Option Trading Blog

I have created this for those who are interested in the mechanism of the Black and Scholes formula, who don’t want to get into the math involved in the original Black and Scholes derivation. This is based on the CRR model (Cox, Ross, Rubinstein).

The biggest revolution in modern finance is the pricing of options. The basic idea is the notion of replication. Suppose we want to write a call option which has 1 month to expire. Let’s assume the strike price, K, is 100$. Let’s also assume the stock, S, is worth currently 100$ and the stock either goes up to 120$, or down to 80$.

The question is how to price the call option. Obviously the call option would be worth either 20$ if the stock goes up, or expire worthless, if the stock goes down to 80.

Let’s assume interest rates for borrowing and lending are set to 20%. If we buy half a stock (suppose it makes sense) and borrow 33.3$ from the bank, for a total cost of 50-33.3=16.7$, we can mimic the payoff of the stock. How come? at the end of the month, if the stock drops to 80, our portfolio is worth 40$ in stocks but we have to return 40$ for the borrowing (33.3 plus the interest), hence 0$, exactly like the payoff of the option which expires worthless. On the other hand, if the stocks rises to 120$, our portfolio would be worth 60-40 = 20$. Again, the same payoff like the option. Check for yourself to see.

This is the revolution of option pricing. We have created a synthetic option which gives the same payoff as the option itself! Hence, it must be that the option is worth the portfolio in the first period, 16.7$! Why is that, you ask? If the call would be worth for instance 20$, meaning that the option is overpriced, we could sell the call option (write a call option for 20$) and buy the portfolio ( buying half a stock and borrowing as we did in our example). We would have 20-16.7=3.3$ in our pocket and since the portfolio we bought would replicate the payoff of the option, we don’t need to worry about the outcome. If the stock goes up to 120$ and we have to pay 20$, the portfolio would be worth 20$! We will be perfectly covered.

Now is the time to make it more general. Suppose the current price of the stock is S. Assume that the stock can go up to or down to , where d < u. We are interested in knowing, how much of stocks, h, to buy and how much of Bonds, B to borrow.

If the stock goes up, the portfolio would be worth . If it goes down it would be worth .

What we want to make sure is that:

and that

Where and are the payoffs of the option in the case the stock goes up or down respectively.

We want to find h, called also the hedging ration and B. If you do the math you will find that

and that

If you do the math, you’d see that for the example in the beginning, you’d get that and that . This is how I knew how to replicate exactly the option.

In the next post I’ll explain the notion of Risk-Neutral probability which is tossed around a lot and make the model more general to make it more realistic and show how it converges to the famous Black and Scholes formula.