In the last few modules, we introduced the volatility surface, and we saw how to price certain types of derivative securities using the volatility surface. However, there are many derivative securities that cannot be priced using the volatility surface, and that is because the prices of these derivative securities depend on the joint risk neutral distribution of the stock price at multiple times. An example of that is a barrier option, and we're going to see in this module an example where a barrier option cannot be priced using the volatility surface. So we will discuss the limitations of the volatility surface, how it can only be used to price certain types of derivative securities, and how we will need to use models to price other types of derivative securities. So in practice, when we need to price more exotic securities, we need to resort to using models. We cannot use the volatility surface that we see in the marketplace. Suppose there are two time periods T_1 and T_2 of interest, and then a non-dividend paying security A has risk-neutral distributions given by the following expressions here. Where in particular Z_1_A and Z_2_A are independent N(0,1) random variables. So you can see here that this is the stock price at time T_1, this is the stock price at time T_2. So we've got some horizon here. Today's date zero, we've got T_1 and we've got T_2, and we see that the stock price at times T_1 and T_2 are logged normally distributed, and that's because Z_1 is a normal random variable, and the sum of two independent normal random variables is also normal. So we see that the stock price at each time T_1 and T_2 is log normally distributed. Note that the value of rho A greater than zero can capture a momentum effect, and a value of rho A less than zero can capture a mean reversion effect. Well, what do I mean by that. I mean the following, so notice the following. Suppose the stock price at time T_1 is known to you, and maybe it's a higher value than you'd previously expected. So in other words, suppose Z_1_A is greater than zero, maybe much greater than zero. Well, in that case, if rho A is positive then rho A times Z_1_A will also be positive, and so on average, this random variable here will also be positive. In other words, the stock price at time T_2 will also be larger than you would have previously expected. So that's what I mean by momentum effect. On the other hand, if rho A is negative, then this term here will be negative, and so you will get a mean reversion type effect. In other words, if the stock price at time T_1 tends to be large, then the stock price at time T_2 will tend to be small and vice versa. So that would be a mean reversion effect. So you can capture a momentum effect if rho A is greater than zero, and a mean reversion effect if rho A is less than zero. Suppose now that there is another non-dividend paying security B, with risk-neutral distributions given by these quantities again, and again, we've got Z_1_B and Z_2_B being independent N( 0,1) random variables. So notice stock B and stock A actually have identical marginal distributions. So notice, we're implicitly assuming here that the initial stock price of stock A is equal to the initial stock price of stock B is equal to 1. We're also assuming that they've got the same volatility parameter. So sigma A equals sigma B equals sigma. So I see sigma appearing here, and it's the same sigma up here. Now here's an observation, if Z_1 and Z_2 are independent N( 0,1) random variables, then for any row in the range minus one to one. Rho times Z_1 plus the square root of 1 minus rho squared Z_2 is also a standard normal random variable. So this is a standard result. It's certainly easy to see that the expected value of this random variable is zero, and the variance of it is indeed one. So given this, we can go back to the previous slide and notice the following. We can see that this is N( 0,1), and this term here is also N( 0,1), and so it is clear that the stock price at time T_1, for the two securities A and B, have identical risk-neutral distributions. Likewise, the stock prices at time T_2 have identical risk-neutral distributions, and that's the point we're making here. Therefore, it follows that European options on A and B with the same strike and maturity must have the same price. After all, they've got the same marginal risk neutral distributions, and it is these marginal risk neutral distributions that we would use to compute these European option prices. So therefore, A and B would have identical volatility surfaces. We've only got two maturities here, but we could price options with many different strikes, and so we can say they've got identical volatility surfaces. But now consider a knock-in put option with strike one, and expiration T_2. In order to knock-in, the stock price at time T_1 must exceed the barrier price of 1.2. So therefore, the payoff function is given to us by the maximum of 1 minus S_T_2 and 0, which is the payoff of a regular put option, times the indicator function which is 1 if the stock price at time T_1 is greater than or equal to 1.2 and 0 otherwise. So you can think of this payoff as follows, or the security as follows: So this is time T here. This is the stock price at time T. You can imagine the stock price starting off a value of one. Maybe this is 1.2. This might be time T_1. This might be time T_2. You can see that in order to get a payoff, what must happen is that the stock price must somewhere or another get above 1.2 at time T_1, and then at time T_2, it must be below one in order to be in the money. So it can still move about, but it's got to come down and be a below the strike level of one. So this is the strike level. This is our K, and 1.2 is our barrier level, and the stock price must be above this at time T_1 and it must be below one at time T_2 in order to get a payoff. So now the question we can ask is as follows: With the knock-in put option on A have the same price as the knock-in put option on B? In other words, does the value of the security depends on whether the underlying security is stock A or stock B? Also, how does your answer depend on rho A and rho B? Well, to answer the first question, we know that they've got the same marginal risk neutral distributions. So we made this point on the previous slide, and we see it again here. So the stock prices at each time T_1 and T_2 have the same marginal risk neutral probabilities. But what about the joint risk neutral distribution? In other words, what is the risk-neutral distribution? Let's call it f_a to stock A of S_T_1_a and S_T_2_b. So this is the joint risk neutral distribution for security A at times T_1 and T_2. Similarly, we've got the risk-neutral distribution, the joint risk neutral distribution for stock B at times T_1 and T_2. We know these, they have the same marginal risk neutral distributions. That was the point in the previous slide. But what about the joint distributions? Remember, if we want to evaluate this option or compute the price of this option, it's going to depend on f_a in the case of security A or f_b in the case of security B. So how does our answer depend on rho A and rho B. Well, if you look at this plot here, what you can see is the stock price needs to go up, and then it needs to fall. So in fact, a small value of rho will actually help you achieve that. In particular, you would like your rho parameter to be as small as possible because, if rho A for example, suppose rho A is very small, less than zero, close to minus one. Well, if the stock price at time T_1 is above the barrier 1.2, that means Z_1_a would have to have been large. But if Z_1_a is large, then rho A times Z_1_a will be negative because rho A is now much smaller than zero. So in that case, this will be negative and therefore, you'll have a much better chance of this entire quantity being negative, and the stock price at time T_2 being below the strike of one. So in fact, a negative value of rho A, and the more negative, the closer to minus one the better, will help the option X bar with a positive payoff. So in fact, the answer to the question is that if rho A is less than rho B, then the price, let's call it P_0_ a, will be more expensive than the price of the knock-in on B. Similarly, if rho B is less than rho A, then the price of the knock-in on B would be more expensive than the price of the knock-in on A. So it certainly does depend on rho A and rho B, and what this tells us is that the volatility surface alone cannot give us enough information to price these options. After all, these two securities, A and B, will have identical volatility surfaces because they have the same risk-neutral distributions, same marginal risk neutral distributions, but they have a very different joint risk neutral distribution, and that joint risk neutral distribution will actually determine the value of this knock-in put option. Depending on the value that rho parameter, we're going to get different prices. So this is really just another way of saying what we said in the previous module, that we can use volatility surfaces to price derivative securities that only depends on the marginal risk-neutral distribution of the stock price at a fixed time T. But anti-security whose value depends on the joint risk neutral distribution cannot be priced just using the information in the implied volatility surface.
