Of course, now we know that the proper discount rate to apply is the return on the (dynamic) portfolio of the stock and the bond that a) maximizes the expected growth rate and, equivalently, b) makes the discounted stock and bond prices martingales under P.

To frame an option pricing model requiring (or, more practically, worked in reverse, implying) a risky discount rate requires imposing some restrictions on utility preferences (as per Rubinstein, Ross, and Carr) or an asset pricing model (e.g. CAPM, APT). I took the latter approach in the late 1980’s, as I was suspicious of aggregating utility from the small to the large.If memory serves me right, he was applying a pretty standard discounted expected cash flow argument to Samuelson’s lognormal stock price model to find the present value of the option. That gives you the right general shape of the resulting valuation formula, but with a risk adjusted discount rate that is hard to determine.

An interesting practical problem is that a formulation with risky discount rates introduces a second unobservable into the valuation equation, which complicates things. Joint estimation of implied volatility and expected return from the same data is possible but messy. I used a 2-step procedure – first imply Black-world vols, and then use that vol in the risky discount rate option model to back out expected return from option price.

A related question if you would humour me – within the assumptions of the model, is the usual risk-neutral Black Scholes implied volatility an unbiased estimate of real world vol, or is somehow affected by the risk-neutral valuation? I vote for the former.

Within the assumptions of the model -- which include the ability to make instantaneous, costless trades -- Black-Scholes implied volatility is an unbiased estimate of real world volatility.A related question if you would humour me – within the assumptions of the model, is the usual risk-neutral Black Scholes implied volatility an unbiased estimate of real world vol, or is somehow affected by the risk-neutral valuation? I vote for the former.

This is true, but in a slightly trivial sense: within the assumptions of the model, real world volatility is a constant and the model implied volatility will equal this constant. Furthermore, observing a time series of prices over an arbitrarily short time period, if sampled with sufficient frequency, will reveal this volatility.

It's not clear to me that such a two-parameter model exists in continuous-time, rendering the arguments about the vol suspect. What exactly are the assumptions of the model?To frame an option pricing model requiring (or, more practically, worked in reverse, implying) a risky discount rate requires imposing some restrictions on utility preferences (as per Rubinstein, Ross, and Carr) or an asset pricing model (e.g. CAPM, APT). I took the latter approach in the late 1980’s, as I was suspicious of aggregating utility from the small to the large.If memory serves me right, he was applying a pretty standard discounted expected cash flow argument to Samuelson’s lognormal stock price model to find the present value of the option. That gives you the right general shape of the resulting valuation formula, but with a risk adjusted discount rate that is hard to determine.

An interesting practical problem is that a formulation with risky discount rates introduces a second unobservable into the valuation equation, which complicates things. Joint estimation of implied volatility and expected return from the same data is possible but messy. I used a 2-step procedure – first imply Black-world vols, and then use that vol in the risky discount rate option model to back out expected return from option price.

A related question if you would humour me – within the assumptions of the model, is the usual risk-neutral Black Scholes implied volatility an unbiased estimate of real world vol, or is somehow affected by the risk-neutral valuation? I vote for the former.