01/01/02Deutsche Bank: Option Greeks & Market Making Copyright (c) 1997-2002 by Marshall, Tucker &...

01/01/02 Deutsche Bank: Option Greeks & Market Making Copyright (c) 1997-2002 by Marshall, Tucker & Associates, LLC

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Deutsche Bank

The Option Greeks and Market Making

Alan L. Tucker, Ph.D.631-331-8024 (tel)631-331-8044 (fax)

[email protected]

Copyright © 1997-2001Marshall, Tucker & Associates,LLC

All rights reserved


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ALAN L. TUCKER, Ph.D.

Alan L. Tucker is Associate Professor of Finance at the Lubin School of Business, Pace University, New York, NY and an Adjunct Professor at the Stern School of Business of New York University, where he teaches graduate courses in derivative instruments. Dr. Tucker is also a principal of Marshall, Tucker & Associates, LLC, a financial engineering and derivatives consulting firm with offices in New York, Chicago, Boston, San Francisco and Philadelphia. Dr. Tucker was the founding editor of the Journal of Financial Engineering, published by the International Association of Financial Engineers (IAFE). He presently serves on the editorial board of Journal of Derivatives and the Global Finance Journal and is a former associate editor of the Journal of Economics and Business. He is a former director of the Southern Finance Association and a former program co-director of the 1996 and 1997 Conferences on Computational Intelligence in Financial Engineering, co-sponsored by the IAFE and the Neural Networks Council of the IEEE.

Dr. Tucker is the author of three books on financial products and markets: Financial Futures, Options & Swaps, International Financial Markets, and Contemporary Portfolio Theory and Risk Management (all published by West Publishing, a unit of International Thompson). He has also published more than fifty articles in academic journals and practitioner-oriented periodicals including the Journal of Finance, the Journal of Financial and Quantitative Analysis, the Review of Economics and Statistics, the Journal of Banking and Finance, and many others.

Dr. Tucker has contributed to the development of the theory of derivative products including futures, options and swaps, and to the theory of international capital markets and trade. He has also contributed to the theory of technology adoption over the life-cycle. The Social Sciences Citation Index shows that his research has been cited in refereed journals on over one hundred occasions.

As a consultant, Dr. Tucker has worked for The United States Treasury Department, the United States Justice Department, Morgan Stanley Dean Witter, Union Bank of Switzerland, LG Securities (Korea), and Chase Manhattan Bank. Dr. Tucker holds the B.A. in economics from LaSalle University (1982), and the MBA (1984) and Ph.D. (1986) in finance from Florida State University. He was born in Philadelphia in 1960, is married (Wendy) and has three children (Emily, 1993, Michael and Matthew, both 1995).


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• Topics addressed in this module include:

– Measurement and economic interpretation of option greeks

– Dynamic delta-gamma-vega hedging for dealers

– An illustration of hedging failure due to jump risk

– Interactions among the greeks


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• Basic option prices are a function of five variables: S, X, r, T - t, and vol

• X is a constant and T - t changes deterministically and thus neither represent a source of price risk to a dealer

• Most option prices are quite insensitive to r

• Thus most of the price risk associated with making markets in options is driven by unexpected changes in two input variables, S and vol

• Since vol is unobservable, it is the keystone input and so making markets in options is very much an exercise in making markets in vol

• Indeed, dealers actually quote vol in some OTC markets



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• Price risk for options is quantified by “risk metrics”, which are just partial derivatives or “greeks”

• Delta is a key greek as it measures the sensitivity of an option price with respect to a small change in S

• Gamma is another key greek as it measures the sensitivity of an option price with respect to a somewhat larger change in S

• Gamma is also a measure of an option’s “convexity risk” as well as a measure of how delta itself changes with S

• Vega is another key greek as it captures the sensitivity of an option price with respect to small changes in vol. (Vanna is sometimes used to refer to the second derivative of the option price with respect to vol.)



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• Example: Suppose that a call option has a delta of 0.50 and that the underlying asset increases in price by $0.25 per share. Then the call price will increase by approximately $0.125.

• Example: Suppose that this same option has a gamma of 0.03. A more exact estimate of the change in the call option price, given the $0.25 increase in S, is given by:

0.50 x $0.25 + (1/2)(0.03)($0.25^2) = $0.1259375

A gamma of 0.03 also indicates that the option’s delta will increase from 0.50 to approximately 0.50 + (.03)(0.25) = 0.5075



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• Example: Let our same call option have a vega of 18. Then if the vol of the underlying asset increases from say 45% to 46% per annum, the call option price will increase by approximately (0.01)(18) = $0.18 per share.

• The next thing you should keep in mind is that the greeks of an option book are simply weighted averages of the greeks of the options that comprise the book, where the weights are represented by the number of shares underlying the option position.



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• Example: Let a dealer have a three-option book:

Option Position Shares Delta Gamma Vega

Short Call 50000 0.55 0.04 17

Short Call 20000 0.48 0.02 14

Short Put 40000 -0.50 0.03 15

The book’s delta is (-50000)(0.55) + (-20000)(0.48) + (-40000)(-0.50) = -17100. The book’s gamma = -3600 while its vega = -1,730,000. This means that if the underlying asset price increases by $1, the dealer will lose about $17,100 on paper. The delta will decrease from -17100 to -20700. Finally, if the relevant vol surface should increase by 1% then the dealer will lose about $17,300 on paper.



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• The next few things you should keep in mind are:

– A dealer tends to be a net writer and therefore generally carries a negative gamma and vega in his or her book. For this reason dealers are said to always look for “cheap ways” to add gamma and vega.

– Carrying a negative vega means that the dealer is “short vol”. In other words, a decline in vol will tend to profit dealers and vice versa. Being short equity vol was generally profitable during 2001, but not during 1995-2000. Being short vol is more dangerous than being long vol as vol can go up without bound but is bounded below at zero.



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• Continuing:

– A dealer will face limits on a book’s greeks. These limits are imposed by the middle office and a middle office report for an options dealer should be able to show, in more or less real time, all of the dealer’s outstanding positions, the key greeks of the book, and whether or not limits are in violation. The protocol for when violation occurs is for the middle office to notify the desk manager who in turn instructs the dealer to rebalance the book. Limits on greeks can vary over time, for example, they come in during merger talks.



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• Continuing:

– Delta, gamma and vega change all the time and so price risk management for dealers is a dynamic exercise. In practice, dealers tend to delta hedge every trading day and sometimes intra-day. Gamma and vega hedging occur less frequently, e.g., on Friday afternoons and before other long market closes.

– “Stress testing” tends to be an after-hours practice wherein the book’s P&L is inspected for robustness with respect to large changes in key variables like S and vol.



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• Quantifying greeks is straightforward for most options, even exotics. They are obtained analytically (also known as “closed form”) by solving for the partial derivatives, or they can be estimated through a variety of numerical methods (e.g., reading the delta off of a binomial tree).

• Software demonstration here.

• A cash asset has a delta of 1 and no gamma or vega. “Linear securities” like stock index futures contracts have no gamma or vega either.



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• An important result from theoretical finance is the following: In order to hedge N sources of risk, then one needs at least N + 1 traded assets. So, to delta hedge only an options book, one needs at least two assets. To delta and vega hedge, one needs three. To delta, gamma and vega hedge, one needs at least four traded assets. This “+1th asset” is always a financing instrument - borrowing or lending capital. In reality, this asset is the house’s credit line. A dealer is charged for using up the house’s credit lines when financing his or her trading activity.



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• Finally, the following matrix is useful to keep in mind:

Delta Gamma Vega

Spot Asset Yes (1) None None

Linear Asset Yes None None

Option Yes Yes Yes



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• Essentially, this matrix reminds us that one can manage delta using any asset - cash , linear, or non-linear (option). On the other hand, managing gamma and vega requires trading options.

• Now you are wise to why dealers loathe to gamma and vega hedge. It requires trading options and therefore paying spreads!

• Armed with all of this information, we are now ready to focus on option dealer price risk management, a.k.a. dynamic delta-gamma-vega hedging. Keep in mind that such hedging is not a panacea. It has shortcomings as the risk metrics are based on partial derivatives only. There is no such thing as hedging against jump risk!



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• We begin with a one-option book and delta hedging only. Once understood, we can quickly expand the analysis to accommodate entire option books and all risk metrics.

• A dealer writes a five-week European-style put on 10,000 shares of a non-dividend paying stock having the following parameter values: S = X = $20, r = 6%, and vol = 15%. The option contract’s ask price is $5,000. Assuming weekly re-balancing and the following weekly share prices and deltas (in boldface), illustrate the dynamics of delta hedging including the Week-5 net cumulative cost of hedging.



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Stock Position Shares Cost of Cumulative Interest

Week Price Delta Bought Shares Cost Cost

0 20.000 0.4415 (4415) (88300) (83300) (102)

1 20.625 0.1918 2487 51500 (31902) (43)

2 20.125 0.3870 (1952) (39284) (76229) (88)

3 19.750 0.6310 (2440) (48190) (124507) (144)

4 19.000 0.9918 (3608) (68552) (193203) (223)

5 18.000 1.0000 (82) (1476) (194902)

At expiration, the dealer is short 10,000 shares, the put is exercised, and so the dealer closes the short position at a cost of $20 = X per share or $200,000. That figure is offset by the cumulative proceeds of $194,902 plus the initial contract ask price of $5,000 as well as the interest earnings on said ask price. Notice that two assets were traded.



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• To illustrate delta hedging for a multiple option book, recall our earlier three-option book that had delta -17,100, gamma = -3600 and vega = -1,730,000. Here the dealer must purchase 17,100 shares of the underlying stock in order to instantaneously delta hedge the book. Re-balancing occurs thereafter in the same fashion as illustrated above for the one-option book.

• Now consider delta and gamma hedging the same three-option book. To gamma hedge, we must trade an option. Suppose there is an option available with delta 0.50, gamma 0.06, and vega 20. Then to gamma hedge, the dealer must purchase this option entailing 60,000 shares. This would add 3600 gamma to the book.



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Continuing, buying the option on 60,000 shares will add 30,000 delta to the book, and so the book’s new delta is 12,900. The dealer must short or sell off from inventory 12,900 shares in order to be delta zero. There are no feedback effects on gamma and so the dealer is both delta and gamma hedge. The dealer traded three assets.

• To vega hedge the book, the dealer will need to trade an option. Let there be another option available that has delta 0.60, gamma 0.03 and vega 25. Recall that the book’s vega is -1,730,000. So the dealer will need to buy this option entailing 69,200 shares, thus adding 1.73MM vega. This transaction will add 41,520 delta to the book, and so the dealer will need to short 24,420 shares to delta hedge. Again, there is no feedback on the vega and three assets were traded.



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• Now consider simultaneously hedging the book’s gamma and vega using our two traded options. We have:

-3600 + w1(.06) + w2(.03) = 0 (gamma equation)

-1,730,000 + w1(20) + w2(25) = 0 (vega equation)

Solving (using matrix algebra and rounding to whole lots), we have w1 = 42,300 and w2 = 35,400. So buying option #1 entailing 42,300 shares and buying option #2 entailing 35,400 shares will do the trick. These trades will add 42,390 delta to the book, so the dealer will need to short or sell off from inventory 25,290 shares in order to delta hedge. A total of four assets were traded to delta-gamma-vega hedge.



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• One can delta hedge with linear contracts such as stock index futures. But because these contracts do not have a delta of 1 - the delta of a cash asset - an adjustment must be made. The idea here is to trade the right number of linear contracts so as to achieve the same delta hedging action as trading the right number of cash assets. The adjustment factor is:

(# of cash assets)/(delta of linear contract)



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• Let us take an example. Suppose that a SP500 options dealer has a book delta of +150,000. Also, let the SP500 (I) be 1100, its dividend yield (y) be 1.5%, and the relevant funding rate (r) be 3% with continuous compounding. Also, the dealer elects to hedge using the nearby SP500 futures contract which has a maturity (T) of 1 month. Then the futures price (F) is Iexp(r - y)(T) or

1100exp(.03 - .015)(1/12) = 1101.38 and the futures delta is exp(.03 - .015)(1/12) = 1.001250782. To delta hedge using the futures contract, the dealer should short 149,812.6171 index units; given a futures contract multiplier of 250, the dealer should short 599 contracts.



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• There is no such thing as hedging against jump risk. Notable jumps have included the worldwide equity crash of October 1987, the devaluation of the Brazil real by 67% in December 1999, the tripling of Russian interest rates from 50% to 150% in August 1997, and the UK Long-Gilt yield moving by 400 basis points in a single trading day in 1992.

• Next I show a “hedge fail” example by introducing a price shock. The point is to demonstrate that an option book that has near-zero greeks can still suffer large losses due to jump risk. (An analogous demonstration which focuses on a jump in vol could be presented.)


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• It is well known that, under the assumptions invoked by the Black-Scholes-Merton framework, the value of an option book, V, will change according to the following equation (ignoring higher order dt terms):

dV = (dS)(Bdelta) + (dvol)(Bvega) + (dt)(Btheta) +

(.5)(dS^2)(Bgamma) + (.5)(dvol^2)(Bvanna)(eq 1)

Now introduce a simple two-option book that, combined with a position in the underlying stock, has nearly zero delta, gamma, vega and vanna. According to the above equation, the change in the value of the book should be about zero for a change in S (or vol). And, indeed, this is correct if the change in S is small. But how much will the book’s value really change by if S jumps?


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• The two options and stock position comprising the book are:

– Option #1: A European call on a non-dividend stock. S = $100 = X, vol = 25.0%, r = 1.5%, and T-t = 0.25. The call price is $5.16, delta = 0.5368 (per share), gamma = 0.0318 (per share), vega = 0.1986 (per percent), and vanna = 0 (for all intents and purposes). The dealer is long the call involving 200,000 shares.

– Option #2: Also a European call on the same stock. X = $92.50, vol = 20.5%, r = 1.7%, and T-t = 0.30. The call price is $9.41, delta = 0.7869, gamma = .0259, vega = .1592, and vanna = 0. The dealer is short the call involving 245,000 shares.

– The dealer buys 85,400 shares.


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• The book’s overall greeks are delta = -30, gamma = +14, vega = +716, and vanna = 0. These are quite small and well within position limits for any equity options dealer. [The theta of the book is irrelevant in this context as the stock price jump (introduced next) is instantaneous.]

• Now let the stock price drop 10% from $100 to $90. The new prices of the two options are $1.39 (#1) and $3.14 (#2). The P&L of the three positions is a loss of nearly $72,000:

(1.39 -5.16)(200,000) = -$754,000

(3.14 - 9.41)(-245,000) = +$1,536,150

(90 - 100)(85,400) = -$854,000

Total = -$71,850


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• Our last topic concerns interaction among the greeks as well as how the greeks change with respect to changes in key variables such as volatility and option maturity.

• There are several ways to go about this task. We consider three approaches. The first invokes equation 1 from Slide #24, which is repeated here:

dV = (dS)(Bdelta) + (dvol)(Bvega) + (dt)(Btheta) +

(.5)(dS^2)(Bgamma) + (.5)(dvol^2)(Bvanna) (eq 1)


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• By using equation 1, we can inspect the interaction among the various greeks. Here is an example. Suppose that the book is delta, vega, and vanna hedged. Then if the book’s theta is highly negative, its gamma tends to be highly positive and vice versa. Thus theta and gamma are proxies for each other.

• The graphs on the next two slides illustrate this interaction among theta and gamma. When gamma is positive, theta tends to be negative and the book’s value declines if there is no change in S over time, but increases if there is a large positive or negative change in S over time. When gamma is negative, theta tends to be positive and the book increases in value if there is no change in S over time but decreases in value if there is a large positive or negative change in S over time. So as the absolute value of gamma increases, the book’s P&L becomes more sensitive to changes in S over time.


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dV dV

dSdS

Slightly positive gamma Large positive gamma


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dV dV

dSdS

Slightly negative gamma Large negative gamma


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• Another way to inspect the interaction among the greeks is of course to compare their analytic formulae, assuming that said formulae exist. For example, for a vanilla equity option (European-style call or put on a non-dividend stock) the gamma and vega equations are:

Gamma = N’(d1)/[(S)(vol)(T-t)] Vega = N’(d1)(S)(T-t)

Since both equations represent scaled values of the standard normal density function N’(d1), then both greeks vary in a similar manner with changes in the option’s moneyness (S/X). In other words, a deep out-of-the-money option will have low gamma and vega, an at-the-money option will have high gamma and vega, and a deep in-the-money option will have low gamma and vega - consistent with the shape of the standard normal density function.


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• To analyze the effects of changes in greeks occasioned by changes in key input variables (e.g., delta v. vol, delta v. t, gamma v. vol), one should simply solve (either analytically or numerically) the partial derivative of the greek with respect to the input variable. If the sign of the derivative is ambiguous - which is commonly the result - then graphical analysis is useful.

• For instance, the delta of a vanilla call is N(d1). To inspect how delta changes with, say, vol, solve dN(d1)/dvol. The sign of this derivative is ambiguous. However, an inspection of the derivative reveals obvious properties, including that delta and vol tend to be negatively related for low vol levels and positively related for high vol levels.


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As another example, consider the relationship between gamma and vol. The sign of dGamma/dvol is unambiguously negative for vanilla options, with gamma dropping off sharply for large values of vol.

• Next consider delta v. time. The sign of dDelta/dt is ambiguous, but its inspection reveals certain obvious properties, such as delta rising for in-the-money options as they unwind and delta falling for out-of-the-money options as they unwind.

• Software demonstration.

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