Riskneutral Valuation

In the binomial lattice frame work, pricing of options and other derivatives was expressed concisely as discounted risk-neutral valuation This concept works in the Ito process framework as well.

For the geometric Brownian motion stock price process dS{t) - pSdt +aSdz (13.17)

we know from Section 117 that

In a risk-neutral setting, the price of the stock at time zero is found from its price at time t by discounting the risk-neutral expected value at the risk-free rate This means that there should hold

It is clear that this formula would hold if H [5(/)] - 5(0)<?rf. From (1,3 17) and (1.3,.18) this will be the case if we define the process dS = rSdt +cxSdz (13 19)

where z is a standardized Wiener process, and we define E as expectation with respect to the z process In other words, starting with a lognormal Ito process with rate p., we obtain the equivalent risk-neutral process by constructing a similar process but having rate /.

This change of equation is analogous to having two binomial lattices for a stock process: a lattice for the real process and a lattice for the risk-neutral process, In the first lattice the probabilities of moving up or down are p and 1 — p, respectively The risk-neutral lattice has the same values as the stock prices on the nodes, but the probabilities of up and down are changed to q and 1 —q. For the Ito process we have two processes—like two lattices Because the probability structures are different, we use z and z to distinguish them.

Once the risk-neutral probability structure is defined, we can use risk-neutral valuation to value any security that is a derivative of 5. In particular, for a call option the pricing formula is

This is analogous to (12 13) in Chapter 12

We know that the risk-neutral distribution of 5(7) satisfying (13.19) is lognormal with E (ln[5(7 )/5(0)]} = >1 - \a2J and var {ln[5(/)/5(0)J} = a2 T We can use this distribution to find the indicated expected value in analytic form. The result will be identical to the value given by the Black-Scholes equation for a call option price. Specifically, writing out the details of the lognormal distribution, we have c = tc~f/ /°V " K)e-lx-lam-r7+al7/2il/i2ai7i±x (13.21)

^2tto2T J\nK

This is the Black-Scholes formula in integral form.

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