The Greeks

I’ve already discussed how to price vanilla options in the BS model. But options traders need to know more than just the price: they also want to know how price changes with the various other parameters in their models.

The way traders make money is just the same way that shop-keepers do – by selling options to other people for a little bit more than they buy them for. Once they sell an option, they have some money but they also have some risk, since if the price of the underlying moves in the wrong direction, they stand to lose a large amount of money. In the simplest case, a trader might be able to buy a matching option on the market for less than she sold the original option to her client for. This would cancel (‘hedge’) all of her risk, and generate a small positive profit (‘PnL’ or Profit & Loss) equal to the difference in the two prices.

This might be difficult to do, however, and it won’t generate as much profit as the trader would like, because whoever she buys the hedging option from will also be trying to charge a premium over the actual price. Another possibility is to try and create a hedged portfolio consisting of several options and the underlying stock as well, so as to minimise the net risk of the portfolio.

Since she has sold an option on a stock (for concreteness, let’s say she has sold a call option expiring at time T on a stock with a spot price S(t) and the option has strike K – because she has sold it, we say she is ‘short’ a call option), the trader will have to pay the larger of zero or ( S(T) – K ) at the option expiry to the client. Clearly, if the stock price goes up too high, she will lose more money than she received for selling the option. One possibility might be for her to buy the stock for S(t). Since she will have to pay a maximum of S(T) – K, but would be able to sell the stock for S(T), she would cover her position in the case that the stock price goes very high (and actually guarantee a profit in this case). But she has over-hedged her position – in the case that the stock falls in price, she will lose S(t) – S(T) on the stock. This is shown in the graph below.

The payoff at expiry of the three portfolios shown in the text. The unhedged option makes the trader money if spot doesn’t rise by more than the premium. The overhedged option is the reverse – now the trader loses money if the stock falls too far below the strike price – this is called a covered call, the payoff is the same as the payoff for an uncovered put option. Finally, a delta hedged option will make money as long as the spot does not move too far in either direction – the trader has taken a directionless bet, she is instead betting on volatility remaining low 

In fact, she can ‘delta-hedge’ the option by buying a fraction \inline \Delta of the stock. One way of deriving the BS equation (which I’ll get to at some point) is to construct a portfolio consisting of one option and a (negative) fraction of the underlying stock where the movement of the option price due to the underlying moving is exactly cancelled out but the movement of the underlying in the portfolio – any small increase in S(t) will increase the price of the option but decrease the value of the negative stock position so that the net portfolio change is zero. The fraction \inline \Delta is called the Delta of the option, mathematically it is the derivative of the option price C with respect to the stock price S

The price of a vanilla call is roughly the same as the payoff at expiry for very high spots and very low spots. Near-the-money, the difference between the option price and its payoff at expiry is greatest as the implicit insurance provided by the option is most useful. The delta of this option is the local gradient of the call price with spot. This is the amount of the underlying that would be required to delta-hedge the portfolio, so that its value is unaffected by small changes in the spot price.

\Delta = {\partial C \over \partial S}

For a call option this will be positive and for a put it will be negative, and the magnitude of both will be between 0 when far out-of-the-money and 1 when far in-the-money.

This graph shows the instantaneous change in PnL due to changes in spot for the three portfolios discussed above. Both the covered and the uncovered calls have some delta – a change in the spot price will have a direct effect in the value of the portfolio. By contrast, the value of the delta hedged portfolio is insensitive to the value of spot for small moves. Unfortunately, it is short gamma, so large moves in either direction will reduce portfolio value, so the trader must be careful to re-hedge frequently. As all of these portfolios are short an option, they are long theta – that is, the value of the portfolio will INCREASE with time if other factors remain constant, as the time value of the option is decaying towards expiry.

Similarly, if the market changes its mind about the implied volatility of an option, this will increase or decrease the price of the trader’s current option portfolio. This exposure can also be hedged, but now she will need to do it by trading options in the stock as the stock price itself is independent of volatility. This time the relevant quantity is the ‘vega’ of the option, the rate of change of price with respect to vol.

These sensitivities of the derivative price are called the Greeks, as they tend to be represented with various greek letters. Some examples are delta (variation with spot), vega (variation with vol), theta (variation with time); and second-order greeks like gamma (sensitivity of delta to spot), vanna (sensitivity of delta to vol, or equivalently sensitivity of vega to spot), and volga (sensitivity of vega to vol).

For vanilla options in the BS model, there are simple expressions for the greeks (see, for example, the Wikipedia page). I’ve updated the PRICERS page to give values for a few of the vanilla greeks of these options along with the price, and there are some graphs of typical greeks below.

The deltas of a long call and long put position – note that these are opposite sign and everywhere separated by a constant value (here 1, but in general the discount factor at the option expiry). Vega is always positive – so increased vol will always increase the price. Increasing spot tends to increase the value of a call, while it decreases the value of a put, but by a progressively smaller amount as spot increases. For these options (and the graph below), spot = fwd = strike = 100; vol = 0.1, expiry = 1 and r = 0.
The Gamma, Vanna and Volga of long vanilla options (these are the same for a call and a put). Gamma is always positive for long options – this means that price is a convex function of spot. Vanna and volga tell us the sensitivity of other greeks to volatility, and are useful in hedging portfolios if vol is changing rapidly.

For exotic options greeks are often intractable analytically, so typically they will be calculated by ‘bump and revalue’, where input parameters are varied slightly and the change in price is observed. For example, a derivative’s \inline \Delta at spot price S could be calculated from its price by ‘bumping’ spot \inline S by a small amount \inline \delta S:

\begin{matrix} C(S+\delta S,\sigma) = C(S,\sigma) + {\partial C \over \partial S}\delta S + {1 \over 2}{\partial^2 C \over \partial S^2} (\delta S)^2 + O[(\delta S)^3] \\ C(S-\delta S,\sigma) = C(S,\sigma) - {\partial C \over \partial S}\delta S + {1 \over 2}{\partial^2 C \over \partial S^2} (\delta S)^2 + O[(\delta S)^3] \end{array}

{C(S+\delta S, \sigma) - C(S-\delta S, \sigma) \over 2 (\delta S) } = {\partial C \over \partial S} + O[(\delta S)^3] \simeq \Delta

which is the derivative delta to a very good approximation for small \inline \delta S.

Since banks will have large portfolios and want to calculate their total exposure fairly frequently, pricing procedures will typically need to be fairly fast so that these risk calculations can be done in a reasonable amount of time, which usually rules out Monte Carlo as a technique here.

These hedges will only work for small changes in the underlying price (for example, delta itself changes with the underlying price according to the second-order greek, gamma). What this means is that the trader will need to re-hedge from time to time, which will cost her some money to do – one of the main challenges for a trader is to balance the need to hedge her portfolio with the associated costs of doing so. Hopefully by buying and selling a wide variety of options to various clients she will be able to minimise many of her greek exposures naturally – this ‘warehousing of risk’ is one of the main functions that banks undertake and a key driver of their profits.

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