On the Convexity and Risk-Sensitivity of the Price of American Interest Rate Derivatives

We consider the form and sensitivity to risk of the price of perpetual American interest rate derivatives for a broad class of one-factor diffusion models of interest rates. We first present, in terms of the infinitesimal coefficients of the underlying interest rate dynamics, a set of usually satisfied conditions under which the value of the contingent claim is convex, at least on the set where exercising the contract is suboptimal. In line with previous parametrized models considering the valuation of perpetual interest rate derivatives, we find that given our general conditions, the convexity of the exercise payoff is preserved under rational valuation. Consequently, we are able to establish a set of typically satisfied conditions under which increased volatility unambiguously increases the price of the claim and postpones rational exercise by expanding the region where exercising the claim is suboptimal.

[1]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[2]  Klaus Sandmann,et al.  A Term Structure Model and the Pricing of Interest Rate Derivative , 1993 .

[3]  Luis H. R. Alvarez.  Does increased stochasticity speed up extinction? , 2001, Journal of mathematical biology.

[4]  Luis H. R. Alvarez.  Solving optimal stopping problems of linear diffusions by applying convolution approximations , 2001, Math. Methods Oper. Res..

[5]  A. Irturk,et al.  Term Structure of Interest Rates , 2006 .

[6]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[7]  Alan G. White,et al.  Pricing Interest-Rate-Derivative Securities , 1990 .

[8]  Robert C. Merton,et al.  An Asymptotic Theory of Growth Under Uncertainty , 1975 .

[9]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[10]  Eduardo S. Schwartz,et al.  A continuous time approach to the pricing of bonds , 1979 .

[11]  Alan G. White,et al.  One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities , 1993, Journal of Financial and Quantitative Analysis.

[12]  T. Björk.  Interest rate theory , 1997 .

[13]  S. Taylor.  DIFFUSION PROCESSES AND THEIR SAMPLE PATHS , 1967 .

[14]  S. Shreve,et al.  Robustness of the Black and Scholes Formula , 1998 .

[15]  Georges Courtadon.  The Pricing of Options on Default-Free Bonds , 1982, Journal of Financial and Quantitative Analysis.

[16]  Eduardo S. Schwartz,et al.  Investment Under Uncertainty. , 1994 .

[17]  D. Heath,et al.  Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation , 1990, Journal of Financial and Quantitative Analysis.

[18]  T. Björk.  Arbitrage Theory in Continuous Time , 2019 .

[19]  F. Black,et al.  A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options , 1990 .

[20]  Stephen A. Ross,et al.  The Term Structure of Interest Rates , 1985 .

[21]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[22]  Stephen A. Ross,et al.  An Analysis of Variable Rate Loan Contracts , 1980 .

[23]  Stephen A. Ross,et al.  Waiting to Invest: Investment and Uncertainty , 1992 .

[24]  Luis H. R. Alvarez,et al.  Wicksellian Theory of Forest Rotation Under Interest Rate Variability , 2001, SSRN Electronic Journal.

[25]  Wolfgang M. Schmidt,et al.  On a general class of one-factor models for the term structure of interest rates , 1996, Finance Stochastics.

[26]  L. Alvarez.  On the form and risk-sensitivity of zero coupon bonds for a class of interest rate models , 2001 .

[27]  F. Black,et al.  Bond and Option Pricing when Short Rates are Lognormal , 1991 .

[28]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[29]  D. Duffie,et al.  A YIELD-FACTOR MODEL OF INTEREST RATES , 1996 .