# Derivative pricing under $\mathbb{P}$

I recently learnt about the Girsanov-Cameron-Martin theorem, which basically says, that if $$(\tilde{B}(t),t\in[0,T])$$ is some Brownian motion with a (possibly stochastic) drift $$\theta(t)$$ defined on $$(\Omega,\mathcal{F},\mathbb{P})$$, then we can construct a new probability measure $$\mathbb{\tilde{P}}$$, that is, assign new weights to the sample paths $$\tilde{B}(\cdot,\omega)$$) by mutiplication with the factor:

$$M(T) = \exp\left[-\int_{0}^{T}\theta(t)d\tilde{B}(t)-\frac{1}{2}\int_{0}^{T}\theta^2(t)dt\right]$$

so that:

$$\mathbb{\tilde{P}}(\mathcal{E})=\mathbb{\tilde{E}}[1_\mathcal{E}]=\mathbb{E}[M(T) 1_\mathcal{E}]$$

for any event $$\mathcal{E} \in \mathcal{F}$$.

The result is that, $$\tilde{B}(t)$$ is a standard brownian motion under $$\mathbb{\tilde{P}}$$.

Moreover, we can write $$d\tilde{B}(t) = dB(t) + \theta(t)dt$$. The expectation under $$\mathbb{\tilde{P}}$$ could be written as:

$$\mathbb{\tilde{E}}[V(T)] = \int_{\Omega}V(T)d\mathbb{\tilde{P}}=\int_{\Omega}V(T)\left(\frac{d\mathbb{\tilde{P}(\omega)}}{d\mathbb{P}(\omega)}\right)d\mathbb{P}(\omega)=\int_{\Omega}V(T)M(T)d\mathbb{P}(\omega)=\mathbb{E}[M(T)V(T)]$$

That means, the risk-neutral pricing formula becomes:

\begin{align} V(0) &= \mathbb{\tilde{E}}\left[\exp\left(-\int_{0}^{T}r(t)dt\right)V(T)\right]\\ &= \mathbb{E}\left[\exp\left(-\int_{0}^{T}r(t)dt\right)M(T)V(T)\right]\\ &= \mathbb{E}\left[\exp\left(-\int_{0}^{T}(r(t)+\frac{1}{2}\theta^2(t))dt-\int_{0}^T \theta(t)dB(t)\right)V(T)\right] \end{align}

Is the above pricing formula correct?

Edit: $$\theta(t)$$ represents the market price of holding the risky asset. For a stock, this is $$\theta(t)=(\mu(t)-r(t))/\sigma(t)$$, $$\mu(t)$$ is the expected return on the stock.

• The $M(T)$ in your first formula should be an exponential. Nov 2, 2023 at 8:29
• Corrected the typo. Nov 2, 2023 at 19:16

Assuming that $$V(t)$$ is the price process of some (perhaps implicitly) traded claim, it is correct. A typical interpretation would be the following: Given that the $$T$$-claim $$\mathcal{X}$$ is replicable in the market $$(S^0, S^1)$$ (with $$S^0$$ locally riskless, since you mentioned one risky asset), by some self-financing portfolio $$h = (h^0, h^1)$$, it has a corresponding price process, given by $$\Pi_t(\mathcal{X}) = V^h_t := h^0_t S^0_t + h^1_t S^1_t.$$ Then indeed your formula would hold for $$V(T) = V^h_T = \mathcal{X}$$, where the last equality is by replication. The reason for this is that $$V^h_t / S^0_t$$ is a martingale under $$\tilde{\mathbb{P}}$$, like $$S^1_t / S^0_t$$.