# 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. 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 Apr 20 '16 at 15:53

## 1 Answer

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.

• Thanks for the response but I still don't get why the model ends up implying compensation only for 'long the market' positions? Apr 20 '16 at 15:55
• The model is in the Riskless Measure, so it doesn't say that. But it is true in the Real World Measure (if you believe that stock market exposure is compensated positively in the long run). Apr 20 '16 at 18:49
• I agree. This is nothing to do with the model for options. it is a property of positions in the underlying stocks.
– dm63
Apr 21 '16 at 2:21