Stuart M. Turnbull

Pricing foreign currency options with stochastic volatility

Abstract This paper investigates the consequences of stochastic volatility for pricing spot foreign currency options. A diffusion model for exchange rates with stochastic volatility is proposed and estimated. The parameter estimates are then used to price foreign currency options and the predictions are compared to observed market prices. We find that allowing volatility to be stochastic results in a much better fit to the empirical distribution of the Canada-U.S. exchange rate, and that this improvement in fit results in more accurate predictions of observed option prices.

A Quick Algorithm for Pricing European Average Options

An algorithm is described that prices European average options. The algorithm is tested against Monte Carlo estimates and is shown to be accurate. The speed of the algorithm is comparable to the Black-Scholes algorithm. A closed-form solution is derived for European geometric average options.

The intersection of market and credit risk

Abstract Economic theory tells us that market and credit risks are intrinsically related to each other and not separable. We describe the two main approaches to pricing credit risky instruments: the structural approach and the reduced form approach. It is argued that the standard approaches to credit risk management – CreditMetrics, CreditRisk+ and KMV – are of limited value when applied to portfolios of interest rate sensitive instruments and in measuring market and credit risk. Empirically returns on high yield bonds have a higher correlation with equity index returns and a lower correlation with Treasury bond index returns than do low yield bonds. Also, macro economic variables appear to influence the aggregate rate of business failures. The CreditMetrics, CreditRisk+ and KMV methodologies cannot reproduce these empirical observations given their constant interest rate assumption. However, we can incorporate these empirical observations into the reduced form of Jarrow and Turnbull (1995b). Drawing the analogy. Risk 5, 63–70 model. Here default probabilities are correlated due to their dependence on common economic factors. Default risk and recovery rate uncertainty may not be the sole determinants of the credit spread. We show how to incorporate a convenience yield as one of the determinants of the credit spread. For credit risk management, the time horizon is typically one year or longer. This has two important implications, since the standard approximations do not apply over a one year horizon. First, we must use pricing models for risk management. Some practitioners have taken a different approach than academics in the pricing of credit risky bonds. In the event of default, a bond holder is legally entitled to accrued interest plus principal. We discuss the implications of this fact for pricing. Second, it is necessary to keep track of two probability measures: the martingale probability for pricing and the natural probability for value-at-risk. We discuss the benefits of keeping track of these two measures.

The Subprime Credit Crisis of 07

The current U.S. financial crisis began in 1997 when the market lost confidence in the creditworthiness of subprime mortgages and the structured financial products tied to them. In attempting to unravel the origins of the current problems, it has become clear that many factors and many market participants have played a role. In this article, the authors dissect the subprime mortgage crisis and clarify its many facets and offer a number of recommendations for changes to the system that could greatly reduce the probability of similar crises in the future.

Modeling the Loss Distribution

In this paper, we focus on modeling and predicting the loss distribution for credit risky assets such as bonds and loans. We model the probability of default and the recovery rate given default based on shared covariates. We develop a new class of default models that explicitly accounts for sector specific and regime dependent unobservable heterogeneity in firm characteristics. Based on the analysis of a large default and recovery data set over the horizon 1980--2008, we document that the specification of the default model has a major impact on the predicted loss distribution, whereas the specification of the recovery model is less important. In particular, we find evidence that industry factors and regime dynamics affect the performance of default models, implying that the appropriate choice of default models for loss prediction will depend on the credit cycle and on portfolio characteristics. Finally, we show that default probabilities and recovery rates predicted out of sample are negatively correlated and that the magnitude of the correlation varies with seniority class, industry, and credit cycle. This paper was accepted by Wei Xiong, finance.

The pricing of foreign currency options

This study examines the assumption that the exchange rate follows a log-normal probability distribution and it tests whether different stochastic specifications translate into important differences in implied option prices. The authors investigate a class of processes, which includes the log-normal probability distribution as a limiting case. None of the models perform particularly well. The main problem appears to be that the volatility estimates from actual exchange rate data are significantly smaller than those implied by observed option prices.

Additional Aspects of Rational Insurance Purchasing

The purchase of insurance provides an individual with the ability to shift risk associated with either timeless or temporal uncertain prospects (see Dreze and Modigliani 1972). Previous insurance decision analysis can be further divided into cases in which the insurance contract is exogenously specified (see Mossin 1968; Smith 1968) and cases in which the optimal form of contract is to be determined (see Raviv 1979; Brennan and Solanki 1981). This note is concerned with extending some of the results for timeless uncertain prospects with the insurance contract exogenously specified. Many of the properties associated with the purchase of insurance for timeless uncertain prospects are analyzed by Mossin (1968) for different forms of contracts (see also Gould 1969). Utilizing the Arrow-Pratt measure of absolute risk aversion, the effect of wealth upon the optimal amount of insurance is determined for the different forms of contracts. Recent work by Ross (1981) demonstrates that in situations where individuals face more than one (timeless) uncertain prospect and they can purchase insurance only for a subset of these prospects, then a more risk-averse individual in the Arrow-Pratt This paper examines the use of the ArrowPratt and Ross measures of risk aversion. It is shown that in situations where the individual faces more than one uncertain prospect and can purchase insurance only for a subset of these prospects. then in many cases the Arrow-Pratt and Ross measures of risk aversion are not sufficient to describe the behavior of the individual in purchasing insurance.

Pricing Credit Default Swaps with Observable Covariates

Observable covariates are useful for predicting default, but several studies question their value for explaining credit spreads. We introduce a discrete-time no-arbitrage model with observable covariates, which allows for a closed-form solution for the value of credit default swaps (CDS). The default intensity is a quadratic function of the covariates, specified such that it is always positive. The model yields economically plausible results in terms of fit, the economic impact of the covariates, and the prices of risk. Risk premiums are large and account for a smaller percentage of spreads for firms with lower credit quality. Macroeconomic and firm-specific information can explain most of the variation in CDS spreads over time and across firms, even with a parsimonious specification. These findings resolve the existing disconnect in the literature regarding the value of observable covariates for credit risk pricing and default prediction. The Author 2013. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com., Oxford University Press.

An Integrated Approach to the Hedging and Pricing of Eurodollar Derivatives

Taking the term structure of Treasury securities and Eurodollar rates as exogenous, this paper provides an integrated approach to the pricing and hedging of LIBOR derivatives. Our approach allows the spread between Eurodollar and Treasury rates to reflect both the credit risk in holding Eurodollar deposits and a convenience yield from holding Treasury securities. This integrated approach includes the models of Babbs [1991], Grinblatt [1994], and Jarrow and Turnbull [1995] as special cases.

Optimal Patenting and Licensing of Financial Innovations

Recent court decisions, starting with the State Street decision in 1998, allow business methods to be patentable and now give financial institutions the option to seek patent protection for financial innovations. This new patentability paradigm and the heterogeneity of characteristics associated with financial innovations pose an immediate decision problem for senior management: what to patent. We present a parsimonious decision framework that answers this question. We show that for innovations with certain characteristics, it is optimal not to patent, even if the option of patenting and licensing is available. Our model emphasizes the role of embedded real options that arise from certain types of financial innovations. The model provides an explanation of observed patenting behavior of financial institutions and the success of a wide class of innovations, including swaps, credit derivatives, and pricing algorithms.