SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins
Evaluating the Expected Value of Functions
by Means of the Characteristic Function
Let p(z) be the probability distribution function of the variable z. The
expected value of a function of z, say g(z) is defined as:
E{g} = ∫-∞∞g(z)p(z)dz
Sometimes the probability distribution p(z) is not known but its
characteristic function is known. The probabililty distribution p(z)can be
found from its characteristic function Φ(ω) by means of the
inverse formula:
p(z) = (1/2π)∫-∞∞exp(-iωz)φ(ω)dω
Instead of carrying out the inversion of the characteristic function and
using the result to compute the expected value consider the substitution
of the inversion formula into the expression for the expected value; i.e.,
E{g} = ∫-∞∞g(z)p(z)dz
= (1/2π)∫-∞∞∫-∞∞g(z)exp(-iωz)φ(ω)dωdz
Reversing the integration operation gives:
E{g} = ∫-∞∞Γ(ω)φ(ω)dω
where
Γ(ω) = (1/2π)∫-∞∞g(z)exp(-iωz)dz
Thus the expected value of g can be computed as the integral of the
product of the
characteristic function of the probability distribution and the characteristic
function of the function g.
Application
Consider a call option, the right to purchase a security at a specified
price, the exercise price, within a specified period of time. Let S be the
market price of the security and X the exercise price. At the time of
expiration of the call option the value C0 is equal to
u(S-X) where the function u(z) is equal to zero if z<0 and z if z≥0.
The value of the call option when there is t time before expiration is:
Ct(S,X,r) = e-rtC0(S0-X)
where S0 is the security price on the expiration day of the
call option and r is the risk-free interest rate.
The expected value of the option with t time until expiration is then:
E{Ct(St,X,r)} = e-rtE{C0(S0-X)}
The distribution of S0 will depend upon the value of the price
t time periods before. The relationship is
0 = St(1+z)
where z is proportional change in price and is a random variable with
distribution p(z).
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