EARLY EXERCISE OPTIONS: UPPER BOUNDS

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1 EARLY EXERCISE OPTIONS: UPPER BOUNDS LEIF B.G. ANDERSEN AND MARK BROADIE Abstract. In this article, we discuss how to generate upper bounds for American or Bermudan securities by Monte Carlo methods. These techniques provide a useful supplement to strategies that provide lower bound estimates (e.g., eqf and eqf13-025, allowing one to both generate valid confidence intervals for the true option price and to test the accuracy to any proposed approximation to the optimal exercise strategy. 1. Setup and Basic Results As usual, we work on a filtered probability space and consider a contingent claim with early exercise rights, i.e., the right to accelerate payment on the claim at will. Let the claim in question be characterized by an adapted, non-negative payout process U(t, payable to the option holder at a stopping time (or exercise policy τ T, chosen by the holder. If early exercise can take place at any time in some interval, we say that the derivative security is an American option; ifexercisecan only take place on a discrete set of dates, we say that it is a Bermudan option. Let the allowed set of exercise dates larger than or equal to t be denoted D(t, and suppose that we are given at time 0 a particular exercise policy τ taking values in D(0, as well as a pricing numeraire N inducing a unique martingale measure Q N.LetC τ (0 be the time 0 value of a derivative security that pays U(τ. Under technical conditions on U(t, we can write the value of the derivative security as ( U(τ (1 C τ (0 = E N, N(τ where E N ( denotes expectationinmeasureq N and where we have assumed, with no loss of generality, that N(0 = 1. Let T (t bethetimet set of (future stopping times taking value in D(t. In the absence of arbitrage, the time 0 value C(0 of a security with early exercise into U is then be given by the optimal stopping problem, (2 C(0 = sup C τ (0 = τ T (0 ( U(τ sup E N τ T (0 N(τ reflecting the fact that a rational investor would choose an exercise policy to optimize the value of his claim. With E N t ( denoting expectation conditional on the information (i.e., the filtration at time t, we can extend (2 to future times t ( U(τ (3 C(t =N(t sup E N t, τ T (t N(τ Date: July 11, Key words and phrases. American option, Bermudan option, Monte Carlo, Doob-Meyer decomposition, duality formulation, upper bound, confidence intervals. 1

2 2 LEIF B.G. ANDERSEN AND MARK BROADIE where sup τ E N t (U(τ/N (τ is known as the Snell envelope of U/N under Q N. Here C(t must be interpreted as the value of the option with early exercise, conditional on exercise not having taken place before time t. To make this explicit, let τ T(0 be the optimal exercise policy, as seen from time 0. We can then write, for 0 <t T, (4 C(0 = E N ( 1 {τ t}c(t/n (t +E N ( 1 {τ <t}u(τ /N (τ, where we break the time 0 value into two components: one from the time t value of the option, should it not have been exercised before time t; and one from the right to exercise on [0,t]. As we can always elect possibly suboptimally to never exercise on [0,t], from (4 we see that C(0 E N (C(t/N (t, which establishes that C(t/N (t isasupermartingale under Q N. This result also follows directly from known properties of the Snell envelope; see [13]. In numerical implementations, it is most relevant to consider the discrete-time (i.e Bermudan case and assume that D(0 = {T 1,T 2,...,T B }, where T 1 0and T B = T. For t<t i+1, define H i as the time t value of the Bermudan option when exercise is restricted to the dates D(T i+1 ={T i+1,t i+2,...,t B }. That is, H i (t =N(tE N t (C(T i+1 /N (T i+1, i =1,...,B 1. At time T i, H i (T i can be interpreted as the holding value of the Bermudan option, that is, the value of the Bermudan option if not exercised at time T i.ifanoptimal exercise policy is followed, clearly we must have at time T i such that C(T i =(U(T i,h i (T i, i =1,...,B, (5 H i (t =N(tE N t ( (U(T i+1,h i+1 (T i+1, i =1,...,B 1. Starting with the terminal condition H B (T =0, (5 defines an iteration backwards in time for the value C(0 = H 0 (0. 2. Option Pricing Bounds In a setting where U(t is a function of a low-dimensional diffusion process, the iteration (5 can often be solved numerically by PDE methods, e.g., the finite difference method (see eqf In many cases of practical interest, however, these methods either do not apply or are computationally infeasible. In such situations, we may be interested in at least bounding the value of an option with early exercise rights. Providing a lower bound is straightforward: postulate an exercise policy τ and compute the price C τ (0 by direct methods, e.g., the Monte Carlo method. From (2, this clearly provides a lower bound (6 C τ (0 C(0. The closer the postulated exercise policy τ is to the optimal exercise policy τ,the tighter this bound will be. Two common strategies for approximation of τ in a Monte Carlo setting are discussed in eqf and eqf13-025, the first based on regression estimates of holding values H in (5, and the second on optimization of parametric rules for the exercise strategy.

3 EARLY EXERCISE OPTIONS: UPPER BOUNDS 3 To produce an upper bound, we can rely on duality results established in [10] and [15]. To present these results here, let K denote the space of adapted martingales π for which sup τ [0,T ] E N π(t <. For a martingale π K,wethenwrite C(0 = ( U(τ sup E N τ T (0 N(τ ( U(τ = sup E N + π(τ π(τ τ T (0 N(τ ( U(τ = π(0 + sup E N τ T (0 N(τ π(τ. In the second equality, we have relied on the Optional Sampling Theorem to tell us that the martingale property is satisfied up to a bounded random stopping time, i.e., that E N (π(τ = π(0. See [12] for details. We now turn the above result into an upper bound by forming a pathwise imum at all possible future exercise dates D(0: (7 C(0 = π(0 + sup τ T (0 E N ( U(τ ( N(τ π(τ π(0 + E N t D(0 ( U(t N(t π(t. With (6 and (7 we have, as desired, established upper and lower bounds for values of options with early exercise rights. Let us consider how to make these bounds tight. As mentioned earlier, to tighten the lower bound we need to pick exercise strategies close to the optimal one. Tightening the upper bound is a bit more involved and requires usage of the Doob-Meyer Decomposition (see eqf02-017, which here can be used to show that (8 C(t/N (t =M(t A(t, where M(t is a martingale and A(0 an increasing, predictable process with A(0 = 0(suchthatC(0 = M(0. Given (8, consider taking π(t =M(t inequation (7, to get ( C(0 C(0 + N(0E N U(t t D(0 N(t M(t ( = C(0 + E N U(t t D(0 N(t C(t N(t A(t C(0. The last inequality follows from the fact that C(t U(t anda(t 0. We have therefore arrived at a dual formulation of the option price ( (9 C(0 = inf (π(0 + E N U(t π K t D(0 N(t π(t, and have demonstrated that the infimum is attained when the martingale π is set equal to the martingale component M of the deflated price process C(t/N (t. 3. Monte Carlo Upper Bound Methods Let us consider how we can use the upper bound results (7 and (9 in an actual Monte Carlo application. According to (7, to generate an upper bound for the true option price, it evidently suffices to simply pick any martingale process adapted to

4 4 LEIF B.G. ANDERSEN AND MARK BROADIE the filtration we work in, and then compute the expectation (7 by Monte Carlo methods. For instance, if the filtration is generated by a vector-valued Brownian motion W (t, we can always set (10 π(t = t 0 σ(t dw (t, for some adapted vector-process σ(t satisfying the usual conditions required for the stochastic integral to be proper martingale. Clearly, however, if σ(t is chosen arbitrarily, the resulting upper bound is likely to be very loose, and probably not very useful. While (9 is of little immediate practical use (since we are do not know the process C(t/N (t, it does suggest that for a chosen martingale π(t in(7to produce a tight upper bound, it needs to be close to M(t. Several strategies have been proposed for constructing a good martingale π(t. When working in a simple model setup on simple payouts, sometimes one can make inspired guesses for what π(t should be. For instance, in a simple one-dimensional Black-Scholes model, [15] shows that using the numeraire-deflated European call option price (which is analytically known as a guess for π(t generates good bounds for a Bermudan call option price. This approach, however, does not easily generalize to settings with more complicated dynamics and/or more complicated exercise payouts The Andersen-Broadie algorithm. A general strategy for generating upper bounds in proposed in [1], which can start from any approximation to the optimal exercise strategy, perhaps generated from either of the methods in eqf or eqf Using a straightforward simulation within a simulation approach, the authors construct an estimate to the value process C τ (t and use its estimated martingale component as π(t in (7. Specifically, working on a discrete timeline, they set ( C τ (T i+1 N(T i+1 π(t i+1 π(t i = Cτ (T i+1 N(T i+1 EN T i ( ( U(τ U(τ (11 = E N T i+1 E N T N(τ i N(τ τ T i+1, where nested simulations are used to estimate both expectations on the right-hand side of the equation. 1 The resulting Monte Carlo estimate of the upper bound is shown to be biased high always, with the bias being a decreasing function in the number of inner simulation trials. As suggested by (9, the upper bound produced by the algorithm in [1] strongly depends on the quality of the exercise strategy: the better the strategy, the tighter the bound. The strategy in [1] is generic, in that it can handle virtually any type of multidimensional process dynamics and security payouts. Although the use of nested simulation makes the algorithm in [1] computationally expensive, 2 it guarantees that the choice of π induces an upper bound estimate that is biased high. Importantly, this key property is not shared by many alternative estimators, such as regression, of the expectations in (11. One exception is discussed in [9] where a special martingale-preserving regression approach is introduced. This algorithm, 1 In cases where U(t is not known in closed form as may be the case for complicated callable securities (see eqf nested simulation can also be used to establish estimates for U(t. 2 But see [3] for techniques to improve speed and accuracy.

5 EARLY EXERCISE OPTIONS: UPPER BOUNDS 5 however, requires strong conditions on regression basis functions, that may be hard to check in practice The Belomestny-Bender-Schoenmakers Algorithm. In the special case where dynamics are driven only by Brownian motions, the usual martingale representation theorems show that the optimal strategy π (t must be an Ito integral, i.e., of the form (10. Starting again from a postulated exercise strategy, [2] use this observation to construct a regression on a set of basis functions to uncover an estimate for the function σ(t. By applying regression techniques this way rather than to compute directly expectations of U(τ/N (τ the authors are able to construct a true martingale process π(t, which can be turned into a valid upper bound through (7. The resulting non-nested simulation algorithm requires careful implementation to yield stable results, in part because the optimal integrand σ(t can be expected to be considerably less regular than π (t itself; this, in turn, requires additional thought in the selection of appropriate basis functions for the regression. One possibility advocated in [2] is to include, whenever available, exact or approximate expressions for the diffusion term in dynamics of several still-alive European options underlying the Bermudan option. This strategy is akin to that of [15], and its feasibility depends on the pricing problem at hand. In cases where it does apply, the authors of [2] demonstrate that their method gives good results, with the upper bound often being nearly as tight as that of the nested algorithm in [1]. They also show how to use their technique to develop a variance-reduced version of the algorithm in [1]. 4. Confidence intervals and Practical Usage Assume that we have used estimated an exercise strategy using, either of the approaches in eqf or eqf Suppose that the Monte Carlo estimate for the lower bound price is Ĉlo(0 with a sample standard deviation of ŝ lo based on N lo Monte Carlo trials. Using, say, the algorithm in [1], we also estimate an upper bound Ĉhi(0 with a sample standard deviation ŝ hi computed from N hi (outer simulation trials. With z x denoting the xth percentile of a standard Gaussian distribution, asymptotically a 100(1 α% confidence interval for the true price C(0 must be tighter 3 than [ ] ŝ lo (12 Ĉ lo (0 z 1 α/2 ; Ĉhi(0 ŝ hi z 1 α/2. Nlo Nhi Most often, upper bound simulation algorithms can be expected to be both more involved and/or more expensive than lower bound simulation methods. In many cases, the role of the upper bound simulation algorithm will therefore be to test whether postulated lower bound exercise strategies are tight or not. Specifically, starting from some guess for the exercise strategy, we can produce confidence intervals using (12 to test whether the lower bound estimate is of good quality, in which case the confidence interval can be made tight by using large values of N lo and N hi (as well as the number of inner simulation trials. In case the lower bound estimator is deemed unsatisfactory, we can iteratively refine it, by altering the choice of basis functions, say, until the confidence interval is tight. Importantly, such tests 3 The confidence interval is conservative because of the low bias in Ĉ lo (0 (i.e., E N (Ĉlo C(0 and the high bias in Ĉ hi (0 which originates in part from the nature of the upper bound, and in part from the earlier mentioned additional high bias introduced by the inner simulations.

6 6 LEIF B.G. ANDERSEN AND MARK BROADIE can often be done at a high level, covering entire classes of payouts and/or models. Once an exercise strategy has been validated for a particular product or model, day-to-day pricing of Bermudan securities can be done by the lower bound method, with only occasional runs of the upper bound method needed (e.g., if market conditions change markedly. If upper bound methods are predominantly used in this fashion, the fact that they may be expensive to compute becomes largely irrelevant. 5. Extensions and Related Work The results (7 and (9 are sometimes known as additive duality results. Jamshidian [8] has introduced alternative multiplicative results. A comparative study of additive and multiplicative duality was undertaken in [7], with the authors concluding that the additive duality results are preferable in applications. Earlier methods for producing lower and upper bounds were proposed in [5] and [6]. Both methods [5] and [6] have the significant feature of producing automatically convergent bounds. However [5] is only practical when the number of exercise dates is a small finite number (e.g., less than five. The method proposed in [6] does not suffer this drawback, but is more challenging to implement and is substantially slower than most lower bounds methods. For a fixed exercise strategy, e.g., determined from one of the lower bound methods, early exercise options are effectively European-style options, so their Greeks can be approximated efficiently using standard simulation methods originally proposed in [4] (see also [14]. An algorithm for computing lower and upper bounds for Greeks of early exercise options is proposed in [11]. However their method is extremely computationally intensive. 6. Related EQF Articles eqf02-017: Doob-Meyer Decomposition eqf02-019: Martingale Representation Theorem eqf05-007: American options eqf12-002: Finite difference: early exercise, American options eqf13-006: Basis function regression methods for Bermudan options eqf11-011: Callable Libor exotics (CIF, CCF, CRANs eqf13-024: Exercise boundary optimization methods References [1] Andersen, L. and M. Broadie (2004, A primal-dual simulation algorithm for pricing multidimensional American options, Management Science, 50, pp [2] Belomestny, D., C. Bender, and J. Schoenmakers (2007, True upper bounds for Bermudan products via non-nested Monte Carlo, forthcoming in Mathematical Finance. [3] Broadie, M., and M. Cao (2008, Improved Lower and Upper Bound Algorithms for Pricing American Options by Simulation, forthcoming in Quantitative Finance. [4] Broadie, M., and P. Glasserman (1996, Estimating Security Price Derivatives Using Simulation, Management Science, 42, pp [5] Broadie, M., and P. Glasserman (1997, Pricing American-Style Securities Using Simulation, Journal of Economic Dynamics and Control, 21, pp [6] Broadie, M., and P. Glasserman (2004, A Stochastic Mesh Method for Pricing High- Dimensional American Options, Journal of Computational Finance, 7, pp [7] Chen, N. and P. Glasserman (2005, Additive and Multiplicative Duals for American Option Pricing, forthcoming in Finance and Stochastics.

7 EARLY EXERCISE OPTIONS: UPPER BOUNDS 7 [8] Jamshidian, F. (2006, The duality of optimal exercise and domineering claims: A Doob- Meyer decomposition approach to the Snell envelope, Stochastics: An International Journal of Probability and Stochastics Processes, 79, pp [9] Glasserman, P. and B. Yu (2005, Pricing American Options by Simulation: Regression Now or Regression Later? in Monte Carlo and Quasi-Monte Carlo Methods, (H. Niederreiter, ed., Springer Verlag. [10] Haugh, M. and L. Kogan (2004, Pricing American options: a duality approach, Operations Research, 52, pp [11] Kaniel, R., S. Tompaidis, and A. Zemlianov, Efficient Computation of Hedging Parameters for Discretely Exercisable Options, forthcoming in Operations Research. [12] Karatzas, I. and S. Shreve (1991, Brownian Motion and Stochastic Calculus, 2nd Edition, Springer Verlag. [13] Lamberton, D. and B. Lapeyre (2007, Introduction to Stochastic Calculus Applied to Finance, 2nd Edition, CRC Press. [14] Piterbarg, V. (2004, Computing deltas of callable Libor exotics in forward Libor models, Journal of Computational Finance, 7, pp [15] Rogers, L.C.G. (2001, Monte Carlo valuation of American options, Mathematical Finance, 12, pp