How can this problem be defined formally?

Question

Let's consider a straightforward example in which I possess a portfolio consisting of two stocks:

$ R(t) = S_{1}(t) \cdot x_1 + S_{2}(t) \cdot x_2, $

Here, $t$ represents the time index, $R(t)$ symbolizes the portfolio's value in terms of dollars, $x_i$ indicates the weight of stock $i \in [1,2]$, and $[S_{1}(t),S_{2}(t)]$ denote two correlated stock prices, exhibiting the following dynamics:

$ \frac{dS_{1}(t)}{S_{1}(t)} = \mu_{1} dt + \sqrt{1-\rho^2} \sigma_{1} dW_{1}(t) + \rho \sigma_{1} dW_{2}(t), \\ \frac{dS_{2}(t)}{S_{2}(t)} = \mu_{2} dt + \sigma_{2} dW_{2}(t). $

Assuming a forecasting horizon $H > T$, I can employ numerical simulations to predict potential future portfolio values $R(h)$ for all $h \in [T,H]$.

Now, let's suppose that at time $T$, I'm privy to insider information suggesting an impending price drop for stock $S_2(h)$. Specifically, I have knowledge that the price $S_2(h)$ will undoubtedly be less than a constant $S_2(h) = K$ for a known time $h$ in the future. By excluding scenarios that don't meet this condition, I can calculate the expected portfolio value given this scenario. While I can perform numerical computations, defining this context formally remains a challenge.

Therefore, my inquiries are as follows:

1. How can I analytically define this scenario?
2. How can I formally define the stock price conditional to this scenario?
3. Would applying Girsanov's theorem and altering probabilities be necessary?
4. How can I define the portfolio value conditional to this scenario?
5. Could you recommend literature that delves into similar problems?

I welcome your input in refining my understanding.

Answer 1

The event that the second stock price is less than a constant $K$ at the time $h$ is mathematically described by $\{S_2(h)< K\}$.

The first stock price, at the time $t\in [T,H]$, given the event is $$S_{1}(t) \cdot \mathbf{1}_{\{S_2(h)< K\}}$$

where $\mathbf{1}_{\{ \}}$ is the indicator function.

The value of the portfolio, at the time $t\in [T,H]$, given the event is $$\left(S_{1}(t) \cdot x_1 + S_{2}(t) \cdot x_2\right)\cdot \mathbf{1}_{\{S_2(h)< K\}}$$

The expected value of the portfolio, at the time $s$ such that $T\le s \le h \le H$ and $T \le s \le t \le H$ , is then equal to $$\begin{align} V(s) &:=\mathbb{E}\left(\left(S_{1}(t) \cdot x_1 + S_{2}(t) \cdot x_2\right)\cdot \mathbf{1}_{\{S_2(h)< K\}}| \mathcal{F}_s\right) \\ &=x_1\mathbb{E}\left(S_{1}(t) \cdot \mathbf{1}_{\{S_2(h)< K\}}| \mathcal{F}_s\right) + x_2\mathbb{E}\left(S_{2}(t) \cdot \mathbf{1}_{\{S_2(h)< K\}}| \mathcal{F}_s\right) \tag{1} \end{align}$$

The second term of $(1)$ is relatively like a put on $S_2$, you can compute it easily for example by changing to the measure $S_2$-neutral.

The calculation of the first term of $(1)$ requires a double integral (it is possible to write the result as a 2-dimensional normal probability distribution). The calculation is a little cumbersome but not difficule, I let you to do it.