Wiener Process A fund manager manages a fund of size S(t). The value of the fund

Question

Wiener Process

A fund manager manages a fund of size S(t). The value of the fund is assumed to satisfy the following stochastic process dS(t) = mu S(t) dt + sigma S(t) dW(t). where {W(t); t greaterthanorequalto 0} is a Wiener process and u the annual expected return) and sigma the annual it volatility are constants. The reward to the manager consists of two components, a fixed percent alpha of the total value of the fund S(t) plus a certain percentage beta of the fund value at the future time T in excess of some threshold value K. The total reward to the fund manager is therefore presented by the random variable R(t, T) = alpha s + beta max(s (T)- K, 0) What is the general expression for the expected value of the fund manager's reward? Find the numerical value for the expected total reward if alpha = 1.5%, beta = 3 7.5%, s(t) =$100m, K = $107.50 m mu = 12.5% sigma = 16.76% T = 1 and the risk free rate r = 3.5% What happens to the reward if the volatility of the fund increases from 16.75% to 25.00%?

Explanation / Answer

Standard (one-dimensional) Brownian motion starting at 0, also called the Wiener
process, is a stochastic process B(t, ) with the following properties:
(1) B(0, ) = 0 for every ;
(2) for every 0 t1 < t2 < t3 < · · · < tn, the increments
Bt2 Bt1
, Bt3 Bt2
, . . . , Btn Btn1
are independent random variables;
(3) for each 0 s < t < , the increment Bt Bs is a Gaussian random
variable with mean 0 and variance t s;
(4) the sample paths B : [0, ) R are continuous functions for every
.
The existence of Brownian motion is a non-trivial fact. The main issue is to
show that the Gaussian probability distributions, which imply that B(t+t)B(t)
is typically of the order
t, are consistent with the continuity of sample paths. We
will not give a proof here, or derive the properties of Brownian motion, but we will
describe some results which give an idea of how it behaves. For more information
on the rich mathematical theory of Brownian motion, see for example [15, 46].
The Gaussian assumption must, in fact, be satisfied by any process with inde-
pendent increments and continuous sample sample paths. This is a consequence of
the central limit theorem, because each increment
Bt Bs =
Xn
i=0

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