Today let’s take a break from economic issues and get back to option pricing. In continuation with my last piece on option pricing (Link) , wherein we explored a much easier way to price options and what was even better was that it allowed for the flexibility to have non-normal probability distribution in pricing. Today I would like to write about how we can develop the concept further to include in the money and out of money options. Well if one understands the basic premise of my last article on this topic it’s fairly simple.
As I wrote before, basically if I buy an “At the money” call and put, the price that I should pay is the expected movement of the stock in that period. So in essence
C+P = E[S] – Eq 1
Now for any option that is “In the money” for the buyer to make any money and for the seller to loose money would be when the Stock crosses the Strike price of the option, so in essence what we really care about are the probabilities wherein the Stock Price crosses the Strike Price of the option.
So the price of the option would be:
C = (S-K) +E[ ∑x=kn px*S]
The above formula gives the flexibility to the reader to assume different probability distributions and use them in the option pricing.
Please Note that If I try to simplify the Black Scholes model i.e. take the assumption of Normal probability distribution I would be only be approximating the pricing, but nevertheless let me put it down anyways.
The call price (assuming interest rates as negligible) c = SN(d1) – KN(d2), the put formula is also accordingly KN(-d2) – SN(-d1),
c= SN(ln(S/K) + σ/2*√t) – KN(ln(S/K) - σ/2*√t), Since We are taking volatility for the entire duration of the option so I am going to replace σ*√t with σ .
Also I would express S in terms of K and σ as that’s what we are concerned about, i.e.
S = K+ K*σ*Ɵ
Putting these values into the Black Scholes formula and expanding the log portion using taylor equation the log portion would be reduced to Ɵ – Ɵ^2* σ/2
Finally adding both these terms and using the Normal expansion and exponential expansion by taylor series
N(x) =[∫-∞0exp^- (x)^2 + ∫0 σ/2exp^- (x)^2]/sqrt(2 π)
The first part is equal to 0.5, to integrate the second part of the expression let me expand it using the Taylor series. The Taylor series expansion of the exponential term is
ex = 1 + x + x^2/2! + x^3/3! ……
Replacing x by Ɵ – Ɵ^2* σ/2+σ/2 and then integrating we get
N(Ɵ – Ɵ^2* σ/2+σ/2) = 0.5+ [0 σ/2 ∑0n (-1) n (Ɵ – Ɵ^2* σ/2+σ/2 )2n+1/2nn!(2n+1)] /sqrt(2 π)
As SD is small number (in decimals) we can ignore the higher power of σ,
Please note that if σ is high the whole notion of Normality falls apart and as does the Black Scholes model.
So expanding this expression and putting it in Eq 1 we get
C = (2* Ɵ - Ɵ^2* σ/2 + σ) * √(1/2π)
Or more simply C = (2* Ɵ + σ) * √(1/2π)
Again this is simplying Black Scholes using mathematical tools what’s more important to understand is that the genesis of any option pricing i.e. using any probability distribution is this formula below:
C = (S-K) +E[ ∑x=kn px*S]
Till Next Time………….