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 paramenters include the underlying security price, the underlying volatility paramenter sigma, as well as the time to maturity. We're going to focus on to 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 striking and expiration capital T. It is given to us by this quantity here d1 and d2 were given over here. And, capital N refers to, 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 assumed to satisfy or follow geometric Brownian motion dynamics for Wt as a Brownian motion. The Greeks refer to the partial mathematical derivatives of a financial derivative security price with respect to the modern parameters. So I emphasize here that we've got the word derivative appearing in two different contexts. Sometimes we refer to it as 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 an awful lot in industry. They refer as I said here, to the partial mathematical derivatives of the financial derivatives 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. And so the delta for European call option price is given to us by this, iIt'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 5, but it actually requires a somewhat tedious calculation. If you just look at this expression, then it does indeed appear to be the case. The delta C, delta S is equal to e to the minus cT, N d1. And indeed that is what we have over here, but don't be fooled by this, actually there is a little bit more work involved, because d1 itself depends on S and therefore, d2 which appears over here also depends on this. 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 a put option is equal to C0 plus Ke to the minus rt, minus S0e to the minus cT. And so therefore delta P, delta S is equal delta C, delta S minus e to the minus cT. So once we know the delta of call option, which is given to us here. We can easily calculate the delta for European put option as well, thatâs 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 state it on the slide here, that the stripe 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 a call, the payoff at 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 and that's why the delta of a call option is greater than zero. 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 zero. 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 thats why I'm using the approximent sign here, this is approximately equal to S minus K. If S is very large, and 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. And by very small, I mean S is much smaller than K. And in particular, it is small enough that the chance is of exercising the option are approximately 0. And then otherwise, where intermediate values of S, well, we can calculate 0 as been just a BlackâScholes formula. The important thing to note here is that delta C delta S is therefore going to be equal to 1 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 a 10 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 of a put options. Well, 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. And that is approximately 0 for S being very large, and for intermediate values of S we would actually use the Black-Scholes formula. So I'll use BS for Black-Scholes formula, and here for intermediate values of S. The important thing to note though is for S very small, then I can use this expression here. And the derivative of this with respect to S is minus 1 and 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. It's 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 over 1. And similarly the delta for a put option, is approximately minus 1or 0. Here, we've 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 two and a half weeks. T equals 0.25 years, so approximately three months and T equals 0.5 years corresponding to the six month expiration. So we see here the deltas. Notice again, as in the previous slide, for S sufficiently large or small, the delta is going to go to 0 or1. But notice that they go to 0 or 1 faster for smaller time's maturity. So in other words, if we'd look at the case where t equals .05 which corresponds to two and half weeks. We see that S doesn't have to be too far away from the strike of 100. Before it's delta goes to 1 or 0 and that's because with only two and a half weeks to maturity. There's not much chance for the strike to either get into the money if its down low, or to fall out of the money if its up high. And 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. And so that's why we see the red curve corresponding to T equals 0.5 years. Moving to 0 or 1 faster than the options with maturity T equals 0.25 years and T equals 0.05 years. And the same argument of course also implies that the T equals 0.25 year option. The delta of this also goes to 0 or 1 faster if you like than the option that 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 have time to maturity down here. So, 1 corresponds having one year to maturity, 0.5 corresponds to six months to maturity and 0 corresponds to having 0 time maturity. We've got three different options. We have at-the-money option, so at-the-money has K equal to the current value off the stock price, ITM stands for In The Money. So a 10% in the money option, means that S0, is equal to 1.1 times K. So therefore, it is in the money and the 10% out of the money, OTM option stands for an option with a strike that satisfies S0 equals 0.9 K. And so, what we are seeing in this case makes sense. We see that for the option that is out of the money, it is 10% out of the money, so 0.9k, the current stock price is less than K. So if the option were to expire today, you'd get nothing. And therefore, what we see is that the delta decreases, and it decreases toward 0 as the time to maturity decreases toward 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 S, T, is equal to 1.1 k, well this would be equal to 1 K, so it's in the money. And what we see here is, that, as the time to majority goes towards 0, the delta of this in the money option, goes towards 1. And that be, and that is because as the time to maturity goes towards 0 would become more and more likely to exercise the option. And 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 in the 10% out of the money case well then this is going to behave like 0. If we're 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 S0. Well, then in that case, and I'm talking approximately here, the chances of exercising approach 50%. So, basically, there's a 50% chance of exercising and 50% chance of not exercising. And it turns out and it can be confirmed by differentiating the Black-Scholes formula, or calculate the expression we saw on the earlier slide that the delta actually approaches 0.5. The gamma of an option is the partial derivative of the options delta with respect to the price of the underlying security. So, the gamma measures the sensitivity of the option delta to the price of the underlying security. The gamma of a call option is therefore given to us by delta 2C delta S squared, and again it's somewhat tedious but it can easily be calculated using basic calculus. We can take the partial derivatives of the BlackâScholes formula to calculate the gamma. If we do that we will find a sequel to this expression here. E to the minus C time T, N of d1 divided sigma S square root T. How about the gamma of the European put option? Well that's easily calculated from put-call parity. So put-call parity is given to us here. So therefore, we can actually say that P, is equal to C, plus e to the minus rt times K, minus e to the minus cT times S. So we can therefore, see the delta 2p, delta s squared is equal to delta 2c delta s squared. Well, plus 0 minus 0 because the second partial derivative of this is equal to 0. And the second partial derivative of this with respect to S is equal to 0. So therefore, we see that, see that the gamma for put option is equal to the gamma for a call option. So, once we know the gamma for European call option we therefore we have the gamma for European put option. And in fact, you can see that this expression is always greater than are equal to 0. So the gamma for European options is always positive. this is due to what's called option convexity. Here is the plot of the gamma for European options is, as time to maturity varies. So, the gamma here is a function of the stock price, and we've got three different times to maturity. 0.05 years, 0.25 years and 0.5 years as we saw before. Notice that the gamma is steepest for the shortest maturity. So, in this case, for T equals 0.05 years i.e approximately two and half weeks to maturity, we see that the gammas are very steep around the strike of K equals 100. But we also see that it falls away to 0 much faster than the option when T equals 0.25 or the option when T equals 0.5 years. The reason is as follows, the delta of the European option when T equals 0.05 years, well, it's going to be a half, or approximately a half when the stock prices at the strike K. But as the stock price goes up, the delta's going to move towards 1 and it's going to move towards 1 much faster than the options with the higher time to maturity. Similarly, as the stock price falls below the strike of 100, the delta of the call option is going to move towards 0. And it's going to move towards 0 much faster than the delta of the options with times to maturity of 0.25 and 0.5 years. So this actually is, another, this plot here is just another way of looking at this plot. In the option, where T equals two and a half weeks or 0.05 years to maturity. We see that when S eqauls 100 that delta is approximately 0.5. But for small moves of, of S above 100 the delta quickly goes towards 1. And for small moves of S below 100 the delta quickly goes towards 0. And does it does much faster than the options with larger times to maturity. So we're seeing without the option which is two and half weeks to maturity has a much higher gamma than the option when T equals 0.5 years or T equals 0.25 years to maturity. And another way of looking at this is to now plot the gamma as a function of time's maturity. So one year to maturity, six months to maturity zero time to maturity. We see that for the option that's out of the money, 10% out of the money or 20% out of the money. I mean this would also be true for options that are in the money, the gamma of those options actually goes towards 0, it falls towards 0. And that's because as the time to maturity goes to 0, we know for sure we're not going to be exercising if we're out of the money. And we know for sure that we are going to be exercising if we're in the money. In other words, delta will be 1 if we're in the money, delta will be 0 if we're out of the money, and so gamma will be 0 in both cases. On the other hand, if we're dealing with an at the money option, where the current stock price is equal to K. So here is where s is equal to K for this blue curve. Well, then as the time to maturity goes to 0, we're going to find that our delta's equal to a half but that the gamma will actually be very, very large, and grow very large. And that's because small moves in s will move the delta to either 1 if S increases, or 0 if S decreases. And so we get a very large gamma for at-the-money options. I should mention as well by the way, that with all of these plots that we're looking at of delta and gamma market practitioners understand this behavior. They understand it at an intuitive level. They know how the delta of a colon European put option behaves, they know how the gamma of these options behave. And so, it's very important, that if you're working with options and practice, that you understand these figures and you understand why they behave, and look the way that they do.