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 and example of 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. And 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, T1 and T2 of interest and then a non-dividend paying security A. Has risk-neutral distributions given by the following expressions here, where in particular Z1A and Z2A are independent N 0,1 random variables. So, you can see here, that this is the stock price at time T1, this is the stock price at time T2. So, you've got some horizon here, today's date 0. We've got T1 and we've got T2. And we see that the stock price at times T1 and T2 are log normally distributed, and that's because Z1A 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 T1 and T2 is log normally distributed. Note, that the value of rho A greater than 0, can capture a momentum effect, and a value of row A less than 0 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 T1 is known to you, and maybe it's a higher value than you'd previously expected. So, in other words, suppose Z1A is greater than 0. Maybe much greater than 0. Well, in that case, if rho A is positive, then row A times Z1A will also be positive. And so on average, this random variable here will also be positive, in other words the stock price at time T2 will also be larger than you will have previously expected. So thatâs what I mean by the 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 of time T1 tends to be large, then the stock price at time T2 will tend to be small and vise versa. So, that would be a mean-reversion effect. So, you can capture a momentum effect, if rho A is greater than 0. And a mean-reversion effect if rho A is less than 0. 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 Z1B and Z2B being independent N 0,1 random variables. 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 and so I see sigma appearing here. And it's the same sigma up here. Now here is an observation, if Z1 and Z2 are independent N 0,1 random variables, then, then for any rho in minus, in the range minus 1 to 1. Rho times Z1 plus the square root of 1 minus rho squared Z2 is also a standard normal random variable. So this is the standard result. It's certainly easy to see that the expected value of this random variable is 0, and the variance of it is indeed 1. 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 T1 for the two securities A and B have identical risk-neutral distributions. And likewise the stock prices at T2 have identical risk mutual 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 1. And expiration T2. In order to knock in, the stock price at time T1 must exceed the barrier price of 1.2. So therefore, the payoff function is given to us by the maximum of 1 minus ST2 and 0, which is the payoff of a regular put option, times the indicator function. Which is 1, if the stock price at time T1 is greater than or equal to 1.2, and 0, otherwise. So, you can think of this payoff as follows, or of the security as follows. So, this is time T here, this is the stock price at time T. You can image the stock price starting off at a value of 1. Maybe this is 1.2. This might be time T1, this might be time T2. And, you can see that in order to get a payoff, what must happen is that the stock price must, some way or another, get above 1.2 at time T1. And then, at time T2, it must be below 1 in order to be in the money. So, it can still move about, but its got to come down, and be below, the strike level of 1. 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 T1, and it must be below 1 at time T2, in order to get a payoff. So now the question we can ask is as follows. Would 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 depend on whether the underlying security is stock A or stock B? And 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 T1 or T2 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 fa for stock A of ST1a and ST2b, so this is the joint risk neutral distribution for security a at times T1 and T2. And similarly we've got the risk neutral distribution, the joint risk neutral distribution for stock b at times T1 and T2. We know these, they have the same marginal risk neutral distributions. That was the point of the previous slide. But what about the joint distributions? Remember, if we evaluate this option or compute the price of this option It's going to depend on fa in the case of security a or fb 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 0, close to minus 1. Well, if z, if the stock price of time T1 is above the barrier 1.2, that means Z1A would have to have been large. But if Z1A is large then rho a times Z1A will be negative because rho A is now much smaller than 0. So in that case, this will be negative and therefore you have a much better change of this entire quantity being negative. And the stock price of time T2 being below the strike of 1. So in fact a negative value of rho A and the more negative, the closer to minus 1 the better, will help the option xbar 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, let's call it P0a 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 will 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 not input option. And depending on the value of 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 any security who's value depends on the joint risk neutral distribution cannot be priced just using the information in the implied volatility surface. Moving on, lets talk about derivatives pricing in practice and in general. So we've seen the dynamic replication theory of Black, Scholes, and indeed Merton. Its' very elegant, we saw it as well in the binomial model. But it is not possible to dynamically replicate and, therefore, price derivative securities in practice. And this is for multiple reasons. First of all security prices don't follow geometric bounding motions. We don't know the particular processes that they do follow, nor do we know the exact parameters that govern those processes. It is also true, that you can not trade at every point in time, continuous trading is not possible in practice, transactions cost would render it impossible and so on. So, dynamic replication really is only something we can hope to do approximately. Instead actually supply and demand is what sets derivative security prices. This is particularly the case with the most liquid securities like European and American options in the fixed income markets. This is also true of caps and floors, swaptions and so on. We have also seen these securities earlier in the course. And indeed volatility is itself an asset class. Well, what do I mean by that? Well, I mean the following, sometimes people want to trade volatility. They have a view that volatility will increase or maybe they have a view that volatility will decrease. In that case they want to buy volatility or they want to sell volatility. And people treat volatility as an asset class. And indeed it is possible to buy volatility by buying a European call option for example or buying a European put option. We know the value of the European call option or put option will increase, as volatility increases. Remember the vega, which is equal to delta c delta sigma is positive, so we know that European call and put prices increase as volatility increases. So by buying a European call put option, your actually buying volatility. And so volatility is viewed as, as an asset class, you can actually buy volatility. It's also true, by the way, of other concepts, like correlation. Correlation can also be viewed as an asset class, and indeed there are mechanisms in securities out there, that will enable you to actually buy correlation or indeed sell correlation. So returning to this point, supply and demand sets the rate of derivative prices in general. And this is true of derivative securities in other markets. Fixed income derivatives, FX derivatives, credit derivatives, commodity derivatives and so on. Most derivative prices in these markets are determined by supply and demand. That having been said, derivatives pricing models are still needed. You need it for two principal reasons. Number one, to price exotic and other less liquid derivative securities. Remember if you have, for example, European call and put option prices, yes you can use that to construct an implied volatility surface. And you can use your implied volatility surface to price some types of derivatives securities. In particularly those securities whose pay off only depends on the underlying stock price at a fixed point in time. But there are other more exotic derivative securities like barrier options for example, who's value depends on the joint risk neutral distribution. You can not see this in the marketplace. You can not determine it from the volatility surface. And so you need models, arbitrage free models to price these more exotic and less liquid derivative securities. We also need derivatives pricing models to risk manage portfolios of derivatives. So, we can do this via the Greeks or via scenario analysis that we saw earlier. So, when I say via the Greeks, I mean the following. I could have a derivatives portfolio and compute the overall delta, delta P, delta S. So, P is the value of my derivatives portfolio. S is the underlying security. I can compute delta P, delta S, and this is the delta from my portfolio. And I can actually hedge this exposure to the underlying security by buying some stock. So, for example, suppose this is equal to $10 million. Well, if I then go out into the marketplace and short $10 million of the underlying security, then my portfolio will have a net delta of 0 and I am delta hedged. You can do similar sorts of hedging with vega risks. Maybe my delta P delta sigma is equal to some quantity. Maybe it's $100,000 say. Well, what I can then do, is, if I want to, I can go out into the marketplace, and buy or sell a security which has a vega of minus $100,000. By adding that to my portfolio, the new net vega in my portfolio would be zero. And so, in fact, that's how people will often use the Greeks in practice. They will use it to hedge away risks that they don't want. So, that's what I mean when I say, we need derivatives models to risk manage portfolios via the Greeks. Because we could, we can compute the Greeks from models, like the Black-Scholes model, for example. We can also risk manage portfolios using scenario analysis. And we saw an example of this in an earlier module. Where if you recall, we had a portfolio of options on the SMP 500 and we considered various scenarios across the top. We were stressing volatility, so we were stressing sigma and down here on the, on this axis we were stressing the underlying security. So, we were able to look at the P and L in the portfolio as the implied volatility or the implied volatile surface is moved to different values. Likewise, as the underlying security is stressed or changed to different values. So, in each of these scenarios here, I need to be able to recompute the value of my portfolio. And that means actually having a model to recompute the prices of the securities in my portfolio and calculating the PNL from this scenario. So, certainly derivatives models are needing in practice to price exotic and other less liquid securities, but also to aid in the risk management process. These models are arbitrage by construction and they are calibrated to liquid security prices. We saw an example of calibration when we were calibrating the Black Derman Toy model to the term structured interest rates. Note however that these models are only an approximation to reality. And generally they are not a great approximation. For example, witness how often they need to be re-calibrated. I mentioned in the past that when you're re calibrating these models, often you have to do so several times a day. Of course, if a model was the correct model you'd only need, you would only need to calibrate it once and you would be done with it. No more calibrations would be required In practice people are always having to re-calibrate their models. And that's just another indicator of how these models are at best only an approximation to the real world. These models also generally completely ignore the endogeneity of markets. What do I mean by the endogeneity of markets? I mean the following, most of these models do not account for the fact that the actual trading of these securities can move the prices of these securities. And if too many people enter into a market and all buy the same security, well, that's going to change the price dynamics of that security. In particular, if a market panic occurs, if people become suddenly very concerned about that security, everybody will try to sell at the same time. And security price will collapse. So, that's what I mean by endogeneity, what the market is actually doing. The trading of that security is going to change the price dynamics of that security, and this can be an extremely important characteristic of the financial markets. Certainly have played a role in the financial crisis of 2008 and beyond, when many people were holding the same types of securities. Many people ran for the exits at the same time, and so the endogeneity of the markets was actually missed by many participants. All of that having been said, the ideas of dynamic replication have not been abandoned and they're still useful. These ideas are still used to partially hedge derivatives portfolios in the same manner as I explained up here. So, just to summarize, the concept of exact dynamic replication is only a theoretical construct. You cannot exactly replicate a security in practice. But the ideas of dynamic replication are indeed still useful and they are still used by participants to perform risk management and so on.
