Stock Price Dynamics under Risk-neutral Probabilities

The process of the stock can be derived from the stochastic process of the return. The cumulative return Y from 0 to t is normally distributed, and follows the normal distribution N(U, a2t). The stock price St is a function of the return: St = S0 exp(7(). The stochastic process for the return is:

Since the stock price is a function of the random Y and since we know the process for Y, Ito lemma can be used for deriving the process of S — S0 exp(Y). The Ito lemma states that when F= St is a function of another process Yt, the process of St is:

The term AY is the above process. The partial derivatives are:

After substitution:

At this point, it is possible to shift the mean from |^ to another value p provided that the diffusion term Az is replaced by a new diffusion term, Az*, because the probabilities change. The new equation is:

Using a value of p — r - a2/2, the drift rate collapses to the risk-free rate, and the stock process becomes:

The same shift of drift that allowed valuing the stock as the expectation of all its future values, using a growth term equal to the risk-free rate and discounting at the risk-free rate, transforms the drift of the stock price process into the risk-free rate.

The risk premium is eliminated and the random drift parameter is the risk-free plus the volatility term, both being known. The transformation of real-world probabilities into risk-neutral probabilities allows eliminating the unknown risk premium. Moreover, we know how to determine such probabilities. We have simply switched the mean to the lower value equal to the risk-free rate. Under risk-neutral probabilities the dynamics of the stock price have a drift equal to the risk-free rate.

The transformation can be performed for all assets, changing their return to the risk-free rate to model the asset process under risk-neutral probabilities. Note the analogy with the technique of forming risk-free portfolio with derivatives for valuing them. The resulting PDE does not depend on risky return, only on the risk-free rate. In fact, this is equivalent to assuming that the stock price grows at the risk-free rate.

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