28.1.1 Portfolio Delta

Suppose you have a portfolio of options consisting of calls and puts (with different strikes and time to maturity) but on the same underlying stock (e.g. Microsoft). The portfolio delta of the options is defined as:

(28.1)

We take if you are long (= buy) a call or put and if you are short (=sell) a call or put.

28.1.2 Gamma

In the following sections we denote the derivatives price as so our remarks will apply to either call or put options. The gamma of an option is defined as:

(28.2)

Gamma is the second derivative of the option price with respect to the stock price.¹ Gamma is also the change in the option's delta as S changes by a small amount , hence:

(28.3)

Gamma (like delta) takes on different values for different values of . The relationship between gamma and the stock price (other variables held constant) is roughly ‘bell shaped’ and the maximum value for gamma occurs when the current stock price is close to the strike price – that is, for a ‘near-ATM’ option (Figure 28.1).

If and the stock price changes by a small amount (e.g. $1), the slope of the Black–Scholes curve (i.e. the option's delta) will not change by very much, say from 0.5 to 0.53, giving a gamma of 0.03. (However, note that the value of gamma for an ATM option which is also ‘close to maturity’ can be much larger than this – see below.)

Clearly Figure 28.1 shows that when a call is either very or very then is small and fairly constant (if the stock price changes by a small amount). This implies that for OTM and ITM options, a delta hedge will work quite well even if you do not rebalance very frequently (e.g. every 5 days), since a small gamma means that delta does not change very much as S changes, from day to day.

In contrast, for an option that is currently ATM (i.e. ) then gamma can be relatively large and vary a great deal as S changes (Figure 28.1). A large gamma alerts an options trader who is delta hedging her position, that to remain delta hedged she will probably have to rebalance her portfolio frequently (e.g. twice per day). Alternatively she can ‘eliminate’ the current large value of gamma by ‘gamma hedging’ – see below.

The value for gamma is obtained from the Black–Scholes equation for European options (on a stock paying continuous dividends):

(28.4)

Gamma is positive for either a long call or a long put. Also the gamma for a call is equal to the gamma for a put option (with the same underlying asset, strike price, and time to maturity). To obtain a ‘negative gamma’ you must sell (write) calls or puts.

Gamma is useful when calculating the change in the option price for a large change in the stock price – using gamma is sometimes referred to as a ‘curvature adjustment’. A delta-hedged position does not give protection against large changes in the stock price. This can be seen by noting that the change in the option premium can be approximated by a second-order Taylor series expansion of the option price (with respect to S):

(28.5)

The change in the option price depends on both delta and gamma – as can be seen in Example 28.3.

28.1.3 Portfolio Gamma

The gamma for a portfolio of options (with different strikes and time to maturity but on the same underlying stock) is defined as:

(28.6)

If Ms Smart's options portfolio is delta hedged (i.e. ) but the gamma of the portfolio of options is large, then the portfolio will not be hedged against ‘large’ changes in the stock price. But if Ms Smart can form a ‘new portfolio’ of stocks and options which makes both the portfolio delta and the portfolio gamma equal to zero, then she is protected from both small and large changes in the stock price – this is ‘delta-gamma’ hedging and is discussed below.

Note that the ‘gamma of a stock’ is zero – this is because the value of a stock is linear in the stock price. If then and .

28.1.4 Theta

The change in the option premium due to the passage of time is non-stochastic and is measured by the option's theta (see Appendix 28):

(28.7)

In the Black–Scholes formula ‘time’ is measured in years or fractions of a year and when discussing theta we can take , so (years) or we can measure theta per calendar day . The option's theta depends on the current stock price, time to maturity, stock volatility, and interest rate.

Theta is usually negative for most long positions in options – both long calls and long puts lose value as they get closer to maturity (all other variables held constant). Theta is most negative for options that are nearly ATM, so ATM options lose time value rather quickly, day by day. Alternatively, if the current stock price is very low then theta is close to zero (for calls and puts). When the time to maturity is short (e.g. few weeks to maturity) the value of theta for ITM, OTM and especially for ATM options can become relatively large – so these options all lose time value relatively quickly (all other variables held constant).

Note that as ‘time’ is deterministic, the theta of the option is not used in any kind of hedging as there is no uncertainty. Theta is useful as an estimate of the change in value of the option over time. There is another more esoteric use for theta. It can be shown (using the Black–Scholes partial differential equation [PDE], see Chapter 48) that for in a portfolio of options on a single underlying asset with current price the following holds, . For a delta-neutral portfolio we have:

This shows that for a delta-neutral portfolio, when theta is large and positive the gamma tends to be large and negative – so here theta is also an indication of the gamma-risk in the options portfolio.

28.1.5 Rho

The change in the option price due to a change in is known as ‘rho’, . For call and put options (for both a dividend paying and non-dividend paying stock):

(28.8a)

(28.8b)

In the Black–Scholes pricing formula, is entered as a decimal. Hence, a change in interest rates of 1% is entered as in the formula .

Theta and rho are the ‘lesser Greek Gods’ since usually the option premium is not particularly sensitive to either changes in or changes in the time to maturity.

28.1.6 Vega

The change in the option's premium for a small change in volatility , is measured by the option's vega :

(28.9)

‘Vega’ is not a Greek letter and ‘vega’ is also referred to as lamda, kappa, or sigma, which are Greek letters. The numerical value of vega is the same for a long call or long put (with same strike, time to maturity, and underlying stock) and has a bell shape with respect to the stock price (Figure 28.2). To obtain a ‘negative vega’ you must sell (write) calls or puts.

If the vega of an option is large this implies the option price is highly sensitive to small changes in volatility . For a European option on a stock, the vega is large when the option is close to being at-the-money (i.e. ) and is zero for options that are well OTM or well ITM (Figure 28.2). The analytic expression for the vega of a European (call or put) option on a stock (or stock index) paying a continuous dividend yield is:

Strictly speaking, using the Black–Scholes vega to hedge against changes in volatility of the underlying asset is inconsistent – as the Black–Scholes model assumes volatility is constant – but this approach seems to work reasonably well in practice (i.e. even when volatility is stochastic). It should be obvious that the vega of a stock is zero – this is because the value of a stock portfolio does not depend on the stock's volatility (if then ).

Explicit expressions for ‘the Greeks’ can be obtained if we have a closed-form solution for the option premium. These are given in Appendix 28 for European calls and puts (on a stock that pays a continuous dividend yield), based on the Black–Scholes equation. Using the Greeks enables an options trader to hedge her existing portfolio of options either from small or large changes in the price of the underlying asset, or from changes in volatility and changes in interest rates.

28.1.7 Approximating Option Price Changes

The Greeks also provide a convenient way of finding the approximate change in the option premium , as all the variables change. A Taylor series expansion f the equation for the (Black–Scholes) option price is:

(28.10)

where = change in the stock price, etc. This is also a useful expression for calculating the approximate change in the option price , once we have any reasonable way of obtaining estimates of the Greeks (e.g. from an approximate closed-form option pricing equation, from MCS or from the BOPM).

Note also, that the values for all the Greeks change over time as the values of and time to maturity , change. For example, a portfolio of options which currently has will not represent a gamma-neutral position some time later because as and change over time, so does the option's gamma, .

28.2.1 Portfolio of Options

Suppose you have a portfolio of options consisting of calls and puts (with different strikes and time to maturity) but on the same underlying stock (e.g. Microsoft). The delta, gamma, and vega of the portfolio of options are:

(28.11)

• if long (buy) a call or put
• if short (sell) a call or put
• for long call (from Black–Scholes)
• for long put (from Black–Scholes)

Also, each long stock has a delta of and a gamma and vega of zero. If you hold (buy) stocks then and if you short-sell stocks .

28.2.2 Gamma Neutral

The gamma for Microsoft stock (or futures on the stock) are both zero. So, the only way you can change the gamma of an existing options portfolio on several Microsoft calls and puts is to take new positions in some other options on Microsoft. Suppose you already have a portfolio-A, of several calls and puts on Microsoft which have a portfolio delta of and gamma of . If you now add a further traded options on Microsoft (with different strike prices and maturity dates) into your existing options portfolio-A, then the gamma of this new portfolio is:

(28.12a)

Your new portfolio is gamma-neutral when:

(28.12b)

But including additional options will also alter the overall portfolio's delta. Hence we then buy or sell a number of Microsoft stocks (or futures contracts on Microsoft stock) to make the overall options portfolio delta neutral. An example is given in Example 28.8.

Note that as time passes S, σ, r and T change, hence so do gamma and delta – therefore periodic rebalancing is required to maintain delta-gamma neutrality.

Kurpiel and Roncalli (1998) find that (in a Black–Scholes constant volatility world) when you just delta hedge a call option with rebalancing over either {5, 1, 1/2} days, the percentage hedging error is ² whereas with a delta-gamma hedge these figures improve to , which are ‘reasonable’ even when rebalancing takes place only every 5 days.

28.2.3 Vega Neutral

The vega of an existing options portfolio on Microsoft can only be altered by adding another traded option on Microsoft to the existing portfolio. Suppose we initially hold a portfolio of Microsoft stocks with a portfolio vega of and another option-X on Microsoft is available with a vega . To obtain a vega-neutral position we require options which satisfy:

(28.13a)

hence

(28.13b)

Our new portfolio will not be delta neutral but we know that if we now take a position in Microsoft stocks, we can also achieve delta neutrality. For daily rebalancing in a stochastic volatility world, Kurpiel and Roncalli (2009) report an improvement in hedge performance as we move from purely delta hedging with , to delta-vega hedging where %HE = 0.135%. However, we still have a potential problem because our delta-vega neutral portfolio may have a non-zero gamma and therefore may not be protected against large changes in the stock price.

28.2.4 Gamma-Vega-Delta Neutral

Agamma-vega-delta neutral position implies that the value of the portfolio is (nearly fully) protected against small or large changes in the stock price and against small changes in the stock's volatility.

Suppose we have a portfolio-A of options on Microsoft which is already vega neutral but has a gamma of . How can we simultaneously achieve gamma and vega neutrality? At first sight it may seem as if we just include new options-X (on Microsoft) to achieve gamma neutrality. However, this is incorrect, as adding the new options-X destroys our initial vega neutrality, because the new options themselves have non-zero vegas. To achieve a delta-gamma-vega neutral portfolio we proceed in two steps:

• First, we include ‘new’ options on Microsoft (with different strike prices and maturity dates and hence different gammas and vegas) to our original portfolio-A (of Microsoft options), so that we simultaneously achieve gamma-vega neutrality.
• Second, we go long or short Microsoft stocks to make our ‘new’ options portfolio delta neutral – this will not alter our ‘new’ gamma-vega neutrality because stocks have a gamma and a vega of zero.

An example of setting up a ‘delta-gamma-vega’ neutral position is given in Example 28.9.

The value of your ‘final options and stocks’ position consists of your initial portfolio-A of several options on Microsoft (with different strikes and time to maturity) plus our new positions in options-Y and options-Z (on Microsoft) which give gamma-vega neutrality, and finally short-selling Microsoft stocks to achieve delta neutrality. Your new positions in options and stocks means that your ‘final portfolio’ is now hedged against small (delta) and large (gamma) changes in S and also against (small) changes in volatility (vega).

28.2.5 Frequency of Rebalancing

A long call or a long put on a stock (with the same strike price and maturity) have the same positive values for either gamma or vega. Short calls and puts have negative gamma and vega. Although options traders might frequently rebalance their portfolios to maintain delta neutrality, they generally do not rebalance often to maintain gamma and vega neutrality, since it may be difficult to find appropriate ‘offsetting options’ at competitive prices. They therefore adjust their options position to give gamma and vega neutrality only after these ‘Greeks’ become unacceptably large. However, their portfolio of options positions will usually be monitored daily and their potential vulnerability to possible changes in either stock prices or volatility (of stock returns) will be calculated – this is the subject of Chapter 46 on risk measurement.

The options trading desks of many banks mainly write calls and puts for their clients and hence build up negative gamma and negative vega positions, in the normal course of trading. They therefore often step in as buyers of options when these are available at competitive prices, in order to reduce their gamma-vega exposures. Because an option's vega and gamma are large when the options are close to ATM, an options dealer may gamma-vega (and delta) hedge a portfolio of ATM options under these circumstances, even when the options have a considerable time to maturity.

It is also the case that the gamma and vega for ATM options which are close to expiration are exceptionally large and hence the options trader may consider a gamma-vega (and delta) hedge, if she holds options that are both ATM and close to expiration. A trader who initially sells ATM calls and puts and delta hedges her position hopes that the options will become well OTM or ITM, since then the vega and gamma become very small (Figure 28.2) and there may be no need to undertake costly gamma-vega hedging.

A portfolio of futures options (e.g. options on AT&T futures contracts) can be used together with futures contracts (on AT&T stocks) in implementing dynamic delta hedging. An existing portfolio of futures options-A (e.g. on AT&T futures contracts) can be combined with other futures options-B (on AT&T futures contracts, but with different strikes and time to maturity) to simultaneously produce a gamma-neutral and vega-neutral portfolio. Finally, one can use futures contracts (on AT&T) to achieve a portfolio which is gamma-vega-delta neutral.

As we have seen, if you hold a portfolio of options on AT&T stocks then you can gamma-vega-delta hedge using other options (written on AT&T stock) together with the AT&T stocks themselves. However, because the stock price and the futures price move closely together you can also gamma-vega-delta hedge a portfolio of options (on AT&T stocks), using options on futures contracts of AT&T – together with futures contracts on AT&T stocks (to ensure delta neutrality).

The Black–Scholes formula for the price of a European call option on a stock which pays a continuous dividend yield at a rate (per annum) can be obtained by taking the Black–Scholes equation for an option on a non-dividend paying stock and replacing by .

Call and Put Premia: European Options, Dividend Paying Stock

For calls and puts on a stock (index) paying continuous dividends:

(28.A.1)

(28.A.2)

Since then:

(28.A.3a)

(28.A.3b)

Delta:

Call Delta:

Put Delta:

Gamma:

Vega:

Rho: (for both a dividend paying and non-dividend paying stock)

Call:

Put:

Theta :

a. Non-dividend paying stock

Note that T = ‘time to maturity’ and therefore as the call gets closer to maturity, T falls. It can be shown that as the option approaches the maturity date, theta increases and the call premium becomes more sensitive to changes in ‘time’ – the call loses time-value more quickly.

b. Dividend paying stock

In the above formulas time is measured in years. For example, if , then the change in price of the option over 2 trading days is (16 cents).

Theta is usually negative for a long position in an option – as time passes the option becomes less valuable – this is ‘time decay’. However, there are some (pathological) cases where theta can be positive – for example, for an ITM European put on a non-dividend paying stock or an ITM European call on a currency that pays a high interest rate.

If we set in any of the above formulas then we will obtain the Black–Scholes ‘Greeks’ for an option on a non-dividend paying stock.