# Björks second $S$ process when introducing martingale measures

When Björk presents the Black-Scholes model and martingale measures he starts off with a process modeling the stock price calling it $$S$$ with some given dynamics w.r.t some measure $$P$$.

Then he demonstrates that the price of any contract should satisfy the Black and Scholes PDE. Then he solves this PDE using Feynman-Kac then a new dynamic $$X$$ makes it's appearance.

Now for some reason he insist on relabeling this new process $$X$$ to $$S$$, what is the point of this relabeling? To me it only looks like he messes things up.

Everthing can be found around page $$103$$ in his third edition of Arbitrage theory in continuous time

I think everything is related to the concept of Risk Neutral measure $$\mathbb{Q}$$. In deriving Black- Scholes equation you use the dynamics $$\begin{equation} dS(t)=\mu S(t)dt + \sigma S dW(t) \end{equation}$$ where $$W$$ is a brownian motion under the $$\mathbb{P}$$ measure, and you get the following PDE for the price $$f$$ of a certain derivative: $$\begin{equation} rS\frac{\partial f}{\partial S} +\frac{\partial f}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial ^2 f}{\partial S^2} =rf \end{equation}$$ Now the Feynamn- Kac formula says that for a PDE of the form $$\begin{equation} \begin{cases} \frac{\partial f}{\partial t}(t,x) + \alpha(t,x)\frac{\partial f}{\partial x}(t,x) + \frac{1}{2}\beta^2(t,x)\frac{\partial^2 f}{\partial x^2}(t,x) = k(t)f(t,x) \\ f(T,x) = \Phi(x) \end{cases} \end{equation}$$ you have a solution that can be written as expectation with respect to a particular measure $$\mathbb{Q}$$ $$\begin{equation} f(t,x) = \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^Tk(s)ds}\Phi(X)\Big|X(t)=x\right] \end{equation}$$ where $$X$$ has SDE: $$\begin{equation} dX(t)=\alpha(t,X)dt + \beta(t,X)d\overline{W}(t) \end{equation}$$ where $$\overline{W}$$ is a $$\mathbb{Q}$$-brownian motion. Now, in order to match Black-Scholes equation to the general PDE that I wrote, you just put $$k(t)=r$$, $$\alpha(t,X)=rX$$,$$\beta(t,X)=\sigma X$$ and you rename $$X$$ by $$S$$. At this point you can forget the original specification of the dynamics of the stock and you procede by calculating the expectation to price your derivative with the new one that is: $$\begin{equation} dS=rSdt + \sigma Sd\overline{W}(t) \end{equation}$$ If you want to go deeper and investigate the relationship between these two dynamics you can observe that we can change from the original SDE to the new one by changing the brownian motion in this way: $$\begin{equation} dW(t)=d\overline{W}(t) - \left(\frac{\mu - r}{\sigma}\right)dt \end{equation}$$ Now the Girsanov theorem says that given this relationship between the two brownian motion the measure $$\mathbb{P}$$ and $$\mathbb{Q}$$ are equivalent.
• I wrote a general case of application of Feynman Kac formula so I called the variable by $X$. When you apply this to Black-Scholes you just put $k(t)=\mu(t,X)=r$ and you name $X$ with $S$. Just think of which condition you have to impose in order to match your Black- Scholes equation with the general PDE I wrote. – ab94 Sep 23 '19 at 20:24