Expected Exposure of a European Put Under the Real-World Measure
Summary
The document considers how to estimate the future discounted exposure of a European stock put under the real-world measure, including whether a Black–Scholes value evaluated at the expected future stock price is an adequate approximation. It explains that the expected option value is the expectation of the Black–Scholes value across future stock prices; generally, this differs from valuing the option at the expected stock price because the option value is nonlinear in the underlying price.
For the real-world measure, the proposed route is to integrate the future Black–Scholes value against the real-world stock-price distribution, potentially by numerical methods. The discussion contrasts this with using a single drifted stock-price path, which is deterministic rather than a stochastic simulation. It also notes that historical drift alone is rarely considered a sufficient model of the underlying. No worked example, numerical comparison, or closed-form exposure formula is provided, so the discussion offers conceptual guidance rather than a validated approximation.
Key ideas
- Expected option value is generally not equal to the option value evaluated at the expected stock price.
- The expected future Black–Scholes value depends on the distribution of future stock prices and the option’s nonlinear payoff value.
- Under the real-world measure, the expectation can be estimated by integrating over the real-world stock-price distribution.
- Drifting the stock price by its expected growth produces a single deterministic path, not a stochastic simulation.
- Historical drift alone may be an inadequate model for real-world exposure analysis.
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Full text
# Analytical formula for discounted exposure of a European Put on a stock in Real-World measure
# 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.
## Answer by Arshdeep (score 0)
https://quant.stackexchange.com/a/76182
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.
## Answer by Parag Biswas (score -3)
https://quant.stackexchange.com/a/81394
[Editing my answer to make it more precise]
Well, the closed-form solution of Black-Scholes equation is the analytical solution for pricing vanilla options.
For the future stock prices, you are simply trying to drift the prices forward in time. It is ok to model a stock price in that way. However, I do not understand why you wrote the expression as an expectation. You can simply try it like this: -
S(T_k) = S(T_0) * exp(mu * T_k)
These will be your simulated stock prices. Since, you are not using random numbers for simulation, you have a single, defined path when you are drifting the prices into the future. Thus, no need for calculating expected stock prices.
Is all above a good approximation in real-world measure? No. It is rare for users to model the underlying with just historical drift, even with frequent (monthly or weekly) updates to the parameter. We try to incorporate as much information available to model the underlying. The regression results should be able to point it out.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.