Efficient Risk Estimation via Nested Sequential
Simulation

by M. Broadie, Y. Du and C. Moallemi
Management Science, 2011, Vol. 57, No. 6, pp. 1172-1194.

Abstract: We analyze the computational problem of estimating financial risk in a nested simulation. In this approach, an outer simulation is used to generate financial scenarios and an inner simulation is used to estimate future portfolio values in each scenario. We focus on one risk measure, the probability of a large loss, and we propose a new algorithm to estimate this risk. Our algorithm sequentially allocates computational effort in the inner simulation based on marginal changes in the risk estimator in each
scenario. Theoretical results are given to show that the risk estimator has a faster convergence order compared to the conventional uniform inner sampling approach. Numerical results consistent with the theory are presented.

Managing Corporate Liquidity: Strategies and Pricing Implications

by A. Asvanunt, M. Broadie and S. Sundaresan
International Journal of Theoretical and Applied Finance, 2011, Vol.14, No.3, 369-406.

Abstract: Defaults arising from illiquidity can lead to private workouts, formal bankruptcy proceedings or even liquidation. All these outcomes can result in deadweight losses. Corporate illiquidity in the presence of realistic capital market frictions can be managed by a) equity dilution, b) carrying positive cash balances, or c) entering into loan commitments with a syndicate of lenders. An efficient way to manage illiquidity is to rely on mechanisms that transfer cash from ``good states'' into ``bad states'' (i.e., financial distress) without wasting liquidity in the process. In this paper, we first investigate the impact of costly equity dilution as a method to deal with illiquidity, and characterize its effects on corporate debt prices and optimal capital structure. We show that equity dilution produces lower firm value in general. Next, we consider two alternative mechanisms: cash balances and loan commitments. Abstracting from future investment opportunities and share re-purchases, which are strong reasons for corporate cash holdings, we show that carrying positive cash balances for managing illiquidity is in general inefficient relative to entering into loan commitments, since cash balances a) may have agency costs, b) reduce the riskiness of the firm thereby lowering the option value to default, c) postpone or reduce dividends in good states, and d) tend to inject liquidity in both good and bad states. Loan commitments, on the other hand, a) reduce agency costs, and b) permit injection of liquidity in bad states as and when needed. Then, we study the trade-offs between these alternative approaches to managing corporate illiquidity. We show that loan commitments can lead to an improvement in overall welfare and reduction in spreads on existing debt for a broad range of parameter values. We derive explicit pricing formulas for debt and equity prices. In addition, we characterize the optimal draw down strategy for loan commitments, and study its impact on optimal capital structure.

Understanding Index Option Returns

by M. Broadie, M. Chernov and M. Johannes
Review of Financial Studies, 2009, Vol.22, No.11, 4493-4529.

Abstract: Previous research concludes that options are mispriced based on the high average returns, CAPM alphas, and Sharpe ratios of various put selling strategies. One criticism of these conclusions is that these benchmarks are ill-suited to handle the extreme statistical nature of option returns generated by nonlinear payoffs. We propose an alternative way to evaluate the statistical significance of option returns by comparing historical statistics to those generated by well-accepted option pricing models. The most puzzling finding in the existing literature, the large returns to writing out-of-the-money puts, are not inconsistent (i.e., are statistically insignificant) relative to the Black-Scholes model or the Heston stochastic volatility model due to the extreme sampling uncertainty associated with put returns. This sampling problem can largely be alleviated by analyzing market-neutral portfolios such as straddles or delta-hedged returns. The returns on these portfolios can be explained by jump risk premia and estimation risk.

The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps

by M. Broadie and A. Jain
International Journal of Theoretical and Applied Finance, 2008, Vol.11, No.8, 761-797.

Abstract: We investigate the effect of discrete sampling and asset price jumps on fair variance and volatility swap strikes. Fair discrete volatility strikes and fair discrete variance strikes are derived in different models of the underlying evolution of the asset price: the Black-Scholes model, the Heston
stochastic volatility model, the Merton jump-diffusion model and the Bates and Scott stochastic volatility and jump model. We determine fair discrete and continuous variance strikes analytically and fair discrete and continuous volatility strikes using simulation and variance reduction techniques and numerical integration techniques in all models. Numerical results show that the well-known
convexity correction formula may not provide a good approximation of fair volatility strikes in models with jumps in the underlying asset. For realistic contract specifications and model parameters, we find that the effect of discrete sampling is typically small while the effect of jumps can be significant.

Growth Options and Optimal Default under Liquidity Constraints: The Role of Corporate Cash Balances

by A. Asvanunt, M. Broadie and S. Sundaresan
Columbia University working paper.

Abstract: In this paper, we develop a structural model that captures the interaction between the cash balance and investment opportunities for a firm that has already some debt outstanding. We consider a firm whose assets produce a stochastic cash flow stream. The firm has an opportunity to expand its operations, which we call a growth option. The exercise cost of the growth option can be financed either by cash or costly equity issuance. In absence of cash, we derive implicit solutions for equity and debt prices when the option is exercised optimally, under both firm value and equity value maximization objectives. We characterize the optimal exercise boundary of the option, and its impact on the optimal capital structure and the debt capacity of the firm. Next, we develop a binomial lattice method to investigate the interaction between cash accumulation and the growth option. In this framework, the firm optimally balances the payout of dividends with the buildup of a cash balance to finance the growth option in the ``good states'' (i.e., high asset value states), and to provide liquidity in the ``bad states'' (i.e., low asset value states). We provide a complete characterization of the firm's strategy in terms of its investment and dividend policy. We find that while the ability to maintain a cash balance does not add significant value to the firm in absence of a growth option, it can be extremely valuable when a growth option is present. Finally, we demonstrate how our method can be extended to firms with multiple growth options.

Implications of Heavy Tails on Simulation-Based Ordinal Optimization

by M. Broadie, M. Han and A. Zeevi
Proceedings of the 2007 Winter Simulation Conference, eds: S.G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J.D. Tew, and R.R. Barton, The Society for Computer Simulation, 439-447.

Abstract: We consider the problem of selecting the best system using simulation-based ordinal optimization. This problem has been studied mostly in the context of light-tailed distributions, where both Gaussian-based heuristics and asymptotically optimal procedures have been proposed. The latter rely on detailed knowledge of the underlying distributions and give rise to an exponential decay of the probability of selecting the incorrect system. However, their implementation tends to be computationally intensive. In contrast, in the presence of heavy tails the probability of selecting the incorrect system only decays polynomially, but this is achieved using simple allocation schemes that rely on little information of the underlying distributions. These observations are illustrated via several numerical experiments and are seen to be consistent with asymptotic theory.

Recent Advances in Simulation for Security Pricing

by P. Boyle, M. Broadie, and P. Glasserman
Proceedings of the 1995 Winter Simulation Conference, eds: Alexopoulos, Kang, Lilegdon, and Goldsman, The Society for Computer Simulation, San Diego, CA,
212-219. This paper was selected as a landmark paper in the four decades of the Winter Simulation Conference and was reprinted in Proceedings of the 2007 Winter Simulation Conference, eds: S.G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J.D. Tew, and R.R. Barton, The Society for Computer Simulation.

Abstract: Computational methods pay an important role in modern finance. Through the theory of arbitrage-free pricing, the price of a derivative security can be expressed as the expected value of its payouts under a particular probability measure. The resulting integral becomes quite complicated if there are several state variables or if payouts are path-dependent. Simulation has proved to be a valualbe tools for the calculations. This paper summarizes some of the recent applications and developments of the Monte Carlo method to security pricing problems.

Improved Lower and Upper Bound Algorithms for Pricing American Options by Simulation

by M. Broadie and M. Cao
Quantitative Finance, 2008, Vol.8, No.8, 845-861.

Abstract: This paper introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal-dual simulation algorithm (Andersen and Broadie 2004), we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance
grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.

Pricing and Hedging Volatility Derivatives

by M. Broadie and A. Jain
Journal of Derivatives, 2008, Vol.15, No.3, 7-24.

Abstract: This paper studies the pricing and hedging of variance swaps and other volatility derivatives, including volatility swaps and variance options, in the Heston stochastic volatility model. Pricing and hedging results are derived using partial differential equation techniques. We formulate an optimization problem to determine the number of options required to best hedge a variance swap. We propose a method to dynamically hedge volatility derivatives using variance swaps and a finite number of European call and put options.

Model Specification and Risk Premia: Evidence from Futures Options

by M. Broadie, M. Chernov and M. Johannes
Journal of Finance, 2007, Vol.62, No.3, 1453-1490.

Abstract: This paper examines model specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a time series test to detect the presence of jumps in volatility, and find strong evidence in support of their presence. Next, using the cross section of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility based on model fit. The evidence points toward economically and statistically significant jump risk premia, which are important for understanding option returns.

Optimal Debt and Equity Values in the Presence of Chapter 7 and Chapter 11

by M. Broadie, M. Chernov and M. Sundaresan
Journal of Finance, 2007, Vol.62, No.3, 1341-1377.

Abstract: In a contingent claims framework with a single issue of debt and full information, we show that the presence of a bankruptcy code with automatic stay, absolute priority rules, and potential debt
forgiveness, can lead to significant conflicts of interest between borrowers and lenders. In the first best outcome (i.e., firm value maximization subject to limited liability requirements), the bankruptcy code can add significant value to both parties by way of higher debt capacity, lower credit spreads, and improvement in the overall value of the firm. If control of the ex-ante timing of entering into bankruptcy and the ex-post decision to liquidate once the firm goes into bankruptcy is given to equity holders, most of the benefits of the code are appropriated by the equity holders at the expense of the debt holders. In our results the debt holders can restore the first best outcome, in large measure, by seizing this control or by
the ex-post transfer of control rights which allows them to decide when to liquidate the firm that has been taken to the Chapter 11 process by the equity holders. Irrespective of who is in control of the bankruptcy and liquidation decision, our model implies, based on the term structure of probabilities of default and liquidation, that firms are more likely to default on average and are less likely to liquidate on average relative to the benchmark model of Leland (1994).

A Binomial Lattice Method for Pricing Corporate Debt and Modeling Chapter 11 Proceedings

by M. Broadie and O. Kaya
Journal of Financial and Quantitative Analysis, 2007, Vol.42, No.2, 279-312.

Abstract: The pricing of corporate debt is still a challenging and active research area in corporate finance. Starting with Merton (1974), many authors proposed a structural approach in which the value of the assets of the firm is modeled by a stochastic process, and all other variables are derived from this basic process. These structural models have become more complex over time in order to capture more realistic aspects of bankruptcy proceedings. The literature in this area emphasizes closed-form solutions that are
derived by either PDE methods or analytical pricing techniques. However, it is not always possible to build a comprehensive model with realistic model features and achieve a closed-form solution at
the same time. In this paper, we develop a binomial lattice method that can be used to handle complex structural models such as ones that include Chapter 11 proceedings of the U.S. bankruptcy code.
Although lattice methods have been widely used in the option pricing literature, they are relatively new in corporate debt pricing. In particular, the limited liability requirement of the equityholders
needs to be handled carefully in this context. Our method can be used to solve the Leland (1994) model and its extension to the finite maturity case, the more complex model of Broadie, Chernov and
Sundaresan (2007).

Exact Simulation of Stochastic Volatility and other Affine Jump Diffusion Processes

by M. Broadie and O. Kaya
Operations Research, 2006, Vol.54, No.2, 217-231.

Abstract: The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating
security prices under these models. However, discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an
acceptable level. This paper suggests a method for the exact simulation of the stock price and variance under Heston's stochastic volatility model and other affine jump diffusion processes. The
sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security. We compare our method with the more
conventional Euler discretization method and demonstrate the faster convergence rate of the error in our method. Specifically, our method achieves an O(1/s^{1/2}) convergence rate, where s is the
total computational budget. The convergence rate for the Euler discretization method is O(1/s^{1/3}) or slower, depending on the model coefficients and option payoff function.

A Double-Exponential Fast Gauss Transform for Pricing Discrete Path-Dependent Options

by M. Broadie and Y. Yamamoto
Operations Research, 2005, Vol.53, No.5, 764-779.

Abstract: This paper develops algorithms for the pricing of discretely sampled barrier, lookback and hindsight options and discretely exercisable American options. Under the Black-Scholes framework, the pricing of these options can be reduced to evaluation of a series of convolutions of the Gaussian distribution and a known function. We compute these convolutions efficiently using the double-exponential integration formula and the fast Gauss transform. The resulting algorithms have computational complexity of O(nN), where the number of monitoring/exercise dates is n and the number of sample points at each date is N, and our results show the error decreases exponentially with N. We also extend the approach and provide results for Merton's lognormal jump-diffusion model.

Exact Simulation of Option Greeks under Stochastic Volatility and Jump Diffusion Models

by M. Broadie and O. Kaya
Proceedings of the 2004 Winter Simulation Conference, eds: R.G. Ingalls, M.D. Rossetti, J.S. Smith, and B.A. Peters, The Society for Computer Simulation, 1607-1615.

Abstract: This paper derives Monte Carlo simulation estimators to compute option price derivatives, i.e., the 'Greeks,' under Heston's stochastic volatility model and some variants of it which include jumps in the price and variance processes. We use pathwise and likelihood ratio approaches together with the exact simulation method of Broadie and Kaya (2004) to generate unbiased estimates of option price derivatives in these models. By appropriately conditioning on the path generated by the variance and jump processes, the evolution of the stock price can be represented as a series of lognormal random variables. This makes it possible to extend previously known results from the Black-Scholes setting to the computation of Greeks for more complex models. We give simulation estimators and numerical results for some path-dependent and path-independent options.

A Primal-Dual Simulation Algorithm for Pricing Multi-Dimensional American Options

by L. Andersen and M. Broadie
Management Science, 2004, Vol. 50, No. 9, pp. 1222-1234.

Abstract: This paper describes a practical algorithm based on Monte Carlo simulation for the pricing of multi-dimensional American (i.e., continuously exercisable) and Bermudan (i.e., discretely-exercisable)
options. The method generates both lower and upper bounds for the Bermudan option price and hence gives valid confidence intervals for the true value. Lower bounds can be generated using any number of primal algorithms. Upper bounds are generated using a new Monte Carlo algorithm based on the duality representation of the Bermudan value function suggested independently in Haugh and Kogan (2004) and Rogers (2002). Our proposed algorithm can handle virtually any type of process dynamics, factor structure, and payout specification. Computational results for a variety of multi-factor equity and interest rate options demonstrate the simplicity and efficiency of the proposed algorithm. In particular, we use the
proposed method to examine and verify the tightness of frequently used exercise rules in Bermudan swaption markets.

Option Pricing: Valuation Models and Applications

by M. Broadie and J. Detemple
Management Science, 2004, Vol. 50, No. 9, pp. 1145-1177.

Abstract: This paper surveys the literature on option pricing, from its origins to the present. An extensive review of valuation methods for European- and Amer\-ican-style claims is provided. Applications to complex securities and numerical methods are surveyed. Emphasis is placed on recent trends and developments in methodology and modelling.

A Stochastic Mesh Method for Pricing High-Dimensional American Options

by M. Broadie and P. Glasserman
Journal of Computational Finance, 2004, Vol.7, No.4, pp. 35-72.

Abstract: High-dimensional problems frequently arise in the pricing of derivative securities -- for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, i.e., where the owner has the right to exercise early, are particularly challenging to price. We introduce a stochastic mesh method for pricing high-dimensional American options when there is a finite, but possibly large, number of exercise dates. The algorithm provides point estimates and confidence intervals; we provide conditions under which these estimates converge to the correct values as the computational effort increases. Numerical results illustrate the performance of the method.

Application of the Fast Gauss Transform to Option Pricing

by M. Broadie and Y. Yamamoto
Management Science, 2003, Vol. 49, No. 8, pp. 1071-1088.

Abstract: In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as the summation of
Gaussians. Though this operation usually requires O(NN') work when there are N' summations to compute and the number of terms appearing in each summation is N, we can reduce the amount of work to O(N+N') by using a technique called the fast Gauss transform. In this paper, we apply this technique to the multinomial method and the stochastic mesh method, and show by numerical experiments how it can speed up these methods dramatically, both for the Black-Scholes model and Merton's lognormal jump-diffusion model. We also propose extensions of the fast Gauss transform method to models with non-Gaussian densities.

Nonparametric Estimation of American Option Exercise Boundaries and Call Prices

by M. Broadie, J. Detemple, E. Ghysels, and O. Torres
Journal of Economic Dynamics and Control, 2000, Vol. 24, Nos. 11-12, pp. 1829-1857.

Abstract: Unlike European-type derivative securities, there are no simple analytic valuation formulas for finite-lived American options, even when the underlying asset price has constant volatility. The early exercise feature considerably complicates the valuation of American contracts. The strategy taken in this paper is to rely on nonparametric statistical methods using market data to estimate the call prices and the exercise boundaries. A comparison is made with parametric constant volatility model-based prices and exercise boundaries. The paper focuses on assessing the adequacy of conventional formulas by comparing them to nonparametric estimates. We use daily market option prices and exercise data on the S&P100 contract, the most actively traded American option contract. We find large discrepancies between the parametric and nonparametric call prices and exercise boundaries. We also find remarkable similarities of the nonparametric estimates before and after the crash of October 1987.

American Options with Stochastic Dividends and Volatility: A Nonparametric
Investigation

by M. Broadie, J. Detemple, E. Ghysels, and O. Torres
Journal of Econometrics, 2000, Vol. 94, pp. 53-92.

Abstract: In this paper, we consider American option contracts when the underlying asset has stochastic dividends and stochastic volatility. We provide a full discussion of the theoretical foundations of American option valuation and exercise boundaries. We show how they depend on the various sources of uncertainty which drive dividend rates and volatility, and derive equilibrium asset prices, derivative prices and optimal exercise boundaries in a general equilibrium model. The theoretical models identify the relevant factors underlying option prices but yield fairly complex expressions which are difficult to estimate. We therefore adopt a nonparametric approach in order to investigate the reduced forms suggested by the theory. Indeed, we use nonparametric methods to estimate call prices and exercise boundaries conditional on dividends and volatility. Since the latter is a latent process, we propose several approaches, notably using EGARCH filtered estimates, implied and historical volatilities. The nonparametric approach allows us to test whether call prices and exercise decisions are primarily driven by dividends, as has been advocated by Harvey and Whaley (1992a, Journal of Financial Economics 30, 33-73; 1992b, Journal of Futures Markets 12, 123-137) and Fleming and Whaley (1994, Journal of Finance 49, 215-236) for the OEX contract, or whether stochastic volatility complements dividend uncertainty. We find that dividends alone do not account for all aspects of option pricing and exercise decisions, suggesting a need to include stochastic volatility.

Connecting Discrete and Continuous Path-Dependent Options

by M. Broadie, P. Glasserman, and S. Kou
Finance and Stochastics, 1999, Vol. 3, No. 1, 55-82.

Abstract: This paper develops methods for relating the prices of discrete- and continuous-time versions of path-dependent options sensitive to extremal values of the underlying asset, including lookback, barrier, and hindsight options. The relationships take the form of correction terms that can be interpreted as shifting a barrier, a strike, or an extremal price. These correction terms enable us to use closed-form solutions for continuous option prices to approximate their discrete counterparts. We also develop discrete-time discrete-state lattice methods for determining accurate prices of discrete and continuous path-dependent options. In several cases, the lattice methods use correction terms based on the connection between discrete- and continuous-time prices which dramatically improve convergence to the accurate price.

Optimal Replication of Contingent Claims Under Portfolio Constraints

by M. Broadie, J. Cvitanic, and M. Soner
Review of Financial Studies, 1998, Vol. 11, No. 1, 59-79.

Abstract: We determine the minimum cost of super-replicating a nonnegative contingent claim when there are convex constraints on portfolio weights. We show that the optimal cost with constraints is equal to the price of a related claim without constraints. The related claim is a dominating claim, that is, a claim whose payoffs are increased in an appropriate way relative to the original claim. The results hold for a variety of options, including some path-dependent options. Constraints on the gamma of the replicating portfolio, constraints on the portfolio amounts, and constraints on the number of shares are also considered.

Monte Carlo Methods for Security Pricing

by P. Boyle, M. Broadie, and P. Glasserman
Journal of Economic Dynamics and Control, 1997, Vol. 21, Nos. 8-9, pp. 1267-1321.

Abstract: The Monte Carlo approach has proved to be a valuable and flexible computational tool in modern finance. This paper discusses some of the recent applications of the Monte Carlo method to security pricing problems, with emphasis on improvements in efficiency. We first review some variance reduction methods that have proved useful in finance. Then we describe the use of deterministic low-discrepancy sequences, also known as quasi-Monte Carlo methods, for the valuation of complex derivative securities. We summarize some recent applications of the Monte Carlo method to the estimation of partial derivatives or risk sensitivities and to the valuation of American options. We conclude by mentioning other applications.

Pricing American-Style Securities Using Simulation

by M. Broadie, and P. Glasserman
Journal of Economic Dynamics and Control, 1997, Vol. 21, Nos. 8-9, pp. 1323-1352.

Abstract: We develop a simulation algorithm for estimating the prices of American-style securities, i.e., securities with opportunities for early exercise. Our algorithm provides both point estimates and error bounds for the true security price. It generates two estimates, one biased high and one biased low, both asymptotically unbiased and converging to the true price. Combining the two estimators yields a confidence interval for the true price. The proposed algorithm is especially attractive (compared with lattice and finite-difference methods) when there are multiple state variables and a small number of exercise opportunities. Preliminary computational evidence is given.

A Continuity Correction for Discrete Barrier Options

by M. Broadie, P. Glasserman, and S. Kou
Mathematical Finance, 1997, Vol. 7, No. 4, pp. 325-349.

Abstract: The payoff of a barrier option depends on whether or not a specified asset price, index, or rate reaches a specified level during the life of the option. Most models for pricing barrier options assume continuous monitoring of the barrier; under this assumption, the option can often be priced in closed form. Many (if not most) real contracts with barrier provisions specify discrete monitoring instants; there are essentially no formulas for pricing these options, and even numerical pricing is difficult. We show, however, that discrete barrier options can be priced with remarkable accuracy using continuous barrier formulas by applying a simple continuity correction to the barrier. The correction shifts the barrier away from the underlying by a factor of exp(beta*sigma*sqrt(dt)), where beta approx 0.5826, sigma is the underlying volatility, and dt is the time between monitoring instants. The correction is justified both theoretically and experimentally.

Click here for pdf file

The Valuation of American Options on Multiple Assets

by M. Broadie and J. Detemple
Mathematical Finance, 1997, Vol. 7, No. 3, pp. 241-286.

Abstract: In this paper we provide valuation formulas for several types of American options on two or more assets. Our contribution is twofold. First, we characterize the optimal exercise regions and provide valuation formulas for a number of American option contracts on multiple underlying assets with convex payoff functions. Examples include options on the maximum of two assets, dual strike options, spread options, exchange options, options on the product and powers of the product, and options on the arithmetic average of two assets. Second, we derive results for American option contracts with nonconvex payoffs, such as American capped exchange options. For this option we explicitly identify the optimal exercise boundary and provide a decomposition of the price in terms of a capped exchange option with automatic exercise at the cap and an early exercise premium involving the benefits of exercising prior to reaching the cap. Besides generalizing the current literature on American option valuation our analysis has implications for the theory of investment under uncertainty. A specialization of one of our models also provides a new representation formula for an American capped option on a single underlying asset.

Click here for pdf file

American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods

by M. Broadie and J. Detemple
Review of Financial Studies, 1996, Vol. 9, No. 4, pp. 1211-1250.

Abstract: We develop lower and upper bounds on the prices of American call and put options written on a dividend-paying asset. We provide two option price approximations one based on the lower bound (termed LBA) and one based on both bounds (termed LUBA). The LUBA approximation has an average accuracy comparable to a l,000-step binomial tree. We introduce a modification of the binomial method (termed BBSR) that is very simple to implement and performs remarkably well. We also conduct a careful large-scale evaluation of many recent methods for computing American option prices.

Estimating Security Price Derivatives Using Simulation

by M. Broadie, and P. Glasserman
Management Science, 1996, Vol. 42, No. 2, 269-285.

Abstract: Simulation has proved to be a valuable tool for estimating security prices for which simple closed form solutions do not exist. In this paper we present two direct methods, a pathwise method and a likelihood ratio method, for estimating derivatives of security prices using simulation. With the direct methods, the information from a single simulation can be used to estimate multiple derivatives along with a security's price. The main advantage of the
direct methods over re-simulation is increased computational speed. Another advantage is that the direct methods give unbiased estimates of derivatives, whereas the estimates obtained by re-simulation are biased. Computational results are given for both direct methods and comparisons are made to the standard method of re-simulation to
estimate derivatives. The methods are illustrated for a path independent model (European options), a path dependent model (Asian options), and a model with multiple state variables (options with stochastic volatility).

American Capped Call Options on Dividend-Paying Assets

by M. Broadie and J. Detemple
Review of Financial Studies, 1995, Vol. 8, No. 1, pp. 161-191.

Abstract: This article addresses the problem of valuing American call options with caps on dividend-paying assets. Since early exercise is allowed, the valuation problem requires the determination of optimal exercise policies. Options with two types of caps are analyzed: constant caps and caps with a constant growth rate. For constant caps, it is optimal to exercise at the first time at which the underlying asset's price equals or exceeds the minimum of the cap and the optimal exercise boundary for the corresponding uncapped option. For caps that grow at a constant rate, the optimal exercise strategy can be specified by three endogenous parameters.