Optimal Exercise Policy Research Articles

Pricing tenure payment reverse mortgages with optimal exercised prepayment options by accounting for house prices, interest rates, and mortality risk

Prepayment options can be exercised to terminate reverse mortgages (RM hereafter) early and receive house prices, minus loan balances, at the expense of future annuity proceeds. Prior RM evaluation studies use probability or intensity models to calibrate option exercise policies with historical prepayment records and may not apply to countries without sufficient historical records. In addition, these models may fail to capture time-varying policies due to changing market conditions. Accordingly, insurers may run the risk of undervaluing option premiums and overestimating fair annuity rates. To find optimal exercise policies that maximize option premiums and establish the most conservative annuity rates, we propose a three-dimensional tree for modeling stochastic house prices, interest rates, and mortality risks. We analyze the gain and loss to exercise the option in each scenario to determine the optimal policy. Fair annuity rates are evaluated to ensure that expected insurer losses (i.e. loan balances exceeding house values) equal gains (i.e. insurance premiums plus house values exceeding loan balances). We find that such non-optimal exercise policies undervalue option premiums and overestimate fair annuity rates. Increasing upfront premiums, insurance premium rates, and early redemption charges reduce prepayment incentives and increase fair annuity rates. We also analyze influences from factors such as the policyholder's age and volatilities of house prices and interest rates.

On the exercise of American quanto options

American option pricing is an important and engaging area of financial economics, particularly so in the presence of negative interest rates. Quanto options offer major international hedging/investment opportunities. We provide a comprehensive description of the optimal exercise policies associated with American quanto options. We show that a non-standard exercise policy characterized by a double continuation region may be optimal in the presence of non-positive domestic interest rates. We study empirical examples of finite-maturity American quanto options for which a double continuation region surrounding a non-empty early exercise region exists even if the infinite-maturity early exercise region is empty and the value of the infinite-maturity option is unbounded. Under the assumptions underpinning such empirical examples, we carefully characterize the existence, the monotonicity properties and the close-to-maturity behaviour of the upper and lower critical prices.

American options and stochastic interest rates

We study finite-maturity American equity options in a stochastic mean-reverting diffusive interest rate framework. We allow for a non-zero correlation between the innovations driving the equity price and the interest rate. Importantly, we also allow for the interest rate to assume negative values, which is the case for some investment grade government bonds in Europe in recent years. In this setting we focus on American equity call and put options and characterize analytically their two-dimensional free boundary, i.e. the underlying equity and the interest rate values that trigger the optimal exercise of the option before maturity. We show that non-standard double continuation regions may appear, extending the findings documented in the literature in a constant interest rate framework. Moreover, we contribute by developing a bivariate discretization of the equity price and interest rate processes that converges in distribution as the time step shrinks. This discretization, described by a recombining quadrinomial tree, allows us to compute American equity options’ prices and to analyze their free boundaries with respect to time and current interest rate. Finally, we document the existence of non-standard optimal exercise policies for American call options on a non-dividend-paying equity.

Evaluating Gaussian process metamodels and sequential designs for noisy level set estimation

We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels. Our focus is on strongly stochastic simulators, in particular with heavy-tailed simulation noise and low signal-to-noise ratio. To guard against noise misspecification, we assess the performance of three variants: (i) GPs with Student-t observations; (ii) Student-t processes (TPs); and (iii) classification GPs modeling the sign of the response. In conjunction with these metamodels, we analyze several acquisition functions for guiding the sequential experimental designs, extending existing stepwise uncertainty reduction criteria to the stochastic contour-finding context. This also motivates our development of (approximate) updating formulas to efficiently compute such acquisition functions. Our schemes are benchmarked by using a variety of synthetic experiments in 1–6 dimensions. We also consider an application of level set estimation for determining the optimal exercise policy of Bermudan options in finance.

Counterparty Credit Risk and American Options

This paper analyzes the effect of counterparty credit risk on optimal early exercise policy and value of American options. In contrast with the existing literature we find that the price of the underlying asset at which it is optimal to exercise an American option can be significantly different when there is counterparty credit risk. We show that it is not always optimal to exercise when a credit event is likely yet it is sometimes optimal to exercise for credit reasons even when the risk of a credit event is fairly remote. Further, we find that discounting the expected payoff on American options at a higher credit-risk-adjusted rate may lead to inaccurate valuation results. Numerical examples illustrate that early exercise can mitigate only approximately one third of the expected credit loss for American options. This result conflicts with the existing literature that claims early exercise can largely eliminate such expected credit loss. The remaining expected credit loss can be attributed to two sources: the write-down of the payoff if financial distress occurs; and the loss of time value if early exercise was motivated by credit concerns.

Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options

AbstractThis paper is devoted to the American option pricing problem governed by the Black-Scholes equation. The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain. The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes. This free boundary is then deformed to a fixed boundary by the front-fixing transformation. The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory), is treated by the perfectly matched layer technique. Finally, the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method. In particular, for Delta, one of the Greeks, we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition. Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.

Randomized Stopping Times and Early Exercise for American Derivatives in Dry Markets

This paper studies the impact of dry markets for underlying assets on the optimal stopping time and optimal exercise policy of American derivatives. We consider that the underlying is transacted at all points in time except for a subset of dates, for which there is an exogenous probability that trading may exist. Using superreplicating strategies, we derive expectation representations for the range of arbitrage-free values of the derivatives. For arbitrary probability, an enlarged filtration jointly induced by the price process and the market existence process makes ordinary stopping times sufficient to characterize such representation. For the deterministic case where the probability is zero, randomized stopping times are required. Several comparisons of the ranges obtained with the two market restrictions are performed. Finally, we conclude that market incompleteness caused by dryness may delay the optimal exercise of American derivatives.

Real Options and American Derivatives: The Double Continuation Region

We study the nonstandard optimal exercise policy associated with relevant capital investment options and with the prepayment option of widespread collateralized-borrowing contracts like the gold loan. Option exercise is optimally postponed not only when moneyness is insufficient, but also when it is excessive. We extend the classical optimal exercise properties for American options. Early exercise of an American call with a negative underlying payout rate can occur if the option is moderately in the money. We fully characterize the existence, the monotonicity, the continuity, the limits, and the asymptotic behavior at maturity of the double free boundary that separates the exercise region from the double continuation region. We find that the finite-maturity nonstandard policy conspicuously differs from the infinite-maturity one.This paper was accepted by Jerome Detemple, finance.

Non-transferable non-hedgeable executive stock option pricing

To value non-transferable non-hedgeable (NTNH) contingent claims and price executive stock options (ESOs), we use a replication argument to translate portfolios with NTNH derivatives into portfolios of primary assets (only) with stochastic portfolio constraints. By identifying stochastic discount factors and finding subjective prices of NTNH European and American ESOs, for block and continuous partial exercise, we derive executives׳ optimal exercise policies, and use these to find objective prices/costs of ESOs to firms. Through numerical simulations, we obtain policy implications regarding ESOs׳ incentivizing efficiency. For the first time, we demonstrate that, unlike under block exercise, subjective prices under continuous partial exercise may be higher than objective ones. Moreover, volatility regimes and executives׳ “other wealth” are important in ESO pricing, and are thus essential to empirical executive compensation studies.

Employee stock option-implied risk attitude under Rank-Dependent Expected Utility

Probability weighting is one of the cornerstones of decision-making theories accommodating gambling preferences. This paper examines its relevance to explaining employee stock option exercise behavior. We characterized the optimal exercise policy for a representative employee with Rank-Dependent Expected Utility (RDEU) preferences. We find that the RDEU framework leads to improved predictions of empirical exercise patterns. The implications from our findings are twofold: (1) probability weighting implies an increase in stock option cost to shareholders; (2) employee exercise behavior-implied sentiment is affected by the firm's stock market risk and performance.

Optimal Exercise for Derivative Securities

This article reviews the literature on American-style derivatives. The presentation stresses some of the major developments in the field. The focus is on the determination of optimal exercise policies and the structure of derivatives’ prices. Illustrative examples highlight the complexity of the optimal exercise decision.

Analytic solution for American barrier options with two barriers

This paper concerns American barrier options with two barriers. Standard American Options are difficult to price but there exist good numerical or analytical approximation methods. The situation is different for American barrier options. These options cease to exist or come into being if some price barrier is hit during the option's life. The paper studies analytic valuation of American barrier options with two barriers where the barriers become active by turns. In this paper, analytic valuation formulas for these options are derived by using both constant and exponential barriers for optimal early exercise policies.

Non-Transferable Non-Hedgeable Executive Stock Options Pricing

To value non-transferable non-hedgeable (NTNH) contingent claims and price executive stock options (ESOs), we use a replication argument to translate portfolios with NTNH derivatives into portfolios of primary assets (only) with stochastic portfolio constraints. By identifying stochastic discount factors and finding subjective prices of NTNH European and American ESOs, for block and continuous partial exercise, we derive executives’ optimal exercise policies, and use these to find objective prices/costs of ESOs to firms. Through numerical simulations, we obtain policy implications regarding ESOs’ incentivizing efficiency. For the first time, we demonstrate that, unlike under block exercise, subjective prices under continuous partial exercise may be higher than objective ones. Moreover, volatility regimes and executives’ “other wealth” are important in ESO pricing, and are thus essential to empirical executive compensation studies.

American Option Sensitivities Estimation via a Generalized Infinitesimal Perturbation Analysis Approach

In this paper, we develop efficient Monte Carlo methods for estimating American option sensitivities. The problem can be reformulated as how to perform sensitivity analysis for a stochastic optimization problem with model uncertainty. We introduce a generalized infinitesimal perturbation analysis (IPA) approach to resolve the difficulty caused by discontinuity of the optimal decision with respect to the underlying parameter. The IPA estimators are unbiased if the optimal decisions are explicitly known. To quantify the estimation bias caused by intractable exercising policies in the case of pricing American options, we also provide an approximation guarantee that relates the sensitivity under the optimal exercise policy to that computed under a suboptimal policy. The price-sensitivity estimators yielded from this approach demonstrate significant advantages numerically in both high-dimensional environments and various process settings. We can easily embed them into many of the most popular pricing algorithms without extra simulation effort to obtain sensitivities as a by-product of the option price. Our generalized approach also casts new insights on how to perform sensitivity analysis using IPA: we do not need path-wise continuity to apply it.

BOUNDARY EVOLUTION EQUATIONS FOR AMERICAN OPTIONS

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.

Does stock return predictability affect ESO fair value?

Executive Stock Options (ESOs) are modified American options that cannot be valued using standard methods. With a few exceptions, the literature has discussed the ESO fair value by assuming unpredictable stock returns which are not supported by the available empirical evidence. In this paper we obtain the fair value of American ESOs when stock returns are predictable and, specifically, driven by the trending Ornstein–Uhlenbeck process of Lo and Wang (1995). We solve the executive’s portfolio allocation problem for a simple buy-and-hold strategy when his wealth can be distributed between a risk-free asset and a market portfolio. This problem is jointly solved with the executive’s optimal exercise policy. We find that executives tend to wait longer the higher the predictability, independently of the composition of executive’s asset menu. We have also analyzed the implications under the FAS123R proposals for the ESO fair value and found that, even for low autocorrelations, there is a meaningful mispricing when unpredictable returns are erroneously assumed.

Real options with synergies: static versus dynamic policies

We develop a model for determining whether a firm should exercise two real options individually or simultaneously. The simultaneous exercise of both options has synergy of cost savings, while the separate exercise of each option benefits from project flexibility. This trade-off determines the optimal exercise policy. We compare static and dynamic management of multiple real options. A firm under static management determines the type of exercise of real options ex ante; on the other hand, a firm under dynamic management makes the decision at the time of exercise. We show that highly correlated projects increase the option values under both styles of management because a firm is more likely to enjoy the synergy gains of joint investment. We also highlight the advantage of dynamic management over static management for weakly correlated projects.

Boundary Evolution Equations for American Options

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.

Real options pricing by the finite element method

Real option pricing problems in investment project evaluation are mostly solved by the simulation-based methods, the lattice methods and by the finite difference method (FDM). Only a few applications of the finite element method (FEM) to these problems have been reported in the literature; although it seems to be an alternative tool for pricing real options. Unlike the existing finite element-based papers, in this paper we use residual formulation and provide a detailed scheme for practical implementations. The FEM is introduced and developed as a numerical method for real options pricing problems. First of all, a partial differential equation (pde) model is defined, then the problem’s domain is discretized by finite elements. The weak formulation of the pde is then obtained, and finally the solution to the real option pricing problem is found by solving an algebraic system. For benchmarking purposes, the FEM is applied to known investment and abandonment option problems found in the literature and the results are compared with those of some traditional methods. These results show a good performance of the FEM and its superiority over the FDM in terms of convergence, and over the simulation-based methods in terms of the optimal exercise policy.

Pricing high-dimensional Bermudan options using the stochastic grid method

This paper considers the problem of pricing options with early-exercise features whose pay-off depends on several sources of uncertainty. We propose a stochastic grid method for estimating the optimal exercise policy and use this policy to obtain a low-biased estimator for high-dimensional Bermudan options. The method has elements of the least-squares method (LSM) of Longstaff and Schwartz [Valuing American options by simulation: A simple least-squares approach, Rev. Finan. Stud. 3 (2001), pp. 113–147], the stochastic mesh method of Broadie and Glasserman [A stochastic mesh method for pricing high-dimensional American option, J. Comput. Finance 7 (2004), pp. 35–72], and stratified state aggregation along the pay-off method of Barraquand and Martineau [Numerical valuation of high-dimensional multivariate American securities, J. Financ. Quant. Anal. 30 (1995), pp. 383–405], with certain distinct advantages over the existing methods. We focus on the numerical results for high-dimensional problems such as max option and arithmetic basket option on several assets, with basic error analysis for a general one-dimensional problem.