets are also options with non-convex payoff functions. Both upper and lower bounds for prices of European call options on the minimum of several assets calculated using copositive relaxation are considered in Section and 3. Some computational results for these call and put options are reported in Section 4. Upper Bounds We consider the European call options written on the minimum of n assets. At maturity, these assets have price X 1,..., X n respectively. If the option strike price is K, then the expected payoff can be calculated as follows: P = E[( min 1 k n X k K) ]. (1) he rational option price can be obtained by discounting this expectation at the risk-free rate under the no-arbitrage assumption. herefore, we can firstly derive bounds for this expected payoff P without discount factor involvement and obtain bounds for the option price later. We do not assume any distribution models for the multivariate nonnegative random variable X = (X 1,..., X n ). Given that first and second moments of X, E[X] = µ and E[XX ] = Q, we would like to calculate the tight upper bound P max = max X (µ,q) E[(min 1 k n X k K) ] and lower bound P min = min X (µ,q) E[(min 1 k n X k K) ]. In this section, we focus on upper bounds while lower bounds will be considered in Section 3. We have, the upper bound P max is the optimal value of the following optimization problem: P max = max f s.t. R(min n 1 k n x k K) f(x)dx R x n k f(x)dx = µ k, k = 1,..., n, R x n k x l f(x)dx = Q kl, 1 k l n, f(x)dx = 1, R n f(x) 0, x R n, () where f is a probability density function.

aking dual of Problem () (see Bertsimas and Popescu []), we obtain the following dual problem: or equivalently, P u = min Y,y,y0 Q Y µ y y 0 s.t. x Y x x y y 0 (min 1 k n x k K), x R n, P u = min Y,y,y0 Q Y µ y y 0 s.t. x Y x x y y 0 0, x R n, x Y x x y y 0 min 1 k n x k K, x R n. (3) Weak duality shows that P u P max, which means P u is an upper bound for the expected payoff P. Under a weak Slater condition on moments of X, strong duality holds and P u = P max, which becomes a tight upper bound (see Bertsimas and Popescu [] and references therein). We now attempt to reformulate Problem (3). he first constraint is equivalent to a copositive matrix constraint as shown in the following lemma: Lemma 1 x Y x x y y 0 0 for all x R n if and only if Ȳ = Y Proof. We have: x Y x x y y 0 = x 1 Y y y y 0 y x. 1 y y 0 is copositive. If the matrix Ȳ is copositive, then clearly x Y x x y y 0 0 for all x R n as (x, 1) R n1 for all x R n. Conversely, if x Y x x y y 0 0 for all x R n, we prove that x Y x also nonnegative for all x R n. Assume that there exists x R n such that x Y x < 0 and consider the function f(k) = (kx) Y (kx) (kx) y y 0. We have: f(k) = (x Y x)k (x y)k y 0, which is a strictly concave quadratic function. herefore, lim k f(k) =, which means there exists z = kx R n such that z Y z z y y 0 < 0 (contradiction). hus we have x Y x 0 for all x R n. It means that z Ȳ z 0 for all z R n1 or Ȳ is copositive. he reformulation makes it clear that finding the (tight) upper bound P u is a hard problem. Murty [10] shows that even the problem of determining whether a matrix is not copositive is NP-complete. In order to tractably compute an upper bound for the expected payoff P, we relax this constraint using a well-known copositivity sufficient condition (see Parrilo [11] and references therein): Remark 1 (Copositivity) If Ȳ = P N, where P 0 and N 0, then Ȳ is copositive. 3

According to Diananda [5], this sufficient condition is also necessary if Ȳ Rm m with m 4. Now consider the second constraint, we will relax it using the following lemma: Lemma If there exists µ R n, n k=1 µ Y k = 1, such that Y µ = (y n k=1 µ ke k ) y n k=1 µ ke k y 0 K copositive, where e k is the k-th unit vector in R n, k = 1,..., n, then x Y xx yy 0 min 1 k n x k K for all x R n. is Proof. he second constraint can be written as follows: min max 1 k n x Y x x y y 0 x k K 0. x R n We have: max 1 k n x k = max z C z x, where C is the convex hull of e k, k = 1,..., n. If we define f(x, z) = x Y x x y y 0 z x K, then the second constraint is min x R n max f(x, z) 0. z C Applying weak duality for the minmax problem min x R n max z C f(x, z), we have: min x R n max z C f(x, z) max z C min f(x, z). hus if max z C min x R n f(x, z) 0 then the second constraint is satisfied. his relaxed constraint can be written as follows: x R n z C : f(x, z) 0, x R n. We have: C = { n k=1 µ ke k µ R n, n k=1 µ k = 1 }, thus the constraint above is equivalent to the following constraint: µ R n, n µ k = 1 : x Y x x y y 0 k=1 n µ k x k K 0, x R n. k=1 Using Lemma 1, we obtain the equivalent constraint: n µ R n Y, µ k = 1 : Y µ = (y n k=1 µ ke k ) k=1 y n k=1 µ ke k y 0 K is copositive. hus we have, x Y x x y y 0 min 1 k n x k K for all x R n if there exists µ R n, n k=1 µ k = 1, such that Y µ is copositive. From Lemma 1 and, and the copositivity sufficient condition in Remark 1, we can calculate an upper bound for the expected payoff P as shown in the following theorem: 4

heorem 1 he optimal value of the following semidefinite programming problem is an upper bound for the expected payoff P : Pu c = min Q Y µ y y 0 s.t. Y y = P 1 N 1, y y 0 (y n Y k=1 µ ke k ) n k=1 µ k = 1, µ 0, y n k=1 µ ke k y 0 K = P N, (4) P i 0, N i 0 i = 1,. Proof. Consider an optimal solution (Y, y, y 0, P 1, N 1, P, N, µ) of Problem (4). According to Remark 1, Ȳ is a copositive matrix. herefore, (Y, y, y 0 ) satisfies the first constraint of Problem (3) following Lemma 1. Similarly, the second constraint of Problem (3) is also satisfied by (Y, y, y 0 ) according to Lemma. hus, (Y, y, y 0 ) is a feasible solution of Problem (3), which means Pu c P u. We have P u P max ; therefore, P c u P max or P c u is an upper bound for the expected payoff P. 3 Lower Bounds he tight lower bound of the expected payoff P is P min = min X (µ,q) E[(min 1 k n X k K) ]. However, due to the non-convexity of the payoff function, it is difficult to evaluate P min. Jensen s inequality for the convex function f(x) = x, we have: Applying max{0, E[ min 1 k n X k K]} E[( min 1 k n X k K) ]. Define P min = min X (µ,q) E[min 1 k n X k K], then clearly, max{0, P min } P min or max{0, P min } is a lower bound for the expected payoff P. We have, Pmin can be calculated as follows: P min = max f s.t. R min (K n 1 k n x k )f(x)dx R x n k f(x)dx = µ k, k = 1,..., n, R x n k x l f(x)dx = Q kl, 1 k l n, f(x)dx = 1, R n f(x) 0, x R n, (5) 5

where f is a probability density function. aking the dual, we obtain the following problem: or equivalently, P l = min Y,y,y0 Q Y µ y y 0 s.t. x Y x x y y 0 K min 1 k n x k, x R n, P l = min Y,y,y0 Q Y µ y y 0 s.t. x Y x x y y 0 x k K 0, x R n, k = 1,..., n. (6) Similarly, P l P min according to weak duality and if the Slater condition is satisfied, P l = P min. Now consider the constraints of Problem (6). Using Lemma 1, each constraint of Problem (6) is equivalent to a copositive matrix constraint: x Y x x y y 0 x k K 0, x R n Y ye k y 0 K ye k is copositive. With Remark 1, we can then calculate a lower bound for the expected payoff P as shown in the following theorem: heorem max{0, Pl c } is a lower bound for the expected payoff P, where P c l = min Q Y µ y y 0 s.t. Y ye k = P k N k, y 0 K ye k k = 1,..., n P k 0, N k 0 k = 1,..., n. (7) Proof. Consider an optimal solution (Y, y, y 0, P k, N k ) of Problem 7. According to Remark 1, the matrix Y ye k is copositive for all k = 1,..., n. Lemma 1 shows that (Y, y, y ye k 0 ) satisfies y 0 K all constraints of Problem 6. hus (Y, y, y 0 ) is a feasible solution of Problem 6, which means P c l P l. We have P min P l and max{0, P min } P min ; therefore, max{0, P c l } P min or max{0, P c l } is a lower bound for the expected payoff P. 6

4 Computational Results 4.1 Call Options on the Minimum of Several Assets We consider the call option on the minimum of n = 4 assets. In order to compare the bounds with the exact option price, we assume that these assets follow a correlated multivariate lognormal distribution. At time t, the price of asset k is calculated as follows: S k (t) = S k (0)e (r δ k /)tδ kw k (t), where S k (0) is the initial price at time 0, r is the risk-free rate, δ k is the volatility of asset k, and (W k (t)) n k=1 is the standard correlated multivariate Brownian motion. We use similar parameter values as in Boyle and Lin [4]. he risk-free rate is r = 10% and the maturity is = 1. he initial prices are set to be S k (0) = $40 for all k = 1,..., n. For each asset k, the price volatility is δ k = 30%. he correlation parameters are set to be ρ kl = 0.9 for all k l (and obviously, we can define ρ kk = 1.0 for all k = 1,..., n). X = (S k ( )) n k=1using the following formulae: and hese values are used to calculate first and second moments, µ and Q, of E[X k ] = e r S k (0), k = 1,..., n, E[X k X l ] = S k (0)S l (0)e r e ρ klδ k δ j, k, l = 1,..., n. he rational option price is e r P, where P is the expected payoff. he exact price is calculated by Monte Carlo simulations of correlated multivariate Brownian motion described in Glasserman [6]. he upper and lower bounds are calculated by solving semidefinite programming problems formulated in heorem 1 and. In this report, all codes are developed using Matlab 7.4 and semidefinite programming problems are solved with SeduMi solver (Sturm [1]) using YALMIP interface (Löfberg [8]). We vary the strike price from K = $0 to K = $50 in this experiment and the results are shown in able 1 and Figure 1. In this example, we obtain valid positive lower bounds when the strike price is less than $40. he lower and upper bounds are reasonably good in all cases. When the strike price decreases, the lower bound tends to be better (closer to the exact value) than the upper bound. 4. Put Options on the Maximum of Several Assets European put options written on the maximum of several assets also have non-convex payoff. he payoff is calculated as P = E[(K max 1 k n X k ) ], where X k is the price of asset k at the maturity. 7

Option price with upper and lower bounds 5.0000 0.0000 15.0000 10.0000 5.0000 0.0000 0 5 30 35 40 45 50 Strike price Figure 1: Prices of call options on the minimum of multiple assets and their upper and lower bounds 8

Strike price 0 5 30 35 40 45 50 Exact option price 18.199 13.7308 9.8097 6.6091 4.340.571 1.5011 Upper bound 3.3489 19.1889 15.1476 11.3819 8.0961 5.545 3.887 Lower bound 15.965 11.4383 6.914.3900 0.0000 0.0000 0.0000 able 1: Call option prices with different strike prices and their upper and lower bounds Similar to call options on the minimum of multiple assets, upper and lower bounds of this payoff can be calculated by solving the following semidefinite programming problems: min Q Y µ y y 0 s.t. Y y = P 1 N 1, y y 0 (y n Y k=1 µ ke k ) n k=1 µ k = 1, µ 0, y n k=1 µ ke k y 0 K = P N, (8) P i 0, N i 0 i = 1,, and min Q Y µ y y 0 s.t. Y y e k = P k N k, y 0 K y e k k = 1,..., n P k 0, N k 0 k = 1,..., n. Solving these two problems using the same data as in the previous section and varying the strike price from $40 to $70, we obtain the results for this put option, which are shown in able and Figure. Strike price 40 45 50 55 60 65 70 Exact option price 1.7419 3.4669 5.8114 8.7931 1.1431 16.0553 0.0943 Upper bound 4.896 6.69 9.0706 1.5363 16.4070 0.5079 4.47 Lower bound 0.0000 0.0000 0.0000 3.853 8.3495 1.8737 17.3979 able : Put option prices with different strike prices and their upper and lower bounds (9) We also have valid positive lower bounds when the strike price is higher than $50. he lower bound is closer to the exact value than the upper bound when the strike price increases. In general, both upper 9

Option price with upper and lower bounds 5.0000 0.0000 15.0000 10.0000 5.0000 0.0000 40 45 50 55 60 65 70 Strike price Figure : Prices of put options on the maximum of multiple assets and their upper and lower bounds 10

and lower bounds are significant as compared to the exact option prices. References [1] D. Bertsimas and I. Popescu. On the relation between option and stock prices: a convex optimization approach. Operations Research, 50():358 374, 00. [] D. Bertsimas and I. Popescu. Optimal inequalities in probability theory: a convex optimization approach. SIAM Journal on Optimization, 15(3):780 804, 005. [3] F. Black and M. J. Scholes. he pricing of options and corporate liabilities. Journal of Political Economy, 81:637 654, 1973. [4] P. Boyle and X. Lin. Bounds on contingent based on several assets. Journal of Financial Economics, 46:383 400, 1997. [5] P. H. Diananda. On non-negative forms in real variables some or all of which are non-negative. Proceedings of the Cambridge Philosophical Society, 58:17 5, 196. [6] P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer, first edition, 004. [7] A. W. Lo. Semi-parametric upper bounds for option prices and expected payoffs. Journal of Financial Economics, 19:373 387, 1987. [8] J. Löfberg. YALMIP : A toolbox for modeling and optimization in MALAB. In Proceedings of the CACSD Conference, aipei, aiwan, 004. [9] R. C. Merton. heory of rational option pricing. Bell Journal of Economics and Management Science, 4:141 183, 1973. [10] K. G. Murty. Some NP-complete problems in quadratic and nonlinear programming. Mathematical Programming, 39:117 19, 1987. [11] P. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of echnology, 000. [1] J. F. Sturm. Using SeDuMi 1.0, a Matlab toolbox for optimization over symmetric cones. Optimization Methods and Software, 11-1:65 653, 1999. [13] L. Zuluaga and J. Peña. A conic programming approach to generalized chebycheff inequalities. Mathematics of Operations Research, 30:369 388, 005. 11