It is time to extend the Black-Scholes-Merton Model. Historically what happened, the first generalization on Black-Scholes-Merton, was stochastic volatility models. It was a natural thing to do. Thinking about Black-Scholes Model in which everything is driven by volatility, which was assumed constant. It was then natural to think, maybe we can get better models, the models that better fit the data, if the model Sigma is not being constant, but is a random process evolving in time. That's what we are going to present here. First, let me tell you the case of complete market models in which there will still be only one source of randomness, meaning only one Brownian motion. Since we have one Brownian motion and once stock, it will be still a complete model that will be unique price for everything and all the claims can be replicated. That's still not going to be necessarily good for fitting data. But it's the first historically step that was done and also the first one that we're going to look at here. Simply what you do, you assume that your parameters may be the interest rate through and the volatility are functions of current time and current stock price. If you do this, you can show that there is still possible to replicate all the claims so that the market will be complete. Roughly speaking, and there's some technical conditions, Brownian motion model is complete if you have the same number of Brownian motions, so the same number of sources of risk, as the number of risky assets. If you have three stocks and three Brownian motions, your model is going to be complete. But if you have two stocks and three Brownian motions, typically the model will not be complete because there is an extra source of randomness that you cannot hedge. Well, here there is only one Brownian emotion and one risky asset. If you write like this, that means that the stock at maturity is of this form. When your parameters depend on time and are random, or whether they are random or not, doesn't really matter as long as they depend on time, what happens wherever we had multiplication by T-t, we will have integrals. Before we had in the Black-Scholes Model, we just had here r minus Sigma squared over two times T minus t, like this. But now instead of that, you integrate r minus sigma squared over two overall time, USUDU. Again it's a simple application of Ito's rule to show that the second equation is in fact the solution to the first equation here. It's exactly the same, it is a rule as we had before for the exponential function. Because it's the same Ito's rule, the partial differential equation is going to look the same. The partial differential equation for the value of a European option, which pays a G or a sub T at maturity, looks exactly the same, except R is a function of T and S and Sigma is a function of T and S. But that doesn't really change much conceptually because everything was a function of TS anyway. This C is also a function of T and S. This derivatives CT and CSS, they are all functions of T and S. You just get them off complicated partial differential equation. It may not always be possible to solve it explicitly but you know how to write it, so you can try to solve it numerically. Or you can use the expected value formula, martingale risk-neutral pricing and try to compute the expected value. Conceptually, nothing too different, in fact, nothing different from Black-Scholes Model except typically we will not have explicit formulas for a call option or a put option. Here's a one, not recent but early model of this type, so-called constant elasticity of variance or CEV model. Where the through the function, it doesn't depend on time directly, it depends on the underlying S in this way, it's some constant called Sigma. Again, over stock price to some power between zero and one. The economic reason for this is empirically in the data, historically usually volatility of the stock market, let's think of maybe S&P 500 large stock index. Typically volatility of the market, it goes up when the times are bad, when the market value goes down, and when the market value is doing well and it's high, and then typically volatility is low. That's what they're trying to capture here in this model, the higher stock price means lower volatility, lower stock price means high volatility. The reason why you put Beta only up to one is mathematical reason. If you put the power higher than one, this stochastic differential equation may not have a solution. It's the same with the ordinary differential equations, if you don't have linear growth, the solution may not exist. That's a mathematical reason why you bound the Beta to be less than one. That was the first historically and early attempt to generalize Black-Scholes model. In fact, if you're thinking about how realistic is Black-Scholes in terms of having normal distribution for the logarithm of your stock prices, there are theorems that show you that pretty much any continuous distribution can be obtained by appropriately choosing function here of S. This is now much more flexible in the sense that you can get all distributions for the stock, it doesn't have to be normal or log-normal, it can be pretty much anything. However, that's the distribution for a single time, but to help properties solve the process how it evolves in time, you're still restricted if you only assume one Brownian motion like we do here. It turns out in the data you typically still you need what are called the multi-factor models. To match the data, you need more than one random factor, in this case, more than one Brownian motion. That leads us to the part with more than one Brownian motion. Before I go to more than one Brownian motion, just to mentioned this fact, it's really going to be also in your homework. That if you have a special case when r and Sigma are deterministic functions of time, all that happens if you have a Black-Scholes formula like you have for a call option, all that happens is, here is the rule, you replace Sigma squared times time to maturity by its integral. Integral from lowercase t to uppercase T of Sigma squared u du, and if r is also deterministic function of time, you replace r times time to maturity with the integral. That's maybe not so surprising for this part with the interest rate. The reason why you also do it for volatility like that is the following mathematical fact which can be useful to know. If you look at the stochastic integral of Sigma u dW where Sigma is deterministic, then that has normal distribution. As a random variable that has a normal distribution with mean zero and the volatility, this integral of Sigma squared u du. The intuition is the sum of normal random variables is a normal random variable, linear combination of normals is a normal random variable. Integrals are just limits of sums, so in the limit, you still have a normal random variable. We know that even if Sigma is not deterministic domain is zero, so that's we had that fact before. We also know we had also formula for the variance, and the formula for the variance was expectation of this integral here. But I'm going to remove this, let's see. Here you would have expectation, but since Sigma is deterministic, then that expectation is not necessary. That's just more of curiosity because if you are going to make your parameters r and Sigma move in time, you might as well make them random because if they're changing, it's unrealistic to assume they're changing according to calendar time. Calendar time doesn't have much of an economic meaning, they probably change because some economic factors are changing, not because a calendar time is moving along. People don't really use models in which r and Sigma are deterministic functions of time. But it's a nice fact to know here.
