In this module we're going to spend a little bit of time discussing how to risk manage options portfolios. We're going to briefly discuss two methods. The first method is based on the Greeks. We're going to use a delta-gamma vega approximation to hedge against relatively small changes in the underlying security price and the volatility parameter sigma. However, that method will not work well, when these changes in the underlying security price on the volatility parameter are substantial. In that situation, we would use scenario analysis instead. And we will spend a little bit of time discussing scenario analysis as well. Once again, we have here the Black-Scholes Formula. I just want to emphasize that the Black-Scholes Formula gives us closed form or analytic expression for the price of European call and put options in the Black Scholes framework. That is where it is assumed the stock price follows a geometric Brownian motion with these dynamics where we can trade continuously in time but no transactions costs. And we're short selling of the stock is allowed. So Black and Scholes, using a replicating argument that we also use in the case of the binomial model, showed how to compute the prices of call and put options in this framework. And indeed they came up with this, The Black-Scholes Formula. In earlier modules, we saw how we can compute the delta. We know that the delta is equal to delta C, the delta s. We also saw how to compute gamma, which is equal to delta 2c delta squared. We also saw how to compute vega. So vega was equal to delta c delta sigma. So these are just partial derivatives of the option price with respect to the parameters s and sigma. We also saw, indeed, how to compute theta, which is equal to the negative of the partial derivative of the option price, with respect to time to maturity. So it's very straightforward to compute these quantities, just by taking derivatives appropriately, inside here. And, indeed, in the case with put option, we can also compute these expressions very easily. Sometimes simply using put-call parity, in fact, by put-call parity, it is easy to see that the gamma and vega of column put options are identical. So at this point we have the Black-Scholes Formula and I'm going to assume that we know how to calculate these quantities as well. These are easy to calculate programmatically, one can do in Excel or indeed an R, Python and any programming language that you like. Let's consider some approximations. We're going to view the option price as a function of s and sigma only. Then a simple application of Taylor's theorem. Now Taylor's theorem is a theorem, I hope you saw in your undergraduate mathematics class. If you haven't, don't worry about it. What it does is the following, it enables us to see what happens to the call option price for small changes in s and small changes in sigma. In particular, suppose we let s go to s plus delta S. So delta S represents the change in the underlying stock price, and sigma goes to sigma plus delta sigma. So delta sigma represents the change in sigma, the volatility parameter. Well, then Taylor's theorem allows us to say that this option price at the new parameters s plus delta S and sigma plus delta sigma is approximately equal to the option price at the original parameters as in sigma plus delta s times delta C delta S. Plus a half delta S squared, times delta 2C, delta S squared plus delta sigma times delta C, delta sigma. So we recognize that delta C, delta S is equal to delta, delta 2 C, delta S squared is equal to our gamma term. And delta C delta sigma is equal to our vega term. So we therefore get that the P&L, remember the P&L will therefore be this term minus C s sigma. So I bring this over to this side and I get the P&L on the left-hand side, P&L standing for profit and loss. So P&L would be the delta times the change in the stock price plus gamma over 2 times the change in stock price squared plus vega times sigma. And so what we have is that the profit and loss on the option price when the stock changes by an amount of delta S and the volatility changes by the amount of delta sigma then that profit and loss is equal to the a delta component which is this. And gamma component which is this and a vega component which is this. Now I should mention as well if I wanted to I could also include time to maturities, another parameter. So I could have T and T plus delta T in here and then I would also have a theta component. So that's perfectly fine as well and, indeed, people do this. But to keep things simple, I just want to stick with delta, gamma and vega here. When in fact, sometimes people just work with delta and gamma. So if I assume delta sigma equals 0, I will obtain a delta-gamma approximation. So the P&L in this case will just be due to delta and gamma. And this is often used, for example, in historical value risk calculations. Now we won't go anymore into value risk for option portfolios here but I know you've seen Value-at-Risk elsewhere in the course.