Ann: Loyalty and Piggyback Options Program, page-450

No worries.

No the volatility of the underlying stock price still matters when valuing the piggyback option, even if the piggyback option is unlisted. The B&S option price I gave above is just the discounted expected payoff of the piggyback option, under the assumption that the underlying stock price S(t) follows a geometric Brownian motion with drift μ=r and volatility σ, where r is the "risk-free" rate.

E[max(S(T_PBO) - K_PBO, 0)] (the expected payoff of the piggyback option once the LO is exercised), will always increase if you increase the volatility σ, even though E[S(T_PBO)] doesn't change.

To get some intuition for why this is, imagine I gave you an option with a expiry date of T_PBO and a strike price of K_PBO=100, on a stock which had a 50% chance of being equal to 100 + σ and a 50% chance of being equal to 100 - σ, on the expiry date.

While the expected value (aka mean value) of the stock price S(T_PBO) minus the strike price K_PBO on the expiry date is always zero (as E[S(T_PBO) - K_PBO) = 0.5*(100 + σ - 100) + 0.5*(100 - σ - 100) = 0), this doesn't matter as we don't have to exercise the option when S(T_PBO) < K_PBO, and so the expected payoff of the option is given by E[max(S(T_PBO) - K_PBO, 0)] = 0.5*(payoff when S(T_PBO) > K_PBO)) + 0.5*(payoff when S(T_PBO) < K_PBO)) = 0.5*(100 + σ - 100) + 0.5*0 = 0.5*σ.

So you can see that as the "volatility" σ of the stock price increases, the value of the option also increases.

When using the Black-Scholes model, the math is a bit more complicated, but essentially the assumption is that the stock price T years in the future is given by the equation S(T) = S(0)*exp(rT)*exp(-σ^2/2)*exp(σW(T)), where S(0) is the current stock price, r is the risk-free rate, and W(T) is a Wiener process (continuous-time Brownian motion) with variance T.

It can be shown that even though E[S(T)] remains the same as σ increases, E[max(S(T) - K, 0)] increases as the volatility σ increases for any value of K > 0.

If you know how to use Python you can play around with the attached script[1] to gain some intuition for how the price of PAROA changes as various B&S model parameters change.

Also note that although the Black-Scholes model above gives E[S(T)] = S(0)*exp(rT), the discounted expected value of S(T) (when setting the discount rate equal to the risk-free rate r and discounting the future value of the stock price back to the present time t=0), is equal to the current stock price S(0), as E[exp(-rT)*S(T)] = exp(-rT)*S(0)*exp(rT) = S(0).

[1] paroa_monte_carlo_simulation.py.txt (remove the .txt extension)