This are some simple option pricing formulae for European Options on futures, stocks and currencies. They are special in that they treat all 3 types of underlying as specializations of a general European Option model. The types of underlying are differentiated by the value of alpha. I also added a fudge factor for pricing American options. These formulae are also good in that they are very fast to calculate. I did this work in the late 1980's when computers were much slower than they are today.
I did not go through the derivation of the formulae, however the derivation is quite simple. The price of the option is the discounted present value of the expected value at expiration. The prices are lognormally distributed, with mean alpha * tau, and variance sigma ^2 * tau. The discount factor is exp( -r * tau).
These are the Black Schole's formulae for European Options. Note that the Put and the Call are intrinsically related because the exercise boundary of the put and call together is a straight line.
These are the derivatives of price with respect to sigma and tau. Time decay is defined as the second order term of the Ito differential of option price. That is, sigma^2 * S^2 / 2 * D2 Option Price / D S ^ 2 .
This is the definition of the exercise boundary for an American call. Note that the intrinsic value of the American call is the greater of the intrinsic value of the European Call, and the value of immediate exercise. In the case where alpha > 0, the boundary has 3 separate line segments. A fudge factor is defined, so that a European call price can be used to approximate the value of the American call. The derivatives of the fudge factor with respect to underlying price, and time are calculated.
This is the definition of the exercise boundary for an American put. Note that the intrinsic value of the American put is the greater of the intrinsic value of the European Put, and the value of immediate exercise. In the case where alpha < 0, the boundary has 3 separate line segments. A fudge factor is defined, so that a European put price can be used to approximate the value of the American put. The derivatives of the fudge factor with respect to underlying price, and time are calculated.
This is the definition of the fudge factor for American Options on Futures. It is different than the previous fudge factor, because the delta's of futures are the change in price with respect to the future price of the underlying, as opposed to the current price of the underlying. Therefore the delta must be changed by cost of carry. Another way of viewing this effect, is to consider the value of a call and put = exp( - r * tau) * (S - K) . The delta of the combo is exp(-r*tau) instead of 1. This fudge factor is the adjustment for the effect of the discount.
This is a diagram of the exercise boundary for an American Call and a European Call. The intersection point is calculated.
This is a graph of a call on a currency option and a currency cash future. The alpha is negative, so that the future is below the cash. Equivalently the foreign interest rate is higher than the domestic rate since alpha = r - r_foreign. The cash call in this diagram was calculated with a binomial model. The futures call was valued with a Black Scholes calculation.
This is a graph of a call on a currency option minus the price of currency cash future. There are two lines. One is for 2 months until expiration. The other is at expiration. Again there alpha is negative, and significantly large relative to the pricing of the options.
