Estimating Expected Value for Put Credit Spreads
Summary
The document examines how to estimate expiration probabilities and expected value for a two-leg put credit spread. It divides outcomes into three price regions: above the short strike, between the strikes, and below the long strike. The author asks whether short-option delta can stand in for the probability of finishing in the money, and whether a lognormal distribution can estimate the probability of landing between strikes.
The text presents these as open questions rather than resolving them. It highlights useful modeling distinctions: option delta is sometimes used as a probability proxy, while probability calculations from a price distribution require evaluating that distribution at the relevant strike prices. It gives no data, worked calculation, or comparison of models. Its proposed expected-value expression is therefore not validated here, and the discussion leaves out matters such as probability assumptions, discounting, and payoff treatment. Readers should treat it as a framing of the problem, not a complete pricing method.
Key ideas
- A put credit spread has distinct expiration payoff regions separated by its two strikes.
- The author asks whether short-put delta approximates the probability of expiring in the money.
- The probability of finishing between strikes requires a price-distribution calculation at both strikes.
- The document raises modeling questions but supplies no validated method or empirical evidence.
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Full text
# Expected value of option spreads
# Expected value of option spreads
I'm attempting to write a tool that will automatically calculate the expected value of arbitrary options positions, and I need to clarify my understanding. I am neither a statistician nor a professional quant so please permit me a bit of (perhaps unnecessary) exposition:
#### Scenario
Suppose I am interested in ticker $\\\$XYZ$ and I open a standard put credit spread by shorting a put at strike $K_{short}$ with delta $\Delta_{short}$ and buying a protective put with strike $K_{long}$ and delta $\Delta_{long}$ for a net credit/premium of $C$. Obviously, I can divide my calculation into 3 cases for ease of computation:
- At expiration the short leg expires out of the money with $\\\$XYZ$ closing above $K_{short}$.
- At expiration the short leg expires in the money with $\\\$XYZ$ closing between $K_{short}$ and $K_{long}$.
- At expiration the short and long legs expire in the money with $\\\$XYZ$ closing below $K_{long}$.
#### The question in 2 parts
Among retail traders it is commonly accepted that an option's Delta approximates the chance of that option expiring in the money. My questions are:
a. Is it valid to use delta in this case? I.e. for case 1 above, is it reasonable to calculate EV by:
$EV = \Delta_{short} \cdot (K_{short}-C) + (1-\Delta_{short}) \cdot K_{short}$
?
Or, given that Black Scholes is primarily applicable to European options, is delta not appropriate here? Would it be better to use a stochastic model and derive probabilities from something like monte carlo trials? Is there some other limitation of which I'm not aware?
b. For case 2 above, some of the sources I've read online suggest "using the lognormal distribution" but don't go into any further detail. I'm familiar with the log-normal distribution in the sense that I understand its basic properties, but it's not clear to me specifically how to calculate/estimate the probability that the price falls between the strikes. If the log-normal CDF was $F(x)$, is it as simple as $F(\Delta_{short}) - F(\Delta_{long})$ ?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.