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The Option Greeks and Market Making
Alan L. Tucker, Ph.D.
631-331-8024 (tel)
631-331-8044 (fax)
[email protected]
Copyright © 1997-2001
Marshall, Tucker & Associates,LLC
All rights reserved
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
1
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).
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
2
The Option Greeks and Market Making
• Topics addressed in this module include:
–
–
–
–
01/01/02
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
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
3
The Option Greeks and Market Making
• 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
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
4
The Option Greeks and Market Making
• 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.)
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
5
The Option Greeks and Market Making
• 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
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
6
The Option Greeks and Market Making
• 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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Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
7
The Option Greeks and Market Making
• Example: Let a dealer have a three-option book:
Option Position
Short Call
Short Call
Short Put
Shares
50000
20000
40000
Delta
0.55
0.48
-0.50
Gamma
0.04
0.02
0.03
Vega
17
14
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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
8
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
9
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
10
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
11
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
12
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
13
The Option Greeks and Market Making
• 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
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
14
The Option Greeks and Market Making
• 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-gammavega 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!
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
15
The Option Greeks and Market Making
• 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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Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
16
The Option Greeks and Market Making
Week
0
1
2
3
4
5
Stock
Price
20.000
20.625
20.125
19.750
19.000
18.000
Position
Delta
0.4415
0.1918
0.3870
0.6310
0.9918
1.0000
Shares
Bought
(4415)
2487
(1952)
(2440)
(3608)
(82)
Cost of
Shares
(88300)
51500
(39284)
(48190)
(68552)
(1476)
Cumulative
Cost
(83300)
(31902)
(76229)
(124507)
(193203)
(194902)
Interest
Cost
(102)
(43)
(88)
(144)
(223)
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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
17
The Option Greeks and Market Making
• 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. Rebalancing 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
18
The Option Greeks and Market Making
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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
19
The Option Greeks and Market Making
• Now consider simultaneously hedging the book’s gamma and vega
using our two traded options. We have:
-3600 + w1(.06) + w2(.03) = 0
-1,730,000 + w1(20) + w2(25) = 0
(gamma equation)
(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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
20
The Option Greeks and Market Making
• 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)
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
21
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
22
The Option Greeks and Market Making
• 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.)
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
23
The Option Greeks and Market Making
• It is well known that, under the assumptions invoked by the BlackScholes-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?
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
24
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
25
The Option Greeks and Market Making
• 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 =
01/01/02
-$71,850
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
26
The Option Greeks and Market Making
• 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)
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
(eq 1)
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The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
28
The Option Greeks and Market Making
dV
dV
dS
Slightly positive gamma
01/01/02
dS
Large positive gamma
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
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29
The Option Greeks and Market Making
dV
dV
dS
Slightly negative gamma
01/01/02
dS
Large negative gamma
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
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30
The Option Greeks and Market Making
• 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-themoney option will have high gamma and vega, and a deep in-themoney option will have low gamma and vega - consistent with the
shape of the standard normal density function.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
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31
The Option Greeks and Market Making
• 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.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
32
The Option Greeks and Market Making
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-themoney options as they unwind.
• Software demonstration.
01/01/02
Deutsche Bank: Option Greeks & Market Making
Copyright (c) 1997-2002 by Marshall, Tucker &
Associates, LLC
33