By S.Lin.
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So far, we have assumed that the riskfree rate is a constant and nonrandom throughtout the entire period 0 T ]: More realistically, the interest rate should be assumed to depend on the information available up to a current point in time. Mathematically, this is equivalent to assuming that the interest rate process rt t = 1 T is a predictable process(a stochastic process X (t) is said to be predictable if X (t + 1) is a stochastic process). 14) for n = 1 N . 13). 3 Valuation In this section, we assume that the market is complete.
W (t) is normal with mean t and variance 2 t. 9) Hence W (s) ; W (t) and W (t) are independent since two normal random variables are independent if and only if their covariance is zero. Thus, W (t) is a Wiener process. Many useful martingales related to W (t) can then be derived from Z (t). Noting the fact that the derivative of a martingale is still a martingale, if it exists, W (t) ; t = @Z@ (t) j =0 is a martingale. is also a martingale. 2 2 W (t) ; t ; 2 t = @ @Z 2(t) j =0 Finally, we state without a proof that 4.
3) It can also be shown that the distribution of a random variable is uniquely determined by its characteristic function. e. 2 that the price of a risky security can be expressed in terms of a random walk. In that case, the price S (t) at time t is S (t) = S (0)eX (t) 0 t T 53 where X (t) is a random walk with length of step , average mean , and average variance 2. Imagine that trading becomes more and more frequent and eventually continuous trading is achieved. This is the case when ! 0. Thus, if X (t) approaches a continuous-time stochastic process, say W (t), the price of the security will be expressed as S (t) = S (0)eW (t) : Obviously, the limiting stochastic process W (t) will inherit the properties that the random walk X (t) possesses.