In the last module we saw Delta and Gamma, in this module we're going to see Vega and Theta. Vega is the sensitivity of the option price with respect to changes in the parameter Sigma, the volatility parameter Sigma. Whereas theta is the sensitivity of the option price with respect to changes in the time to maturity. So we're going to discuss Vega and Theta in this module, we'll see how they behave as a function of the underlying security price and indeed as a function of the underlying term to maturity. It's very important that we understand how all the Greeks work, and the Vega and Theta are very important Greeks in practice. Again we have here the Black-Scholes formula, we're going to use the Black-Scholes formula in this module to compute the Vega and Theta of an option. So just to remind ourselves, the Black-Scholes model assumes that the stock price follows a geometric Brownian motion, so that the stock price at time Lt is equal to S0e to the R minus C minus Sigma squared over two times t, plus Sigma Wt, where Wt is a Brownian motion under the risk neutral probability distribution of Q. So this is the formula here, we saw in the last module what the Delta of an option is, we also saw what the Gamma of an option is, and we could calculate the Delta and Gamma by taking the appropriate derivatives of this expression here for a call option, likewise we could do the same for a put option, or if we like we could use put-call parity to compute those expressions for put options. So first let's deal with the Vega. The Vega of an option is the partial derivative of the option price with respect to the volatility parameter Sigma. So the volatility parameter is this parameter over here. Now if you stop and think about it for a moment, you might think that as Sigma increases, the value of the option will increase, and indeed that is true. For example, the payoff of a call option capital t, CT is equal to the maximum of zero and ST minus K. Well, it makes sense the sigma gets larger, the value of the security will also increase. An easy way to see that perhaps is imagine that S0, the initial stock price, is much less than the strike K. Well in that case, if Sigma is very small, the chances of the stock price growing enough so that the option ends up in the money, will be zero or approximately zero. On the other hand, as sigma gets sufficiently large, the probability that ST will be greater than K, will actually increase in which case the option value will be non-zero. So it makes sense that the call price, the initial price of the option C0, should be increasing and Sigma, and we will see that that is indeed the case. The Vega of an option is the partial derivative of the option price with respect to the volatility parameter Sigma. Vega therefore measures the sensitivity of the option price to Sigma, and using the Black-Scholes formula it can easily be calculated Vega is equal to Delta-C, Delta-Sigma, which turns out to be E to the minus CT, as square root of time to maturity times Phi of d1, where Phi is the probability density function of a standard normal random variable. Now, we can also compute the Vega for European put option by using put-call parity. So this is put-call parity here, remember, so it implies that the put price is equal to the call plus E to the minus RT times K, minus E to the minus CT times S. So therefore, we can actually compute Delta-P, Delta-Sigma, we see it's equal to well, Delta-C, Delta-Sigma, that's the first term here. Then these other two terms don't depend on Sigma at all, so it's plus zero or minus zero, and so we see Delta P Delta-Sigma equals Delta-C, Delta-Sigma. So the Vega of a European put option is the same as the Vega of a European call option. Here's a question for us, is the concept of Vega inconsistent in anyway with the Black-Scholes model? The answer is yes. If you recall, the Black-Scholes math model assumes that St, the stock price of any time T is equal to S0, E to the Mu minus Sigma squared over two times T, plus Sigma times Wt where Wt is a standard Brownian motion. Mu and Sigma are constants in this model, they are not assumed to change, and that indeed was the assumption of the Black-Scholes model. They assumed continuous trading, they assumed that there were no transactions costs and that short sales were allowed, and that borrowing or lending at the risk-free interest rate r was also possible. Using these assumptions, they constructed a self-financing trading strategy that replicated the payoff of the option, and that is indeed how they obtained the Black-Scholes formula. Nothing in their model allowed Sigma to change, Sigma was a known constant, and yet when we're talking about Delta-C, Delta-Sigma, we're implicitly recognizing the fact that Sigma can change and indeed in the marketplace, Sigma does change. So in that sense, mathematically one can always define Delta-C, Delta-Sigma, there's no problem with that. Within the economics of the Black-Scholes model, it is inconsistent to talk about Sigma changing, because we obtained the Black-Scholes option price under the assumption that Sigma could not change. Here are some plots of Vega for European options as a function of the stock price at time T equal to zero, and as the time to maturity varies. So we've got three different times to maturity, T equals 0.05 years, T equals 0.25 years, and T equals 0.5 years. There are probably two things to notice first. The first observation is as follows. Note that if I pick any one of these options, the Vega goes to zero as the stock price moves away from the strike which was $100 here. So what is going on here? Well it's very simple. So again returning to what we did in previous modules, we know the following. Let's take a call option as our example. We know that the call option price will be approximately equal to, and again ignoring interest rate factors and so on, it will approximately be equal to S0 minus K, for S0 being very large. By very large, I mean much larger than K and sufficiently large that I'm almost certain I'm going to be exercising the option. It will be equal to zero for S0 being very small, and very small here means much smaller than K and indeed small enough that the chances of exercising the option are approximately zero. Well we can see here that Delta-C, Delta-Sigma, therefore must be equal to zero in this situation because the partial derivative of S0 minus K with respect to Sigma is zero, and also zero down in this situation as well. Therefore for S0 very large which is up here, or S0 very small which is down here, we see that Delta-C, Delta-Sigma goes to zero. Indeed that's what we see for each of these three options. So that's the first observation. The second observation, so let's call this observation one. The second observation here is that the Vega increases in time to maturity capital T. So we see that the option where T equals 0.5 years, the blue curve here is larger than the Vega for the option when T equals 0.25 years and so on. In fact this is not surprising because if we go back to the Black-Scholes formula over here, we can see that every place where sigma appears, we find that together with the square root of T, or if you like when Sigma squared appears in the Black-Scholes formula, I see a T appearing. So I've got Sigma square root T or Sigma squared T appearing here. So basically every time I see a Sigma, I'm multiplying it by the square root of T, and therefore the impact of a change in Sigma, i.e. the Vega, it will be amplified by the square root of T. It is therefore the case and by the way maybe I should have mentioned, or it was assumed to be equal to C, it was assumed to be equal to zero in these plots here. We can see that the blue curve is a factor of square root of T, which is approximately equal to 1.4 times higher than the green curve. The green curve is a factor of the square root of five which is approximately equal to 2.2 something, greater than the red curve. That's no surprise, because 0.5 years divided by 0.25 years is equal to two. So the square root of two is approximately 1.4 and indeed we see the green curve reaches a peak of 20 here, 1.4 times 20 is 28, and that's roughly the peak of the blue curve. Likewise, down here we have the red curve reaching maybe a peak of approximately 8.5 or nine multiplied by 2.2, brings us up towards approximately 20. So in fact this behavior is entirely predictable, the change that the Vega for the option is magnified by the square root of the time to maturity. Another way of seeing this is by looking at this figure here where we have plotted the Vegas for three options and not the money option, at 10 percent out of the money option, at a 20 percent out of the money option, and in all three cases we see that the Vega converges to zero as the time to maturity goes to zero. Indeed this would also be true if I showed a 10 percent in the money option, or a 20 percent in the money option.
