Showing papers by "Marti G. Subrahmanyam published in 1998"

The Valuation of American Barrier Options using theDecomposition Technique

Abstract: In this paper, we propose an alternative approach for pricing and hedging non-standard American options. In principle, the proposed approach applies to any kind of American-style contract for which the payoff function has a Markovian representation in the state space. Specifically, we obtain an analytic solution for the value and hedge parameters of barrier options, an important example of path-dependent options. The solution includes standard American options as a special case. The analytic formula also allows us to identify and exploit two key properties of the optimal exercise boundary - homogeneity in price parameters and time-invariance - for American options. In addition, some new put-call ``symmetry" relations are also derived. These properties suggest a new, efficient and integrated approach to pricing and hedging a variety of standard and non-standard American options. From an implementation perspective, this approach avoids the current practice of repetitive computation of option prices and hedge ratios. Our implementation of the analytic formula for barrier options indicates that the proposed approach is both efficient and accurate in computing option values and option hedge parameters. In some cases, our method is substantially faster than existing numerical methods with equal accuracy. In particular, the method overcomes the difficulty that existing numerical methods have in dealing with prices close to the barrier, the case where the barrier matters most.

An Empirical Examination of the Convexity Bias in the Pricing of Interest Rate Swaps

Abstract: This paper examines the convexity bias introduced by pricing interest rate swaps offthe Eurocurrency futures curve and the market's adjustment of this bias in prices over time. The convexity bias arises because of the difference between a futures contractand a forward contract on interest rates, since the payoff to the latter is non-linear in interest rates. Using daily data from 1987-1996, the differences between market swap rates and the swap rates implied from Eurocurrency futures prices are studied for the four major interest rate swap markets - $, £, DM and ¥. The evidence suggests thatswaps were being priced off the futures curve (i.e. by ignoring the convexity adjustment) during the earlier years of the study, after which the market swap rates drifted below the rates implied by futures prices. The empirical analysis shows that this spread between the market and futures-implied swap rates cannot be explained by default risk differences, liquidity differences or information asymmetries between theswap and the futures markets. Using alternative term structure models (one-factorVasicek, Cox-Ingersoll and Ross, Hull and White, Black and Karasinski, and the two factor Heath, Jarrow and Morton), the theoretical value of the convexity bias is found to be related to the empirically observed swap-futures differential. We interpret these results as evidence of mispricing of swap contracts during the earlier years of the study, with a gradual elimination of that mis pricing by incorporation of a convexity adjustment in swap pricing over time.

Coupon Effects and the Pricing of Japanese Government Bonds An Empirical Analysis

Abstract: In many markets, the term structure of interest rates implied by coupon Treasury bonds provides a key input for pricing and hedging interest rate-sensitive securities. Previous studies in the Japanese market, however, suggest that the prices of the Japanese Government Bonds (JGB's) were significantly affected by regulatory and liquidity factors. Consequently, it has been argued that term structure modelling in the Japanese context based on interest rate factors could lead to misleading results. Since the previous studies, there have been significant structural changes in the regulatory environment, and in the liquidity of the Japanese bond market in the 1990's. In this light, we examine the effect of these changes on the JGB prices during the period between 1990 and 1996 by analyzing the term structure of interest rates in the JGB market over time. Specifically, we use the B-spline method to fit the term structure of interest rates using weekly prices of "non-benchmark" ten-year JGB's. We also use a non-linear econometric model to examine the significance of the "coupon" effects, which are the results of regulatory, accounting and liquidity factors. Our empirical analysis shows that it is possible to closely fit the term structure of interest rates in the JGB market, with fitted price errors only slightly larger than those found in similar studies of the U.S. Treasury bond market. Furthermore, the fitted price errors diminish over our sample period, suggesting that the effect of non-present value factors became somewhat muted over time. Our empirical results also indicate that the coupon of a bond in the JGB market has a highly nonlinear effect on the prices due to the "par-bond" effect and the "high-coupon" effect, although the "par-bond" effect is more pronounced in the recent period. Further analysis shows that three factors (level, slope and curvature) explain a substantial proportion of the variation in the JGB spot rates, as in the case of the U.S. Treasury market. Overall, these results indicate that the efficiency of the JGB markets has improved over time. Hence, the time-series movement of the JGB's can be captured to a substantial degree by common interest rate factors, although care should be taken to incorporate the special characteristics of individual bonds.

Who Buys and Who Sells Options: The Role and Pricing of Options in an Economy with Background Risk

Abstract: In this paper, we drive an equilibrium in which some investors buy call/put options on the market portfolio while others sell them. Also, some investors supply and others demand forward contracts. Since investors are assumed to have similar risk-averse preferences, the demand for these contracts is not explained by differences in the shape of utility functions. Rather, it is the degree tow which agents face other, non-hedgeable, background risks that determines their risk-taking behavior in the model. We show that investors with low or no background risk have a concave sharing rule, i.e., they sell options on the market portfolio, whereas investors with high background risk have a convex sharing rule and buy these options. A general increase in background risk in the economy reduces the forward price of the market portfolio. Furthermore, the prices of put options rise and the prices of call options fall. Investors without background risk then react by choosing a sharing rule with higher slope and concavity.

The risk of a currency swap: a multivariate‐binomial methodology

Abstract: In general, the risk of a financial instrument on a future valuation date depends on several stochastic variables. In the case of a currency swap, its value on a future date, can be modelled as a function of five stochastic variables. These represent the factors that determine the term structure of interest rates in the two currencies, and the foreign exchange rate between the currencies. The joint-probability distribution of the relevant variables on the horizon date is approximated by a multivariate-binomial distribution. The proposed methodology provides a fast and flexible alternative to Monte-Carlo simulation of the swap value. The distributions of value produced by the method can be employed to assist with both market and credit risk management.

Arbitrage Restrictions and Multi-Factor Models of the Term Structure of Interest Rates

Abstract: In this paper we investigate models of the term structure where the factors are interest rates. As an example, we derive a no-arbitrage model of the term structure in which any two futures (as opposed to forward) rates act as factors. The term structure shifts and tilts as the factor rates vary. The cross-sectional properties of the model derive from the solution of a two-dimensional autoregressive process for the short rate, which exhibits mean reversion and a lagged memory parameter. We show that the correlation of the factor rates is restricted by the no-arbitrage conditions of the model. Hence in a multiple-factor model it is not valid to independently choose both the mean reversion, volatility and correlation parameters, contrary to the approach of some models in the literature. The term-structure model, derived here, can be used to value options on bonds and swaps or to generate term structure scenarios for the risk management of portfolios of interest-rate derivatives.

An Arbitrage-free Two-factor Model of the Term Structure of Interest Rates: A Multivariate Binomial Approach

Abstract: We build a no-arbitrage model of the term structure, using two stochastic factors on each date, the short-term interest rate and the forward premium. The model is essentially an extension to two factors of the lognormal interest rate model of Black-Karazinski. It allows for mean reversion in the short rate and in the forward premium. The method is computationally efficient for several reasons. First, interest rates are defined on a bankers' discount basis, as linear functions of zero-coupon bond prices, enabling us to use the no-arbitrage condition to compute bond prices without resorting to cumbersome iterative methods. Second, the multivariate-binomial methodology of Ho-Stapleton-Subrahmanyam is extended so that a multi-period tree of rates with the no-arbitrage property can be constructed using analytical methods. The method uses a recombining two-dimensional binomial lattice of interest rates that minimizes the number of states and term structures. Third, the problem of computing a large number of term structures is simplified by using a limited number of bucket rates in each term structure scenario. In addition to these computational advantages, a key feature of the model is that it is consistent with the observed term structure of volatilities implied by the prices of interest rate caps and floors. We illustrate the use of the model by pricing American-style and Bermudan-style options on interest rates. Option prices for realistic examples using forty time periods are shown to be computable in seconds.

The Term Structure of Interest Rates: Alternative Approaches and Their Implications for the Valuation of Contingent Claims

Abstract: One of the most active areas of research in financial economics has been the modeling of the term structure of interest rates and its relationship to the pricing of contingent claims. There is a vast array of issues in the area, as well as a variety of perspectives, ranging from theoretical to practical. This paper provides a general framework for the analysis of issues in the modeling of the term structure. Specifically, this paper provides an overview of the conceptual issues and the empirical evidence in the area based on an examination of five seminal models by: Black-Scholes-Merton, Vasicek, Cox-Ingersoll-Ross, Ho- Lee and Heath-Jarrow-Morton. The paper provides a synthesis of the area and suggests directions for future research.

The Size of Background Risk and the Theory of Risk Bearing

Abstract: We establish a necessary and sufficient condition for the risk aversion of an agent's derived utility function to increase with independent, zero-mean background risk. This condition is weaker than standard risk aversion. For small risks, the condition is that the ratio of the third to the first derivative of the utility function is decreasing in income. In a market with state-contingent marketable claims, an increase in background risk, which raises the agent's derived risk aversion, reduces the slope of the agent's optimal sharing rule. Under a weak aggregation condition, an increase of background risk for many agents in the economy raises the prices of marketable claims in states with a low level of marketable aggregate income relative to the prices in states with a higher level of such income.

Why are Options Expensive

Abstract: Many valuation models in financial economics are developed using the pricing kernel approach to adjust for risk through the equivalent martingale representation. Often it is assumed, explicitly or implicitly, that the pricing kernel exhibits constant elasticity with respect to the price of the market portfolio. In a representative agent economy this would be close to assuming that the representative agent has constant proportional risk aversion. The elasticity of the pricing kernel has also implications for the pricing of options. This paper shows, first, that given the forward price of the market portfolio, all European options would have higher prices if the elasticity of the pricing kernel was declining instead of constant. Moreover, a volatility smile-effect is generated. Second, the paper shows that the standard geometric Brownian motion underlying the Black/Scholes model requires constant elasticity of the pricing kernel . Third, if the price of the market portfolio at the expiration date of an option is lognormally distributed, then declining elasticity of the pricing kernel implies a stochastic process which is characterized by higher volatility and negative autocorrelation. Thus, declining elasticity of the pricing kernel can explain several empirical findings.

Why are Options Expensive

Abstract: Many valuation models in financial economics are developed using the pricing kernel approach to adjust for risk through the equivalent martingale representation. Often it is assumed, explicitly or implicitly, that the pricing kernel exhibits constant elasticity with respect to the price of the market portfolio. In a representative agent economy this would be close to assuming that the representative agent has constant proportional risk aversion. The elasticity of the pricing kernel has also implications for the pricing of options. This paper shows, first, that given the forward price of the market portfolio, all European options would have higher prices if the elasticity of the pricing kernel was declining instead of constant. Moreover, a volatility smile-effect is generated. Second, the paper shows that the standard geometric Brownian motion underlying the Black/Scholes model requires constant elasticity of the pricing kernel . Third, if the price of the market portfolio at the expiration date of an option is lognormally distributed, then declining elasticity of the pricing kernel implies a stochastic process which is characterized by higher volatility and negative autocorrelation. Thus, declining elasticity of the pricing kernel can explain several empirical findings.