Lookback options and diffusion hitting times: A spectral expansion approach

Lookback options have payoffs dependent on the maximum and/or minimum of the underlying price attained during the options lifetime. Based on the relationship between diffusion maximum and minimum and hitting times and the spectral decomposition of diffusion hitting times, this paper gives an analytical characterization of lookback option prices in terms of spectral expansions. In particular, analytical solutions for lookback options under the constant elasticity of variance (CEV) diffusion are obtained.Received: 1 October 2003, Mathematics Subject Classification: 60J35, 60J60, 60G70JEL Classification: G13The author thanks Phelim Boyle for bringing the problem of pricing lookback options under the CEV process to his attention and for useful discussions and Viatcheslav Gorovoi for computational assistance. This research was supported by the U.S. National Science Foundation under grants DMI-0200429 and DMS-0223354.     

A functional approach to pricing complex barrier options

In this article, a new method for pricing contingent claims, which is particularly well suited for options with complex barrier and volatility structures, is introduced. The approach is based on a high-precision approximation of the Feynman–Kac equation with distributed approximating functionals. The method is particularly well suited for long maturity valuation problems, and it is shown to be faster and more accurate than conventional solution schemes.     

A simple approach for pricing barrier options with time-dependent parameters

Abstract

In this paper we present a simple and easy-to-use method for computing accurate estimates (in closed form) of Black-Scholes barrier option prices with time-dependent parameters. This new approach is also able to provide tight upper and lower bounds (in closed form) for the exact barrier option prices.     

Static versus dynamic hedges: an empirical comparison for barrier options

We conduct an empirical comparison of static versus dynamic hedges of barrier options. Using more than five years of data, we compare a number of static hedges from the literature with dynamic hedges based on the local volatility model. The main result is that the variability of profit-and-loss distributions from certain static hedges is significantly smaller than that of dynamic hedges and robust to changing market scenarios. Furthermore, these static hedges are able to provide a robust tracking of barrier options’ sensitivities. This article reflects the authors’ personal opinion and not necessarily the opinion of their employers.     

Technology and values: problems and options

Science, technology and industry have come to form a positive feedback system of accelerating change, greatly extending the political choice and load carried by the democratic process, and having a potentially massive impact on culture and values. Three possible options are considered for dealing with this situation: limiting research, limiting dissemination, and reforming social institutions and culture. Only one aspect of the third option is considered, ie, the development of an appropriate epistemology to deal with the interaction of science, technology and values. The expansionist claim that science itself is developing to explain the sources of ethics, and will increasingly subsume the traditional areas of social science and the humanities has its problems. The views of the scientific restrictionist, Sir Geoffrey Vickers, a member of the Futures Advisory Board who died recently, deserve wider attention.     

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.     

Static hedges for reverse barrier options with robustness against skew risk: an empirical analysis

We conduct an empirical evaluation of a static super-replicating hedge of barrier options. The hedge is robust to uncertainty about the future skew. Using almost seven years of current data on the DAX, we evaluate the performance of the hedge and compare it with those of both a dynamic and a static replicating hedge. The main result is that the robustness of the static super-replicating portfolio is also empirically confirmed in practice such that the hedge sets an upper bound for the price of skew risk for barrier options.     

Analytical pricing of discretely monitored Asian-style options: Theory and application to commodity markets

We compute an analytical expression for the moment generating function of the joint random vector consisting of a spot price and its discretely monitored average for a large class of square-root price dynamics. This result, combined with the Fourier transform pricing method proposed by Carr and Madan [Carr, P., Madan D., 1999. Option valuation using the fast Fourier transform. Journal of Computational Finance 2(4), Summer, 61–73] allows us to derive a closed-form formula for the fair value of discretely monitored Asian-style options. Our analysis encompasses the case of commodity price dynamics displaying mean reversion and jointly fitting a quoted futures curve and the seasonal structure of spot price volatility. Four tests are conducted to assess the relative performance of the pricing procedure stemming from our formulae. Empirical results based on natural gas data from NYMEX and corn data from CBOT show a remarkable improvement over the main alternative techniques developed for pricing Asian-style options within the market standard framework of geometric Brownian motion.     

Auto-static for the people: risk-minimizing hedges of barrier options

We present and test a method for computing risk-minimizing static hedge strategies. The method is straightforward, yet flexible with respect to the type of contingent claim being hedged, the underlying asset dynamics, and the choice of risk-measure and hedge instruments. Extensive numerical comparisons for barrier options in a model with stochastic volatility and jumps show that the resulting hedges outperform previous suggestions in the literature. We also demonstrate that the risk-minimizing static hedges work in an infinite intensity Levy-driven model, and a number of controlled experiments illustrate that hedge performance is robust to model risk.     

Expensing stock options: a fair-value approach

Now that companies such as General Electric and Citigroup have accepted the premise that employee stock options are an expense, the debate is shifting from whether to report options on income statements to how to report them. The authors present a new accounting mechanism that maintains the rationale underlying stock option expensing while addressing critics' concerns about measurement error and the lack of reconciliation to actual experience. A procedure they call fair-value expensing adjusts and eventually reconciles cost estimates made at grant date with subsequent changes in the value of the options, and it does so in a way that eliminates forecasting and measurement errors over time. The method captures the chief characteristic of stock option compensation--that employees receive part of their compensation in the form of a contingent claim on the value they are helping to produce. The mechanism involves creating entries on both the asset and equity sides of the balance sheet. On the asset side, companies create a prepaid-compensation account equal to the estimated cost of the options granted; on the owners'-equity side, they create a paid-in capital stock-option account for the same amount. The prepaid-compensation account is then expensed through the income statement, and the stock option account is adjusted on the balance sheet to reflect changes in the estimated fair value of the granted options. The amortization of prepaid compensation is added to the change in the option grant's value to provide the total reported expense of the options grant for the year. At the end of the vesting period, the company uses the fair value of the vested option to make a final adjustment on the income statement to reconcile any difference between that fair value and the total of the amounts already reported.     

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value φ(S _T ) at expiry T if neither of the continuous time-dependent barriers b _±:[0,T]→ℝ₊ have been hit during the time interval [0,T]. Using a probabilistic approach, we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions φ, barrier functions b _± and linear diffusions (S _t ) _t∈[0,T]. We show that the barrier premium can be expressed as a sum of integrals along the barriers b _± of the option’s deltas Δ _±:[0,T]→ℝ at the barriers and that the pair of functions (Δ ₊,Δ ₋) solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.     

Analytical pricing formulae for vulnerable vanilla and barrier options

Review of Quantitative Finance and Accounting - This paper proposes analytically vulnerable vanilla option pricing formulae that simultaneously consider the premature default, the correlation...     

Optimal discrete hedging of American options using an integrated approach to options with complex embedded decisions

In order to solve the problem of optimal discrete hedging of American options, this paper utilizes an integrated approach in which the writer’s decisions (including hedging decisions) and the holder’s decisions are treated on equal footing. From basic principles expressed in the language of acceptance sets we derive a general pricing and hedging formula and apply it to American options. The result combines the important aspects of the problem into one price. It finds the optimal compromise between risk reduction and transaction costs, i.e. optimally placed rebalancing times. Moreover, it accounts for the interplay between the early exercise and hedging decisions. We then perform a numerical calculation to compare the price of an agent who has exponential preferences and uses our method of optimal hedging against a delta hedger. The results show that the optimal hedging strategy is influenced by the early exercise boundary and that the worst case holder behavior for a sub-optimal hedger significantly deviates from the classical Black–Scholes exercise boundary.     

Pricing European-type,early-exercise and discrete barrier options using an algorithm for the convolution of Legendre series

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: http://www.chebfun.org/) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.     

Valuing American options by simulation: a simple least-squares approach

This article presents a simple yet powerful new approach forapproximating the value of American options by simulation. Thekey to this approach is the use of least squares to estimatethe conditional expected payoff to the optionholder from continuation.This makes this approach readily applicable in path-dependentand multifactor situations where traditional finite differencetechniques cannot be used. We illustrate this technique withseveral realistic examples including valuing an option whenthe underlying asset follows a jump-diffusion process and valuingan American swaption in a 20-factor string model of the termstructure.     

Free boundary and optimal stopping problems for American Asian options

We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.      

Organisational change and performance in long-lived small firms: a real options approach

This paper supports two key principles of real options reasoning: (a) the value of waiting and (b) the value of staging. It tests whether real options logic applies to small firms implementing significant changes (e.g. in technology) in a model of small firm performance, estimated on data collected by interviews with entrepreneurs. We found that to achieve a higher value by waiting, a delicate balance of precipitators of change against time until exercise is necessary (e.g. if there were just one or two precipitators, then waiting would certainly raise the value). Similarly, to achieve a higher value by staging, the entrepreneur needs to balance embedding against investment time. Thus, provided that investment time is less than 1¼ years, we found that embedding will raise the value. Overall, this implies that strategic flexibility in investment decisions is necessary for good long-run performance of small firms.     

Efficient willow tree method for European-style and American-style moving average barrier options pricing

Moving average options are widely traded in financial markets, but exiting methods for pricing this type of option are too slow. This paper proposes two efficient willow tree methods for pricing European-style and American-style moving average barrier options (MABOs). We first solve the finite-dimensional partial differential equation model for discretely monitored MABOs by willow tree methods, and then compute the value of continuously monitored MABOs by Richardson’s two-point extrapolation. Our new willow tree method employs the interpolation error minimization technique to reduce complexity. The corresponding convergence rate and error bounds are also analyzed. It shows that our proposed methods can provide the same accuracy as the binomial tree approach and Monte Carlo simulation, but require much less computing time. The numerical experiments support our claims.