Sharp Upper and Lower Bounds for Basket Options

Transcription

1 Sharp Upper and Lower Bounds for Basket Options Peter Laurence Dipartimento di Matematica, Università di Roma 1 Piazzale Aldo Moro, 2, I Roma, Italia laurence@mat.uniroma1.it and Tai-Ho Wang Department of Mathematics, National Chung Cheng University 160 San-Hsing, Min-Hsiung, Chia-Yi 621, Taiwan thwang@math.ccu.edu.tw Abstract Given a basket option on two or more assets in a one period static hedging setting we consider the problem of maximizing and minimizing the basket option price subject to the constraints of known option prices on the component stocks and consistency with forward prices and treat it as an optimization problem. We derive sharp upper for general n-asset case and sharp lower bounds for 2-asset case, both in closed forms, of the price of the basket option. In the case n = 2 we give examples of discrete distributions attaining our bounds. We also derive hedge ratios for optimal sub and super replicating portfolios consisting of the options on the individual underlying stocks and the stocks themselves. Keywords: basket option, duality, sharp bound 1

2 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS PETER LAURENCE AND TAI-HO WANG Abstract. Given a basket option on two or more assets in a one period static hedging setting we consider the problem of maximizing and minimizing the basket option price subject to the constraints of known option prices on the component stocks and consistency with forward prices and treat it as an optimization problem. We derive sharp upper for general n-asset case and sharp lower bounds for 2-asset case, both in closed forms, of the price of the basket option. In the case n = 2 we give examples of discrete distributions attaining our bounds. We also derive hedge ratios for optimal sub and super replicating portfolios consisting of the options on the individual underlying stocks and the stocks themselves. 1. Introduction We derive sharp upper and lower bounds for the price of a basket option on several underlying stocks in a one period static arbitrage setting. Our bounds are distribution free, in other words, independent of the probability distributions underlying the stock prices. We focus on a particular kind of basket, the average (or index) option, whose payoff at time T is given by ( + Φ = w i S i (T) K), where S i (t), 0 t T, are the prices of the stocks, K is the strike of the basket and w i s, i = 1,,n, are positive weights. We place ourselves in the position of an investor who at time t = 0 knows the prices of European options of the same maturity T of given strikes K i s and the discount rate r prevailing over the time period [0,T]. Since the true distributions of the stocks (risk-neutral or real world ) are unknown and markets are in general incomplete, the investor cannot calculate a unique no-arbitrage price B of the basket option, but we show that there is a range B L B B U within which the price of the call option on the average must lie, independently of the dynamics underlying the stock prices. These bounds are sharp. That is, they 2

3 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 3 cannot be improved unless more information is provided about the statistical process describing the stocks prices or concerning the prices of additional options with other strikes or maturities. A corresponding range for a put option on the basket follows immediately from put-call parity. 1 The bounds obtained here are derived in a one-period setting. Such bounds can be refined, in a multi-period setting, using the knowledge of the stock prices, S j (t i ),j = 1,,n, at intermediate times t i, 1 i m, the spot prices and new option prices. The bounds could also be sharpened in the future by inputting (as constraints) the prices of options with other strikes. We plan to address these and other refinements in future work. We obtain the bounds by solving optimization problems for families of risk-neutral measures µ = µ(s 1,,S n ) subject to constraints. The basket option s price at time t = 0 is given by ] ( + (1) e rt E µ [( w i S i K) + = e rt w i S i K) dµ R n is our objective functional. We impose constraints on the individual option prices e rt E µi [(S i K 1 ) + ] = c i, i = 1,,n, where µ i,i = 1,,n are the marginal distributions of the unknown measure µ. To ensure risk-neutrality, we also impose the conditions (2) e rt E µi [S i ] = S 0 i, where S 0 i are the spot prices at time t = 0. A distinction to be emphasized, as Andrew Lo (Lo, 1987) made clear in his paper on distribution free bounds, is that a bound for the expectation of the option s payoff is not equivalent to a bound for the price of the option. A necessary condition that a distribution must satisfy to be a pricing measure is risk neutrality. In order to bound options prices, we must consider distributions which also satisfy the conditions (2). These are necessary conditions for the probability distribution to be a pricing measure. Note that, implicit in the constraint (2), is the assumption that the short rate is known and constant during the period [0,T]. As an alternative we may, instead of the 1 Put call parity is well known for European options, but extends naturally to basket options. Since we couldn t locate a reference for this fact in the literature, we note that it follows immediately by using the identity: ( w i S i K) + (K w i S i ) + = w i S i K.

4 4 PETER LAURENCE AND TAI-HO WANG risk neutral measure µ, use the T-forward measure µ T F and assume only the knowledge at time 0 of the price P0 T of a zero-coupon bond with maturity T. In this the form of all constraints and that of the basket price remain unchanged provided we substitute P0 T for e rt and also µ T F for µ. Also all results in this paper can, by means of such a dictionary, be transferred over into this alternative setting. As we will see below, a different level of computational difficulty is involved in deriving upper as opposed to lower bounds. Define D = K n w i K i. This parameter will play an important role throughout this paper. Note that since in the market options on S i, i = 1,,n, of many different strikes K i are traded, only some will in fact, for a basket with strike K, give rise to a zero value of the parameter D. In this important subcase D = 0, the sharp upper bound, which reads n w i c i can be derived by means of an elementary argument. In fact, in this case it was already derived in 1973 by Merton (Merton, 1973), but Merton did not prove that his bound was sharp. We will see that this is the case by establishing the following: the bound is attained provided that the statistical processes describing the stock prices possess an additional property, referred to as Condition P. In the more general case D > 0, it is also possible to give an elementary proof that n w i c i is an upper bound. But in this case we know of no simple argument that establishes that this is a sharp bound. We prove it is so by using a duality argument and a study of the resulting optimization problems. The form of the upper bound in the case D < 0 is very different than in the case D 0. To our knowledge it appears in its present form, for the first time in the short version of our article [13]. In independent work d Aspremont and El-Ghaoui (d Aspremont and El-Ghaoui, 2003) presented an upper bound with a completely different proof. Their bound can be shown to be equivalent to ours, but the form presented in this paper is, in our opinion, much simpler. We derive the sharp lower bound on the price of the basket subject to the constraint of prescribed forward prices, only in the case of two assets. The case where forward prices are constrained is significantly harder than the case where they are not. Also, it is much more relevant financially, since spot prices are readily available and in the absence of their specification, the range of allowable basket prices becomes far too wide. As the reader will see, the two dimensional case already presents a considerable amount of computational complexity. As in the case of the upper bound, our derivations is

5 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 5 by means of the analysis of an optimization problem in the space of distributions satisfying the constraints and its dual and by finding explicit solutions for the value of the dual problem. The same method applies in principle to the general case, case of a basket on n stocks, given knowledge of the prices of n input options and indeed in 2 we formulate the primal and dual problems in this more general setting. Only in the case of 2 stocks are we able to calculate the lower bound in subject to prescribed forward prices in closed form. For n > 2 we show that the price of a forward contract on the basket with delivery price K provides a lower bound and we note that this lower bound can be shown to be sharp, for certain values of the input parameters c i,s i 0 and K i, i = 1,,n, provided distributions satisfying the constraints and a simple additional condition we refer to as Condition Q can be found. An additional contribution of this paper is the derivation of optimal hedge ratios. It is not possible in general to perfectly hedge the basket option by buying or selling a portfolio of options on the individual assets. In the absence of a perfect hedge, in incomplete markets, the next best thing is the least expensive super-replicating strategy involving a portfolio of the individual assetss i, the individual options C i and cash that super-replicates the basket option. Let (u, ν,v ), where u = (u 1,,u n) and v = (v 1,,v n), correspond to the solution of the dual to problem P U in which an infimum is sought. Let V U be the value of this super-replicating strategy. Then B U corresponds to a reasonable ask price for the seller of the basket option, since if he sells at this price and acquires the portfolio with, for i = 1,,n, u i units of call option c i, v i units of asset S i and ν units of cash, the cash flows from his portfolio at time T will more than cover any possible losses from his short position in the basket option. No cheaper super-replicating strategy involving portfolios consisting of the C i, S i and cash would super-replicate the basket option in a model independent framework. Similarly the optimal (most expensive) sub-replicating strategy V L corresponds to a reasonable bid price for the investor seeking to buy the option. An immediate consequence of the duality formulation is that V U = B U and V L = B L, where B U, B L are the sharp upper and lower bounds for the constrained basket option price. Following pioneering work by Robert Merton (Merton, 1973), Andrew Lo (Lo, 1987) derived the following distribution free upper bound for the price C of a European call option on one asset give that the variance V and the short rate r (assumed constant

6 6 PETER LAURENCE AND TAI-HO WANG over the period under consideration) are known S(T) Ke rt + S 0 V e 2rT if S(T) 1 + V C e 2e rt 2rT K 1+V e 2rT 1 [S(T) Ke rt + ] (Ke 2 rt S(T)) 2 + S 2 (T)V e 2rT if S(T) < 2e rt K 1+V e 2rT Although it is difficult to determine empirically the variance of the unobserved risk neutral distribution, Lo showed the usefulness of this bound by comparing it with explicit theoretical prices obtained under the assumption that the distribution of S(T) is known, such as the lognormal distribution and Merton s mixed diffusion jump process. Lo s approach uses an earlier result by Scarf (Scarf, 1958) which shows that the upper bound is attained by a two point distribution. In a recent paper Bertsimas and Popescu (Bertsimas and Popescu, 2002), consider option bounds from a more general standpoint. They treat, in addition to many other interesting problems, that of deriving upper and lower bounds for options with general payoff functions, given knowledge of moments of order n of the underlying s distribution or given prices of other options on the same underlying but with different strike prices. They use a different approach than Lo s, an approach, closely related to the one in our paper, based on semi-indefinite programming and consideration of dual problems. In their Theorem 3, Bertsimas and Popescu recover Lo s result by this method. In Theorem 4, they also resolve a question posed in an earlier paper by Grundy (Grundy, 1991) by deriving the following result, very much in the same spirit as the one in this paper, but limited to the case of an option on one underlying stock. Given prices q i = q(k i ) = E[max(0,X K i )], i = 1,,n, of call options with strikes 0 K 1 K n on a stock X, the range of all possible valid prices for a call option with strike price K where K (K j,k j+1 ) for some j = 0,,n is q (K), q + (K) where ( ) q K K j 1 K j K K j+2 K K K j+1 (K) = max q j + q j 1,q j+1 + q j+2, K K j 1 K j K j 1 K j+2 K j+1 K j+2 K j+1 q + K j+1 K K K j (K) = q j + q j+1. K j+1 K j K j+1 K j A closely related result had also been obtained earlier by Ryan (Ryan, 2000) (see Theorems A1 and A3). Bertsimas and Popescu s result has the appealing feature that is expressed only in terms of observable quantities, i.e., the observed prices of options

7 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 7 on the same underlying with different strikes. Thus its verification by an investor who holds or is considering acquiring an option with illiquid strike K is immediate. In contrast, Lo s result, as mentioned earlier, involves the variance of S under the unobservable risk neutral distribution, whose determination in an incomplete market requires the knowledge of the investor s utility function. As mentioned earlier a short version of the present results appeared in [13]. In the meantime we have learned of results obtained independently by d Aspremont and El- Ghaoui (d Aspremont and El-Ghaoui, 2003). D Aspremont and El-Ghaoui (hereafter DA-EG), succeed, as we do, in providing upper and lower bounds in closed analytic form. In addition when, as in the case of lower bounds with forward prices prescribed, they do not reach this goal, they provide numerical methods that allow the numerical determination of such bounds. Here is a comparison of our results: (i) Both DA-EG and ourselves derive upper bounds in closed form in important setting where forward prices are prescribed. The bounds turn out to be equivalent. But their equivalence is by no means obvious from the form of the result. In our opinion the form we give for the upper bound is considerably simpler. Whereas the DA-EG s form is expressed by means of the sum of minima of n quantities, ours involves one single simple expression. The key ingredient in deriving such a simple expression, is the introduction of a key auxiliary indicator, associated to these bounds, the ordering of the indices according to the order of the slopes Si 0 C i(k i ) K i where S i 0 is the spot price of the i-th stock and C i (K i ) is the call price of the option on asset i with striking price K i. The proofs in DA-EG s papers are very different, though both DA-EG and we consider dual problems. (ii) DA-EG derives an optimal lower bound in closed form in the case where forward prices are not prescribed for all n 1. We did not derive such a lower bound. (iii) DA-EG do not derive a lower bound in closed form when forward prices are prescribed. We have derived such a lower bound in the case n = 2. We have also derived a lower bound for all n, when forward prices are prescribed that is optimal under certain assumptions on the distribution. (see Proposition 4 and Condition Q) (iv) We have derived optimal hedge ratios in closed form for optimal super-replicating portfolios for all n 1 and for optimal subreplicating portfolios in the case n = 2.

8 8 PETER LAURENCE AND TAI-HO WANG (v) In the case n = 2 we have given examples of non-trivial (ie. with support bounded away from (0, 0) discrete distributions that attain our lower and upper bounds. These cannot be viewed as realistic in the case of real financial markets, since their support is discontinuous, but properly smoothed, they could be a candidate for processes dominated by a jump component. (vi) In the important case D = 0 we give necessary and sufficient conditions (Condition P) for distributions to be optimal. Many other authors have made contributions to distribution free option bounds in the case of an option on a single asset. These include the papers by Ritchken (Ritchken, 1985), Perrakis and Ryan (Perrakis and Ryan, 1984), El-Karoui et al.(el- Karoui et al., 1998). In the case of basket options, the only other contributions of which we are aware are the aforementioned paper of Bertsekas and Popescu, those of Gotoh and Konno (Gotoh and Konno, 2002a, 2002b) who find efficient algorithms to deal with a relaxed form of optimization problems for options on several assets with general payoffs. We have also learned but not seen the conference proceedings of Natarajan, Bertsimas and Teo (Natarajan, Bertsimas and Teo, 2003). A paper by Boyle and Lin s (Boyle and Lin 1997) contribution is to provide a solution to a generalization of Lo s problem to the multi-asset setting for the special case of the option on the maximum. We make a remark about the n-point (n 4) distributions that we derived in the two asset case, which attain our upper and lower bounds. Apart from Scarf s result (Scarf, 1958) which Lo used in his paper (Lo, 1987), we know of no other explicit constructions in the literature of this kind, the difficulty lying in the fact that duality principles while leading us and others to bounds cannot easily be exploited to yield explicit solutions. We conclude this introduction with a brief discussion of the long and short term implications of the bounds and hedge ratios obtained in trading index options. As illustrated by the numerical results in 6.1 for the case of an index with two assets, the gap between the optimal upper and lower bound is very wide. This fact is by no means surprising as upper bounds are, modulo some provisos, associated with co-monotonic components whereas lower bounds are associated with anti-monotonic components (for more than two assets the latter is a well defined concept only taken pairwise). But the components in real indices lean far more towards being positively correlated then towards being negatively correlated. Thus it is the upper bound and associated

9 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 9 super-replicating strategy that will tend to be of much greater practical use. As the experiment with real data in 6.2 demonstrates, with a judicious choice of strikes for each component, the super-replicating strategy is more expensive than the actual index option by a factor of about 20 45%. Thus the super-replicating strategy, in its present form, probably seldom provides a means of detecting an arbitrage opportunity by selling the index option and buying the portfolio of individual options and stock. This is to be expected since the optimization problem we have solved does not take into account information concerning the correlations between the assets. Future work should envision incorporating restrictions on these that are realistic in dealing with the particular index under consideration, some indices being much more homogeneous than others. However the gap between the upper bound and the true price can be monitored and when this gap is at a peak, this may well be a signal that the index is undervalued and a signal to buy the index. 2. Notation and Reduction to normalized form Let R n + denote the positive cone in R n R n + = {(x 1,x 2,,x n ) R n : x i 0,i = 1,,n}. The primal problem, in the general setting of the basket option on n assets, is the following: Minimize (or maximize) the T-period discounted price of the basket option (3) e rt ( w i S i K) + dµ(s 1,,S n ) R n + over all probability measures µ on R n + satisfying the following constraints: Known Prices of Options on Individual Single Assets (4) e rt (S i K i ) + dµ(s 1,,S n ) = c i R + n Since the integrand in the n constraints (4) depends only on S i, the latter can also have been expressed as (5) e rt (S i K i ) + dµ i (S i ) = c i, i = 1,,n, R + 1

10 10 PETER LAURENCE AND TAI-HO WANG where µ i is the marginal of µ with respect to S i. Therefore the constraint (4) incorporates our assumed knowledge at time zero, of the market price of vanilla options, trading on assets S i,i = 1,,n, with maturity T and strike K i,i = 1,,n. Risk Neutral Probability Measure To this we add (6) e rt S i dµ i (S i ) = S0, i i = 1, n R n + which expresses the fact that all candidate pricing measures must be risk-neutral measures. We introduce the following space of admissible measures: (7) M = {µ M 1 : E µ [ψ(s 1,,S n )] < +, and µ satisfies (4), (6)}, where M 1 denotes the set of all probability measure on R n + and ( + (8) ψ(s 1,,S n ) = w i S i K). It will be convenient in what follows below to normalize our problem in order to simplify ensuing computations. To this end, introduce the following abbreviations x i = w i S i, K i = w i K i, i = 1,,n, n Γ = w i, ν(x 1,..,x n ) = 1 Γ d µ(x 1,..,x n ) M 1, where µ is the measure (not a probability measure) induced on the (x 1,,x n ) variables, by the change of variables (S 1,,S n ) (x 1,,x n ) and the factor Γ corresponds to the Jacobian of the transformation. With this notation our objective function is e rt ( x i K) + dν(x 1,,x n ). R n + Since the factor e rt does not affect the optimal choice of ν we can adjust for it later and take (9) Λ(ν) := ( x i K) + dν(x 1,,x n ), R n +

11 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 11 as objective functional. The constraints become (10) (x i K i) + dν(x 1,..,x n ) = c i, i = 1,,n, (11) R n + x i dν(x 1,..,x n ) = m i, i = 1,,n, R n + and (12) dν = 1, R n + where we have introduced the abbreviations, for i = 1,,n, c i = e rt w i c i, m i = e rt w i Si 0. Also, we may then define the appropriate space of measures corresponding to the reduced variables by (13) M R = {ν M 1 : Λ(ν) < + and ν satisfies (10), (11)}. Thus we consider two primal problems: and Primal Problem P U corresponding to Upper bound sup{λ(ν) : ν subject to constraints (10), (11), (12)} Primal Problem P L corresponding to Lower bound inf{λ(ν) : ν subject to constraints (10), (11), (12)}. It is easily seen, from the definitions of m i and c i that these must satisfy the following compatibility conditions in order that problems P U and P L admit a solution. Let m = (m 1,,m n ), c = (c 1,,c n) (14) C = {(m,c) R 2n + : m i c i 0 & m i c i K i 0, i = 1,,n}. Peter Carr pointed out to us that these compatibility conditions are already present in Merton s paper (Merton 1973). The Dual Problems The dual problems to respectively P U and P L are

12 12 PETER LAURENCE AND TAI-HO WANG Dual to P U (15) inf { c iu i + ν + m i v i : (u, ν,v) R 2n+1 } subject to the constraints (16) u i (x i K i) + + ν + v i x i ( x i K) +, x i 0, i = 1,,n. Dual to P L (17) sup { c iu i + ν + m i v i : (u, ν,v) R 2n+1 } subject to the constraints (18) u i (x i K i) + + ν + v i x i ( x i K) +, x i 0, i = 1,,n. The new objective functionals (15) and (17) are affine functions on the space R n +. The constraints (16) and (18) of the dual problems constitute a set of piecewise linear inequalities. From the paper of Isii (Isii, 1963) we can deduce the following result of strong duality. Proposition 1. (Strong Duality) If (m,c) C (the interior of C) then the values of the dual and primal problems coincide. The strong duality follows from Theorem 1, page 187 in Isii s paper. However to ensure that the hypotheses of his theorem are verified we must ensure that B U < in the case of the upper bound, and similarly B L >. Since our payoff function is nonnegative the condition for the lower bound is trivially satisfied. In the case of the upper bound it follows from the following argument. [( ) + ] [( ) + ] E ν x i K = E ν i K i) (x + K i K E ν [(x i K i) + ] + ( K i K) +, where the last inequality follows from the elementary inequality (a + b) + a + + b +. Therefore, for any probability measure ν, with respect to which the payoff

13 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 13 ( n + x i K) is integrable and satisfying the constraints (10), we have ] E ν [( x i K) + c i + ( K i K) + < +. Taking the sup over all such ν we have established the necessary condition required in Isii s theorem. 3. Main Results Before stating the main result, let s fix the notation (19) D K w i K i which will play an important role in the following discussion. In case D < 0, for simplicity, let us reorder the indices i according to the order of F i, which is defined by F i = e rt Si 0 c i = m i c i, i = 1,,n, K i K i as 0 F 1 F 2 F n 1 and for the indices that share the same F i s we further reorder those indices such that their corresponding K i s are in nondecreasing order. Finally, let ī be the (newly reordered) smallest index such that w 1 K 1 + w w ī 1 K ī 1 D, w 1 K 1 + w w ī K ī D. Here we remark that such ī is uniquely determined. In the case of the upper bound we can give the following result for the n asset problem. Proposition 2. (Sharp Upper Bound) Let (m,c) C. The upper bound of (3) subject to the constraint that µ M is given by (20) w i c i if D 0, (21) ī 1 w i c i + e rt w i K i F i e rt (K w i K i )F ī if D < 0. i=ī

14 14 PETER LAURENCE AND TAI-HO WANG Note that in the case of D < 0, by the definition of ī, we find that K w i K i = w 1 K 1 + w w ī 1 K ī 1 + D 0 i=ī which indicates that the upper bound of a basket option is larger in case D < 0. Remark 1. In the case D = 0 any probability measure in M supported on the set (22) S 1 S 2 where (23) (24) S 1 = {x R n + : x = y + K,y R + n }, S 2 = {x R n + : x = K y,y R + n }, where K = (w 1 K 1,w 2,,w n K n ), attains the upper bound. We refer to this condition as Condition P. Proposition 3. (Sharp lower bound for n = 2) Let (m,c) C. If n = 2 and K > max{w 1 K 1,w 2 }, then the lower bound of (3) subject to the constraint that µ M is given by For D 0, (25) max { A 1 + e rt w 2 F +,A 2 + e rt w 1 K 1 F +,A 1 + A 2 + e rt KF +, 0 }, For D 0, (26) max { A 1 + e rt (K w 1 K 1 )F +,A 2 + e rt (K w 2 )F +,A 1 + A 2 + e rt KF +, 0 }, where (27) (28) (29) A i = w i c i ( K w ik i )(S0 i c i ) for i = 1, 2, K i F = F 1 + F 2 1, F + = max{f, 0}. Proposition 4. (Sharp lower bound under special conditions) Let (m,c) C. A sharp lower bound of (3) subject to the constraint that µ M in the case K < min{w 1 K 1,,w n K n } is given by (30) w i S0 i e rt K.

15 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 15 Proposition 5. (Lower bound for all n 2, conjectured to be sharp) (30) is still a lower bound even if K min{w 1 K 1,,w n K n }. Moreover, it is sharp provided there exists a joint distribution µ M supported in the region {(x 1,,x n ) R n + : n w ix i K 0}. We refer to this condition as Condition Q. Remark 2. The hard part in showing this lower bound is sharp is exhibiting a distribution compatible with the constraints that also satisfies Condition Q. In the case n = 2 this can be done using a discrete distribution, as we shall give in 5. We conjecture this is also the case for any n 2. The proofs of Proposition 2 for D 0, Proposition 3 and the sharpness in Proposition 4 are given in Appendix I and Appendix II respectively. Both of the proofs make use of the reduced version to simplify the computations. But for the special case D = 0 of the upper bound the following simple argument can be used. Proof of the upper bound when D 0. For D 0 we have ( + (31) w i S i K) = (32) = ( w i (S i K i ) + w i K i K ( + w i (S i K i ) D) w i (S i K i ) + Therefore, for any measure µ M, taking expectations and multiplying by the discount factor e rt we obtain e rt E µ [ψ] w i c i, as required. In the case D = 0 this proof is due to Merton [16]. Note that this proof only establishes that n w i c i is an upper bound, but does not establish that it is the best possible upper bound. However in the case D = 0 it is easily seen that, provided that the joint distribution µ M of S = (S 1,,S n ) is supported on the set S 1 S 2 defined in (23) and (24), all inequalities in (32) become equalities. Proof of a lower bound. ) +

16 16 PETER LAURENCE AND TAI-HO WANG A lower bound can be obtained by the following simple argument. Consider the following inequality, for every (S 1,,S n ) R n +, ( + w i S i K) w i S i K. Therefore, for any measure µ M, taking expectations and multiplying by the discount factor e rt we obtain e rt E µ [ψ] w i S0 i e rt K. 4. Optimal sub and super replicating portfolios Let us recall first that, in portfolio (u, ν,v), the first n entries u = (u 1,,u n ) represent the holding of units of calls C 1,,C n respectively, the entry ν represents holding of units in cash and the last n entries v = (v 1,v n ) represent holding of shares of stocks S 1,,S n respectively. Super-replicating portfolio In case D = K w i K i 0, a super-replicating portfolio (u, ν,v ) is given by (33) (u 1,,u n, ν,v 1,,v n) = (w 1,,w n, 0, 0,, 0), i.e., u i = w i, ν = vi = 0, i = 1,,n. In the case D < 0, a super-replicating portfolio (u, ν,v ) is given by (34) (u 1,,u ī,,u n, ν,v1,,v ī,,v n) = (0,, 0, K n i=ī+1 w n ik i i=ī,w ī+1,,w n, 0,w 1,,w ī 1, w ik i K, 0,, 0), K ī i.e., u i = 0, vi = w i, for i ī 1, u ī = K n i=ī+1 w n ik i, v ī = i=ī w ik i K, K ī u i = w i, v i = 0, for i ī + 1, ν = 0. K ī K ī

17 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 17 Sub-replicating portfolio A subreplicating portfolio in the 2-asset case is given by (u, ν,v ), where u = (u 1,u 2 ) and v = (v 1,v 2 ), and for brevity, we denote the cases where in (25) and (26) is equal to one of A 1, A 2, A 1 + A 2, A 1 + A 2 + KF, A 1 + (K w 1 K 1 )F, A 2 + (K w 2 )F, A 1 + w 2 F, A 2 + w 1 K 1 F, which, taken in the same order, we refer to as Cases 1 : 8. Case u 1 u 2 ν v 1 v 1 K 1 K K w 1K 1 K 1 0 K K w 2 K K 3 K 1 0 K w 1K 1 K 1 K w e rt K w 1 w 2 5 w 1 K w 1K 1 e rt (K w 1 K 1 ) 0 K w 1K 1 6 K w 2 K 1 w 2 e rt (K w 2 ) K w 2 K w 1 + D K 1 w 2 e rt w 2 D K 1 w 2 8 w 1 w 2 + D e rt w 1 K 1 w 1 D 5. Extremizing Discrete Distributions in the case n = 2 The approach we have taken throughout this paper wherein dual problems and duals of these dual problems have been introduced yields bounds and hedge ratios. But it does not appear to easily yield extremizing distributions. In fact, knowledge of extremizing distributions in optimization problems in finance seems to be considerably poorer than knowledge of optimal bounds. The only result we know of in which an extremizing distribution has been found is Lo s bound (Lo, 1987), discussed in the introduction, which uses an earlier result of Scarf. The extremizing distribution of Scarf (Scarf, 1958) is a two point distribution. In the multidimensional context, in the work of Boyle and Lin (Boyle and Lin 1997) on max options, distributions that are good approximations to optimizing distributions are given, but not in closed form. Our approach to finding extremizing distributions is a natural generalization of that of Scarf. We seek these extremizing distributions in the set of 7-point distributions. In practice, however, we find that our distributions are supported on three or four points. Each point lies in one of seven regions in which the plane is divided by the lines x + y = K, x = K 1, y = x 0,y 0, see Figure 1 below. We refer to these

18 18 PETER LAURENCE AND TAI-HO WANG regions as regions R i, i = 1,, 7. When D = 0, these seven regions degenerate into six. S 2 R 6 R 5 R 4 D = 0 R 3 R 1 R 2 S 2 D > 0 S 2 S 1 D < 0 R 6 R 5 R 7 R 4 R 6 R 5 R 4 R1 1 R 2 R 3 R 1 R 7 R 2 R 3 S 1 S 1 Figure 1. Regions R i. Distributions attaining lower bound Distribution attaining lower bound in Case D < 0 Proposition 6. An optimal feasible solution which attains the lower bound A 1 + A 2 + KF + is given by (K ( 1,K K 1 ) ) with prob. c (S 1,S 2 ) = K c,k 1 Q 1 +Q Q 1 +Q 2 D D (K, ) with prob. where Q i = K i (F i 1),i = 1, 2. Q 2 D in R 1, with prob. 1 Q 1+Q 2 D in R 4, Q 1 D in R 5, Proposition 7. An optimal feasible solution which attains the lower bound Ā 1 + F + is given by (0, ) with prob. c 2 K Q 1 K 1 in R 1, (K, 0) with prob. Q 2 ( in R 3, (S 1,S 2 ) = K 1 + c K K 1+Q 1 ) 2 F 1 +F 2 1, with prob. F 1 + F 2 1 in R 4, c (0,K) with prob. 2 K in R 5.

19 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 19 Proposition 8. An optimal feasible solution which attains the lower bound Ā 2 + K 1 F + is given by (K 1, 0) with prob. Q 2 c 1 K K 1 in R 1, c (K, 0) with prob. 1 K K (S 1,S 2 ) = 1 in R 2, ((K 1, ) ) with prob. F in R 4, 0, + c 2K 1 Q 1 with prob. 1 F 1 in R 5. Distribution attaining lower bound in Case D 0 Proposition 9. An optimal feasible solution which attains the lower bound A 1 + A 2 + KF + in the case D > 0 is given by ( ) K 1 + c 1 (Ā1+ F),K Q Q 2 Q with prob. 2 +ɛ Q 2 +ɛ 2 + ɛ 2 in R 3, ( ) 2 (S 1,S 2 ) = K 1 + Ā1+ F F ɛ 1 ɛ 2, + Ā2+K 1 F F ɛ 1 ɛ 2 with prob. F ɛ 1 ɛ 2 in R 4, ( ) K 1 + Q 1,K Q c 2 (Ā 2 +K 1 F) Q with prob. 1 +ɛ K 1 Q 1 +ɛ 1 K 1 K 1 + ɛ 1 in R 5. 1 Remark: It is interesting to note that the solution given above is valid in the case D = 0. On the other hand, it is not straightforward to see the limiting form of the solution attaining the same lower bound in the case D < 0, since D appears in the denominator in several expressions. Distributions attaining the upper bounds D = 0. This case was already dealt with in Remark 1. D < 0. Let β i,i = 1, 2,ξ be three parameters such that 0 β i 1, 0 ξ 1 and β 1 + β 2 + β 3 = 1, but otherwise arbitrary. Proposition 10. An optimal feasible solution is given by the following four point distribution ( ) )) Q (K Q max( Q 1, Q 2, (1 + 1 max( Q 1, Q 2 with prob. max( Q 1 ) K 1, Q 2 ) in R 1, ( ) ξ K c 1 β (S 1,S 2 ) =,K 1 min(f 1,F 2 ) 2 with prob. β 1 min(f 1,F 2 )inr 3 ( ) )) (K (1 ξ 1)c 1 β 2 min(f 1,F 2,K ) 2 (1 + (1 ξ 2)c 2 β 2 min(f 1,F 2 with prob.β ) 2 min(f 1,F 2 ) in R 4, ( )) (K 1, 1 + with prob.β 3 min(f 1,F 2 ) in R 5 ξ 2 c 2 β 3 min(f 1,F 2 ) D > 0. For ɛ n 0 let us indicate by (S ɛn 1,S ɛn 2 ) a feasible solution with the property that the value of the objective functional on this solution is

20 20 PETER LAURENCE AND TAI-HO WANG (S ɛn 1,S ɛn 2 ) = w 1 c 1 + w 2 c 2 Dɛ n. Then we have ( K 1 ) Q 1,K Q max( Q 1, Q with prob. p ) max( Q 1, Q 2 2 in R 1, ) K 1 K 1 (K ( 1, ) ) with prob. 1 ɛ n p 2 in R 7, K 1 + c 1 ɛ n, + c 2 ɛ n with prob. ɛ n in R 4, where p 2 = max( Q 1 K 1, Q 2 ). The value of the objective functional on this convergent subsequence is c 1 + c 2 Dɛ n. Note that when D is small ɛ n does not need to be very small in order for us to be close to the upper bound and correspondingly (S 1,S 2 ) does not need to be very large in region R Simulated Data. 6. Numerical Results We carried out Monte-Carlo simulations in order to test the upper and lower bounds in the cases n = 2 using two families of distributions, the multivariate lognormal distribution, and the multivariate student-t distribution with three degrees of freedom. We tested the n dimensional upper bounds for n = 2, 5, 10, 30, 50, 100 for the lognormal distribution. The results were similar to those for the two dimensional situation in the upper bound case and can be found in [13] (see Table 4). Numerical results in the case n = 2 In the case n = 2 we proceeded as follows: Given the input option prices and the spot prices, for a fixed value of the short rate r, maturity T, strikes K 1, we back out an implied volatility using the univariate log-normal, student-t distribution with 3 degrees of freedom, and thus determine σ i (r,t,s i 0,K i ),i = 1, 2. In the examples illustrated in the accompanying graphs the strikes were taken to be K 1 = = 10, when the maturities T is is equal to.1, and when T =.3 and K 1 = 30, = 20, when the maturity ia T = 3. Once we have backed out the implied volatilities σ i,i = 1, 2, both multivariate distributions are uniquely determined by the specification of the only remaining element of the variance-covariance matrix, the correlation. We consider correlations ρ =.95,.5, 0,.5,.95. In the case of the lognormal distribution ρdt actually corresponds to the correlation of the returns E[ ds 1 ds 2 S 1 S 2 ] and this can be shown to be related to the actual asset correlation ρ T A at time T by the following

21 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 21 formula 2 ρ T A = e σ 1σ 2 ρt 1 e σ 2 1 T 1 e σ2 2 T 1. This formula shows that when the instantaneous correlation ρ tends +1, and when in addition σ 1 = σ 2 (for instance), the asset correlation ρ T A also tends towards +1. But when the instantaneous correlation ρ tends to 1 the asset correlation stays bounded away from 1. The results of our simulations in the cases n = 2 are summarized in Figures 2 and 3 for the lognormal distribution, Figure 4, for the student-t distribution with three degrees of freedom Conparisons T = 0.1 MC UB LB MC Price rho=.95 rho=.5 rho= 0 rho=.5 rho=.95 Figure 2. Dependence of the upper bound on correlation ρ in multivariate Black-Scholes setting when the time to expiration is T =.1 These results show, for all three distributions that when ρ reaches +.95 the upper bound is very close to being reached especially when T = 1, and T = 3. Similarly, when ρ is.95 the lower bound is very close to being reached, especially for the times T = 1 and T = 3. Contrast this with the situation for the lower bound. When ρ =.95 the lower bound is far from being attained when T is small and gets very close to being attained when T = 1 or T = 3. 2 Details available on request

22 22 PETER LAURENCE AND TAI-HO WANG 21 Comparisons T= MC Price UB LB rho=.95 rho=.5 rho=0 rho=.5 rho=.95 Figure 3. Dependence of the upper bound on correlation ρ in multivariate Black-Scholes setting when the time to expiration is T = 3 11 Comparisons T= UB MC Price 4 3 LB 2 rho=.95 rho=.5 rho=0 rho=.5 rho=.95 Figure 4. Dependence of the upper bound on correlation ρ when joint distribution is a multivariate student-t distribution with three degrees of freedom and time to expiration is T =.3 Our numerical results in the case of the lower bound also indicate that when K 1 = the lower bound is nearly attained for long maturities T = 1, 3 for negative asset correlations ρ T A of.95 in the normal and student-t cases and.80 in the log-normal case. We do not, at this point, have a theoretical explanation for this difference, but

23 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 23 it again shows that correlation is not the only ingredient determining when the lower bound is attained Real Data: Illustration from CBOE data on the DJX. In this section we illustrate the use of the upper bound derived in the n-asset case on real data taken from CBOE. The index under consideration is one of the Dow Jones indices, the DJX. We downloaded data on May 17th on the July 2004 contract expiring on Jun 18th. The Dow Jones index is a thirty component price weighted index. On each component options of 8-13 different strikes were trading on May 17th all with the same Jun 18 th maturity. To evaluate the performance of our upper bound, we need as inputs for each component, the spot price and the strike price under consideration. There are two possible ways to use our upper bound. The first use is that of the investor who, for each component of the index, has already chosen one (non-zero) strike among the available ones and wishes to invest in options of that strike. He then simply calculates the hedge ratios provided in (34) and (33) to calculate the weights in each call and possibly stock in his least expensive super-replicating portfolio. The second, more likely use, is to take a double-pronged approach. Instead of using the hedge ratios for a fixed set of strikes, in a first stage he uses a numerical procedure to search, among the available strikes for the set that minimizes (or makes as small as possible) the value of our analytic upper bound. Then, at the second stage, he computes the hedge ratios for that, optimized set of strikes. The table below illustrates the results of this two stage procedure.

24 24 PETER LAURENCE AND TAI-HO WANG Component Component Strike Option MidMarket Price Portfolio weight Stock Price Portfolio weight Ticker Index Given Component Option/given component option chosen Component Stock SBC HON MCD HPQ INTC DD AA JPM WMT IBM PFE VZ UTX AIG C MMM HD AXP KO CAT MO GE MFST XOM DIS PG JNJ MR BA GM Table 1. DJX Data on May 17 th on June contract with 31 days to expiration. The DJX spot was The strike of the basket chosen is 92. The true DJX price of this option was 7.5. The upper bound obtained by our method using a basket of calls is In these calculations the short rate r, based on one year LIBOR was taken to be.018 and time to maturity was 31 days on May 17 th, till the common maturity of all options June 04, which is third Friday of June. ī in the definition of (21) is 14. Ie. only in the case of IBM does the optimal portfolio have both an option and a stock component. In the table below we show the upper bound calculated for other values of the basket strike K = 90, 94, 96, 98, 99, 100, while spot was K DJX N.O. LB O. UB

25 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 25 Table 2. CBOE data on May 17 th on June contract, with 31 days to maturity. The first column is the index option s strike. The second column is the true midmarket price, the third column is the not necessarily optimal lower bound (Equation (30)) and the fourth column is the optimal upper bound for a optimized choice of the optioms/strikes on the components with the same maturity. The not optimal lower bound performs well when the index is in the money and its performance decreases substantially in and out of the money. The optimal upper bound (after searching for optimized strikes to apply it to) also performs better in the money than at and out of the money. Part of its distance from the true price can be explained by the fact that the DJX s components are not co-monotonic. The resulting upper bound on the DJX is more than a theoretical tool. If the DJX price happens to be larger than this price, the approach discussed in this paper, gives the investor the possibility of selling the index and going long one single name option per asset for all components with index i ī and long stock for all i ī. In this way he is assured a riskless profit at maturity. In practice, such a mispricing will be rare, but since our upper bound is model independent, we believe it provides a valuable diagnostic tool to detect mispricing and possible arbitrage strategies. Moreover when the relative distance of the price to the optimal upper bound is large, this may well be a sign that the index is undervalued. 7. Conclusion We have derived upper bounds in closed form and optimal super-replicating hedge ratios for basket options on n-assets, for arbitrary n. We also have derived an optimal lower bound and subreplicating strategy in the case n = 2. The form of our lower bound in the case n = 2 shows that there are for both D < 0 and D 0 four regimes and in each of these regimes the sharp lower bound takes a different form, according to the maximum of four expressions. In the case n > 2 we have found a-conjecturedto-be-sharp, for certain regimes of parameters, lower bound. We have discovered and illustrated that perfect correlation or co and anti-monotonicity do not in of themselves provide a key to saturating our upper and lower bounds. The gap between our upper and explicit lower bound in the case n = 2 is large in many ranges of D. This is due to two factors. The first is the richness of the family of all risk-neutral distributions

26 26 PETER LAURENCE AND TAI-HO WANG compatible with our constraints. Thus distributions such as the discrete one in the previous section can satisfy these bounds while being legions away from the continuous distributions used under normal market conditions. The second reason is that in our optimization problem we not exploit all market information, concentrating on calibrating to input options with a single maturity and a single strike. Including prices of options with other strikes and maturities as inputs, when these are available, will no doubt significantly narrow the gap and make our bounds more useful in practice. Such extensions will require a significant amount of additional work. Taken in their present state the primary use of the bounds and hedging strategies presented here may be to provide model independent estimates and hedging strategies in highly stressed markets when standard parametric models break down. Our results may also be used to bound new experimental parametric models and provide a sanity check for their output. Moreover our upper bounds for general n, especially in the case D < 0, give a particularly useful model independent no-arbitrage upper bound on the basket. To summarize, in its present form the upper bound may be used for the following purposes Provide a sanity check on the output of parametric methods Risk Management for assessing the model risk in using a given parametric model. One may use the upper bound as a model independent index of the dependence of the components. When the gap, in percentage terms, between the actual price of the index and the optimal upper bound (for judiciously choses strikes) becomes unusually large, it is a model independent sign that there is a lack of comonoticity in the index compared with its historical behaviour and so the index is probably undervalued at that time and is unusually cheap. So it may be a good time to buy the index 3. References [1] D Aspremont, A. and El-Ghaoui, L. (2003) Static Arbitrage Bounds on basket options,arxiv:math.oc/ , [2] Bertsimas, D. and Popescu, I. (2002) On the relation between option and stock prices: A convex optimization approach, Operations Research, 50(2), This strategy has not been put to any exhaustive tests so we are not recommending it as anything other than a reasonable conjecture at this point

27 SHARP UPPER AND LOWER BOUNDS FOR BASKET OPTIONS 27 [3] Bertsimas, D. and Tsitsilkis, J. N. (1997) Introduction to linear optimization, Athena Scientific. [4] Boyle, P. and Lin, X. S. (1997) Bounds on contingent claims based on several assets, J. Financial Econ., 46, [5] Cherubini, U. and Luciano, E. (2002) Bivariate option pricing with copulas, Applied Math. Finance, 9(2), [6] Courtant, S., Durrleman, V., Rapuch, C. and Roncalli, T. (2002) Copulas, multivariate risk-neutral distributions and implied dependence functions, Working paper, [7] El Karoui, N., Jeanblanc-Picqué, M. and Shreve, S. E. (1998) Robustness of the Black and Scholes formula, Math. Finance, 8(2), [8] Gotoh, J. and Konno, H. (2002) Bounding option prices by semidefinite programs: A cutting plane algorithm, Management Science, 48(5), [9] Gotoh, J. and Konno, H. (2002) A cutting plane algorithm for semi-definite programming problems with applications to failure discriminant analysis, J. Computational and Applied Math., 146(1), [10] Grundy, B. (1991) Option prices and the underlying asset s return distribution, J. Finance, 46(3), [11] Isii, K. (1963) On sharpness of Chebyshevtype inequalities, Ann. Inst. Stat. Math., 14, [12] Joe, H. (1997) Multivariate Models and Dependence Concepts, Chapman and Hall, London. [13] Laurence, P., Wang, Tai-Ho, What s a basket worth? Risk Magazine, February [14] Lo, A. (1987) Semi-parametric bounds for option prices and expected payoffs, J. Financial Econ. 19, [15] Lunenberg, D. G. (1984) Linear and Nonlinear Programming, 2nd edition, Addison-Wesley, Reading, MA. [16] Merton, R. (1973) The Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4, [17] Natarajan, K., Bertsimas, D. and Teo, C.P. (2003) Upper bounds on the prices of a class of multi-asset options with convex payoffs, INFORMS 2003, Atlanta. [18] Nelsen, R. B. (1999) An Introduction to Copulas, Lectures Notes in Statistics, 139, Springer- Verlag, New York. [19] Perrakis, S. and Ryan, P. J. (1984) Option Pricing Bounds in Discrete Time, J. Finance, 39(2), [20] Rapuch, G. and Roncalli, T. (2002) Pricing multi-asset options and credit derivatives with copulas, Crédit Lyonnais working papers, [21] Ritchken, P. (1985) Option Pricing Bounds, J. Finance, 40(4), [22] Ryan, P. (2000) Tighter option Bounds from multiple Exercise Prices, Review of Derivatives Research, 4, [23] Scarf, H. (1958) A min-max solution of an inventory problem. In Studies in the mathematical theory of inventory and production, , Stanford University Press, Stanford, CA, K.J. Arrow, S. Karlin and H. Scarf, eds. 4 First version submitted to Risk February 2003.

28 28 PETER LAURENCE AND TAI-HO WANG 8. Appendix I - Upper Bound To make the notation simpler, given that in the dual problems only the normalized versions of the optimization problems enter our discussion, we will in the sequel denote c i and K i by c i and K i respectively in this section. The dual problem for the upper bound in n-asset case is to find subject to the constraints min u, ν,v c i u i + ν + u i (x i K i ) + + ν + m i v i v i x i ( x i K) +, x R n +. Let us introduce the following transformation, for i = 1,,n, x i = x i K i, λ i = u i + v i, µ i = K i v i. Then the problem is transformed, in the new variables, into (35) min λ,µ, ν c i λ i + ν + subject to the (implicit) constraints on λ, ν and µ (36) (K i λ i µ i )( x i 1) + + ν + Define the function f : R n + R by m i c i µ i K i µ i x i ( K i x i K) + 0 x R n +. f( x) = (K i λ i µ i )( x i 1) + + ν + µ i x i ( K i x i K) +. Remark 3. Notice that f is a piecewise affine function, it attains its local extrema at the points {(x 1,,x n ) R n +} described below which we group into 2 categories as follows. Category I {(x 1,,x n ) : x i = 0, 1}.