Expected Values for options

Make an assumption about the distribution of the stocks price in 1 year (log normal) - and then integrate this against the payout function for the call.


Now the real (and subtle) question - If we take this expected value and take its present value (using the 1 yr spot rate) should it equal the value of the option today?

No, you shouldn't discount an option, which is not risk free with the 1 year risk free rate.

You said,


"If we take this expected value and take its present value (using the 1 yr spot rate) should it equal the value of the option today?"


Should the present value equal the value today? Yes, by definition.


What the heck is the "1 yr spot rate?"


If we take a synthetic stock - by purchasing a call and selling a put (K=K,t=t), and investing our spare Reg T margin money in T-Bills to mature at option expiration - and then we remove the Reg T margin investment, and then remove the put, the expected rate of return on the call can be expressed in an equation including,


1) the risk-free rate,

2) the expected rate of return on the stock,

3) the spot price for the stock, and

3) the price paid for the call.


The expected rate on the call should be quite a bit greater than the expected rate on the stock but, of course, the expected rate on a short stock future should also be negative, for whatever that's worth. Also, notice the short put also expects to make more than the risk-free rate, because it is unduly likely to expire worthless, so they cooperate to reflect the supernormal returns of the stock.


eLROY