We're now going to return to a question that I raised in an earlier module. That question concerned the following situation: we had a one period binomial model, and we had two securities that were absolutely identical except they had different probabilities of an up move. The first security had a probability of 0.99 of an up move, the second security had a probability of 0.01 of an up move. We argued however that both of these securities had the same option prices and we found that to be somewhat surprising. Well, we're going to address that issue in this module as well as some other questions that are also of interest. So if you recall we had the following situation. We had two securities, stock ABC, and stock XYZ. The two stocks were identical in every way except they had different probabilities of going up and going down. Stock ABC had a probability of going up with 0.99, and the probability of going down of 0.01 where stock XYZ had these probabilities flipped. The probability of going up was 0.01, the probability of going down was 0.99. Then we talked about a call option with payoff equal to the maximum of S1 minus 100 and zero. We wanted to know how much a call option is worth on each of these two stocks. Well, the strike is 100, so the payoff here would be 10 or zero in each case. So this is the payoff of the call option C1. Probability of a payoff here is 0.99, and down here it's 0.01. So it stands to reason that the value of a call option on this security with a payoff of 10 with probability 0.99 should be worth a hell of a lot more than a payoff of 10 with probability 0.01. Well, if you believe the theory that we've developed so far, then that is simply not true. Assuming you can invest or borrow in the cash account at a gross risk-free rate of r and that you can buy or short sell the stock, then in that case, the call option both cases must be identical and it turns out to be approximately $4.8. So C0 is approximately $4.8 here, C0 being the fair value of the option price at term one, and C0 down here is also equal to $4.8. So this seems very strange indeed. A lot of people get upset with this idea, they feel something is wrong with the theory when you can have a situation where 10 with probability 0.99 is worth $4.8, but also 10 with probability 0.01 is also worth $4.8. So it seems like there's some contradiction or something strange going on here. We're going to resolve that issue by reminding ourselves first of all of our one-period theory. In our one-period theory, we saw that the fair value of the option C0 is equal to 1 over R times the risk-neutral expected value of the payoffs. So that's this quantity here, where the risk-neutral probabilities are given to us by that quantity and this quantity, u minus R over u minus d equals 1 minus q. Of course, we see that the true probability p, well it just doesn't seem to appear everywhere. The only assumptions we made about P was that P is greater than 0, and 1 minus p is greater than 0. Likewise, with no arbitrage, we know that d is less than or is less than u, so that q is greater than 0 and 1 minus q is greater than 0. So it appears that p does not matter, and this is the source of confusion for a lot of people. But in fact, it only appears surprising because we are asking the wrong question. So the question we are asking is why is an option in this model worth $4.8 and it's also worth $4.8 here? That's not the right question to ask. The right question to ask is why would you find a situation like this in an economy where there's a stock with probability of 0.99 growing by 10 percent, and in the same economy that you would find another stock which would have a probability of only 0.01 of going up by 10 percent? These stocks are incredibly different. You would never expect to see two securities like this in the same economy. So your problems shouldn't be with the option pricing theory and why you get these two option prices being equal to one another, your question should be why would you ever expect to see two securities like this in the same economy. This is the source of the problem, assuming that you could find two stocks like this in the same economy. I've never seen the situation myself, I've never seen any situation like this, and I can't imagine a situation like it happening. But if you did find yourself in an economy where you had two stocks ABC with this characteristic where these characteristics, and stock XYZ with these characteristics, then indeed, you will see that the option prices on the two securities will be the same. So the source of confusion is resolved by focusing on why you would see these two securities, and not by focusing on the option pricing theory. The option pricing theory simply states that if you do have two securities like this in the economy, then they will have the same option price. On this slide, I want to discuss another interesting example. Consider the following three period binomial model, which we have here. The stock price starts off at S0 equal to 100, and it goes up by a factor of u equals 1.06, or it decreases by a factor of d in every period. The gross risk-free rate is 1.02, and we want to price a European call option with strike K equals 95. So the way we do that as usual is we determine the payoff of the option at maturity, so the maturity is at t equals 3, and this is the payoff of the option. It's 24.10 if the stock prices is 119.1, 11 if the stock price is 106, and zero on the other two states. So note that this cash flow is non-negative and it's strictly positive in these two states up here. We can price this option using our knowledge learned to date. We can find out that the option price is equal to 11.04. Let's price the same option again. But this time, we're actually going to change the risk-free rate. We're going to change it from 1.02-1.04. Everything else about the option is identical. We've got the same set of cash flows at maturity at t equals 3. We can work backwards in the lattice computer's price and if we do so, we'll find the price is equal to 15.64. So in the first case, when R equals 1.02, we find the option price is 11.04. In the second case, when R equals 1.04, we find the option price equals 15.64. Now this is a little bit surprising because what we've done here is we've computed the value of this cash flow by computing its values is a t equals 0. We've discovered that when we increase the interest rate, the option price is increased. That is totally against what you would see in a deterministic world. In a deterministic world when you increase the interest rate, what you do is you decrease the present value of the cash flow. But here the cash flow hasn't changed, but we haven't decreased its present value. Its fair value is 15.64, which is an increase on 11.04. So this should tell you that option pricing can produce and threw up some interesting and surprising results. It is very different from the deterministic world which you are probably more familiar with. Before we end this module, I want to spend a couple of minutes discussing the existence of risk-neutral probabilities and their implications for no-arbitrage. First of all, recall our analysis of the binomial model. We saw there the no-arbitrage is equivalent to d being less than nor being less than u. We also saw that any derivative security with a time capital T pay off C_T can be priced using risk-neutral pricing as follows: for q and 1 minus q the risk-neutral probabilities, and n is the number of periods. Just to relate n and t to you as follows, if you assume that delta t is the length of a period, then capital T equals n times delta t. So that's the link between t here and n over here. So this is what we saw in the binomial model. When there's no arbitrage, there are risk-neutral probabilities that are strictly positive such that all cash flows and derivative securities can be priced using this representation here. This representation is actually more general. In fact, you can show that if there exists a risk-neutral probability or risk-neutral distribution, Q, such that for holes in any model than arbitrage cannot exist. Why is this the case? Just to see a simple example, consider the following situation. Suppose we've got a model with m states, omega one, omega i down to omega m. Suppose we want to consider some security which has a payoff in each of these states. We'll assume that payoff is non-negative in all of these states. Let's suppose that there's actually one state where the payoff is strictly positive. Now if we have risk-neutral probabilities, strictly positive risk-neutral probabilities Q1 so on down to Qi down to Qm, such that this situation is satisfied, then notice that the expected value of C_T, so this is C_T here. Then the expected value of C_T must be greater than zero. Why is that? Well, all of these probabilities are strictly positive. You're multiplying them by a cash flow that's non-negative, and at least in one state strictly positive. Therefore, the expected value of C_T is strictly positive. Well, that means is that you can't get a type B arbitrage. That's not possible. Similarly, you can show that you can't get a type A arbitrage. You just replace this with a greater than or equal to sign and show then that the price of this according to the representation for is greater than or equal to zero, which means it can't have a negative cost, which means there can't be a type A arbitrage. So in fact, any model which has this representation in other words, we can compute the value of C_T using a set of risk-neutral probabilities, then in such a model there cannot be an arbitrage. The reverse actually is also true. If there is no arbitrage, then a risk-neutral distribution exists. We've shown that in the binomial model, but it's actually true more generally. Together, these two last statements are often called the first fundamental theorem of asset pricing. That is the existence of risk-neutral probabilities, and no-arbitrage are equivalent.