# 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?

• A call is a Bullish instrument, a put is a Bearish instrument. The natural stock market drift (in the Real Measure) is Bullish. – noob2 Apr 19 '16 at 21:51
• Could you please add a reference for Allan Malz paper? thanks – sets Apr 20 '16 at 13:12
• @sets Allan Malz - Financial Risk Management Models, History, and Institutions. Section 6.7 Merton Model – Jay Na Apr 20 '16 at 15:53