Let phi be the normal density function (i.e. the Gaussian), and let F be the forward price, K the strike, V the volatility and T time to expiry. Define d1 and d2 as follows
d1 = (LN(F / K) + V ^ 2 * T / 2) / (V * SQRT(T))
d2 = (LN(F / K) - V ^ 2 * T / 2) / (V * SQRT(T))
The odd result is that
F phi(d1) = K phi(d2)
It's easily proved by substituting the standard exponential formula for the Gaussian. Then there is a load of cancellation and you find the ratio of phi(d1) to phi(d2) = exp(ln(K/F)) = K/F
Is there any textbook reference for this, or has it been noted by others?
Even I can't remember that one .. 4th episode of season 2. It was always on TV on Friday afternoons around 4 pm.It doesn't -- small joke -- very small. Neg. vol => mirror world, get it?Not with you I'm afraid. Where does Black Scholes feature in the old Star Trek?I think that was the basis of several old Star Trek episodes -- Mirror World
So long as it's not@comply,
If it was my paper, I would just say,
"familiar computations show F phi(d1) = K phi(d2)".
Take a call spread option PDE, for example. Taking [$]\sigma1 = -0.4, \sigma2 = -0.2[$] gives the same result as [$]\sigma1 = 0.4, \sigma2 = 0.2[$]. Not surprising I suppose because PDE only 'sees' [$]\sigma^2[$].Lots of unusual results have been derived. For example, if you put a negative vol into the BS call price formula you get the negative of the put price.