Paul Wilmott on Quantitative Finance, Chapter 7, The Greeks #Finance

in chapter 7 I learned about the Greeks so Greeks so what are the Greeks the Greeks are Greek letters and some other names that represent various mathematical quantities so one of them we’ve seen before Delta that was partial of V with respect to S we saw this in the uh black Sholes

black Sholes so when we dve black shols we saw Delta and what was Delta Delta was the amount of stock we had to short so this is the um we short the asset by a factor Delta to balance out the long option and that was the instantaneous time step as we derive black shols all right so what does this

represent this represents the sensitivity to changes in the underline so as the asset price changes the option price changes and the Delta represents you know the ratio there so if if Delta is three the asset price goes up one then the option price should go up three but that’s just a linear

model that’s not the whole story so then we come to the next one gamma this is the second derivative with respect to S so gamma is a sensitivity of Delta to changes in the under line so what happens s changes then that makes Delta change and we have to re hedge right so if the asset price if

if we’re perfectly hedged and shorting the asset you know following the black shs model so we we’re long an option and short the asset and then asset price changes Delta will change by an amount gamma so let’s draw a little diagram trying to make this visual okay here is s and

first I’m going to draw the payoff here’s the payoff this is a call option so here’s the strike price here now I’ll draw the value of the option before the payoff it’s not very good I’ll do that again so this is V of s so it’s sort of a a curve here so if

we look at a specific point so up here what’s Delta Delta equals 1 down here Delta is about zero and around here maybe Delta equals a half and so this is how much stock we have to short to to hedge changes in the option price this is the the option as a function of the asset price what about

Gamma so if you think about the second derivative if the second derivative is positive you get something like this second derivative is negative you get something like this just parabas so which way does the second derivative go it opens upwards like this and so gamma is greater than zero if gamma

is greater than zero you can say that we’re long Gam gamma so if we’re doing the black sh strategy and we’re long the option and short the stock for a call option we’re long gamma what does that mean that means as we re hedge every time we re hedge we have a

Profit but that’s not all es because if the asset asset price is not changing then we’re we’re losing money basically we’re not gaining enough money to make up for the cost of our position so you might imagine that gamma zero so we could be neutral gamma and

so neutral gamma is nice because that means that we don’t have to re hedge very often so it saves hene costs so if they are transaction cost like it costs money to buy and sell over and over again to hedge every little time period if we’re gamma neutral that means that as s changes

Delta won’t change very much I mean it might change a little bit because this is not a perfect um this is a linear model of how fast Delta will change and there’s higher order terms but this will make make Delta change not as fast as it would otherwise which means that our hedging will

be simplified and so what’s an example we might have an exotic option and so we Delta hedge by shorting the underlying and then gamma Hedge by buying and selling vanilla options and so this means that we’ve made it so that the value of our exotic option as the asset price changes we we

don’t change the value of the Exotic option but then our Delta hedge might change but then because we gamma hedged that made gamma zero and that means that our Delta doesn’t change very much as the price changes and so this means we’re almost entirely hedged so that the asset

price change doesn’t change any you know anything in our position but it’s not the whole story there’s more so there’s speed this is the third derivative oops of V with respect to S and so this is just another higher order term so this is the sensitivity of gamma to changes

in the underline all right so that’s the most that’s the highest order derivative that’s commonly used but you could imagine there’s even you know higher order stuff but that becomes ridiculous so what else do we have so another Greek is Capital Theta so

Capital Theta is the change with respect to T so it’s the sensitivity of the option price to time so for a call option this will generally be negative you’re losing value over time and for other other types of options it could be positive or negative all right now we

have another section I I think of this as parameter risk so first we have Vega this is definitely not a Greek letter but Vega is a change of V with respect to volatility so this is the sensitivity to volatility volatility so this is a little bit weird because the volatility is a parameter of the

black sches model so it it doesn’t you can’t really take the derivative to it and have it be that meaningful but if we sort of you know don’t think about that too much we can see that if you change the volatility the value of the option right now changes and so by by plotting out

Vega we can see how much risk we have for using the wrong volatility in our calculations and you might think that if you make Vega smaller there’s less risk to using the wrong volatility or misestimating volatility but there’s a little bit of a danger here so one little caveat is this

only makes sense if gamma has a single sign so all was positive or all negative if gamma changes sign then the Vega starts not having a an understandable interpretation it stops being useful so another Greek is row so row equals change of V with respect to R this is sensitivity to interest rate and

so again R was a parameter of our model so you know this is you have to have a little bit of faith that this makes sense but you can take the derivative with respect to R and think that as as our assumptions about the interest rate change how does that affect our Valuation of the

option and you can think if this is very large you might be at risk for having a bad estimate of the interest rate so pretty cool so the Greeks are a way to talk about option pricing and they actually give you a way to think about about how to implement Hedges so that’s pretty neat