Phelim P. Boyle

OPTIONS: A MONTE CARLO APPROACH

This paper develops a Monte Carlo simulation method for solving option valuation problems. The method simulates the process generating the returns on the underlying asset and invokes the risk neutrality assumption to derive the value of the option. Techniques for improving the efficiency of the method are introduced. Some numerical examples are given to illustrate the procedure and additional applications are suggested.

Monte Carlo methods for security pricing

The Monte Carlo approach has proved to be a valuable and flexible computational tool in modern finance. This paper discusses some of the recent applications of the Monte Carlo method to security pricing problems, with emphasis on improvements in efficiency. We first review some variance reduction methods that have proved useful in finance. Then we describe the use of deterministic low-discrepancy sequences, also known as quasi-Monte Carlo methods, for the valuation of complex derivative securities. We summarize some recent applications of the Monte Carlo method to the estimation of partial derivatives or risk sensitivities and to the valuation of American options. We conclude by mentioning other applications.

A Lattice Framework for Option Pricing with Two State Variables

A procedure is developed for the valuation of options when there are two underlying state variables. The approach involves an extension of the lattice binomial approach developed by Cox, Ross, and Rubinstein to value options on a single asset. Details are given on how the jump probabilities and jump amplitudes may be obtained when there are two state variables. This procedure can be used to price any contingent claim whose payoff is a piece-wise linear function of two underlying state variables, provided these two variables have a bivariate lognormal distribution. The accuracy of the method is illustrated by valuing options on the maximum and minimum of two assets and comparing the results for cases in which an exact solution has been obtained for European options. One advantage of the lattice approach is that it handles the early exercise feature of American options. In addition, it should be possible to use this approach to value a number of financial instruments that have been created in recent years.

Discretely adjusted option hedges

Abstract This paper analyses the distribution of returns on a hedged portfolio, consisting of a European call option and its associated stock, when the portfolio is rebalanced at discrete time intervals. Under the assumptions of the Black-Scholes model this distribution is particularly skew. In tests of the average return on a hedged portfolio this skewness leads to biased t -statistics. The paper explores the nature and extent of this bias and suggests procedures for overcoming it. Other aspects of discrete hedging are also discussed.

Pricing Lookback and Barrier Options under the CEV Process

This paper examines the pricing of lookback and barrier options when the underlying asset follows the constant elasticity of variance (CEV) process. We construct a trinomial method to approximate the CEV process and use it to price lookback and barrier options. For look-back options, we find that the technique proposed by Babbs for the lognormal case can be modified to value lookbacks when the asset price follows the CEV process. We demonstrate the accuracy of our approach for different parameter values of the CEV process. We find that the prices of barrier and lookback options for the CEV process deviate significantly from those for the lognormal process. For standard options, the corresponding differences between the CEV and Black-Scholes models are relatively small. Our results show that it is much more important to have the correct model specification for options that depend on extrema than for standard options.

Guaranteed Annuity Options

Under a guaranteed annuity option, an insurer guarantees to convert a policyholders accumulated funds to a life annuity at a fixed rate when the policy matures. If the annuity rates provided under the guarantee are more beneficial to the policyholder than the prevailing rates in the market the insurer has to make up the difference. Such guarantees are common in many US tax sheltered insurance products. These guarantees were popular in UK retirement savings contracts issued in the 1970s and 1980s when long-term interest rates were high. At that time, the options were very far out of the money and insurance companies apparently assumed that interest rates would remain high and thus that the guarantees would never become active. In the 1990s, as long-term interest rates began to fall, the value of these guarantees rose. Because of the way the guarantee was written, two other factors influenced the cost of these guarantees. First, strong stock market performance meant that the amounts to which the guarantee applied increased significantly. Second, the mortality assumption implicit in the guarantee did not anticipate the improvement in mortality which actually occurred.The emerging liabilities under these guarantees threatened the solvency of some companies and led to the closure of Equitable Life (UK) to new business. In this paper we explore the pricing and risk management of these guarantees.

Fertility trends, excess mortality, and the Great Irish Famine

This paper has developed estimates of the age-specific mortality rates prevailing during the Great Irish Famine and has analyzed fertility trends during the 25 years before the Famine. Our calculations confirm that 1 million Irish people perished as a result of this disaster. This figure does not include the deaths among the 1.3 million emigrants who left Ireland during the Famine period. The Famine produced a significant drop in the fertility rate, and we estimate that more than 300,000 births did not take place as a result of the Famine. The effects were especially severe on the very young and the very old, a result echoed in the findings of demographic analyses of other famines. Our procedure permits a reconstruction of the Irish population by age and sex during the period 1821-1841. In addition, it yields year-by-year estimates of the birth rate over this period. We estimate that the rate fell by about 14 percent, a result robust to our assumptions regarding emigration. Economic historians have debated this issue, and we hope that our evidence, although preliminary, will be of assistance. Our analysis also permits year-by-year reconstruction of Irish population totals for the period 1821-1851. Two years are of particular interest. Virtually all recent writers, with the notable exception of Lee (1981), have suggested that the 1831 census returns overestimated the actual population resident in Ireland at that date. Our reconstruction supports the validity of the 1831 census figure. We obtain a total of 7,847,000, which is in good agreement with the disputed census figure of 7,767,000. But perhaps the most interesting figure is the population total for the end of 1845, the highest ever achieved in Ireland. We estimate that the population on the eve of the Great Famine was 8,525,000. Throughout the paper we have tried to highlight those areas in which the data are unreliable, unavailable, or distorted. We have tried to devise cross-checks for consistency and to test the sensitivity of the results to a range of assumptions. A case in point concerns the age-sex profile and volume of emigration to England, Scotland, and Wales. Additional work at the micro level would be helpful here. More solid evidence on Famine births would also be helpful. The parish registers we have sampled certainly provide a clue to trends, but we have only made a start in that respect. A much more comprehensive survey is needed to convey the national picture.(ABSTRACT TRUNCATED AT 400 WORDS)

Keynes Meets Markowitz: The Trade-Off Between Familiarity and Diversification

We develop a model of portfolio choice to nest the views of Keynes, who advocates concentration in a few familiar assets, and Markowitz, who advocates diversification. We use the concepts of ambiguity and ambiguity aversion to formalize the idea of an investors “familiarity” toward assets. The model shows that for any given level of expected returns, the optimal portfolio depends on two quantities: relative ambiguity across assets and the standard deviation of the expected return estimate for each asset. If both quantities are low, then the optimal portfolio consists of a mix of familiar and unfamiliar assets; moreover, an increase in correlation between assets causes an investor to increase concentration in familiar assets (flight to familiarity). Alternatively, if both quantities are high, then the optimal portfolio contains only the familiar asset(s), as Keynes would have advocated. In the extreme case in which both quantities are very high, no risky asset is held (nonparticipation). This paper was accepted by Brad Barber, Teck Ho, and Terrance Odean, special issue editors.

The impact of variance estimation in option valuation models

Abstract This paper examines some implications of using an estimate of the variance in option valuation models. This procedure produces biased option values. It is shown that the magnitude of this bias is not large. The dispersion induced in the option price is more significant particularly for parameter values of practical interest. The nature and extent of this dispersion is examined by numerical examples. The paper suggests how a Bayesian approach could be used to cope with the estimation error.

Bounds on contingent claims based on several assets

Abstract In 1987, Lo derived an upper bound on the price of a European call option on a single asset. Los bound depends only on the mean and variance of the terminal asset price and is termed a semi-parametric bound. This paper derives similar semi-parametric bounds on a European call on the maximum of any number of assets. A distribution-free bound for the price of this option is obtained. The bound depends only on the means and covariance matrix of the returns on n underlying assets. The bound is obtained by optimizing over the entries of a positive definite matrix A . This can be accomplished by a technique known as semidefinite programming. We demonstrate the methodology using two specific applications. The first concerns the valuation of a European call option on the maximum of several assets. This is known as an outperformance option and is of some practical interest. The second application concerns the valuation of a discretely adjusted lookback option. These lookback options are of interest in connection with certain equity annuity insurance products.