[944] Error in my `quick solution' of the Q2 in the 2016 final

[Gautam Iyer]

Hi All,

I tried the "cheap trick" today to solve Q2 on last years final, by
saying that tilde(S) is a geometric Brownian motion (with some other
volatility and mean return rate). Now I use Black-Scholes formula for
this, and compute the price.

The algebra was OK, but the logic wasn't correct: The reason for this is
that in order for me to apply the Black-Scholes formula here, I need to
be able to construct a replicating portfolio that trades both tilde(S)
and cash. I can't do this! I can only trade S and cash, not tilde S. So
I can NOT use the Black Scholes formula directly here.

One method to solve this correctly would be to use the risk neutral
pricing formula, and directly compute the solution. This is cumbersome
and will lead to the correct answer. A shorter way is outlined in the
solutions posted online. One point in this solution might be a bit
confusion: the solution says "Thus we need to discount by 2r + sigma^2
instead of r to use the Black-Scholes formula".

Let me explain this a bit: If I have a geometric Brownian motion with
return rate r, then

    E (e^{-r(T-t)} (S(T) - K)^+ | F_t)

is given explicitly by the Black-Scholes formula. Forgetting the
underlying financial meaning, I can treat this as a explicit computation
of the conditional expectation of a geometric Brownian motion,
discounted by its mean return rate. That is, if I have a geometric
Brownian motion with a different mean return rate, say b, then

    E (e^{-b(T-t)} (tilde S(T) - K)^+ | F_t)

is given by the same Black-Scholes formula (but with the interest rate
b, of course). In the second line in the computation of hat c, the
conditional expectation is exactly this, with b = 2r + sigma^2. Using
the Black-Scholes formula to evaluate this conditional expectation will
give you the final answer.

GI

-- 
Alternative definitions of terms from Math Lectures:
SKETCH OF A PROOF: I couldn't verify all the details, so I'll break it
down into the parts I couldn't prove.