# How can this problem be defined formally?

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.

• Are you required to keep $S_2$ in the portfolio? Aug 29 at 13:53
• yes, I keep it in the portfolio anyway. Aug 29 at 14:02
• There is a name for this technique, where you posit a stochastic process, but then you remove all trajectories meeting a specified condition, resulting in another stochastic process, which is a kind of biased version of the first. Aug 29 at 16:23
• @nbbo2 do you know the name of such technique? Can you give me some references? Aug 30 at 7:54
• The only time I saw it mentioned was in the article Survival by Stephen J. Brown, William N. Goetzmann and Stephen A. Ross, The Journal of Finance, Jul 1995. It may just be called "conditioning" but that is a commonly used term with many other meanings. Aug 30 at 8:52

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.

• Thank you so much for your considerate reply. I apologize for any confusion – I have actually made an edit to my question. There was a typo: the scenario involves the second stock price being less than a constant, rather than the second stock price being less than the first stock price. Aug 31 at 21:07