Moving on, let's talk about derivatives pricing in practice, and in general. So we've seen the dynamic replication theory of Black-Scholes, and indeed, Merton, it's 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 cannot trade at every point in time, continuous trading is not possible in practice. Transactions costs 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 for the most liquid securities, like European and American options, in the fixed income markets, it's 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'll 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 and put option, you're actually buying volatility. And so volatility is viewed 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 and securities out there that enable you to actually buy correlation, or indeed, sell correlation. So returning to this point, supply and demand sets derivative security prices in general. And this is true of derivatives 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, they're needed 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 derivative securities. In particular, those securities whose payoff 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, whose value depends on the joint risk mutual distribution. You cannot see this in the marketplace, you cannot 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. We can do this via the Greeks, so we can actually hedge by trading. 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 stocks. 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 risk. 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 of my portfolio will be 0. 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 S&P 500, and we considered various scenarios across the top. We were stressing volatility, so we were stressing sigma. And down here on this axis, we were stressing the underlying security. So we were able to look at the P&L in the portfolio as the implied volatility, or the implied volatility 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 P&L from this scenario. So certainly, derivatives models are needed in practice to price exotic and other less liquid securities, but also to aid in the risk management process. These models are arbitrage-free 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 structure of 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 recalibrated. I mentioned in the past that when you're recalibrating these models, often you have to do so several times a day. Of course, if a model was the correct model, 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 recalibrate their models. And that's just another example, or 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? Well, 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 the 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, it played a role in the financial crisis of 2008 and beyond, when many people were holding the same types of securities, the ABS, CDO markets, which we'll talk about soon. Many people ran for the exits at the same time, and so the endogeneity of the markets was actually missed by many participants. They didn't take it into account when they were trading these instruments. 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.