Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-15

August 2000

Some theorems on the price censor

R.O. Davies and A.J. Ostaszewski

Abstract

The optimality condition associated with hedging the purchase at some future date  T  of a non-resellable raw material requires finding a censor (or cap)  X  on the future price which satisfies the `censor equation' 

\int0X b q(b) db  +  \intX\infty X q(b) db  =  1,
where  q(b)  is the probability density function for the future price of the raw material. If for times  t  prior to  T  the price of the raw material  bt  follows a geometric Brownian motion, the distribution for  bt  at the time of the next purchase is log-normal and the quest for a censor transforms to finding the solution  W  of the equation
e-\mu  =   \Phi(W-\sigma) + e\sigma W-\sigma2/2 \Phi(-W),
where  \Phi  denotes the standard cumulative normal distribution function,  \mu  is the drift and  \sigma  the standard deviation per unit time (both assumed positive) of the price  bt.  We show that  W = W(\mu,\sigma)  is increasing with  \sigma  and decreasing with  \mu,  discuss the monotonicity of  W(t) = W(\mu t,\sigma t1/2)  and derive a number of asymptotic formulas for fixed  \mu  and  \sigma  small or large, e.g.  W(\mu,\sigma) = -\mu/\sigma + \sigma/2 + o(\sigma)  as  \sigma --> 0+  and  W(\mu,\sigma) = \sigma - m - {1+O(1)}/(\sigma-m)  as  \sigma --> \infty,  where  m = \Phi-1(1-e-\mu).  These formulas are used to derive the dependence of the expected profit on the waiting period  T,  drift and volatility.

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