In this module, we're going to use the Black-Scholes formula to compute the sensitivity of option prices to the underlying parameters. The underlying parameters include the underlying security price, the underlying volatility parameter sigma, as well as the time to maturity. We're going to focus on two of the so-called Greeks in this module. The Delta of an option and the Gamma of an option. The Delta of an option is the sensitivity of the option price with respect to the price of the underlying security. The Gamma is the sensitivity of the Delta with respect to the price of the underlying security. So we're going to discuss the Delta and Gamma in this module and see how they behave as time to maturity changes and as the underlying security price changes as well. So recall the Black-Scholes formula for the price of a European call option with strike K and expiration capital T, is given to us by this quantity here. D1 and d2 are given over here, and capital N refers to the cumulative distribution function of a standard normal random variable. R is the risk-free interest rate, c is the dividend yield, and the stock price St is seemed to satisfy or follow geometric Brownian motion dynamics where Wt is a Brownian motion.The Greeks refer to the partial mathematical derivatives of a financial derivative security price with respect to the model parameters. So I emphasize here that we've got the word derivative appearing in two different contexts. Sometimes, it refers to a security, a derivative security price, and sometimes it's going to refer to the mathematical derivative. So the Greeks are very important part of derivatives. They're used in awful lot in the industry. They refer, as I said here, to the partial mathematical derivatives of the financial derivative security price with respect to the model parameters. The first Greek we're going to consider is Delta. The Delta of an option is the partial derivative. Again, it's the partial mathematical derivative of the option price with respect to the price of the underlying security. The Delta measures the sensitivity of the option price to the price of the underlying security. So the Delta of a European call option price is given to us by this. It's Delta C, Delta S. Now, given the Black-Scholes formula over here, we can easily calculate Delta C, Delta S. It comes out to be e to the minus c times capital T times N of d1, where if you recall N is the cumulative distribution function for standard normal random variable. Now, this follows from five, but it actually requires a somewhat tedious calculation. If you just look at this expression, then it does indeed appear to be the case, that Delta C, Delta S is equal to e to the minus cT and d1, and indeed, that is what we have over here. But don't be fooled by this. Actually, there's a little bit more work involved, because d1 itself depends on S and therefore d2, which appears over here, also depends on S. So in order to calculate Delta C, Delta S, we actually have to take derivatives within this N(d1) term and indeed within this N(d2) term as well. If we do that using the chain rule and so on and then simplify everything down, it turns out that indeed we get Delta C, Delta S is equal to just this expression here. The Delta of a European put option is also easily calculated. One way to do this is to use Put-Call parity. So if you recall, Put-Call Parity implies that P0, the initial price for put option is equal to C0 plus Ke to the minus rT minus S0e to the minus CT. So therefore, Delta P, Delta S is equal to Delta C, Delta S minus e to the minus cT. So once we know the Delta of a call option, which is given to us here, we can easily calculate the Delta for European put option as well. That's given to us over here. So here, in this figure, we have plotted the Delta for a call option and a put option. So we've assumed although, it doesn't stated on the slide here, that the strike is K equal to 100. The first thing to note is that the call Delta is always between 0 and 1, and the Delta of a put is always between minus 1 and 0. Now, if you think about it, this makes sense because the payoff for a call option. So for call, the payoff of time t is equal to the maximum of St minus K and 0, and the payoff for a put option at time t, is equal to the maximum of K minus St and 0. So a call price, a call option, the value of a call option clearly increases as S increases. That's why the Delta of a call option is greater than 0. Similarly, the value of a put option will increase as S decreases, and that's why the Delta of a put option is less than 0. Something else to keep in mind here is the following. Note that as the stock price, the current stock price moves away from the strike, then the Delta moves towards either 1 or 0 in the case of a call option or towards minus 1 or 0 in the case of a put option. Now, what's going on here? Well, the easiest way to see why this is happening is the following. We know the value of the call option is given to us by the Black-Scholes formula. However, it is also true, and actually one can check this mathematically with the Black-Scholes formula, that the following is also true, it is equal to, and this is approximately, and I'm ignoring interest rates and so on here. So that's why I'm using the approximate sign here. This is approximately equal to S minus K if S is very large. By very large, I mean, it is bigger than K and much bigger than K, and indeed, it is so much bigger than K that it becomes very unlikely that you wouldn't exercise the option at maturity. Likewise, it is equal to 0 if S is very small. By very small, I mean S is much smaller than K. In particular, it is small enough that the chances of exercising the option are approximately zero. Then otherwise for intermediate values of S, well we can calculate C0 being just the Black-Scholes formula. The important thing to note here is that Delta C, Delta S is therefore going to be equal to one, which is Delta S, Delta S. For S very large and it's equal to 0 for S very small, and indeed, that's what we have here. It's 0 for S very small, and attend towards 1 for S very large. The exact same argument also holds true for the put price. We know that the put price, of course, is given to us by the Black-Scholes formula for put options. But we can also write this as being approximately, and again, ignoring interest rate factors and so on. This is approximately equal to K minus S, for S being very small. It is approximately 0 for S being very large. For intermediate values of S, we would actually use the Black-Scholes formula. So I'll use BS for Black-Scholes formula in here for intermediate values of S. The important thing to note though is that for S very small, then I can use this expression here and the derivative of this with respect to S is minus 1, and so that's why I'm getting minus 1 down here. Similarly, for very large values of S, the put price is 0 or approximately 0. Its partial derivative with respect to S will be 0, and indeed, that's what I have up here. So we see that for extreme values of S, the Delta of a call option is either 0 or 1, or approximately 0 or 1, and similarly, the Delta for put option is approximately minus 1 or 0. Here, we have plotted the Delta for three different European call options. The three options, all have the same strike K equal to 100, but they have different times to maturity, T equals 0.05 years, so approximately 2.5 weeks. T equals 0.25 years, so approximately three months, and T equals 0.5 years corresponding to a six-month exploration. So we see here the Deltas. Notice again, as in the previous slide, for S sufficiently large or small, the Deltas go to 0 or 1. But notice that they go to 0 or 1 faster for smaller times to maturity. So in other words, if we'd look at the case where T equals 0.05, which corresponds to 2.5 weeks, we see that S doesn't have to be too far away from the strike of a 100 before its Delta goes to 1 or 0. That's because with only 2.5 weeks to maturity, there's not much chance for the strike to either get into the money if its download or to fall out of the money if it's a Pi. Therefore, the Delta quickly goes to 0 or 1, depending on whether or not the current stock price is below the strike or above the strike. So that's why we see the red curve corresponding to T equals 0.05 years, moving to 0 or 1 faster than the options with maturity T equals 0.25 years and T equals 0.5 years. The same argument, of course, also implies that the T equals 0.2 five-year option, the Delta of this also goes to 0 or 1 faster, if you like, than the option with T equals 0.5 years. Another way of seeing this is looking at the Delta not as a function of the stock price, as we did on the previous slide, but as a function of time to maturity. So now, we've got time to maturity down here. So one corresponds to having one year to maturity, 0.5 corresponds up to six months to maturity, and zero corresponds to having zero time to maturity. We've got three different options. We have an at-the-money option. So an at-the-money option has K equal to the current value of the stock price. ITM stands for In The Money. So a 10 percent in-the-money option means that S0 is equal to 1.1 times K. So therefore, it is in the money. The 10 percent out-of-the-money, OTM option, stands for an option with the strike that satisfies S0 equals 0.9 K. So what we're seeing in this case makes sense. We see that for the option that is out of the money, it is 10 percent out of the money. So 0.9 K, the current stock price, is less than K. So if the option were to expire today, you would get nothing. Therefore, what we see is that the Delta decreases, and it decreases towards 0 as the time to maturity decreases towards 0. Similarly, the in-the-money option where S0 equals 1.1K, so therefore, remember the payoff of the option is equal to the maximum of St minus K and 0. So if St is equal to 1.1K, well then this will be equal to 0.1K, so it's in-the-money. What we see here is that as the time to maturity goes towards 0, the Delta of this in-the-money option goes towards 1. That is because at the time to maturity goes towards 0, would become more and more likely to exercise the option. So the option behaves more and more like St minus K, because this maximum is going to be equal to St minus K, as the time to maturity goes to 0, and of course, the partial derivative of this expression here is equal to 1, and that's why the Delta goes to 1. On the other hand down here, into 10 percent out-of-the-money case, well then, this is going to behave like 0. If we are out-of-the-money, it's going to behave like 0 as the time to maturity goes to 0, and therefore, the partial derivative of this will be equal to, 0 and that's what we're seeing here. Perhaps, the more interesting case is when the option is at the money and K equals 0. Well then in that case, and I'm talking approximately here, the chances of exercising approach 50 percent. So basically, there's a 50 percent chance of exercising, a 50 percent chance of not exercising, and it turns out, and it can be confirmed by differentiating the Black-Scholes formula or calculating the expression we saw on the earlier slide, that the Delta actually approaches 0.5.
