We're not going to discuss option pricing in the 1-period binomial model. We'll see that option pricing in this model amounts to no more then of solving a series of two linear equations and two unknowns. The great advantage of working with the 1-period binomial model is that we will see it easily extends to pricing options in the multiperiod binomial model. So this is the one period binomial model, we are assuming that the stock price starts off at S0 equals $100. The stock price at time, t equals 1, will either have grown to uS 0, which is $107 in this case, so here I am assuming u equals 1.07. R will have fallen to $93.46, which is d times S0 and d is equal to 1 over u. Okay. So this is the 1-period binomial model. Stock price goes up by a factor of u or it falls by a factor of d. We have a probability p, which is the probability of an up move and we have a probability 1 minus p, which is the probability of a down move. Okay. So we also assume that we have a cash account available, which enables us to borrow or lend at a gross risk free rate of R. So, if we invest $1 in the cash account at time t equals 0. It would be worth R dollars at time t equals 1. Similarly, if we were to borrow $1 at time t equals 0, we would have to pay back R dollars at t equals 1. We've also seen that short sales are allowed. What that means is, if I want to short sell a stock, what I can do is I can borrow the stock. I assume I can borrow the stock at no cost and sell it in the marketplace. Then later on I can buy the stock back and return to the person who lent it to me. So that's how short sales work. If I want to short sell a stock. I borrow it, I sell it in the marketplace, later on, I buy it back, and then return it to the person who lent it to me in the first place. We're also implicitly going to be assuming that there are no transactions costs. Okay, so basically, I can buy and sell and borrow and lend with no transactions costs. Okay. So two questions, two simple questions to begin with. The first question is, how much is a call option that pays the maximum of S1 minus 107 and 0 at t equals 1, 1 worth? So in this case, 107 is the strike, and this is a call option. And the second question is, how much is a call option that pays the maximum of S1 minus 92 and 0 at t equals 1 worth? So in this case, 92 is the strike. Well, in fact, here, we'll see that we can answer these questions very easily. In the case of question one, well, if we go back to our 1-period model, we see the maximum security price at time one is 107. So therefore, in fact, the maximum of S1 minus 107, 0 is always going to actually be zero at t equals 1, because in our binomial model, the stock price is never greater than 107. So the call price at time 0, we'll call it C0, must be zero. Okay, for question two, it's actually also quite straight-forward. Again, with the, the strike here is $92, if I go back to my binomial model here, I see that the smallest possible at time t equals 1 is 93.46. So no matter what, in fact, I can write the maximum of S1 minus 92 and 0. Well, that is going to be S1 minus 92. And that is because the stock price at time 1 is always greater than or equal to 92. So the max will always occur at S2 minus 92. So if I'm going to get S1 minus, minus 92 at time t equals 1, how much is this payoff worth today at time 0? Well, linear pricing tells me it must be worth S0 minus 92 over R. Okay. That's because what I'm getting is, I'm getting S1 dollars at t equals 1. Well, S1 dollars at t equals 1 must be worth S0 dollars today. I'm also getting minus $92 at t equal 1. Well, 92 is just a fixed determinant to cash growth, so I just discount its value back to t equals 0 to see if this is a total value of the option price at time 0. Okay. So that was two simple questions that we could answer. But what happens if the strike lies in between 107 and 93.46? Okay, in order to answer that, I actually need to introduce, first of all, some ideas of arbitrage, Type A and Type B arbitrage. And we're going to need these more general definitions, these are more general than the earlier definitions of weak and strong arbitrage that we used in the deterministic world. We're going to need them because we're introducing randomness Into our models. So, we have the two following definitions. A Type A arbitrage is a security or portfolio that produces immediate positive reward at t equals 0 and it has a nonnegative value at t equals 1. So, that is a security with initial cost V0 less than 0. So if its cost is negative, that means, if we buy, it we actually receive money. Okay, it's a little bit confusing. But when you have a negative initial cost, you actually receive money when you buy something. So a Type A arbitrage is a security with initial cost V0 less than 0 and time t equals 1 value V1 greater than or equal to 0. So an example of this type of arbitrage would be maybe finding $10 in the street. So if you find $10 in the street right now, well, you're going to receive a positive amount, it's like having a negative cost of $10 and at time 1 you'd have V1 equals 0. So you find the money right now, you get $10, and you have no liability at time t equals 1. Okay. A type B arbitrage, is a security or portfolio that has a non-positive initial cost, has positive probability of yielding a positive payoff at t equals 1 and zero probability of producing a negative payoff then. So let's translate them. The type B Arbitrage is a security with initial cost less than or equal to 0. So in other words, if you enter or you purchase this security, you're not going to pay anything for it and indeed you might receive something. You receive something if this is strictly negative and its terminal value times t equals 1 is nonnegative and it's not equal to 0. So , an example of such a security would be maybe someone coming up to you in the street and handing you a free lottery ticket. A free lottery ticket means you pay nothing at times 0. Okay, so we pay nothing for the free lottery ticket. And a time one, maybe you'll win something. Okay. But you definitely won't lose anything. So V1 is greater than or equal to 0, because only the lottery ticket means you're never going to have to actually pay out. And there'll be a chance, maybe a small chance, but there will be a chance that you'll actually receive something, so it's not equal to zero. So, over here, you can think of this as being the different states of the lottery. Maybe there's one state where you win. Maybe this is a positive payoff for you. And all the other states are zero. So this is an example of a security which is greater than or equal to 0, but it's not strictly equal to 0. Okay, so that's a Type B arbitrage. Now, let's return to our 1-period binomial model and discuss what conditions must hold in order to have no arbitrage in the model. Recall that we can borrow or lend at gross risk free rate, R per period. We're also assuming that short sales are allowed. So we have the following theorem. The theorem states that there is no arbitrage if and only if d, which is this quantity here, is less than the gross risk free rate, which in turn is less than u. So let's see how we might prove this. So consider the first situation here. Suppose R is less than d is less than u, then we can construct the following portfolio. Let's borrow S0 dollars and invest in the stock. This will actually gives us a Type B arbitrage. How do I know that? Well, let's see. Let's look at the cash flows. So, our cash flows R, so this is under condition one, so we have R less than d less than u, and we're going to follow this portfolio here. So, at t equals 0, we borrow S0 dollars, so that gives us plus S0 dollars, but we invested in the stock. So we're going to buy one unit of stock, so that means we're going to spend the S0 dollars we got from borrowing it and we're going to spend it on the stock, so the net cash flow times 0 is 0. What happens at t equals 1? At t equals 1, so this is from our borrow, and this is our position in the stock, and at t equals 1, we have to pay back our borrowing. So we have to pay back S0, but we have to pay it with interest. So we're going to be paying back S0 times R, okay? But we own one unit of stock, okay? Remember, we invested SS0 dollars into stock at time t equals zero. That's going to be worth us 0 or dS0 at time t equals 1. And the fact we're going to use these proceeds to pay back the borrowings at t equals 1. The net cash flow here then is u minus RS0 if the stock price went up or it's d minus RS0 if the stock price went down. But by assumption, d is greater than R and u is greater than R so this component is positive and this component is positive. So regardless of whether the stock went up or not, we're going to have made money at time t equals 1 and that is an example of a Type B arbitrage. Okay, so that's case one. How about case two? Well, we could do something very similar. In this case, if R is too large, then it suggests that it might be a good idea to short sell the stock, take S0 dollar, dollars in from short selling the stock, invested in the cash account and earn R. And then the time t equals 1, you can buy back the stock for either d times S0 or u times S0, but in either case, you're going to have less than S0 times R. So in fact, you can do the same sort of argument that I did up here to show that you would also get a type B arbitrage in case two. Okay, so we're always going to assume, later on, that d is less than R is less than u. And, that's because we don't want arbitrage to exist in our models. It's a standard economic assumption that we assume there's no arbitrage and the reason is if there was an arbitrage then market forces would act very quickly to dispel that arbitrage. Supply and demand would drive the arbitrage away. So we're always going to be assuming there's no arbirtrage.
