# Analytical formula for discounted exposure of a European Put on a stock in Real-World measure

Is there an analytical formula to approximate the discounted exposure for a European Put on a Stock in the Real-World measure? This is just an initial phase to be able to assess the accuracy of using Longstaff-Schwartz regression method, using a simple example. I would like to compare the regression results with analytical solution for discounted exposure for a European Put on a Stock.

Also, is it fine to calculate the expected stock price at future points as $$E[S(T_{k})] = S(T_{0}) * exp (\mu * T_{k})$$ where $$T_{k}$$ are future time points for $$k = 1, 2, .. , M$$, with $$\mu$$ being the real-world drift of the stock, and subsequently, use Black-Scholes analytical formula for valuing a put using $$S(T_{k})$$ calculated above - this is in order to calculate the approximate expected exposure at a future time point, $$t_{k}$$?

Thanks in advance for any insight into this.

• Please use MathJax for formatting in the future. Commented Jul 20, 2023 at 12:16
• How do you define discounted future exposure. Commented Jul 20, 2023 at 17:25
• Calculate exposure as conditional expected value at a future time point, t, and then discount it to today, say, t0. Commented Jul 21, 2023 at 9:18

## 1 Answer

The put price at any point in the future $$t$$, with stock price $$S(t)$$, is just $$BS(T-t,S(t),K)$$, i.e. the black-scholes price. In american options context the "continuation value" is always the black-scholes price. This is the value I would want the longstaff algorithm to be able to approximate.

The expected stock price at any point in the future $$t$$ is $$S(t)=S(0)*exp(mu+vol^2/2)$$.

To answer your comment, "can I use the expected stock price at a future time t to plug into BS to get the put price at this future time, t? Is it correct to do so?"

Short answer, no.

You need $$E(BS(T-t,S(t))$$ which is different than $$BS(T-t,E(S(t))$$.

The magnitude of this difference depends on how convex BS function is w.r.t S(t), moneyness, time to expiry, pretty much everything, so I don't think I would be comfortable with this approximation.

In the risk neutral measure the former expectation is a martingale so it's (obviously discounted) expectation equals the BS price today!

To get it in the real world measure you can integrate this BS price against the real world density maybe numerically, I'm not sure there's a closed form solution available.

• Thank you, yes, but to use $BS(T−t,S(t),K)$, I would need to simulate S(t). My goal is to have an approximation using just closed form solution - is it possible, especially, in the real-world-measure? My question was, can I use the expected stock price at a future time t to plug into BS to get the put price at this future time, t? Is it correct to do so? Again, this is only to have an approximation i.e., a guide to use as a benchmark. Commented Jul 21, 2023 at 9:16
• Stock price expectation: quant.stackexchange.com/questions/49197/… Commented Jul 21, 2023 at 9:36
• Please see the edited solution @Rhoyourway Commented Jul 21, 2023 at 9:47