Why the Black–Scholes Pricing Process Is Relabeled Under the Risk-Neutral Measure
Summary
The document explains why a process introduced as the stock price under the real-world measure may later be written with a new name and then relabeled as the stock price in a Feynman–Kac representation. In the Black–Scholes PDE, the drift appearing in the pricing equation is the risk-free rate. Matching that PDE to Feynman–Kac gives a process with that drift under a pricing measure, which is used to express the derivative value as a discounted expectation.
The answer distinguishes this pricing process from the stock’s original dynamics under the physical measure and sketches their connection through a Brownian-motion change associated with Girsanov’s theorem. The relabeling is therefore a notational convenience: it emphasizes that the expectation is taken under risk-neutral dynamics. The explanation is limited to the standard Black–Scholes setting and its stated assumptions; the brief claim about equivalent measures does not spell out the theorem’s technical conditions.
Key ideas
- Feynman–Kac represents the solution to the Black–Scholes pricing PDE as a discounted expectation under a suitable measure.
- The process in that expectation has risk-free-rate drift under the risk-neutral measure.
- Relabeling the auxiliary process as the stock price signals that it follows pricing-measure dynamics.
- Girsanov’s theorem relates the physical and risk-neutral Brownian motions under appropriate conditions.
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Full text
# Björks second $S$ process when introducing martingale measures
# 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
## Answer by ab94 (score 1, accepted)
https://quant.stackexchange.com/a/48860
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.