Learn, Think & Do

In the part-1 of this post we talked about arbitrage, replicating call option using portfolio of the stocks and the money market. In this we will see that the initial wealth is the correct pricing for the call option in the problem under consideration where we left off, we show that otherwise there is an arbitrage in the model. Before doing that let me give you a very very short introduction to buying and selling of options. There are two kinds of options, call and put. With them you can have four kinds of transactions buy call, sell call, buy put and finally, sell put. Each of them have a payoff function associated with them depending on the underlying asset, maturity time and the strike price. To make matters simple assume that the underlying asset is a stock with stock price at time T and K is the strike price at maturity time T for both the call and put options. Here we are assuming options to be European that is you can exercise at the maturity time not before that, and if you can exercise option at any time before maturity time it is called an American option, and if you can exercise option at some fixed time before maturity time then it is a Bermuda option. Anyway moving on to payoff function: when buying call option our hope is long that is we hope that the stock price is going to go up so payoff in this case is , when selling call option our view is short that is we believe that the stock is going to go down so by selling we earn the premium of the option but the payoff in this case is we want our stock price in this case to be below the strike price otherwise there is a loss. Similarly when buying put we are long on put option that is our view is that the stock price is going to go down, payoff in this case is and when selling put our view is short on put that is we believe that the stock is going to go up and so we sell put option to earn money from premium of the option, payoff in this case is , we want our stock price in this case to be above the strike price otherwise there is a loss. I know all this can be quite confusing at first but if you want to understand it you need to read, reread and think about it, the following graphs can help.

Consider that we could sell the option for more than 1.20, say for 1.21 units. Then we can replicate the option in the same way as before with the 1.20 units and we can invest remaining 0.01 units in the money market at the rate of 0.25. At time one, we would be able to pay off the option, irrespective of how the coin tossing turned out, and we will have the 0.0125 resulting from the investment in the money market. This is an arbitrage because as the seller of the option we do not need any initial money, and we have made 0.0125 units without any risk. Now suppose we could buy the option for less than 1.20, say 1.19. In this case we would adopt the reverse of the portfolio strategy we adopted for replicating call option. That is we short half the stocks at time zero and from the 2 units of money we get from it, we buy the option at 1.19 and invest the rest of 0.81 in the money market. At time one, we need to pay back half the stocks, suppose we get head on the toss, then we need 4 units for that we get 3 from exercising the call option and the remaining 1 unit we can take from 0.81 invested in the money market which leaves us with 0.0125 units of money. Now in case tail shows up on the toss, we need 1 unit to pay back the stock. As our option expires worthless, we can take money from the 0.81 invested in the money market and it still leaves us with the 0.0125 units of money. So in both the cases we made a profit without any initial wealth and zero probability of loss, so this is an arbitrage. This shows that if the price of the call option is not 1.20 in the given scenario there will be an arbitrage. This means no arbitrage implies that the price must be 1.20. Now we need to show the converse that is, if the price is 1.20 then there is no arbitrage. We would be doing it in the similar way we did in the first part of this blog for the inequality . Suppose the call option is sold at 1.20. Let denotes the number of call options bought at time zero by the investor and as before the number of shares of the stock bought at time zero, we also have so initial cash position for the investor will be . At time one the investor’s portfolio of stock, option, and money market assets is, . We have assumed that head and tail both have positive probability of occurring so after computing we get and . Both have opposite signs for any values of and with positive probabilities so there can not be any arbitrage.

In general one-period model depends on the following assumptions which we have already defined in parts:

1. one can buy fraction of the shares of the stocks;
2. the interest rate for borrowing and investing is the same;
3. the purchase price of the stock is same as its selling price;
4. at any time, the stock can take only two possible values.

First assumption is not that restrictive because in real stock market we have plenty of shares outstanding and we can buy there fractions. The second assumption is not true in general for individuals but it is close to being true for large institutions. The third assumption is more serious and far from reality but for very small interval we can assume it to be true or in case there is not a lot of trading taking place. Th fourth assumption is limited to binomial models, in other more sophisticated and complicated models the stock can take infinitely many values. In general a derivative security is a financial instrument that depends on other underlying assets, for example options are derivative securities, in our case depending on the price of the stocks. Let and be the value of some derivative security at time one depending on the results of the coin toss head and tail respectively. We want to determine the price at time zero of the given derivative security which is the right pricing, that is, it does not give rise to arbitrage. As we have discussed already, the value of the portfolio at time one is , where the notations have the same meaning as defined earlier at time zero and at time one. As per our discussion we want to replicate the payoff for the given derivative security using the stock and money market investment which would give us the right price. For that we want and and this can be achieved by choosing values of which satisfy it since these are the only two quantities which can be changed, all other quantities are either known at time zero or at time one. Assuming and we can write,

Which can be further written as,

Solving the above equations we get,

where , . These are also called risk-neutral probabilities, notice that . Note that these risk-neutral probabilities are different than actual probabilities. In simple terms these are the simplified probabilities we get while simplifying the above equations. We also have ,

This is called as the delta-hedging formula. This tells us the number of shares to buy in order to hedge a short position in the derivative security which is what we did. So if we have given by above formula and $\Delta_0$ given by the delta-hedging formula, then by the way we have solved the equations, the portfolio value at time one is if we get head otherwise it is . Therefore the right price at time zero is given by,

The terms and are called the risk-neutral probabilities, because we have

That is, they make stock growth rate in the money market same as the mean growth rate at time one and this is achieved in solving multiperiod binomial model as well as in continuous models (not to get too ahead but the continuous time models using Brownian motions are basically the limit of multiperiod binomial models where number of periods go to infinity). Therefore these change of variables are important in option pricing.

Reference:

Steven Shreve, Stochastic Calculus for Finance I, The Binomial Asset Pricing Model, Springer