Showing papers by "Marti G. Subrahmanyam published in 1999"

When are Options Overpriced? The Black-Scholes Model and Alternative Characterisations of the Pricing Kernel

Abstract: This paper examines the convexity bias introduced by pricing interest rate swaps off the 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 contract and 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 that swaps 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 the swap and the futures markets. Using alternative term structure models (one-factor Vasicek, 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 mispricing by incorporation of a convexity adjustment in swap pricing over time.

When are Options Overpriced? The Black-Scholes Model and Alternative Characterisations of the Pricing Kernel

Abstract: An important determinant of option prices is the elasticity of the pricing kernel used to price all claims in the economy. In this paper, we first show that for a given forward price of the underlying asset, option prices are higher when the elasticity of the pricing kernel is declining than when it is constant. We then investigate the implications of the elasticity of the pricing kernel for the stochastic process followed by the underlying asset. Given that the underlying information process follows a geometric Brownian motion, we demonstrate that constant elasticity of the pricing kernel is equivalent to a Brownian motion for the forward price of the underlying asset, so that the Black-Scholes formula correctly prices options on the asset. In contrast, declining elasticity implies that the forward price process is no longer a Brownian motion: it has higher volatility and exhibits autocorrelation. In this case, the Black-Scholes formula underprices all options.

When are Options Overpriced? The Black-Scholes Model and Alternative Characterizations of the Pricing Kernel

Abstract: An important determinant of option prices is the elasticity of the pricing kernel used to price all claims in the economy. In this paper, we first show that for a given forward price of the underlying asset, option prices are higher when the elasticity of the pricing kernel is declining than when it is constant. We then investigate the implicaitons of the elasticity of the pricing kernel for the stochastic process followed by the underlying asset. Given that the underlying information process follows a geometric. Brownian motion, we demonstrate that constant elasticity of the pricing kernel is equivalent to a Brownian motion for the forward price of the underlying asset, so that the Black-Scholes formula correctly prices options on the asset. In contast, declining elasticiy implies that the forward price process is no longer a Brownian motion: It has higher volatility and exhibits autocorrelation. In this case, the Black-Scholes formula underprices all options.

Standard Risk Aversion and the Demand for Risky Assets in the Presence of Background Risk

Abstract: We consider the demand for state contingent claims in the presence of a zero-mean, non-hedgeable background risk An agent is defined to be generalized risk averse if he/she reacts to an increase in background risk by choosing a demand function for contingent claims with a smaller slope We show that the conditions for standard risk aversion: positive, declining absolute risk aversion and prudence are necessary and sufficient for generalized risk aversion We also derive a necessary and sufficient condition for the agent's derived risk aversion to increase with a simple increase in background risk

A Two-factor Lognormal Model of the Term Structure and the Valuation of American-Style Options on Bonds

Abstract: A Two-factor Lognormal Model of the Term Structure and the Valuation of American-Style Options on Bonds We build a no-arbitrage model of the term structure, using two stochastic factors, the shortterm interest rate and the premium of the forward rate over the short-term interest rate. The model extends the lognormal interest rate model of Black and Karasinski (1991) to two factors. It allows for mean reversion in the short rate and in the forward premium. The method is computationally e cient for several reasons. First, we use Libor futures prices, enabling us to satisfy the no-arbitrage condition without resorting to iterative methods. Second, the multivariate-binomial methodology of Ho, Stapleton and Subrahmanyam (1995) is extended so that a multiperiod 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 over time. Third, the problem of computing a large number of term structures is simpli ed 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 oors. We illustrate the use of the model by pricing American-style and Bermudan-style options on bonds. Option prices for realistic examples using forty time periods are shown to be computable in seconds. A two-factor lognormal model of the term structure 1

The Valuation of American-style Swaptions in a Two-factor Spot-Futures Model1

Abstract: We build a no-arbitrage model of the term structure of interest rates using two stochastic factors, the short-term interest rate and the premium of the futures rate over the short-term interest rate. The model provides and extension of the lognormal interest rate model of Black and Karasinski (1991) to two factors, both of which can exhibit mean-reversion. The method is computationally efficient for several reasons. First, the model is based on Libor futures prices, enabling us to satisfy the no-arbitrage condition without resorting to iterative methods. Second, we modify and implement the binomial approximation methodology of Nelson and Ramaswamy (1990) and Ho, Stapleton and Subrahmanyam (1995) to compute a multiperiod tree of rates with the no-arbitrage property. The method uses a recombining two-dimensional binomial lattice of interest rates that minimizes the number of states and term structures over time. In addition to these computational advantages, a key feature of the model is that it is consistent with the observed term structure of futures rates as well as the term structure of volatilities implied by the prices of interest rate caps and floors. These prices are shown to be highly sensitive to the existence of the second factor and its volatility characteristics.