Risk Neutral versus True (Actual) Probability that an option expires in the money

Both the Cox-Ross-Rubinstein (discrete-time) and Black-Scholes-Merton (continuous-time) option pricing equations are based on risk neutral as opposed to “true” probabilities. The primary difference between the CRR and BSM equations is that CRR is based on binomial risk neutral probabilities, whereas BSM is based on normal risk neutral probabilities.  However, in the limit, CRR and BSM prices converge since CRR binomial probabilities converge into BSM normal probabilities.

Keeping this in mind, consider the BSM equation for the price of a call option:

C = SN\left( {{d_1}} \right) - K{e^{ - rT}}N({d_2}),

where {d_1} = \displaystyle\frac{{\ln \left( {S/K} \right) + \left( {r + .5{\sigma ^2}} \right)T}}{{\sigma \sqrt T }}, and {d_2} = {d_1} - \sigma \sqrt T = \displaystyle\frac{{\ln \left( {S/K} \right) + rT}}{{\sigma \sqrt T }}.

In this equation, N\left( {{d_1}} \right) corresponds to the option’s “delta”, whereas N\left( {{d_2}} \right) corresponds to the risk neutral probability that the option will expire in-the-money.

In order to determine the “true” probability of expiring in-the-money, all one has to do is replace the riskless rate of interest “r”  in the expression for d_2 with the continuously compounded expected return on the underlying asset for the option, which is {\mu - .5{\sigma ^2}}.  Keep in mind that the riskless rate of interest “r” corresponds to the expected return on the underlying asset in a risk neutral economy, whereas {\mu - .5{\sigma ^2}} corresponds to the continuously compounded expected return on the underlying asset in a risk averse economy.  Thus, the “true” probability that the call option will expire in the money is given by N\left( {\displaystyle\frac{{\ln \left( {S/K} \right) + \left( {\mu - .5{\sigma ^2}} \right)T}}{{\sigma \sqrt T }}} \right).

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