An approximate moving boundary method for American option pricing

We present a method to solve the free-boundary problem that arises in the pricing of classical American options. Such free-boundary problems arise when one attempts to solve optimal-stopping problems set in continuous time. American option pricing is one of the most popular optimal-stopping problems considered in literature. The method presented in this paper primarily shows how one can leverage on a one factor approximation and the moving boundary approach to construct a solution mechanism. The result is an algorithm that has superior runtimes-accuracy balance to other computational methods that are available to solve the free-boundary problems. Exhaustive comparisons to other pricing methods are provided. We also discuss a variant of the proposed algorithm that allows for the computation of only one option price rather than the entire price function, when the requirement is such.     

American Call Options Under Jump‐Diffusion Processes – A Fourier Transform Approach

We consider the American option pricing problem in the case where the underlying asset follows a jump‐diffusion process. We apply the method of Jamshidian to transform the problem of solving a homogeneous integro‐partial differential equation (IPDE) on a region restricted by the early exercise (free) boundary to that of solving an inhomogeneous IPDE on an unrestricted region. We apply the Fourier transform technique to this inhomogeneous IPDE in the case of a call option on a dividend paying underlying to obtain the solution in the form of a pair of linked integral equations for the free boundary and the option price. We also derive new results concerning the limit for the free boundary at expiry. Finally, we present a numerical algorithm for the solution of the linked integral equation system for the American call price, its delta and the early exercise boundary. We use the numerical results to quantify the impact of jumps on American call prices and the early exercise boundary.     

A closed-form analytic correction to the Black–Scholes–Merton price for perpetual American options

This is a complementary study of a recent work by Yoon et al. (2013) [1] [J.-H. Yoon, J.-H. Kim, S.-Y. Choi, Multiscale analysis of a perpetual American option with the stochastic elasticity of variance, Appl. Math. Lett. 26 (7) (2013)] which excludes a certain level of the elasticity of variance. A second-order correction to the Black–Scholes option price and optimal exercise boundary for a perpetual American put option is made under the stochastic elasticity of variance of a risky asset. Contrary to the case of Yoon et al. (2013) [1], it is given by an explicit closed-form analytic expression so that one can access clearly the sensitivity of the option price and the optimal exercise boundary to changes in model parameters as well as the impact of the presence of a stochastic elasticity term on the option price and the optimal time to exercise.     

A self-exciting threshold jump–diffusion model for option valuation

A self-exciting threshold jump–diffusion model for option valuation is studied. This model can incorporate regime switches without introducing an exogenous stochastic factor process. A generalized version of the Esscher transform is used to select a pricing kernel. The valuation of both the European and American contingent claims is considered. A piecewise linear partial-differential–integral equation governing a price of a standard European contingent claim is derived. For an American contingent claim, a formula decomposing a price of the American claim into the sum of its European counterpart and the early exercise premium is provided. An approximate solution to the early exercise premium based on the quadratic approximation technique is derived for a particular case where the jump component is absent. Numerical results for both European and American options are presented for the case without jumps.     

American Options Exercise Boundary When the Volatility Changes Randomly

The American put option exercise boundary has been studied extensively as a function of time and the underlying asset price. In this paper we analyze its dependence on the volatility, since the Black and Scholes model is used in practice via the (varying) implied volatility parameter. We consider a stochastic volatility model for the underlying asset price. We provide an extension of the regularity results of the American put option price function and we prove that the optimal exercise boundary is a decreasing function of the current volatility process realization. Accepted 13 January 1998     

American put options with a finite set of exercisable time epochs

The ordinary American put option assumes that investors can exercise their right at any time epoch. However, due to limitations in actual trades, they are not totally free to exercise in time. In this paper, motivated by this practical situation, we consider American put options with a finite set of exercisable time epochs. Assuming that the underlying stock price process follows a discrete-time Markov process, the put option premium is derived. It is shown that, as for the ordinary American put, the option premium is decomposed into the corresponding European put premium plus the early exercise premium under the stationary independent increments assumption. Moreover, the option premium converges to the ordinary American put premium from below as the number of exercisable time epochs increases under regularity conditions. Some lower bound of the option premium is also obtained.     

On modified Mellin transforms, Gauss-Laguerre quadrature, and the valuation of American call options

We extend a framework based on Mellin transforms and show how to modify the approach to value American call options on dividend-paying stocks. We present a new integral equation to determine the price of an American call option and its free boundary using modified Mellin transforms. We also show how to derive the pricing formula for perpetual American call options using the new framework. A result due to Kim (1990) [24] regarding the optimal exercise price at expiry is also recovered. Finally, we apply Gauss-Laguerre quadrature for the purpose of an efficient and accurate numerical valuation.     

American put option with regime‐switching volatility (finite time horizon)—Variational inequality approach

We study the fair price of American put option with regime‐switching volatility. Assuming that volatility σ(t) takes two different values σ₁ and σ₂, applying Δ hedging technique we obtain a system of evolutionary variational inequalities, which possesses two free boundaries (optimal exercise boundaries). The following are the main results of this paper.

• 1. Two free boundaries are monotonic and infinitely differentiable.
• 2. The optimal exercise boundary of American put option with regime‐switching volatility in the bearish (or bullish) market is smaller (or higher) than the one of standard American put option. And the price of American put option with regime‐switching volatility in the bearish (or bullish) market is higher (or smaller) than the one of standard American put option.
• 3. The solution of problem (1) is unique.

These results are original in the option pricing with regime‐switching volatility, the proof is technical. Copyright © 2008 John Wiley & Sons, Ltd.     

The American Foreign Exchange Option in Time-Dependent One-Dimensional Diffusion Model for Exchange Rate

The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent case is the continuity in time argument of the value function of the American put option and the regularity properties of the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for the American puts with different strikes, maturities and volatilities. Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value function of the American foreign exchange put option as the sum of the European put option value function and the early exercise premium. The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise boundary which can be studied by numerical methods.     

An optimal insurance strategy for an individual under an intertemporal equilibrium

In this paper, we discuss how a risk-averse individual under an intertemporal equilibrium chooses his/her optimal insurance strategy to maximize his/her expected utility of terminal wealth. It is shown that the individual’s optimal insurance strategy actually is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price. Since the cost of avoiding risk can be seen as a risk measure, the put option premium can be considered as a reasonable risk measure. Jarrow [Jarrow, R., 2002. Put option premiums and coherent risk measures. Math. Finance 12, 135-142] drew this conclusion with an axiomatic approach, and we verify it by solving the individual’s optimal insurance problem.     

The British Put Option

Abstract

We present a new put option where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ‘best prediction’ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British put option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. The practical implications of this protection feature are most remarkable as not only can the option holder exercise at or above the strike price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive higher returns at a lesser price. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British put option that leads to the conclusions above and shows that with the contract drift properly selected the British put option becomes a very attractive alternative to the classic American put.     

A Markov-modulated model for stocks paying discrete dividends

We extend the model in [Korn, R., Rogers, L.C.G., 2005. Stock paying discrete dividends: modelling and option pricing. Journal of Derivatives 13, 44–49] for (discrete) dividend processes to incorporate the dependence of assets on the market mode or the state of the economy, where the latter is modeled by a hidden finite-state Markov chain. We then derive the resulting dynamics of the stock price and various option-pricing formulae. It turns out that the stock price jumps not only at the time of the dividend payment, but also when the underlying Markov chain jumps.     

求解美式回望期权的有限元方法

期权作为一种金融衍生产品,在欧美国家一直很受欢迎.由于其规避风险的特性,期权也吸引了中国投资者的兴趣.基于市场的需求,2015年初,上海证券交易所推出了中国首批期权产品,期权定价问题的研究热潮正席卷全球.本文研究的美式回望期权,是一种路径相关的期权,其支付函数不仅依赖于标的资产的现值,也依赖其历史最值.分析回望期权的特点,不难发现:1)这类期权空间变量的变化范围为二维无界不规则区域,难以应用数值方法直接求解;2)最佳实施边界未知,使得该问题变得高度非线性.本文的主要工作就是解决这两个困难,得到回望期权和最佳实施边界的数值逼近结果.现有的处理问题1)的有效方法是采用标准变量替换、计价单位变换以及Landau变换将定价模型化为一个[0,1]区间上的非线性抛物问题,本文也将沿用这些技巧处理问题1).进一步,采用有限元方法离散简化后的定价模型,并论证了数值解的非负性,提出了利用Newton法求解离散化的非线性系统.最后,通过数值模拟,验证了本文所提算法的高效性和准确性.     

Exercise Boundary Near Maturity for an American Option on Several Assets

We consider an American put option on a linear function of d dividend-paying assets. The value function of this option is given as the solution of a free boundary problem. When d = 1, the behavior of the free boundary near the maturity of the option is well known. In this article, we extend to the case d > 1 the study of the free boundary near maturity. A parameterization of the stopping region at time t is given. That enables us to define and give a convergence rate for this region when t goes to the maturity.     

Pricing perpetual American catastrophe put options: A penalty function approach

The expected discounted penalty function proposed in the seminal paper by Gerber and Shiu [Gerber, H.U., Shiu, E.S.W., 1998. On the time value of ruin. North Amer. Actuarial J. 2 (1), 48-78] has been widely used to analyze the joint distribution of the time of ruin, the surplus immediately before ruin and the deficit at ruin, and the related quantities in ruin theory. However, few of its applications can be found beyond except that Gerber and Landry [Gerber, H.U., Landry, B., 1998. On the discount penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Math. Econ. 22, 263-276] explored its use for the pricing of perpetual American put options. In this paper, we further explore the use of the expected discounted penalty function and mathematical tools developed for the function to evaluate perpetual American catastrophe equity put options. We obtain the analytical expression for the price of perpetual American catastrophe equity put options and conduct a numerical implementation for a wide range of parameter values. We show that the use of the expected discounted penalty function enables us to evaluate the perpetual American catastrophe equity put option with minimal numerical work.     

Optimal valuation of American callable credit default swaps under drawdown of Lévy insurance risk process

This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at a fixed rate when the asset price is above a pre-specified level and continuously pays whenever the price increases. This payment scheme is in favour of the buyer as she only pays the premium when the market is in good condition for the protection against financial downturn. Under this framework, we look at an embedded option which gives the issuer an opportunity to call back the contract to a new one with reduced premium payment rate and slightly lower default coverage subject to paying a certain cost. We assume that the buyer is risk neutral investor trying to maximize the expected monetary value of the option over a class of stopping time. We discuss optimal solution to the stopping problem when the source of uncertainty of the asset price is modelled by Lévy process with only downward jumps. Using recent development in excursion theory of Lévy process, the results are given explicitly in terms of scale function of the Lévy process. Furthermore, the value function of the stopping problem is shown to satisfy continuous and smooth pasting conditions regardless of regularity of the sample paths of the Lévy process. Optimality and uniqueness of the solution are established using martingale approach for drawdown process and convexity of the scale function under Esscher transform of measure. Some numerical examples are discussed to illustrate the main results.     

Pricing American currency options in an exponential Lévy model

In this article the problem of the American option valuation in a Lévy process setting is analysed. The perpetual case is first considered. Without possible discontinuities (i.e. with negative jumps in the call case), known results concerning the currency option value as well as the exercise boundary are obtained with a martingale approach. With possible discontinuities of the underlying process at the exercise boundary (i.e. with positive jumps in the call case), original results are derived by relying on first passage time and overshoot associated with a Lévy process. For finite life American currency calls, the formula derived by Bates or Zhang, in the context of a negative jump size, is tested. It is basically an extension of the one developed by Mac Millan and extended by Barone‐Adesi and Whaley. It is shown that Bates' model generates pretty good results only when the process is continuous at the exercise boundary.     

A Note on a Moving Boundary Problem Arising in the American Put Option

We consider an American put option, under the Black–Scholes model. This corresponds to a moving boundary problem for a PDE. We convert the problem to a nonlinear integral equation for the moving boundary, which corresponds to the optimal exercise of the option. We use singular perturbation methods to compute the moving boundary, as well as the full solution to the PDE, in various asymptotic limits. We consider times close to the expiration date, as well as systems where the interest rate is large or small, relative to the volatility of the asset for which the option is sold.     

Pricing perpetual options using Mellin transforms

We have derived the expression for the free boundary and price of an American perpetual put as the limit of a finite-lived option.     

Pricing perpetual American options under a stochastic-volatility model with fast mean reversion

In this paper, we present a “correction” to Merton’s (1973) well-known classical case of pricing perpetual American puts by considering the same pricing problem under a general fast mean-reverting SV (stochastic-volatility) model. By using the perturbation method, two analytic formulae are derived for the option price and the optimal exercise price, respectively. Based on the newly obtained formulae, we conduct a quantitative analysis of the impact of the SV term on the price of a perpetual American put option as well as its early exercise strategies. It shows that the presence of a fast mean-reverting SV tends to universally increase the put option price and to defer the optimal time to exercise the option contract, had the underlying been assumed to be falling. It is also noted that such an effect could be quite significant when the option is near the money.