Forward-neutral valuation relationships for options on zero coupon bonds

Abstract

This paper extends the literature on Risk-Neutral Valuation Relationships (RNVRs) to derive valuation formulae for options on zero coupon bonds when interest rates are stochastic. We develop Forward-Neutral Valuation Relationships (FNVRs) for the transformed-bounded random walk class. Our transformed-bounded random walk family of forward bond price processes implies that (i) the prices of the zero coupon bonds are bounded below at zero and above at one, and (ii) negative continuously compounded interest rates are ruled out. FNVRs are frameworks for option pricing, where the forward prices of the options are martingales independent of the market prices of risk. We illustrate the generality and flexibility of our approach with models that yield several new closed-form solutions for call and put options on discount bonds.

Keywords:

• Derivatives pricing
• Fixed income derivatives
• Risk neutral distributions
• Pricing kernel

Acknowledgements

I dedicate this paper to the memory of my husband António Câmara, Associate Professor of Finance and Watson Family Chair in Commodity and Financial Risk Management, Spears School of Business, Oklahoma State University, Stillwater, OK 74078. He would have liked to thank San-Lin Chung and the two anonymous referees for their excellent comments. Financial support from the Rackham Faculty Fellowship of the University of Michigan is gratefully acknowledged.

Notes

†Equation (1) was first derived by Jamshidian (1987) in a diffusion context, who introduced the T-forward measure. See also Geman et al. (1995) on the T-forward measure.

†The transformed-normal distribution and its probability density function are well known. See, for example, Camara (2003).

‡The four-parameter geometric random walk has a four-parameter log-normal distribution at the maturity of the option. In general, as pointed by Aitchison and Brown (1957, p. 17), the four-parameter log-normal variate X is confined to the range τ < x < β such that X′ = (X − τ)/(β − X) ∼ Λ(μ, σ²), i.e. ln(X′) ∼ N(μ, σ²). In our application of their model, we have the lower bound τ = 0, the upper bound β = 1, the location parameter μ = g(F _B (t, T, S)) + α, and the volatility σ = ν.

§The inverse sech random walk is constructed on the inverse secant hyperbolic function. The inverse hyperbolic function sech is defined for 0 < x ≤ 1.

†The probability measure is equivalent to the actual probability measure P on ℱ _T .

‡The distributions of the risk adjustment factor m _2,T /m _2,t and of the transformed forward bond price process g(F _B (T, T, S)) are only specified under the actual forward probability measure , and left unspecified under the actual probability measure P.

†The probability measure is equivalent to the probability measure on ℱ _T .

‡See Camara (2003, corollary).

†The volatility ν in models A and B is not the standard deviation of the logarithm of the forward bond price since the distributions of the forward bond price are not log-normal.

†It should be noted that this natural boundary is often ignored by other models because, given alternative reasonable parameter values, defects of this boundary usually turn out to be small.

†The conditional risk adjustment factor is obtained as a conditional expectation under the actual forward probability measure .

‡As defined in the paper, ℱ _Ft is the σ-algebra generated by the forward price process, for delivery at date T of an S-maturity zero coupon bond. Then ℱ _FT contains all the information about the shock to this specific forward bond price up to the expiration date of the option at date T.

†Otherwise, we would obtain equal to , which is a contradiction. Furthermore, as candidate for the forward-neutral location parameter leads to the violation of basic arbitrage bounds for option prices. For example, under such a candidate for the forward-neutral location parameter, the option price is not the asset price when K = 0.