Risk-Neutral and Actual Exercise Probabilities in Black–Scholes

Summary

The document compares put exercise probabilities under the Black–Scholes–Merton framework. It distinguishes a probability calculated using the stock’s expected return from the risk-neutral probability derived using the risk-free rate, and asks why the risk-neutral put probability is larger when the expected return exceeds the risk-free rate. It then raises the apparent opposite result for calls.

The accepted answer frames the difference through aggregate exposure to market risk: investors are collectively long equities and require compensation for bearing that risk. Long-market positions, including short puts, are compensated, while bearish positions such as short calls bear a cost. This gives an intuition for the pricing asymmetry, but the text does not derive the formulas, specify model assumptions in detail, or discuss empirical evidence. The probability comparison should be understood within the stated model setup and its assumptions about expected returns and risk pricing.

Key ideas

• Actual and risk-neutral exercise probabilities use different return assumptions.
• The document states that the risk-neutral put exercise probability exceeds the actual probability when expected return is above the risk-free rate.
• The answer attributes option pricing asymmetry to compensation for aggregate market risk exposure.
• Long-market positions are described as compensated, while bearish positions are described as penalized.
• The response gives intuition rather than a derivation or empirical test.

Tags

# BSM Model - Actual probability


# BSM Model - Actual probability












Actual probability of exercise of put option under BSM model is:

`PD = N(-d2(u))` (using expected return of stock, u)

Risk-neutral equivalent is

```
PD = N(-d2)
```

The latter is always bigger (assuming u > risk free rate), and thus value/price of the put option is greater than the actual expected payoff of put option. Author Allan Malz explains that this is so, because there's compensation to the put writer for taking on the risk.

What I don't get is that it's opposite for call option(underpriced rather than overpriced). Shouldn't the call writer be compensated for the risk like put writer? How come actual expected payoff of call is greater than the price of call?

## Answer by dm63 (score 0, accepted)

https://quant.stackexchange.com/a/25500

Any position that is long the market. Eg long stocks, short puts on stocks etc, is being compensated for taking risk. Any position that is bearish eg short the market, or short calls on the market, is being penalized for taking the risk. There's no contradiction. Investors overall are long stocks, and they need to get paid to take the risk. That's what drives the pricing. Within that there are some longs and shorts, but the overall pricing is determined by the net position (long) of investors.

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.