Introduction:
Option is a contract between the buyer and the seller that gives the right, not an obligation, to buy or sell a particular asset at a later date on a fixed price. For buying a call option is used and to sell a Put option is used. Though this practice exists for centuries but was properly introduced in exchange in 1973. The first stable equilibrium model was presented by Fischer Black and Myron Scholes. In 1979 John Cox, Stephen Ross and Mark Rubinstein published their most famous paper on option pricing. This paper is based on its simplicity, as it deals with hedging with no-arbitrage argument. CRR main objective was to switch the complex mathematical tools for calculating the options with much simpler method, which they did and can be seen in the form of different sections.
The Basic Idea:
The first thing we need to know how option pricing and riskless arbitrage opportunity are related. Riskless arbitrage opportunity basically means to make profit without committing any money and without taking a risk of losing money. The paper begins with a simple hedging example. Suppose stock price is 50, so at the end of a period it can be either 25 or 100. There is a call option available with a strike price of 50. And you can also borrow or lend at 25% interest rate. The only thing left is the current value of call, C. So if there is no arbitrage involved in the above example, C can be calculated very easily. For which a levered hedge must be formed as shown in table 1(appendix), thus giving the equation 3C – 100 + 40 = 0. The value of C comes out to be 20 with zero profit. If C would have been 25, there was an arbitrage in affect with a profit of 15 and if the value of C was 15, same profit could be achieved but by buying 3 calls, selling short 2 shares and lending 40.
The binomial option pricing formula:
CRR start off with 1 period expiration call option. The rate of return on the stock has two possible values either u-1 or d-1 with q and q-1 probabilities respectively. If the current stock price is S, then price of the stock at the end of the period will be either uS or dS. Couple of assumptions are made, such as interest rate is constant, no taxes and zero transaction costs. CRR denoted (r) as the riskless interest rate over 1 period and riskless arbitrage must hold for u>r>d. C will be the current value of call, Cu will be its value at the end of period if stock price goes to Us and Cd if it goes to ds. Using the term of contract and rational exercise policy and substituting the portfolio containing ∆ share of stock and B amount of riskless bonds in it and solve for ∆ and B, CRR came up with the hedging portfolio and , where the value of C can’t be less than it, if no riskless arbitrage needs to be achieved. If that the case then must be true and can be simplified in the form of, where p and p-1 act as a probability and r > 1.CRR then extend the call option with two periods remaining, where Cuu is the value of call with two periods from the current time if the stock price moves upward each period and Cdu and Cdd have analogous definition. Same process of portfolio with containing ∆ shares of stock and B amount of riskless bonds is applied, where the end value of B will be Cu if stock price goes to uS and Cd if it goes to dS. In the case two periods to expiry, there is a chance for the stock price minus the exercise price to be greater than the hedging portfolio, so to counter that CRR had to put a maximisation equation, keeping in mind that Cud=Cdu. Thus with the help of little algebra CRR managed to achieve the equation for any number of steps.
Riskless trading strategies
In this section CRR talk about two different cases through which profit could be made. The first case is if the market price is greater than value of the call calculated; hedge it to make profit from it. The first thing is to take the remainder left after subtracting the value of call from the market price and put it in the bank. Start by taking n=3, now using the money in hand which should be equal to that of the call price, begin the process of buying and selling, where at each step there will be a new value of ∆ and a fixed interest rate will be implemented on borrowing and lending. The main idea is, when the call expires the value you are left with is zero. The amount that was deposited in the bank will be given back to you after discounted at the same fixed interest used. Thus providing you with more profit than the one you initially began with. For this first case the number of calls was kept constant and made adjustments by buying and selling stocks and bonds. For our second case we keep the shares of stock constant and adjust by buying and selling calls and bonds. This could be very risky, if while adjusting the call becomes overpriced, we could end up losing money. Then to remain in the hedge we would need to buy back the call, which depends on their value and we will be completely blinded by the return on our hedge. Worst case scenario we might end up losing everything.
Limiting case
The main thing achieved in the limiting case is to show that rather than having one specific period like a day or even an hour, we can achieve the same objective with even smaller intervals for stock changes. Thus coming with a formula h=t/n, where h is the elapsed time between stock price change, t is the fixed length of calendar time to expiration and n is the number of periods before expiration. Due to increase in frequent trading h will become closer and closer to zero, resulting in n->∞.So, r, u and d needed to be changed as they depend on n. For the interest rate they covert r to r^=rt/n but to calculate u and d, they had to use continuous compounding to calculate mean and the variance in the form of logarithm; and respectively. As n->∞, with the help of algebra they achieve the values for u and d; and respectively. The other thing CRR achieve in this section is to show their limiting case converges with the Black-Scholes formula, and the similarities are quite visible in the final derivation of their Cu and Cd equations. In the same section they also tried converging their derived equation to that of the stochastic jump process.
Dividend and put pricing
In the previous sections CRR assumed that sock pays no dividend, but in this case they do and with a specific dividend policy, it maintains a constant yield δ. At the end of the period, ex dividend date, the stock price will be u(1- δ)v or d(1- δ)vs. We follow the same steps by selecting the a portfolio and change S by (1- δ)vS for Cu and Cd, where v=1. There is one more change besides the no dividend policy; we can exercise the option at any time now. So in the case of δ and S being large, early exercise may be more favourable. Hedging portfolio will be affected by the increase or decrease of dividend received or paid respectively. Even though there won’t be any problems to exercise before the expiration date, but it will prohibit a simple closed form solution for the value of call, with x periods to go. But this can get complicated depending on the dividend policy, yet the basic steps are the same; at the expiration date calculate all the possible prices of stock, and use that information to back one period and calculate the value of call and hence forth. We can also apply the binomial formulation for the put option. Same procedure for choosing the portfolio and the outcome equations is the same as, with minor changes. In calculating puts the chances of things getting messy are very high, to the fact that no one prefers the hold the put option for more than one period. But we can value put the same way we value call options, by reversing the difference between the stock price and the strike price at each stage.
Conclusion
CRR started with a simple hedging example and applied it to the one time step period, keeping in mind the no-arbitrage condition. Then they extended the one period to two periods and ended up with a equation for n number of steps. In the next section they showed two cases of riskless trading, where their main focus was on the difference in the market price and the call calculated. From there they moved to show that their binomial method could converge with Black-Scholes method as, n->∞. For the final section they took off the no-dividend policy and also replaced the call option to put.
Appendix
(1)
Write 3 calls at C each
(2)
Buy 2 shares at $50 each, and
(3)
Borrow $40 at 25% to be paid back at the end of the period.
Table 1
