Pricing Parisian down-and-in options

In this paper, we price American-style Parisian down-and-in call options under the Black–Scholes framework. Usually, pricing an American-style option is much more difficult than pricing its European-style counterpart because of the appearance of the optimal exercise boundary in the former. Fortunately, the optimal exercise boundary associated with an American-style Parisian knock-in option only appears implicitly in its pricing partial differential equation (PDE) systems, instead of explicitly as in the case of an American-style Parisian knock-out option. We also recognize that the “moving window” technique developed by Zhu and Chen (2013) for pricing European-style Parisian up-and-out call options can be adopted to price American-style Parisian knock-in options as well. In particular, we obtain a simple analytical solution for American-style Parisian down-and-in call options and our new formula is written in terms of four double integrals, which can be easily computed numerically.     

Electricity swing options: Behavioral models and pricing

Electricity swing options are supply contracts for power, which give the owner the right to change the required delivery on short time notice. It gives more flexibility than fixed base load or peak load contracts. The name “option” is a bit misleading, since it gives the owner multiple exercise rights at many different time horizons with exercise amounts on a continuous scale. We look at the problem to determine a rational ask price for such a contract from the viewpoint of the contract seller. The pricing of these contracts differs drastically from the pricing of financial options. First, peculiar properties arise from the non-storability of the underlying (the energy) and therefore the impossibility to hedge with the underlying, hedging is only possible with some future contracts. Second, the behavior of the owner plays an important role. Based on some behavioral model for the option holder, we develop a game-theoretic model, which allows to identify the equilibrium price. Besides some theoretical results, we present some numerical results which clarify the dependence of the asked price on the amount of flexibility offered in the swing option.     

Swing options in commodity markets: a multidimensional Lévy diffusion model

We study valuation of swing options on commodity markets when the commodity prices are driven by multiple factors. The factors are modeled as diffusion processes driven by a multidimensional Lévy process. We set up a valuation model in terms of a dynamic programming problem where the option can be exercised continuously in time. Here, the number of swing rights is given by a total volume constraint. We analyze some general properties of the model and study the solution by analyzing the associated HJB-equation. Furthermore, we discuss the issues caused by the multi-dimensionality of the commodity price model. The results are illustrated numerically with three explicit examples.     

利用机制设计解决技术定价问题

本研究以机制设计的观点来试图解决技术定价的问题。本研究将实务上常用的创业投资分次筹资的方式视为随着时间的进行，所取得的有关技术定价的讯息逐渐增多，故透过多次的筹资，逐渐的将技术的真实价格逐步反映的一种经济机制。接着，本研究依循机制设计的观点，设计了根据未来某一时点的条件实现与否来重新补偿创业者的持股，以使得创业者的报酬可以达到该项技术真实价格下创业者应有的报酬。特别是（ML〈MT〈MH）形式下的条件选择权机制可以将事前无法确定的技术真实价格在日后加以表明。透过本研究所建议的方法和思路，可以使得过去因技术出价的差距过大，使得新进投资不能进行，而使得社会丧失一次增加社会财富及社会福利的机会的问题得以解决。     

Approximate valuation of average options

This paper concerns the valuation of average options of European type where an investor has the right to buy the average of an asset price process over some time interval, as the terminal price, at a prespecified exercise price. A discrete model is first constructed and a recurrence formula is derived for the exact price of the discrete average call option. For the continuous average call option price, we derive some approximations and theoretical upper and lower bounds. These approximations are shown to be very accurate for at-the-money and in-the-money cases compared to the simulation results. The theoretical bounds can be used to provide useful information in pricing average options.     

The discontinuous Galerkin method for discretely observed Asian options

Asian options represent an important subclass of the path-dependent contracts that are identified by payoff depending on the average of the underlying asset prices over the prespecified period of option lifetime. Commonly, this average is observed at discrete dates, and also, early exercise features can be admitted. As a result, analytical pricing formulae are not always available. Therefore, some form of a numerical approximation is essential for efficient option valuation. In this paper, we study a PDE model for pricing discretely observed arithmetic Asian options with fixed as well as floating strike for both European and American exercise features. The pricing equation for such options is similar to the Black-Scholes equation with 1 underlying asset, and the corresponding average appears only in the jump conditions across the sampling dates. The objective of the paper is to present the comprehensive methodological concept that forms and improves the valuation process. We employ a robust numerical procedure based on the discontinuous Galerkin approach arising from the piecewise polynomial generally discontinuous approximations. This technique enables a simple treatment of discrete sampling by incorporation of jump conditions at each monitoring date. Moreover, an American early exercise constraint is directly handled as an additional nonlinear source term in the pricing equation. The proposed solving procedure is accompanied by an empirical study with practical results compared to reference values.     

Further calculations for Israeli options

Recently Kifer introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment not less than the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a stochastic saddle point problem associated with Dynkin games. Kyprianou, A.E. (2004) "Some calculations for Israeli options", Fin. Stoch. 8, 73-86 gives two examples of perpetual Israeli options where the value function and optimal strategies may be calculated explicity. In this article, we give a third example of a perpetual Israeli option where the contingent claim is based on the integral of the price process. This time the value function is shown to be the unique solution to a (two sided) free boundary value problem on (0, ∞) which is solved by taking an appropriately rescaled linear combination of Kummer functions. The probabilistic methods we appeal to in this paper centre around the interaction between the analytic boundary conditions in the free boundary problem, Itô's formula with local time and the martingale, supermartingle and submartingale properties associated with the solution to the stochastic saddle point problem.     

Real options with synergies: static <Emphasis Type="Italic">versus</Emphasis> 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.     

Infinite reload options: Pricing and analysis

Infinite reload options allow the user to exercise his reload right as often as he chooses during the lifetime of the contract. Each time a reload occurs, the owner receives new options where the strike price is set to the current stock price. We consider a modified version of the infinite reload option contract where the strike price of the new options received by the owner is increased by a certain percentage; we refer to this new contract as an increased reload option. The pricing problem for this modified contract is characterized as an impulse control problem resulting in a Hamilton–Jacobi–Bellman equation. We use fully implicit timestepping and prove that the discretized equations are monotone, stable and consistent, implying convergence to the viscosity solution. We also derive a globally convergent iterative method for solving the non-linear discrete equations. Numerical examples show that both the exercise policy and the option value are very sensitive to the percentage increase in the reload strike.     

Priority option pricing in an M/M/m queue

We study a system where the service provider offers priority options. We identify the optimal option pricing policy, by deriving the optimal number a customer would buy and the customer’s exercise policy as a function of system congestion, options remaining, time to expiration and possibility of balking.     

Valuation of guaranteed minimum maturity benefits in variable annuities with surrender options

We present a numerical approach to the pricing of guaranteed minimum maturity benefits embedded in variable annuity contracts in the case where the guarantees can be surrendered at any time prior to maturity that improves on current approaches. Surrender charges are important in practice and are imposed as a way of discouraging early termination of variable annuity contracts. We formulate the valuation framework and focus on the surrender option as an American put option pricing problem and derive the corresponding pricing partial differential equation by using hedging arguments and Itô’s Lemma. Given the underlying stochastic evolution of the fund, we also present the associated transition density partial differential equation allowing us to develop solutions. An explicit integral expression for the pricing partial differential equation is then presented with the aid of Duhamel’s principle. Our analysis is relevant to risk management applications since we derive an expression of the delta for the sensitivity analysis of the guarantee fees with respect to changes in the underlying fund value. We provide algorithms for implementing the integral expressions for the price, the corresponding early exercise boundary and the delta of the surrender option. We quantify and assess the sensitivity of the prices, early exercise boundaries and deltas to changes in the underlying variables including an analysis of the fair insurance fees.     

Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model

We study the viscosity solutions of integro-differential Hamilton–Jacobi–Bellman equations of degenerate parabolic type. These equations are from the pricing problem for the European passport options in a jump-diffusion model. The passport option is a call option on a trading account. We discuss the mathematical model for pricing problem. We prove the comparison principle, uniqueness and convexity preserving for the viscosity solutions of related pricing equations.     

A model and some evidence on pricing compound call options

The most widely accepted option pricing model, derived by Black and Scholes (B-S), studies single priced options. Nevertheless, it has important implications for the relative pricing of compound call options. Compound options are two or more option contracts on a given security with different striking prices but with each expiring on the same day.Studying the relative pricing of compound options provides insight into the efficiency of generally accepted option pricing models. Comparing prices of compound options enables us to analyze factors in option pricing that would remain hidden in studies of single options.We are not primarily concerned with efficiency of option pricing, although some of our results may bear on this issue. Our primary concerns are: (1) to determine the implications of the B-S model for compound options and (2) to explain compound option prices by a number of variables, and thus come to conclusions about option pricing generally.We found difficulty with the B-S model when attempting to explain the relative pricing of compound options. Further, from empirical tests, we found that the most important factor in explaining the relative pricing of compound options is the relative degree of leverage which is operative between the various components of a compound option set.     

Differential quadrature method for pricing American options

In this article, differential quadrature method (DQM), a highly accurate and efficient numerical method for solving nonlinear problems, is used to overcome the difficulty in determining the optimal exercise boundary of American option. The following three parts of the problem in pricing American options are solved. The first part is how to treat the uncertainty of the early exercise boundary, or free boundary in the language of the PDE treatment of the American option, because American options can be exercised before the date of expiration. The second part is how to solve the nonlinear problem, because the problem of pricing American options is nonlinear. And the third part is how to treat the initial value condition with the singularity and the boundary conditions in the DQM. Numerical results for the free boundary of American option obtained by both DQM and finite difference method (FDM) are given and from which it can be seen the computational efficiency is greatly improved by DQM. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 711–725, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10028.     

The early exercise region for Bermudan options on two underlyings

This paper investigates the early exercise region for Bermudan options on two underlying assets. We present a set of analytical validation results for the early exercise region which can be used as a means of validating pricing techniques. When all strike prices are identical we show the existence of an intersection point such that for any asset price pair below this point early exercise is always optimal. We develop an approximation to this point in the two asset put case. When the strike prices are not all equal, we show that three separate cases exist for the early exercise region. For a Bermudan put on two assets we present these cases and show that there exists a critical point in which the boundaries of the two asset early exercise region bifurcate. Comparisons are drawn between the Bermudan results presented and the corresponding American option results.     

Compact finite difference method for American option pricing

A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size h   is large (h?0.1)$(h?0.1)$. Compact difference method two must solve an algebraic nonlinear equation obtained by Pantazopoulos (1998) at every time step. This method can obtain second order accuracy for space x and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put, 2003], and this method can obtain high accuracy for space x. The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.     

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.     

An efficient algorithm for Bermudan barrier option pricing

An efficient option pricing method based on Fourier-cosine expansions was presented by Fang and Oosterlee for European options in 2008,and later,this method was also used by them to price early-exercis...     

Discrete-Time Pricing and Optimal Exercise of American Perpetual Warrants in the Geometric Random Walk Model

An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional linear programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put.     

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.