Real options maximizing survival probability under incomplete markets

By integrating the survival problem into the theory of real option valuation under incomplete markets, we analyze an entrepreneurial firm's optimal survival probability and the joint decisions of business investments and portfolio choices when the business investment opportunity has undiversifiable idiosyncratic risks. Based on the theory of stochastic control, we derive the semi-closed-form solutions for the firm's optimal survival probability, its investment thresholds and the implied option value. The results show that the goal of maximizing the survival probability greatly changes the entrepreneur's business investment strategies, the pattern of asset allocation and the correlation between the option value and the project risks. The comparative statics analysis shows that public authorities should subsidize entrepreneurs and maintain stabile financial markets in order to encourage entrepreneurship.     

American Parisian options

In this article, we describe the various sorts of American Parisian options and propose valuation formulae. Although there is no closed-form valuation for these products in the non-perpetual case, we have been able to reformulate their price as a function of the exercise frontier. In the perpetual case, closed-form solutions or approximations are obtained by relying on excursion theory. We derive the Laplace transform of the first instant Brownian motion reaches a positive level or, without interruption, spends a given amount of time below zero. We perform a detailed comparison of perpetual standard, barrier and Parisian options.     

Pricing options in incomplete equity markets via the instantaneous Sharpe ratio

We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. First, we apply our method to price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and Saá-Requejo (J Polit Econ 108:79–119, 2000) and of Björk and Slinko (Rev Finance 10:221–260, 2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque et al. (Derivatives in financial markets with stochastic volatility. Cambridge University Press, 2000).     

Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise boundary for early-exercise options. Our approach is based on the Spitzer identities for general Lévy processes and on the Wiener–Hopf method. Our direct calculation of the price of α-quantile options combines for the first time the Dassios–Port–Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Lévy processes.     

Attainability of European path-independent claims in incomplete markets

In this paper we consider the question which path-independent claims are attainable through self-financing trading strategies in an incomplete market. For continuous-time stochastic volatility models we show that only affine payoffs can be replicated. We provide a simple proof for this proposition based on the requirement that, for replication, the stock and the claim must be locally perfectly correlated, and based on the partial differential equation that any path-independent claim has to satisfy. Moreover, we show that this result does not carry over to discrete setups.     

A valuation algorithm for indifference prices in incomplete markets

A probabilistic iterative algorithm is constructed for indifference prices of claims in a multiperiod incomplete model. At each time step, a nonlinear pricing functional is applied that isolates and prices separately the two types of risk. It is represented solely in terms of risk aversion and the pricing measure, a martingale measure that preserves the conditional distribution of unhedged risks, given the hedgeable ones, from their historical counterparts.Received: 1 September 2003, Mathematics Subject Classification: 93E20, 60G40, 60J75JEL Classification: C61, G11, G13The second author acknowledges partial support from NSF Grants DMS 0102909 and DMS 0091946.     

Valuation of American options in the presence of event risk

This paper studies the valuation of American options in the presence of external/non-hedgeable event risk. When the event occurs, the American option is terminated and a rebate is paid instead of the promised pay-off profile. Consequently, the presence of event risk influences the exercise strategy of the option holder. For the financial market in a diffusion setting, the probabilistic structure in terms of equivalent martingale measures is briefly analysed. Then, for a given equivalent martingale measure the optimal stopping problem of the American option is solved. As a main result, no-arbitrage bounds for American option values in the presence of event risk are derived, as well as hedging strategies corresponding to the no-arbitrage bounds.Received: May 2004, Mathematics Subject Classification: 90C47, 60H30, 60G40JEL Classification: G13, D52, D81The author thanks John Gould and Ross Maller for useful discussions. The author is also grateful to a referee for helpful comments. This research was partially supported by University of Western Australia Research Grant RA/1/485.     

A test of the beta model on Eurodollar futures options

The paper reports empirical tests of the beta model for pricing fixed-income options. The beta model resembles the Black–Scholes model with the lognormal probability distribution replaced by a beta probability distribution. The test is based on 32 817 daily prices of Eurodollar futures options and concludes that the beta model is more accurate than alternative option pricing models.     

An improved convolution algorithm for discretely sampled Asian options

We suggest an improved FFT pricing algorithm for discretely sampled Asian options with general independently distributed returns in the underlying. Our work complements the studies of Carverhill and Clewlow [Risk, 1990, 3(4), 25–29], Benhamou [J. Comput. Finance, 2002, 6(1), 49–68], and Fusai and Meucci [J. Bank. Finance, 2008, 32(10), 2076–2088], and, if we restrict our attention only to log-normally distributed returns, also Ve?e? [Risk, 2002, 15(6), 113–116]. While the existing convolution algorithms compute the density of the underlying state variable by moving forward on a suitably defined state space grid, our new algorithm uses backward price convolution, which resembles classical lattice pricing algorithms. For the first time in the literature we provide an analytical upper bound for the pricing error caused by the truncation of the state space grid and by the curtailment of the integration range. We highlight the benefits of the new scheme and benchmark its performance against existing finite difference, Monte Carlo, and forward density convolution algorithms.     

Aggregate implications of indivisible labor, incomplete markets, and labor market frictions

We study the impact of tax and transfer programs on steady-state allocations in a model with search frictions, an operative labor supply margin, and incomplete markets. In a benchmark model that has indivisible labor and incomplete markets but no trading frictions we show that the aggregate effects of taxes are identical to those in the economy with employment lotteries, though individual employment and asset dynamics can be different. The effect of frictions on the response of aggregate hours to a permanent tax change is highly nonlinear. There is considerable scope for substitution between “voluntary” and “frictional” nonemployment in some situations.     

Numerical evaluation of the critical price and American options

An approximate solution to the American put value is proposed and implemented numerically. Relaxation techniques enable the critical price to be determined with high accuracy. The method uses a modification of the quadratic approximation of MacMillan and Barone-Adesi and Whaley which gives an expression for the critical price. Numerical experimentation and iterative methods quickly provide highly accurate solutions.     

Risk measure pricing and hedging in incomplete markets

This article attempts to extend the complete market option pricing theory to incomplete markets. Instead of eliminating the risk by a perfect hedging portfolio, partial hedging will be adopted and some residual risk at expiration will be tolerated. The risk measure (or risk indifference) prices charged for buying or selling an option are associated to the capital required for dynamic hedging so that the risk exposure will not increase. The associated optimal hedging portfolio is decided by minimizing a convex measure of risk. I will give the definition of risk-efficient options and confirm that options evaluated by risk measure pricing rules are indeed risk-efficient. Relationships to utility indifference pricing and pricing by valuation and stress measures will be discussed. Examples using the shortfall risk measure and average VaR will be shown. The work of Mingxin Xu is supported by the National Science Foundation under grant SES-0518869. I would like to thank Steven Shreve for insightful comments, especially his suggestions to extend the pricing idea from using shortfall risk measure to coherent ones, and to study its relationship to utility based derivative pricing. The comments from the associate editor and the anonymous referee have reshaped the paper into its current version. The paper has benefited from discussions with Freddy Delbaen, Jan Večeř, David Heath, Dmitry Kramkov, Peter Carr, and Joel Avrin.     

High-order accurate implicit finite difference method for evaluating American options

A numerical method is presented for valuing vanilla American options on a single asset that is up to fourth-order accurate in the log of the asset price, and second-order accurate in time. The method overcomes the standard difficulty encountered in developing high-order accurate finite difference schemes for valuing American options; that is, the lack of smoothness in the option price at the critical boundary. This is done by making special corrections to the right-hand side of the differnce equations near the boundary, so they retain their level of accuracy. These corrections are easily evaluated using estimates of the boundary location and jump in the gamma that occurs there, such as those developed by Carr and Eaguet. Results of numerical experiments are presented comparing the method with more standard finite difference methods.     

Pricing American options written on two underlying assets

This paper extends the integral transform approach of McKean [Ind. Manage. Rev., 1965, 6, 32–39] and Chiarella and Ziogas [J. Econ. Dyn. Control, 2005, 29, 229–263] to the pricing of American options written on more than one underlying asset under the Black and Scholes [J. Polit. Econ., 1973, 81, 637–659] framework. A bivariate transition density function of the two underlying stochastic processes is derived by solving the associated backward Kolmogorov partial differential equation. Fourier transform techniques are used to transform the partial differential equation to a corresponding ordinary differential equation whose solution can be readily found by using the integrating factor method. An integral expression of the American option written on any two assets is then obtained by applying Duhamel’s principle. A numerical algorithm for calculating American spread call option prices is given as an example, with the corresponding early exercise boundaries approximated by linear functions. Numerical results are presented and comparisons made with other alternative approaches.     

Hedging American Options in Merton's Model: A Locally Risk Minimizing Approach

An interesting problem, related to American options in incomplete markets, is the possibility to select a preferable equivalent martingale measure in order to compute the prices. With this in mind, we consider a particular option that may be viewed as a finite collection of suitable European options and for which the minimal martingale measure permits the minimization of the local risk. Since this option is an approximation of the American put, the stability result presented, concerning the portfolio decomposition, also suggests an argument in favor of the minimal martingale measure in the American case.     

Pricing derivatives of American and game type in incomplete markets

In this paper the neutral valuation approach is applied to American and game options in incomplete markets. Neutral prices occur if investors are utility maximizers and if derivative supply and demand are balanced. Game contingent claims are derivative contracts that can be terminated by both counterparties at any time before expiration. They generalize American options where this right is limited to the buyer of the claim. It turns out that as in the complete case, the price process of American and game contingent claims corresponds to a Snell envelope or to the value of a Dynkin game, respectively.On the technical level, an important role is played by -sub- and -supermartingales. We characterize these processes in terms of semimartingale characteristics.Received: June 2003, Mathematics Subject Classification (2000):   91B24, 60G48, 91B16, 91A15, 60G40JEL Classification:   G13, D52, C73The authors want to thank PD Dr. Martin Beibel for the idea leading to the proof of Proposition A.4 and both anonymous referees for many valuable comments. The second author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg Angewandte Algorithmische Mathematik at Munich University of Technology and by the Fonds zur Förderung der wissenschaftlichen Forschung at Vienna University of Technology.     

Truncation and acceleration of the Tian tree for the pricing of American put options

We present a new method for truncating binomial trees based on using a tolerance to control truncation errors and apply it to the Tian tree together with acceleration techniques of smoothing and Richardson extrapolation. For both the current (based on standard deviations) and the new (based on tolerance) truncation methods, we test different truncation criteria, levels and replacement values to obtain the best combination for each required level of accuracy. We also provide numerical results demonstrating that the new method can be 50% faster than previously presented methods when pricing American put options in the Black–Scholes model.     

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.