Optimization based Option Pricing Bounds via Piecewise Polynomial Super- and Sub-Martingales

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1 28 American Control Conference Westin Seattle Hotel, Seattle, Wasington, USA June 11-13, 28 WeA1.6 Optimization based Option Pricing Bounds via Piecewise Polynomial Super- and Sub-Martingales James A. Primbs Abstract In tis paper we first prove sufficient conditions for a continuous function of a diffusion process to be a super- or sub-martingale. Tis result is ten used to create piecewise polynomial super- and sub-martingale bounds on option prices via a polynomial optimization problem. Te polynomial optimization problem is solved under a sum-ofsquares paradigm and tus uses semi-definite programming. Te results are tested on a Black-Scoles example were a piecewise polynomial function of degree four in bot te stock value and time is used to compute upper and lower bounds. I. INTRODUCTION Te connections between martingales and option pricing teory are well establised. In particular, te fundamental teorem of asset pricing asserts tat for no arbitrage to exist, te discounted price of an option must be a martingale under te so-called risk neutral measure [6]. Tis result allows options to be priced by computing expectations, and as led to te successful use of Monte-Carlo metods in option pricing teory. In recent years researcers ave developed alternative computational metods tat compute ard bounds on option prices via optimization metods (as opposed to soft bounds tat accompany Monte-Carlo metods). Work in tis area includes [13], [1], [2], [7], [11]. Tese results sare a common root in generalized Cebysev bounds, and most can be interpreted as static arbitrage bounds using special securities wit monomial s. An exception is [11] wic allowed certain discrete time dynamic strategies. Tis paper presents a new optimization based ard bound approac tat uses martingale teory to construct bounds. We first derive conditions under wic a continuous function (not necessarily C 2 ) is a super- or sub-martingale. Tis result allows us to construct super- and sub-martingale bounds on an option price via piecewise polynomial functions. Te use of piecewise polynomial functions allows us to optimize te bound via a polynomial programming problem tat is replaced by a sum-of-squares formulation using te software package SOStools [1]. Tus, semi-definite programming is ultimately used to compute te bound. Te paper is organized as follows. In Section II we set up te problem and review basic facts from martingale pricing. In Section III we develop some teoretical preliminaries. Section IV presents te main teorem tat gives sufficient conditions for a continuous function to be a super- or submartingale. Section V uses te main teorem to justify te use of piecewise polynomial functions in an optimization J. A. Primbs is wit te Management Science and Engineering Department, Stanford University, japrimbs@stanford.edu problem tat computes ard upper or lower bounds. Numerical results for te Black-Scoles setup are given in Section VI. Conclusions are discussed in Section VII. II. PROBLEM SETUP AND MARTINGALE PRICING For simplicity of exposition, we present te results for an option on a single underlying asset. However, it may be easily verified tat all te results in tis paper are directly applicable to options on multiple underlying assets. We consider te risk neutral pricing problem were under te risk neutral measure Q, te underlying asset S(t), t [,T] satisfies te stocastic differential equation ds = a(s,t)dt + b(s,t)dz(t) (1) were z(t) is a standard Brownian motion. We assume tat (i) a(s,t) and b(s,t) are continuous on R [,T]. (ii) Tere exists an M suc tat a(s,t) M(1+ S ) and b(s,t) M(1+ S ) for all (S,t) R [,T]. (iii) For eac c, tere exists K c suc tat a(s 1,t) a(s 2,t) K c S 1 S 2 and b(s 1,t) b(s 2,t) K c S 1 S 2 wenever S 1 c, S 2 c. Under tese conditions, (1) as a patwise unique solution wic is a Markov diffusion process (See Capter 5, [4]). A. Martingale Pricing Let c(s,t) be te value of a European call option wit expiration T and strike price K. By te fundamental teorem of asset pricing [3], te value of tis option at time wit S() = S is given by c(s,) = e rt E,S [ ( K) + ]. (2) were E t,st [ ] denotes te expectation under Q conditional upon time t and price S(t) = S t. In particular, te quantity e r(t t) c(s(t),t) (3) is a Q-martingale. Furtermore, assuming tat c(s,t) C 2,1, ten by te Feynman-Kac teorem [8], c(s,t) satisfies te Black-Scoles partial differential equation c t + a(s,t)c S b2 (S,t)c SS rc =, c(s,t) = (S K) + (4) were c t = c t, c S = c S, and c SS = 2 c. S 2 For simplicity (and witout loss of generality) in wat follows we assume tat r =. Tus, c(s,t) is a Q-martingale /8/$ AACC. 363

2 B. Approac via Super- and Sub-Martingales To explain our basic approac, consider te problem of computing an upper bound on c(s,). Now, any supermartingale satisfying V(S,T) (S K) + will provide an upper bound via V(S,) E,S [V(,T)] E,S [ ( K) + ] = c(s,). A similar argument sows tat any sub-martingale satisfying V(S,T) (S K) + will give a lower bound. Our computational approac will be to searc for superand sub-martingales tat are piecewise polynomial. But first we need convenient caracterizations of tese super- and sub-martingales tat allow for computation and optimization. In particular, our piecewise polynomial super- and submartingales will not be everywere C 2,1. Tus, we prove a result tat provides sufficient conditions for a continuous function to be a super- or sub-martingale. Tis condition is ten used as a constraint in our optimizations, wic are conveniently solved using a sum-of-squares polynomial programming approac. Te next two sections develop te teory beind our computational approac. We begin wit preliminary facts and results tat are used in te proof of te main teoretical result in Section IV. III. THEORETICAL PRELIMINARIES As preliminaries for te main teoretical result, we review conditions for Dynkin s formula to old, provide a generalization of te standard Newton-Leibnitz calculus formula, and prove a simple result on te extension of functions. A. Diffusions and Dynkin s Formula Following te development on page of [5], from te assumptions of (i), (ii), and (iii) in Section II, we can furter assert tat for eac m = 1,2,..., tere exists a constant C m (depending on m and T ) suc tat E,S S(t) m C m (1+ S m ), t T (5) Let C 2,1 p denote te space of f C 2,1 suc tat f, f t, f x, f xx satisfy te polynomial growt condition f(x,t) k(1+ x m ), (x,t) R [,T] (6) for some constants k and m. Ten it follows tat f(s(t),t) C 2,1 p satisfies Dynkin s formula, [ t2 ] E t1,s 1 [ f(s(t 2 ),t 2 )] f(s 1,t 1 ) = E t1,s 1 A f(s(τ),τ)dτ, t 1 were A is te differential operator (7) A f = f t + a(s,t) f x b(s,t)2 f xx. (8) See [5] page 129 for more details. Now, by (6), for f C 2,1 p, A f(x,t) k(1+ x m ) for some k and m. Coupling tis wit (5) guarantees te absolute integrability of te rigt and side of (7). Terefore Fubini s teorem [12] justifies an excange of te expectation and integral. Hence, one may assert from (7) tat te generator is given by E t,s(t) [ f(s(t + ),t + )] f(s(t),t) lim = A f (9) for all f C 2,1 p. Te caracterization of te generator in (9) is important since we will use a condition based on te generator A to caracterize super- and sub-martingales in te main teorem. B. Generalization of Newton-Leibnitz Formula Since we will be dealing wit functions tat are only continuous, we will need te following result from [15]. Lemma 3.1: Let g C[,T]. Extend g to (,+ ) wit g(t) = g(t) for t > T, and g(t) = g() for t <. Suppose tere is a ρ L 1 (,T) suc tat Ten g(t + ) g(t) limsup ρ(t), a.e. t [,T]. (1) β g(β) g(α) α for α β T. limsup + g(τ + ) g(τ) dτ, (11) For a proof of tis result, see [15], page 27. C. Result on extension of functions Te final preliminary result is on te extension of functions. We will need to lower and upper bound our continuous function by a C 2,1 p function tat lies in te domain of te generator. Te following lemma allows us to do tis in a global manner. We give only te upper bound extension, wit a lower bound extension following in exactly te same manner. Lemma 3.2: Let V(x,t) C p for (x,t) R [,T] (tat is, V(x,t) satisfies te bound in (6)). Extend V(x,t) to (x,t) R R by V(x,τ) = V(x,T) for τ > T and V(x,τ) = V(x,) for τ <. Let f C 2,1 suc tat f V in an open set O containing (x,t ). Ten tere exists an open set O containing (x,t ) wit O O and a function f C 2,1 p suc tat f = f in O and f(y,τ) V(y,τ) for all (y,τ). Proof: Define te function g by g(x,t) = M + k(1+ x m ) (12) were k and m 2 correspond to te bound in (6) for V(x,t), M = max (y,τ) B2 (x,t ) V(y,τ), and B 2(x,t ) is te closed ball of radius 2 centered around (x,t ). Tus g(x,t) V(x,t) for all (x,t), and g(x,t) C 2,1 p. Now, let b(x,t) be a C cutoff function tat is 1 on te compact set B ε (x,t ) and zero outside B 2ε (x,t ) were ε > is cosen so tat B 2ε (x,t ) O. Ten letting f(x,t) = f(x,t)b(x,t)+g(x,t)(1 b(x,t)) (13) wit B ε (x,t ) satisfies te teorem. We are now ready to prove te main teoretical result. 364

3 IV. MAIN THEORETICAL RESULT Te following teorem gives sufficient conditions for a continuous function to be a super-martingale (te corresponding result for a sub-martingale follows easily by flipping inequalities). It justifies te computational approac taken later in te paper. Teorem 4.1: Let S(t) satisfy (1) and te conditions (i), (ii), and (iii), and assume tat V(x,t) C p. Let te support of S(t), t [,T] be contained in te set A R. Assume tat for any point (x,t) A, tere exists a function f C 2,1 suc tat f(x,t) = V(x,t) and for some open set O containing (x,t) we ave wit f(y,τ) V(y,τ), (y,τ) O (14) A f(x,t). (15) Ten V(S(t),t) is a super-martingale and satisfies V(S,) E,S [V(,T)]. (16) Proof: At any point (S t,t) A consider te C 2,1 function f(s t,t) from te teorem satisfying V(S t,t) = f(s t,t) and f V in some open set O. By Lemma 3.2 we can replace f by a C 2,1 function f tat globally satisfies f V. Note tat at (S t,t), f satisfies [ E t,st f(s(t + ),t + ) ] f(s t,t) lim = A f = A f. (17) Furtermore, since globally f V, we ave tat E t,s t [V(S(t+),t+)] V(S t,t) E t,st[ f(s(t+),t+)] f(s t,t). Tus, combining (17) and (18), gives (18) E t,st [V(S(t + ),t + )] V(S t,t) limsup. (19) Now, by Fatou s lemma [12] for t < t, limsup E t,st [V(S(t+),t+)] E t,st [V(S(t),t)] E t,s t [limsup E t,s(t) [V(S(t+),t+)] V(S(t),t) ]. Finally, letting g(t) = E t,s t [V(S(t),t)], one may note tat g(t) satisfies te assumptions of Lemma 3.1. Tus E t,s t [V(S(t),t)] V(S t,t ) (2) sowing tat V(S(t),t) is a super-martingale. Setting t = gives (16). Wit tis teoretical result in and, we now proceed to developing an optimization based bounding approac tat uses piecewise polynomial functions. V. A NUMERICAL OPTIMIZATION APPROACH We can use Teorem 4.1 to construct upper and lower bounds on option prices via piecewise polynomial super- and sub-martingales. As before, we present te case of S(t) being one dimensional wit te multidimensional case following along similar lines. Our approac is to searc over piecewise polynomial V(S,t) tat satisfy te conditions of Teorem 4.1. To construct suc a piecewise polynomial V(S,t), we first select break points for S, denoted a 1 < a 2 <... < a n+1 and let te support of S(t) be a subset of [a 1,a n+1 ]. V(S,t) is ten pieced togeter as V(S,t) = f (i) (S,t), S [a i,a i+1 ], t [,T] (21) for i = 1,...,n were eac f (i) is a polynomial in S and t. Now, in order to make V(S,t) continuous, we require tat f (i 1) (a i,t) = f (i) (a i,t), t [,T], i = 2,...,n. (22) Additionally, following te conditions of Teorem 4.1 we need to guarantee tat at every point a C 2,1 function exists tat is locally an upper bound on V. At points were V(S,t) is twice continuously differentiable, we can use te polynomial f (i) (S,t) = V(S,t) itself. At boundary points, we can require te derivative condition f x (i 1) (a i,t) > f x (i) (a i,t), t [,T], i = 2,...,n (23) wic guarantees tat bot f (i 1) or f (i) are locally upper bounds. Finally, as long as A f (i) (S,t), S [a i,a i+1 ], t [,T], i = 1,...,n (24) ten one can easily verify tat V(S,t) defined in tis manner satisfies te conditions of Teorem 4.1 and is a supermartingale. Hence, as long as V(S,T) (S K) +, ten V(S,) is an upper bound on te option price. Our computational procedure for an upper bound is simply based on minimizing over piecewise polynomial functions satisfying te preceding conditions and upper bounding te function of te option at expiration. Tat is, define te optimization problem P u as minv(s,) V(S,T) (S K) + V(S,t) = f (i) (S,t) S [a i,a i+1 ], i = 1...n P u = A f (i) (S,t), S [a i,a i+1 ], i = 1...n f (i 1) (a i,t) = f (i) (a i,t), t [,T], i = 2,...,n. (a i,t) > f (i) (a i,t), t [,T], i = 2,...,n. f (i 1) x x f (i) a polynomial, i = 1,...,n (25) Tis is a polynomial program tat computes an upper bound on te call option price c(s,). In a completely analogous manner, a lower bound optimization P l can also be formulated as a polynomial optimization. Wile tis optimization problem is difficult, te inequalities can be replaced by a more restrictive sum-of-squares condition tat ten allows for semi-definite programming 365

4 metods to be used [9]. Additionally, te freely available software package SOStools [1] automates tis process. Details can be found in te SOStools manual [1]. In te following section, we solve te optimization problems P u and P l and test te effectiveness of tis super- and sub-martingale based optimization approac on te familiar Black-Scoles example VI. BLACK-SCHOLES NUMERICAL EXAMPLE In tis section, we use a Black-Scoles example wit te underlying asset following geometric Brownian motion. We solve te upper bound optimization P u as well as te corresponding lower bound problem P l and explore te tigtness of te bounds compared to te actual Black-Scoles solution. A. Model Let S(t) satisfy ds = σsdz (26) wit σ =.3. We considered te pricing of a European call options wit strike price K = 1 and expiration time T =.4. Te optimization P u was used to compute upper and lower bounds. Tis optimization was solved using SOStools [1] wic linked to te semi-definite programming solver SeDuMi [14]. Specifically, we constructed V(S,t) by piecing togeter four polynomials: f (1) (S,t) on S [,.9], f (2) (S,t) on S [.9,1], f (3) (S,t) on S [1,1.1], and f (4) (S,t) on S [1.1, ). Eac of te polynomials was fourt order in S and t. (Tat is, terms suc as S 4 t 4 were allowed, but not S 5 t.) B. Upper Bounds Results We first computed te upper bound for an at te money option. Tat is, we selected S() = 1. Tis is te most difficult selection for S() in terms of te tigtness of te bounds. Tus, te results of tis section sow te bounds under te most callenging scenario. Solving te optimization resulted in te polynomial functions f (i), i = 1,2,3,4. At expiration, tese functions (blue, magenta, black, and cyan) along wit te of te option (red) are sown in Figure 1. One can see tat wen pieced togeter at te break points.9, 1, 1.1, te functions f (i), i = 1,2,3,4 form a fairly tigt upper bound on te function, especially around S() = 1. However, note tat eac polynomial f (i) individually need not be an upper bound on te. Figure 2 sows te upper bound piecewise polynomial function at time. Tus, te piecewise polynomial function is an upper bound on te Black-Scoles price of te option (wic is given by te green line). For S() = 1, te upper bound and te Black-Scoles price are quite close, indicating tat te bound is fairly effective. Again, we empasize tat pricing an at-te-money option leads to te loosest bound and oter initial conditions only lead to tigter bounds. For reference, te upper bound value at-te-money is UB =.7996 wile te Black-Scoles price is Fig. 1. Piecewise polynomial (fourt order in S and T ) upper bound on te call option (red). Te upper plot sows te entire polynomial functions (blue, magenta, black, and cyan) tat are pieced togeter to make up te upper bound super-martingale V(,T). Te lower plot is a zoomed-in version of te piecewise polynomial function V(,T) were vertical lines sow te breakpoints (.9, 1, and 1.1) at wic te polynomial functions from te upper plot are pieced togeter. C. Lower Bound Results Lower bound results are given in Figures 3 and 4. Tis time Figure 3 sows te piecewise polynomial function creating a lower bound on te of te option. Figure 4 sows te bound on te price at time given by te piecewise polynomial function in reference to te Black-Scoles price sown in green. Again, we see tat te piecewise polynomial function provides a reasonably tigt lower bound. For reference, te lower bound price at-te-money is LB =.6721 wile te Black-Scoles price is.756. D. Discussion To compute tese examples, we only used up to fourt order polynomials in S and t. Higer order polynomials lead to tigter bounds but require increased computational effort. SOStools was able to solve te problem using fourt order polynomials in a matter of a few seconds. Tus, for one dimensional problems, computation time was not an issue. Rater tan using polynomials, one may also consider using oter functional forms. We tested polynomials in S 366

5 Approx. of Call by Piecewise Polynomial Function S() Approx. of Call by Piecewise Polynomial Function S() Fig. 2. Upper bound on te price given by te piecewise polynomial (fourt order in S and t) function V(S(),). In bot plots, te green line is te Black-Scoles price of te option. Te upper plot sows te entire polynomial functions (blue, magenta, black, and cyan) tat are pieced togeter to make up te upper bound super-martingale V(S(),). Te lower plot is a zoomed-in version of te piecewise polynomial function V(S(),) were vertical lines sow te breakpoints (.9, 1, and 1.1) at wic te polynomial functions from te upper plot are pieced togeter. Te red line is te function for te option and is provided for reference. and exponentials in t. Te results were similar to simply using a polynomial in t, so we did not report tose results in tis paper. As mentioned previously, tis approac also applies to pricing options on more tan a single underlying asset, altoug we do not report results ere. Of course, using iger dimensions leads to increased computational times and questions regarding efficient metods for piecing polynomials togeter. VII. CONCLUSIONS In tis paper we first derived sufficient conditions for a continuous function to be a super- or sub-martingale. Tis caracterization allowed us to use piecewise polynomial super- and sub-martingales in an optimization to bound te price of an option. By using polynomials, te optimization was formulated as a polynomial programming problem. We used te software package SOStools wic replaces te Fig. 3. Piecewise polynomial (fourt order in S and T ) lower bound on te call option (red). Te upper plot sows te entire polynomial functions (blue, magenta, black, and cyan) tat are pieced togeter to make up te lower bound sub-martingale V(,T). Te lower plot is a zoomed-in version of te piecewise polynomial function V(,T) were vertical lines sow te breakpoints (.9, 1, and 1.1) at wic te polynomial functions from te upper plot are pieced togeter. problem wit a sum-of-squares program, allowing it to be solved as a semi-definite program. Te results were tested on a Black-Scoles example using four polynomials pieced togeter to create an upper and lower bound. Te results sowed tat we acieved reasonably tigt bounds wit four fourt order polynomials, even for an at-te-money option. VIII. ACKNOWLEDGMENTS Te autor would like to tank Muruan Ratinam for elpful discussions. REFERENCES [1] D. Bertsimas and I. Popescu. On te relation between option and stock prices: A convex optimization approac. Operations Researc, 5(2): , 22. [2] A. d Aspremont and L. El Gaoui. Static Arbitrage Bounds on Basket Option s. Matematical Programming, Series A, October 25. [3] D. Duffie. Dynamic Asset Pricing Teory. Princeton, second edition, [4] S. N. Etier and T. G. Kurtz. Markov Processes: Caracterization and Convergence. Wiley,

6 .35 Approx. of Call by Piecewise Polynomial Function S().3 Approx. of Call by Piecewise Polynomial Function S() Fig. 4. Lower bound on te price given by te piecewise polynomial (fourt order in S and t) function V(S(),). In bot plots, te green line is te Black-Scoles price of te option. Te upper plot sows te entire polynomial functions (blue, magenta, black, and cyan) tat are pieced togeter to make up te lower bound sub-martingale V(S(),). Te lower plot is a zoomed-in version of te piecewise polynomial function V(S(),) were vertical lines sow te breakpoints (.9, 1, and 1.1) at wic te polynomial functions from te upper plot are pieced togeter. Te red line is te function for te option and is provided for reference. [5] W. H. Fleming and H. M. Soner. Controlled Markov Processes and Viscosity Solutions. Springer, 2nd edition, 26. [6] Jon Hull. Options, Futures, and Oter Derivatives. Prentice Hall, fourt edition, 2. [7] J. B. Lasserre, T. Prieto-Rumeau, and M. Zervos. Pricing a Class of Exotic Options via Moments and SDP Relaxations. Matematical Finance, To Appear. [8] Bernt Oksendal. Stocastic Differential Equations: an introduction wit applications. Springer, New York, fift edition, [9] P.A. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Matematical Programming Ser. B, 96(2):293 32, 23. [1] S. Prajna, A. Papacristodoulou, P. Seiler, and P. A. Parrilo. SOS- TOOLS: Sum of squares optimization toolbox for MATLAB, 24. [11] J.A. Primbs. Option pricing bounds via semidefinite programming. In Proceedings of te 26 American Control Conference, pages , Minneapolis, Minnesota, 26. [12] H. L. Royden. Real Analysis. Macmillan, New York, [13] J. E. Smit. Generalized cebycev inequalities: Teory and applications in decision analysis. Operations Researc, 43(5):87 825, September-October [14] Jos Sturm. SeDuMi: version 1.5, October 24. [15] J. Yong and X. Y. Zou. Stocastic Controls. Springer,