General Restrictions on Prices of Financial Derivatives Written on Underlying Diffusions

Yaacov Z. Bergman

It is shown that in any diffusive one-factor model of the term structure, the prices of bonds and of term structure puts decrease as the short-term interest rate increases. However, these prices need not be monotone in the short-term rate, if that rate can experience jumps. An important comparative statics implication of the monotonicity resuly for diffusive models is that to a higher short-term interest rate corresponds a yield curve that lies uniformly above the curve that corresponds to a lower short-term rate. Furthermore, if the diffusion that describes the short-term rate is also homogeneous, then two yield curves that are measured at dfferent dates cannot intersect when drawn from the same time origin. If empirically they do intersect, then the short-term rate cannot be described by a one-factor homogeneous diffusion. It is also shown that if the second partial derivative w.r.t. to the short-term interest rate of the drift of the one-factor diffusion describing that rate is less than or equal to 2 - special cases being the linear drift models-then the prices of deterministic-coupon bonds and term structure puts are convex in that rate. The last result is derived using probabilistic representations of solutions to parabolic partial differential equations. The same methodology is used to derive restrictions on prices of European, American, and Asian options when the underlying price follows a stochastic volatility diffusion. Bounds, asymptotic results, and representations are derived for different linear differential transformations of derivative price functions like option`s delta, rho, and theta. An example from these results is the fact that the rho of a European call written on a stochastic volatility underlying asset is equal to the price of a digital call with the same exercise price, the same time to expiration, and the same underlying asset as the call, multiplied by the time to expiration and by the exercise price. The methodology is described in sufficient detail to allow for its ready application in a variety of situations.

January, 1998
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