Box Models I
In this toy world, we can determine what the value is of a call option with any given strike price. Let’s look first at a call option with strike price 100. Corresponding to each of the two possible prices of the underlying at expiration, this call option will have two possible values at expiration: 0 and 10.
If we are only concerned with the value of the call option at expiration, we could simply replace the stock values in the box with the corresponding option values, and draw the option value directly from the box:
And we can consider that we are simply engaged in a game, whose outcome is determined by the single draw from this box on expiration day. How much should you pay for this call option?
You can answer this question by asking how much this call option is worth on expiration, on average. Half the time it is worth nothing, and the other half of the time it is worth $10. How much would you pay for an item like that?
If you played this game 100 times, you would expect to have an item worth $10 around 50 times, and worth 0 around 50 times. So in 100 draws you would collect $10 x 50 = $500. $500/100 plays = $5 per play. So if you repeatedly paid more than $5 for this option, you would come out behind in the long run. And if you paid less than $5 for it, you would come out ahead.
The value of this call option under Model A is $5, with one proviso: If you paid $5 for it repeatedly, you would in the long run break even, but you would be paid nothing for letting your $5 sit in someone else’s (the call seller’s) pocket for the time between now and expiration. You naturally would like some payment for that, and the natural payment is the amount that your $5 would earn in the bank, or in T-bills or some other safe instrument, for that time. Thus, you should discount the $5 which you will on average receive at expiration, and the natural discount rate is the ” safe” interest rate.
This is the basic procedure to understand. It is the essence of all option pricing calculations and underlies even the most sophisticated algorithms. If you understand it, you are well- positioned to understand the Black Scholes formulation and the evaluation of any option position. More next time.