.2 the stock price falls by the end of the first week to $48.12. The delta declines to0.458, and 6,400 of shares are sold to maintain the hedge. The strategy realizes $308,000 in cash,and the cumulative borrowings at the end of Week | are reduced to $2,252,300. During the secondweek the stock price reduces to $47.37, delta declines again, and so on. Toward the end of the lifeof the option, it becomes apparent that the option will be exercised and delta approaches 1.0. ByWeek 20, therefore, the hedger has a fully covered position. The hedger receives $5 million for thestock held, so that the total cost of writing the option and hedging it is $263,300.
Table 14.3 illustrates an alternative sequence of events such that the option closes out of themoney. As it becomes clear that the option will not be exercised, delta approaches zero. By Week 20the hedger has a naked position and has incurred costs totaling $256,600.
In Tables 14.2 and 14.3 the costs of hedging the option, when discounted to the beginning of theperiod, are close to but not exactly the same as the Black-Scholes price of $240,000. If the hedgingscheme worked perfectly, the cost of hedging would, after discounting, be exactly equal to the Black-Scholes price for every simulated stock price path. The reason for the variation in the cost of deltaThe Greek Letters 307
hedging is that the hedge is rebalanced only once a week. As rebalancing takes place more frequently,the variation in the cost of hedging is reduced. Of course, the examples in Tables 14.2 and 14.3 areidealized in that they assume that the volatility is constant and there are no transactions costs.
Table 14.4 shows statistics on the performance of delta hedging obtained from 1,000 random stockprice paths in our example. As in Table 14.1, the performance measure is the ratio of the standarddeviation of the cost of hedging the option to the Black-Scholes price of the option. Clearly deltahedging is a great improvement over a stop-loss strategy. Unlike a stop-loss strategy, the performanceof a delta-hedging strategy gets steadily better as the hedge is monitored more frequently.
Delta hedging aims to keep the value of the financial institution’s position as close to unchangedas possible. Initially, the value of the written option is $240,000. In the situation depicted inTable 14.2, the value of the option can be calculated as $414,500 in Week 9. Thus, the financialinstitution has lost $174,500 on its option position. Its cash position, as measured by thecumulative cost, is $1,442,900 worse in Week 9 than in Week 0. The value of the shares heldhas increased from $2,557,800 to $4,171,100. The net effect of all this is that the value of thefinancial institution’s position has changed by only $4,100 during the nine-week period.
Where the Cost Comes From
The delta-hedging scheme in Tables 14.2 and 14.3 in effect creates a long position in the optionsynthetically. This neutralizes the short position arising from the option that has been written. Thescheme generally involves selling stock just after the price has gone down and buying stock just
Table 14.2 Simulation of delta hedging (option closes in the money; cost of hedging is $263,300)
 
Cost of shares Cumulative costShares purchased including interest Interest costWeek Stock price Delta purchased (3000) (3000) (3000)0 49,00 0.522 52,200 2,557.8 2,557.8 251 48.12 0.458 (6,400) (308.0) 2,252.3 2.22 47.37 0.400 (5,800) (274.7) 1,979.8 1.93 50.25 0.596 19,600 984.9 2,996.6 2.94 51.75 0.693 9,700 502.0 3,471.5 3.33 53.12 0.774 8,100 430.3 3,905.1 3.86 53.00 0.771 (300) (15.9) 3,893.0 3.77 51.87 0.706 (6,500) (337.2) 3,559.5 3.48 51.38 0.674 (3,200) (164.4) 3,398.5 3.39 53.00 0.787 11,300 598.9 4,000.7 3.810 49.88 0.550 (23,700) (1,182.2) 2,822.3 2.711 48.50 0.413 (13,700) (664.4) 2,160.6 2.112 49.88 0.542 12,900 643.5 2,806.2 2.713 50.37 0.591 4,900 246.8 3,055.7 2.914 52.13 0.768 17,700 922.7 3,981.3 3.815 51.88 0.759 (900) (46.7) 3,938.4 3.816 52.87 0.865 10,600 560.4 4,502.6 4.317 54.87 0.978 11,300 620.0 5,126.9 4918 54.62 0.990 1,200 65.5 5,197.3 5.019 55.87 1.000 1,000 55.9 5,258.2 5.120 57.25 1.000 0 0.0 5,263.3308 CHAPTER 14
Table14.3 Simulation of delta hedging (option closes out of the money; cost of hedging is $256,600)
 
Cost of shares Cumulative costShares purchased including interest Interest costWeek Stock price Delta purchased ($000) ($000) ($000)0 49.00 0.522 52,200 2,557.8 2,557.8 2.51 49.75 0.568 4,600 228.9 2,789.2 2.72 52.00 0.705 13,700 712.4 3,504.3 3.43 50.00 0.579 (12,600) (630.0) 2,877.7 2.84 48.38 0.459 (12,000) (580.6) 2,299.9 2.25 48.25 0.443 (1,600) (77.2) 2,224.9 2.16 48.75 0.475 3,200 156.0 2,383.0 2.37 49.63 0.540 6,500 322.6 2,707.9 2.68 48.25 0.420 (12,000) (579.0) 2,131.5 2.19 48.25 0.410 (1,000) (48.2) 2,085.4 2.010 51.12 0.658 24,800 1,267.8 3,355.2 3.211 51.50 0.692 3,400 175.1 3,533.5 3.412 49.88 0.542 (15,000) (748.2) 2,788.7 2.713 49.88 0.538 (400) (20.0) 2,771.4 2.714 48.75 0.400 (13,800) (672.7) 2,101.4 2.015 47.50 0.236 (16,400) (779.0) 1,324.4 1316 48.00 0.261 2,500 120.0 1,445.7 1417 46.25 0.062 (19,900) (920.4) 526.7 0.518 48.13 0.183 12,100 582.4 1,109.6 1.119 46.63 0.007 (17,600) (820.7) 290.0 0.320 48.12 0.000 (700) (33.7) 256.6
after the price has gone up. It might be termed a buy-high, sell-low scheme! The cost of $240,000comes from the average difference between the price paid for the stock and the price realized for it.Delta of a PortfolioThe delta of a portfolio of options or other derivatives dependent on a single asset whose price is Sisonaswhere [1 is the value of the portfolio.
The delta of the portfolio can be calculated from the deltas of the individual options in the.portfolio. If a portfolio consists of a quantity w, of option i (1 <i <n), the delta of the portfolio
Table 14.4 Performance of delta hedging. (The performance measure is the ratio of the standarddeviation of the cost of writing the option and hedging it to the theoretical price of the option.)
Time between hedge rebalancing (weeks) 5 4 2 1 0.5 0.25
Performance measure 0.43 0.39 0.26 0.19 0.14 0.09The Greek Letters 309
is given bynA=) wAi=1
where A; is the delta of ith option. The formula can be used to calculate the position in the
underlying asset or in a futures contract on the underlying asset necessary to carry out deltauncenying asset a es © act on the uncerlying asset necessar y to carry out ceita
hedging. When this position has been taken, the delta of the portfolio is zero, and the portfolio isreferred to as being delta neutral.
Suppose a financial institution in the United States has the following three positions in optionson the Australian dollar:
1. A long position in 100,000 call options with strike price 0.55 and an expiration date in threemonths. The delta of each option is 0.533.
2. A short position in 200,000 call options with strike price 0.56 and an expiration date in fivemonths. The delta of each option is 0.468.
3. A short position in 50,000 put options with strike price 0.56 and an expiration date in twomonths. The delta of each option is —0.508.
The delta of the whole portfolio is100,000 x 0.533 — 200,000 x 0.468 — 50,000 x (~0.508) = — 14,900
This means that the portfolio can be made delta neutral with a long position of 14,900 Australiandollars.
A six-month forward contract could also be used to achieve delta neutrality in this example.Suppose that the risk-free rate of interest is 8% per annum in Australia and 5% in the UnitedStates (r= 0.05 and ry = 0.08). The delta of a forward contract maturing at time T on oneAustralian dollar is e~#? or e~°-°8*° — 09,9608. The long position in Australian dollar forwardcontracts for delta neutrality is therefore 14,900/0.9608 = 15,508.
Another alternative is to use a six-month futures contract. From equation (14.4), the longposition in Australian doilar futures for delta neutrality is
14,9002710-.09-0.08)x0.5 _ 15 195
Transactions Costs
Maintaining a delta-neutral position in a single option and the underlying asset, in the way thathas just been described, is liable to be prohibitively expensive because of the transactions costsincurred on trades. For a large portfolio of options, delta neutrality is more feasible. Only onetrade in the underlying asset is necessary to zero out delta for the whole portfolio. The hedgingtransactions costs are absorbed by the profits on many different trades,
14.5 THETA
 
 
The theta of a portfolio of options, ©, is the rate of change of the value of the portfolio withrespect to the passage of time with all else remaining the same. Theta is sometimes referred to asthe time decay of the portfolio. For a European call option on a non-dividend-paying stock, it can310 CHAPTER 14
be shown from the Black-Scholes formula that
_ SN (dio2./T
where d; and d, are defined as in equation (12.20) and
@= rKe N(d)
Ni) = mee (14.5)
For a European put option on the stock,
_ SoN "(dio2/7
For a European call option on an asset paying a dividend at rate q,
O= +rKe~™" N(—d>)
SyN'(d;)oe 7"
2MT
where d, and d, are defined as in equation (13.4), and, for a European put option on the asset,
e= + @SypN(de" — rKe™ N(d,(
_ SoN (di)oe"”
2/T
When the asset is a stock index, these last two equations are true with g equal to the dividend yieldon the index. When it is a currency, they are true with g equal to the foreign risk-free rate, rp. Whenit is a futures contract, they are true with g =r.
In these formulas time is measured in years. Usually, when theta is quoted, time is measured indays so that theta is the change in the portfolio value when one day passes with all else remainingthe same. We can either measure theta “per calendar day” or “per trading day”. To obtain thetheta per calendar day, the formula for theta must be divided by 365; to obtain theta per tradingday, it must be divided by 252. (DerivaGem measures theta per calendar day.)
e= — gSpN(—d)e 7 +rKe™ N(—d>)
Example14.3 Consider a four-month put option on a stock index. The current value of the indexis 305, the strike price is 300, the dividend yield is 3% per annum, the risk-free interest rate is 8% perannum, and the volatility of the index is 25% per annum. In this case Sp = 305, K = 300, g = 0.03,r= 0.08, o = 0.25, and T = 0.3333. The option’s theta is
SypN (doe2/TThe theta is —18.15/365 = —0.0497 per calendar day or —18.15/252 = —0.0720 per trading day.
— gSpN(—d\ Je" + rKe™"™ N(—d3) = —18.15
Theta is usually negative for an option.® This is because as the time to maturity decreases withall else remaining the same, the option tends to become less valuable. The variation of © withstock price for a call option on a stock is shown in Figure 14.5. When the stock price is verylow, theta is close to zero. For an at-the-money call option, theta is large and negative. As the
5 An exception to this could be an in-the-money European put option on a non-dividend-paying stock or an in-the-money European call option on a currency with a very high interest rate.The Greek Letters 311
 
Theta
      
Stock price—>-
 
Figure 14.5 Variation of theta of a European call option with stock price
stock price becomes larger, theta tends to —rKe~’’. Figure 14.6 shows typical patterns for thevariation of © with the time to maturity for in-the-money, at-the-money, and out-of-the-moneycall options.
Theta is not the same type of hedge parameter as delta. There is uncertainty about the futurestock price, but there is no uncertainty about the passage of time. It makes sense to hedge againstchanges in the price of the underlying asset, but it does not make any sense to hedge against theeffect of the passage of time on an option portfolio. In spite of this, many traders regard theta as auseful descriptive statistic for a portfolio. This is because, as we will see later, in a delta-neutralportfolio theta is a proxy for gamma.
A Theta
Time to maturity—>-
 
   
Out of the money
   
In the money
At the money
 
/
Figure 14.6 Typical patterns for variation of theta of a European call optionwith time to maturity312 CHAPTER 14
14.6 GAMMA
 
 
The gamma of a portfolio of options on an underlying asset, I, is the rate of change of theportfolio’s delta with respect to the price of the underlying asset. It is the second partialderivative of the portfolio with respect to asset price:
en~~ as?
If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need tobe made only relatively infrequently. However, if gamma is large in absolute terms, delta is highlysensitive to the price of the underlying asset. It is then quite risky to leave a delta-neutral portfoliounchanged for any length of time. Figure 14.7 illustrates this point. When the stock price movesfrom S$ to S$’, delta hedging assumes that the option price moves from C to C’, when in fact itmoves from C to C”. The difference between C’ and C” leads to a hedging error. This errordepends on the curvature of the relationship between the option price and the stock price. Gammameasures this curvature.’
Suppose that 5S is the price change of an underlying asset during a small interval of time, dt, and671 is the corresponding price change in the portfolio. If terms of higher order than 6¢ are ignored,Appendix 14A shows that, for a delta-neutral portfolio,
sm = Ot + AP 68? (14.6)
where © 1s the theta of the portfolio. Higure 14.8 shows the nature of this relationship between6M and 8S. When gamma is positive, theta tends to be negative. The portfolio declines in value ifthere is no change in S, but increases in value if there is a large positive or negative change in S.When gamma is negative, theta tends to be positive and the reverse is true; the portfolio increasesin value if there is no change in S but decreases in value if there is a large positive or negativechange in S. As the absolute value of gamma increases, the sensitivity of the value of theportfolio to S increases.
Callprice
     
 
   
Stock price
t
I‘'I—L —»-5
Figure 14.7 Hedging error introduced by curvature, or gamma
° Indeed, the gamma of an option is sometimes referred to as its curvature by practitioners.The Greek Letters 313
8S 8s
(a) (b)
po\
Figure 14.8 Alternative relationships between 81 and 8S for a delta-neutral portfolio: (a) slightlypositive gamma, (b) large positive gamma, (c) slightly negative gamma, and (d) large negative gamma
\
 
 
(c) (d)
Example 14.4 Suppose that the gamma of a delta-neutral portfolio of options on an asset is—10,000. Equation (14.6) shows that if a change of +2 or —2 in the price of the asset occurs over ashort period of time, there is an unexpected decrcase in the valuc of the portfolio of approximately0.5 x 10,000 x 27 = $20,000.
Making a Portfolio Gamma Neutral
A position in the underlying asset itself or a forward contract on the underlying asset both havezero gamma and cannot be used to change the gamma of a portfolio. What is required is a positionin an instrument such as an option that is not linearly dependent on the underlying asset.Suppose that a delta-neutral portfolio has a gamma equal to [’, and a traded option has agamma equal to I’;. If the number of traded options added to the portfolio is wr, the gamma ofthe portfolio iswl + T
Hence, the position in the traded option necessary to make the portfolio gamma neutral is -'/I'y.Including the traded option is likely to change the delta of the portfolio, so the position in theunderlying asset then has to be changed to maintain delta neutrality. Note that the portfolio is314 CHAPTER 14
 
gamma neutral only fora short period of time. As time passes, gamma neutrality can be maintainedonly if the position in the traded option is adjusted so that it is always equal to —I'/Ty.
Making a delta-neutral portfolio gamma neutral can be regarded as a first correction for the factthat the position in the underlying asset cannot be changed continuously when delta hedging isused. Delta neutrality provides protection against relatively small stock price moves betweenrebalancing. Gamma neutrality provides protection against larger movements in this stock pricebetween hedge rebalancing. Suppose that a portfolio is delta neutral and has a gamma of —3,000.The delta and gamma of a particular traded call option are 0.62 and 1.50, respectively. Theportfolio can be made gamma neutral by including in the portfolio a long position of
3,000
—— =~?Ts 000
However, the delta of the portfolio will then change from zero to 2,000 x 0.62 = 1,240. A quantity1,240 of the underlying asset must therefore be sold from the portfolio to keep it delta neutral.
Calculation of GammaFor a European call or put option on a non-dividend-paying stock, the gamma is given by
_ Na)Spo VT
where d, is defined as in equation (12.20) and N(x) is as given by equation (14.5). The gamma isalways positive and varies with S, in the way indicated in Figure 14.9. The variation of gamma withtime to maturity for out-of-the-money, at-the-money, and in-the-money options is shown inFigure 14.10. For an at-the-money option, gamma increases as the time to maturity decreases.Short-life at-the-money options have very high gammas, which means that the value of the optionholder’s position is highly sensitive to jumps in the stock price.
For a European call or put option on an asset paying a continuous dividend at rate g,
 
pa Nae”
Soo VT
Gamma
K Stock price
 
Figure 14.9 Variation of gamma with stock price for an optionThe Greek Letters 315
A Gamma
\
  
Out of the money
     
At the money
  
In the money
 
 
Time to maturity>
Figure 14.10 Variation of gamma with time to maturity for a stock option
where d, is as in equation (13.4). When the asset is a stock index, q is set equal to the dividend yieldon the index. When it is a currency, q is set equal to the foreign risk-free rate, r,. When it is afutures contract. g =r.
Example14.5 Consider a four-month put option on a stock index. The current value of the indexis 305, the strike price is 300, the dividend yield is 3% per annum, the risk-free interest rate is 8% perannum, and volatility of the index is 25% per annum. In this case, Sy = 305, K = 300, g = 0.03,r = 0.08, o = 0.25, and T = 4/12. The gamma of the index option is given by
Nae"
+ _— = 0.00866SoVT
Thus, an increase of | in the index (from 305 to 306) increases the delta of the option by approxi-mately 0.00866. .
14.7, RELATIONSHIP BETWEEN DELTA, THETA, AND GAMMA
 
 
The price of a single derivative dependent on a non-dividend-paying stock must satisfy thedifferential equation (12.15). It follows that the value of a portfolio of such derivatives, IT, alsosatisfies the differential equation
 316 CHAPTER 14
Since
am an an@=—, Az=—, P=at as as?it follows thatO+rSA+he°SP = rl (14.7)
Similar results can be produced for other underlying assets (see Problem 14.19).For a delta-neutral portfolio, A = 0 and
O+ to’S *PartThis shows that, when © is large and positive, gamma tends to be large and negative, and vice
versa. This is consistent with the way in which Figure 14.8 has been drawn and explains why thetacan be regarded as a proxy for gamma in a delta-neutral portfolio.
14.8 VEGA
 
 
Up to now, we have implicitly assumed that the volatility of the asset underlying a derivative isconstant. In practice, volatilities change over time. This means that the value of a derivative isliable to change because of movements in volatility as well as because of changes in the asset priceand the passage of time.
The vega of a portfolio of derivatives, V, is the rate of change of the value of the portfolio withrespect to the volatility of the underlying asset:!°
_an~ 80
If vega is high in absolute terms, the portfolio’s value is very sensitive to small changes in volatility.If vega is low in absolute terms, volatility changes have relatively little impact on the value of theportfolio.
A position in the underlying asset has zero vega. However, the vega of a portfolio can bechanged by adding a position in a traded option. If V is the vega of the portfolio and Vr is the vegaof a traded option, a position of —V/Vr in the traded option makes the portfolio instantaneouslyvega neutral. Unfortunately, a portfolio that is gamma neutral will not in general be vega neutral,and vice versa. If a hedger requires a portfolio to be both gamma and vega neutral, at least twotraded derivatives dependent on the underlying asset must usually be used.
Example 14.6 Consider a portfolio that is delta neutral, with a gamma of —5,000 and a vega of—8,000. A traded option has a gamma of 0.5, a vega of 2.0, and a delta of 0.6. The portfolio can bemade vega neutral by including a long position in 4,000 traded options. This would increase delta to2,400 and require that 2,400 units of the asset be sold to maintain delta neutrality. The gamma of theportfolio would change from —5,000 to —3,000.
To make the portfolio gamma and vega neutral, we suppose that there is a second traded optionwith a gamma of 0.8, a vega of 1.2, and a delta of 0.5. If w; and w are the quantities of the two
10 Vega is the name given 10 one of the “Greek letters” in option pricing, but it is not one of the letters in the Greekalphabet.The Greek Letters 317
traded options included in the portfolio, we require that
—5,000 + 0.5w, + 0.8w2 = 0
—8,000 + 2.0u,; + 1.2w, =0The solution to these equations is w, = 400, w2 = 6,000. The portfolio can therefore be madegamma and vega neutral by including 400 of the first traded option and 6,000 of the second tradedoption. The delta of the portfolio after the addition of the positions in the two traded options is
400 x 0.6 + 6,000 x 0.5 = 3,240. Hence, 3,240 units of the assct would have to be sold to maintaindelta neutrality.
For a European call or put option on a non-dividend-paying stock, vega is given byV = SyVT N'(d\)
where d, is defined as in equation (12.20). The formula for N’(x) is given by equation (14.5). ForaEuropean call or put option on an asset providing a dividend yield at rate g,
V = SyVT Neo"
where d, is defined as in equation (13.4). When the asset is a stock index, g is set equal to thedividend yield on the index. When it is a currency, q is set equal to the foreign risk-free rate, ry.When it is a futures contract, g =r.
The vega of a regular European or American option is always positive. The general way in whichvega varies with Sy is shown in Figure 14.11.
Example14.7 Consider a four-month put option on a stock index. The current value of the indexis 305, the strike price is 300, the dividend yield is 3% per annum, the risk-free interest rate is 8% perannum, and the volatility of the index is 25% per annum. In this case, Sy = 305, K = 300, g = 0.03,r= (0.08, o = 0.25, and T = 4/12. The option’s vega is given by
SyVT