Valuing employee reload options under the time vesting requirement

Upon the exercise of an employee stock option, the embedded reload provision entitles the holder to receive additional units of new options from the employer. The number of units of new options received is equal to the number of shares tendered as payment of strike and the new strike is set at the prevailing stock price. The reload provision may be subject to a time vesting requirement, that is, after each exercise, the employee is prohibited from exercising the reload until the end of a vesting period. In this paper, we construct an efficient numerical algorithm that computes the market value of the employee reload options under a time vesting requirement. Also, we explore the analytic properties of the price functions and optimal exercise policies of the employee reload options.     

Valuing executive stock options: performance hurdles, early exercise and stochastic volatility

Accounting standards require companies to assess the fair value of any stock options granted to executives and employees. We develop a model for accurately valuing executive and employee stock options, focusing on performance hurdles, early exercise and uncertain volatility. We apply the model in two case studies and show that properly computed fair values can be significantly lower than traditional Black–Scholes values. We then explore the implications for pay-for-performance sensitivity and the design of effective share-based incentive schemes. We find that performance hurdles can require a much greater fraction of total compensation to be a fixed salary, if pre-existing incentive levels are to be maintained.     

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.     

Some calculations for Israeli options

Recently Kifer (2000) 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 exceeding the holders claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a saddle point problem associated with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.Received: December 2002, Mathematics Subject Classification: 90A09, 60J40, 90D15JEL Classification: G13, C73I would like to express thanks to Chris Rogers for a valuable conversation.     

Valuing debt options: Empirical evidence

A two-factor model using the instantaneous rate of interest and the return on a consol bond to describe the term structure of interest rates — the Brennan-Schwartz model — is used to derive theoretical prices for American call and put options on US government bonds and treasury bills. These model prices are then compared with market prices. The theoretical model used to value the dept options also provides hedge ratios which may be used to construct zero-investment portfolios which, in theory, are perfectly riskless. Several trading strategies based on these ‘riskless’ portfolios are examined.     

Valuing qualitative options with stochastic volatility

We find a closed-form formula for valuing a time-switch option where its underlying asset is affected by a stochastically changing market environment, and apply it to the valuation of other qualitative options such as corridor options and options in foreign exchange markets. The stochastic market environment is modeled as a Markov regime-switching process. This analytic formula provides us with a rapid and accurate scheme for valuing qualitative options with stochastic volatility.     

Valuing futures and options on volatility

This paper presents simple closed-form expressions for volatility futures and option prices and examines their implications for the characteristics of these securities. We show that the properties of these volatility derivatives are fundamentally different from those of conventional option and futures contracts. This analysis also provides insights into the role that volatility derivatives may play in managing and hedging volatility risk in financial markets.     

The analysis and valuation of interest rate options

This paper provides a simple, alternative model for the valuation of European-style interest rate options. The assumption that drives the hedging argument in the model is that the forward prices of bonds follow an arbitrary two-state process. Later, this assumption is made more specific by postulating that the discount on a zero-coupon bond follows a multiplicative binomial process. In contrast to the Black-Scholes assumption applied to zero-coupon bonds, the limiting distribution of this process has the attractive features that the zero-bond price has a natural barrier at unity (thus precluding negative interest rates), and that the bond price is negatively skewed. The model is used to price interest rate options in general, and interest rate caps and floors in particular. The model is then generalized and applied to European-style options on bonds. A relationship is established between options on swaps and options on coupon bonds. The generalized model then provides a computationally simple formula, closely related to the Black-Scholes formula, for the valuation of European-style options on swaps.     

Valuing American put options using Gaussian quadrature

This article develops an efficient and accurate method for numericalevaluation of the integral equation which defines the Americanput option value function. Numerical integration using Gaussianquadrature and function approximation using Chebyshev polynomialsare combined to evaluate recursive expectations and producean approximation of the option value function in two dimensions,across stock prices and over time to maturity. A set of suchsolutions results in a multidimensional approximation that isextremely accurate and very quick to compute. The method isan effective alternative to finite difference methods, the binomialmodel, and various analytic approximations.     

Modelling exchange-traded barrier options traded in the Australian options market

Barrier options traded in the Australian market vary considerably in terms of the extent to which the barrier is monitored and in terms of the location of the barrier level relative to the exercise price. This paper examines the impact of these differences on prices and also on deltas and gammas. We find that it is not possible to generalize results concerning hedge parameter values to all barrier options. We find that options examined by Easton et al. (2004) do not display discontinuity of deltas at the barrier levels and that their apparent overpricing cannot be attributed to hedging difficulties.     

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.     

Efficient,exact algorithms for asian options with multiresolution lattices

Asian options are a kind of path-dependent derivative. How to price such derivatives efficiently and accurately has been a long-standing research and practical problem. This paper proposes a novel multiresolution (MR) trinomial lattice for pricing European- and American-style arithmetic Asian options. Extensive experimental work suggests that this new approach is both efficient and more accurate than existing methods. It also computes the numerical delta accurately. The MR algorithm is exact as no errors are introduced during backward induction. In fact, it may be the first exact discrete-time algorithm to break the exponential-time barrier. The MR algorithm is guaranteed to converge to the continuous-time value. This revised version was published online in June 2006 with corrections to the Cover Date.     

Investment options with debt-financing constraints

A contingent claims model is used to study the impact of debt-financing constraints on firm value, optimal capital structure, the timing of investment and other variables, such as credit spreads. The optimal investment trigger follows a U shape as a function of exogenously imposed constraint. Risky, equity-financed R&D growth options increase firm value by increasing the option value on unlevered assets, while their impact on the net benefits of debt is small.     

Double knock-out Asian barrier options which widen or contract as they approach maturity

Barrier options are considered for Asian options using a differential equation method. Solutions are obtained in the form of Fourier series for barriers which expand or contract as they approach maturity. Rigorous bounds are obtained. It is shown that by differentiating with respect to a parameter, solutions for more general payoffs can be obtained.     

Buying real options - Valuing uncertainty in infrastructure planning

For the last two decades real options thinking has been heralded as a new approach for handling uncertainty in investment decisions. However, application of the approach in infrastructure investment decision-making is negligible thus far. In this contribution we address the question: what are the barriers for the implementation of the real option approach (ROA) in practice? We focus on the experiences in several infrastructure-bound sectors: spatial planning and transport, ports, and energy.We conclude that the ROA maturity levels of these different sectors are quite different, and we ascribe these differences to the political setting, the institutional setting and the organisational flexibility of the sectors and their stakeholders. We suggest that the same issues apply to other advanced, quantitative methods.     

Incentive Efficiency of Stock versus Options

This paper examines the relative incentive costs of using stockversus options in management incentive contracts that use market priceas the performance measure. We establish that if the manager'seffort has little or no effect on a firm's operating risk, thenthe cost of incentive risk is less using stock rather than options.However, this result is reversed if the manager's effort has asignificant impact on the firm's operating risk.     

Valuing Fixed-Income Options and Mortgage-Backed Securities with Alternative Term Structure Models

To value mortgage-backed securities and options on fixed-income securities, it is necessary to make assumptions regarding the term structure of interest rates. We assume that the multi-factor fixed parameter term structure model accurately represents the actual term structure of interest rates, and that the values of mortgage-backed securities and discount bond options derived from such a term structure model are correct. Differences in the prices of interest rate derivative securities based on single-factor term structure models are therefore due to pricing bias resulting from the term structure model. The price biases that result from the use of single-factor models are compared and attributed to differences in the underlying models and implications for the selection of alternative term structure models are considered.     

Real options at the interface of finance and operations: exploiting embedded supply-chain real options to gain competitiveness

Exploiting embedded supply-chain real options creates powerful opportunities for competitive manufacturing in high-cost environments. Rather than seeking competitiveness through standardization as is common to lean production, real-options reasoning explores opportunities to use supply-chain variability as a strategic weapon. We present an illustrative case study of a Swiss manufacturer of cable extrusion equipment supported by a formal real-options model that aids in valuing the embedded options that make up supply-chain flexibility: postponement, contraction, expansion, switching, and abandonment. Real-options reasoning provides a plausible retrospective rationale for the case firm's use of supply-chain flexibility that provided protection against competition from low cost, but less responsive competitors. Their intuitive real-options reasoning facilitated incorporating fuller information concerning volatility, flexibility, and control into choosing what products to make, in what quantity, and with work allocated to which supplier. The case study also highlights how competing through exploiting embedded real options requires a different managerial skill set than does competing through cost reduction. Skills such as customer communication, supplier management, and ability to ensure a smooth flow of production join the ability to reduce and control lead times as key sources of competitive advantage.     

The pricing of Bermudan-style options on correlated assets

In this paper, we present a methodology for approximating a correlated multivariate-lognormal process with a recombining or “simple” multivariate-binomial process. The method represents an extension and implementation of previous work by Nelson and Ramaswamy (1990) and Ho, Stapleton and Subrahmanyam (1995) on diffusion approximation. The general method is illustrated by pricing a Bermudan-style put option on the minimum of three asset prices, and by pricing Bermudan-style options on bonds, where the value of the bond at a point in time depends upon the interest rate in two currencies and the foreign exchange rate. This type of structure, known as the “Power Reverse Dual” is a popular product in the case of Japanese Yen-US Dollar currencies. This revised version was published online in June 2006 with corrections to the Cover Date.