The Institute says, in its reply to CP 13/18 (p.10), that
Using the Black-Scholes formula in pricing NNEG will affect the cost of the guarantee, since allowance is not made for the features of mean reversion, momentum and jumps described above. Under geometric Brownian motion the volatility increases with the square root of time while for other models it does not; the value for long term derivatives such as NNEG could materially differ from that assumed under the Black-Scholes model.
This second article on strange distributions discusses the mean reversion claim.
The chart above shows (green line) an underlying asset that follows the price series 95, 96, 95, 96 … I.e. the series has a mean of 95.5 to which it continually reverts.
The blue line shows the price of a put option struck at 90, modelled by the standard option formula. The red line shows the value of a hedge, constructed from a derivative of the same formula (‘delta’). You see they end up in approximately the same place. In practice, the option trader would shade the input volatility in his or her favour, to avoid this kind of tracking error.
Yet the series is mean reverting. Nor is it drawn from anything that even resembles a normal distribution. It is true that under a geometric Brownian motion the volatility increases with the square root of time while for other models it does not, but this clearly doesn’t matter. The longer the sampling period for the strange distribution above, the lower the sampled volatility: the prices series is going exactly nowhere. Yet the standard option pricing works perfectly. The actuaries’ statement that ‘the value for long term derivatives such as NNEG could materially differ from that assumed under the Black-Scholes model’ is as false as you could possibly get.
The family of Black pricing models are among the most practical and robust models of reality that science possesses. What on earth have they done to attract the ire of the august Institute of Actuaries?