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I don't understand how to express the stock dynamics in the stock numéraire

I have $dS_t/S_t = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs

$$S_T f(S_T)$$

I don't get how to get the stock distribution with the new numéraire

Thanks a lot

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1 Answer 1

Let $\text{d}S_t = \mu S_t \text{d}t +\sigma S_t\text{d}W_t$. under the real-world measure

With $S_t$ being numeraire, then $e^{rt}/S_t$ must be a martingale under the equivalent martingale measure.

Under the real world measure, $\frac{e^{rt}}{S_t}= \exp(rt -\mu t-\sigma W_t+\frac{1}{2}\sigma^2t)$, where $W_t$ is a Brownian motion under this measure.

Now you need to make a Cameron-Martin-Girsanov transform to make $\frac{e^{rt}}{S_t}$ a martignale. This essential comes down to $r-\mu+\frac{1}{2}\sigma^2 = -\frac{1}{2}\sigma^2$, or $\mu = r+\sigma^2$.

so under the risk-neutral measure with $S_t$ being numeraire, $S_t=S_0\exp(r+\sigma^2t-\frac{1}{2}\sigma B_t)$, where $B_t$ is a Brownian motion under the risk-neutral measure.. To find time $t<T$ value $V_t$ of an asset with pay off $S_TF(S_T)$, then

$\frac{V_t}{S_t} = \mathbb E[\frac{S_TF(S_T)}{S_T}|\mathcal{F}_t]=E[F(S_T)|\mathcal{F}_t]$

Note, for example, if $F(\cdot) = (K-\cdot)^+$, you can still use Black-Scholes formula though you need to figure out the appropriate parameter and might need to multiply by a factor. Essentially, this is because $S_t$ is still log-normal distributed.

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