REPLICATING S&P500 INDEX FUTURES OPTIONS BY HEDGING THE GREEKS
Tarunkumar Maheshchandra Mishra
B.E., Hemchandracharya North Gujarat University, India, 2005
PROJECT
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF BUSINESS ADMINISTRATION
in
FINANCE
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
FALL
2009
REPLICATING S&P500 INDEX FUTURES OPTIONS BY HEDGING THE GREEKS
A Project
by
Tarunkumar Maheshchandra Mishra
Approved by:
______________________________, Committee Chair
Dr. Peter Sharp, PhD
__________________________
Date
ii
Student: Tarunkumar Maheshchandra Mishra
I certify that this student has met the requirements for format contained in the University
format manual, and that this Project is suitable for shelving in the Library and credit is to
be awarded for the Project.
__________________________
Dr. Monica Lam, Ph.D.
Associate Dean for Graduate and External Programs
College of Business Administration
iii
___________________
Date
Abstract
of
REPLICATING S&P500 INDEX FUTURES OPTIONS BY HEDGING THE GREEKS
by
Tarunkumar Maheshchandra Mishra
Statement of the Problem:
Replication of an option is used to realize the same payoff that could have been realized
by holding a position in the option itself when the target option is not available. In order
to ensure that a portfolio’s market value does not decrease beyond a certain level, it is
necessary to insure it by hedging with options. The project discusses the basic replication
strategy, followed by its application on S&P 500 index futures September 2009 contract.
The Greeks (delta, gamma, rho and kappa) of the option are used, in this replication
strategy, to replicate the payoff.
Sources of Data:
The historical prices, for June and September 2009 contracts on S&P 500 index futures
and the options on these contracts, are retrieved from Bloomberg.com and Reuters.com.
The project does not list the data obtained from those sources because of their license
restrictions. The Eurodollar historical prices data from November 2008 to September
iv
2009 are obtained from the website of Federal Reserve Statistical Release, H.15 which is
released everyday (unless holiday) at 2:30pm. The yield curves historical data from June
2000 to October 2009 are retrieved from the website of United States –Department of the
Treasury, in the Interest Rates Statistics section. For analysis purposes, during the project,
information is obtained from websites of Yahoo! Finance and CME Group/Chicago
Board of Trade.
Conclusions Reached:
The results of the project lead to a conclusion that an option pay-off can be replicated by
hedging the Greeks (delta, gamma, rho and kappa). The basic strategy of replication,
which provides the hedge to the delta of an option, shows that investing a proportion of
total investment, equal to the delta of the option, in the underlying asset of an option, will
provide the same pay-off that could be realized by holding 100% position in the option
itself. The hedging for gamma can be executed by investing a portion in another option
with a higher gamma or a more liquid option. This proportion should be a fraction needed
to neutralize the gamma of the target option payoff curve. Investing in more than one
option, with shorter maturity period or higher gamma as compared to the target option,
will reduce the gamma for replication of the target option. The importance of rho
hedging is realized when there is a change in carrying cost; in this project, it is the
interest rate. Whenever there is a high change in price along with the change in the
interest rate, the replication is adjusted for risk if it is hedged by a proper interest rate
addressing instrument like Eurodollar futures or other interest rate futures. After
v
empirical analysis, it is found that replication by hedging for delta, gamma and rho, all at
a time, provide a better replication to that the delta hedge or delta-gamma hedge by itself.
The project shows that replication of September 2009 contract for S&P 500 index futures.
The net replication error from a delta-gamma neutral hedge is less as compared with the
replication error from delta-neutral hedge. The project calculates the replicating error
difference between delta-gamma neutral hedge and delta-neutral hedge. Delta-neutral
hedge was $0.6187, $0.8773, and $0.1937 for September’09; call option with strike price
of $1025 on January 15, 2009, January 27, 2009, and April 9, 2009, respectively. The
calculations also provide an explanation for theta and rho exposures hedge in the
replication.
________________________________, Committee Chair
Dr. Peter Sharp, Ph.D.
_______________________
Date
vi
ACKNOWLEDGEMENTS
I would like to show my gratitude to Dr. Peter Sharp, Professor at California State
University, Sacramento, under whose guidance I am able to work on my project and who
provided me from the basics to detailed knowledge in financial markets. I would like to
thank him for providing me an opportunity and assistance with resources and directed me
to derive the results for this project. I would like to personally thank Mr. Ho Ho,
Quantitative Portfolio Manager at California Public Employees Retirement System
(CalPERS), Sacramento, who provided timely assistance for working strategies of
replicating an option. Along with that I am thankful to my immediate supervisor and
Investment Officer III at CalPERS, Sacramento, Ms. Lee Ann Nation and Division Chief,
Mr. Matthew Flynn, for providing me resources and encouragement for analysis and
research works that helped me concluding my project. And last but not least, I am
indebted to Mr. Rafael Garcia (Investment Officer II), Mr. Christian Cardeno (Investment
Officer III) and Investment Operations staff at CalPERS, Sacramento for providing me
help with questions and analysis related to options and equity market.
vii
TABLE OF CONTENTS
Acknowledgements ........................................................................................................ vii
List of Tables ....................................................................................................................x
List of Figures ................................................................................................................. xi
Chapter
1. INTRODUCTION ........................................................................................................1
Sources Review .....................................................................................................2
Why Option Replication? .....................................................................................3
2. METHODOLOGY .......................................................................................................6
Basic Replication Strategy ....................................................................................6
Hedging the Delta .................................................................................................6
Combined Hedge for Delta and Gamma .............................................................11
Gamma Hedge Ratio in the Replication .............................................................16
3. IMPLEMENTATION OF METHODOLOGY...........................................................18
Example of Replicating September S&P500 Index Futures Call Option using the
Greeks Hedging Strategy ....................................................................................18
Hedging the Error Related to Delta for the Target Call Option ..........................18
Hedging for Delta and Gamma ...........................................................................23
4. HEDGING WITH OTHER GREEKS AND VOLATILITY CONSIDERATIONS ..30
Role of Carrying Cost on Replication of a Target Option ..................................30
Rho Hedging .......................................................................................................32
Hedging for Kappa ..............................................................................................34
Hedging for Theta ...............................................................................................35
Project Limitations .............................................................................................37
5. CONCLUSION ...........................................................................................................38
Appendix (Rates from Eurodollar Futures versus US Treasury Bills) ...........................40
viii
References .......................................................................................................................44
ix
LIST OF TABLES
Table 1 Change in Delta for Sep ’09 and June ’09 Call Options Prices ........................... 12
Table 2 Delta-Neutral Hedge on September ’09 S&P500 Index Futures Call Option for
April 9, 2009, January 15, 2009 and January 27, 2009 ............................................. 20
Table 3 Values of Greeks for June 2009 Call Option ....................................................... 23
Table 4 Calculations: Multiplier for June’09 Call Option for Gamma Neutral Hedge .... 24
Table 5 Calculations: Delta and Gamma Neutral Hedge Pay-offs and Errors in
Replication ................................................................................................................. 26
Table 6 Replication without Rho Hedging ....................................................................... 31
Table 7 Delta-gamma-rho Hedging .................................................................................. 33
Table 8 Theta Hedge ......................................................................................................... 36
x
LIST OF FIGURES
Figure 1 Historical Prices - September 2009 Futures Contract on S&P500 Index ............. 5
Figure 2 Historical Prices - September 2009 Futures Contract on S&P500 Index ............ 5
Figure 3 Delta Hedge: Tangents to Historical Price Curve of Target Call Option on
September ’09 S&P 500 Index Futures Contract ........................................................ 8
Figure 4 Trend Lines showing Curvature Pay-offs for September and June 2009 S&P 500
Index Futures Contract Call Options ......................................................................... 11
Figure 5 Chart showing Change in Delta for Sep’09 and June’09 Call Options for a
Specific Period ........................................................................................................... 13
Figure 6 Percent change in option delta to percent change in underlying index prices .... 14
Figure 7 Delta and Gamma Hedge on Target Option Pay-off Curvature ......................... 15
Figure 8 Comparison of Change in Option Price with Delta-Neutral Hedge Pay-off ...... 22
Figure 9 Comparison of Change in Option Price with Delta and Gamma Neutral Hedge
Pay-off ....................................................................................................................... 27
Figure 10 Comparison of Delta Hedge Error with Delta-Gamma Hedge Error ............... 28
Figure 11 Yield curves for July-Dec 2000, July’07, Oct ’07, July’08, Oct’08 ................ 41
Figure 12 TED spread larger period (Jan’00 – Oct ’09) ................................................... 42
Figure 13 TED spread for shorter period (Jan ’08 -Oct ’08) ............................................ 42
xi
1
Chapter 1
INTRODUCTION
Stock options provide a comparable alternative to direct investment in the underlying
stock. To decide what to choose as the alternative requires some important facts to be
explained. Those facts are: the valuation of the option over time, its anticipated rate of
return over the specific period, the risk introduced in its pricing as a result of volatility in
the underlying price, the margin requirements, the transaction costs for the option as
compared to the transaction costs of the underlying asset, the buyers, the sellers, and how
to get an option off, if the option position is not available.
Modern portfolio theory says that, “the price behavior of an option is very similar to a
portfolio of the underlying stock and cash that is revised in a particular way over time”,
(Rubinstein and Leland). So there exists a replicating portfolio strategy, involving stock
and cash only, that creates returns identical to those of an option. Option replication can
be defined as a strategy or technique to create an option like payoff pattern through a
series of transactions in the underlying or in related futures contracts. (Gastineau and
Kritzman, 1996). The relevance is similar to traditional portfolio insurance of dynamic
hedging using replicating strategies. It is also similar to creating a long-term option by
using series of short-term options (Gastineau and Kritzman, 1996). The project focuses
on replicating a longer maturity period option on S&P 500 index futures using another
option, with shorter maturity period, on S&P 500 index futures.
2
Sources Review
The project uses the book Option Pricing and Investment Strategies, 3rd edition, by
Richard M. Bookstaber, as a major source for the replication strategy used to replicate a
call option on S&P 500 index futures September ’09 contract. The chapter on option
replication technology discusses the hedging techniques for the Greeks: delta, gamma,
theta and kappa of the target option. The book explains the basic replication strategy as
delta hedge, which is basically an investment in the underlying asset in a proportion equal
to the delta of the target option. The gamma hedge during replication of an option can be
created by introducing an option with higher gamma or more liquid option with less time
to expiration, as compared to the target option. The delta-gamma neutral hedge has less
replication error than the delta hedge.
A second major resource for this project was the book Financial Options from Theory to
Practice, 1st edition, 1992 by Stephen Figlewiski, William Silber, and Marti
Subramanyam. The concept of index option replication is explained with the example of
dynamic hedging and portfolio insurance. The book talks about the need for replication
and general strategies on replication of an option pay-off.
In addition, the project used the concept of implied volatility consideration for option
pricing and added uncertainty with addition of securities in a replication portfolio. The
risk-free interest rate for option pricing can be U.S treasury bills or Eurodollar futures.
3
The article, “Trading Interest Rate Inefficiencies: the stability and predictability of
Eurodollar Futures calendar spreads makes them attractive for interest rate trading
strategies. But to become fluent in Eurodollar forecasting, you'll need to learn a little
about U.S. Treasuries.”, by Paul Cretien discusses the accurate prediction of the interest
rates associated with the Eurodollar futures using the U.S. treasury yield curves and TED
spread. In their article, “Can a dynamic strategy replicate the returns of an option?” by
Michael Asay and Charles Edelsburg, the problems with dynamic hedging as change in
the hedge ratios requires adjusting the hedge each and every time. The anticipated price
gap for adjustment in hedge cannot be ideally observed as the price changes in the
market. Moreover, the adjustment in positions can be done when prices change by at
least a minimal amount to reach the hedge. Also, this adjustment involves adjustment in
number of contract, which can lead to change in the number of contracts outstanding.
Why Option Replication?
Portfolio insurance is one of the important reasons that lead to a decision of pay-off
replication. A portfolio can be insured by another portfolio that replicates the same payoff function equivalent to a protective put strategy (Figlewiski, Silber and
Subramanyam,1992). Alternatively, an option replication technique can be used to
duplicate the pay-off of an option and realize a desirable pay-off for a longer period
beyond the life of the target option; and this pay-off is comparable to the derived price
4
from the option pricing model for that prolonged period. For instance, in order to receive
a pay-off similar to an option on a portfolio, so as to insure the portfolio from a drop in
market value; replication strategy can be useful, once a protective put strategy is decided,
that involves an option on the portfolio. This protective put strategy should at least
determine these factors: the minimum allowed portfolio market value to be maintained,
the volatility of the portfolio and the duration for the insurance. Based on these decisions,
an option on the portfolio can be designed and a similar pay-off to that option on the
portfolio can be obtained using replication strategy. Replication can also be helpful in
creating or hedging exotic options and some of lookback options.
The reason behind considering an option on the portfolio, as a protective put strategy, is
that the portfolio of options is expensive than options on portfolio (Markowitz, 2009). It
is because the volatility of a portfolio of assets overall is less than the sum of volatilities
of individual options in the portfolio, provided the assets in the portfolio are not
correlated to each other (Armbruster, 2009). Based on this reasoning, the price
associated to the option on portfolio, as determined by Black-Scholes option pricing
model, turns out to be less than the sum of prices of options on individual assets in the
portfolio.
The project shows the replication of an S&P 500 index futures option expiring on
September 18th, 2009. This replication method is based on hedging the Greeks,
specifically delta and gamma, obtained from Black-Scholes option pricing model. Later,
5
there will be a short description about hedging the rho and kappa for the option during
replication.
Figure 1 Historical Prices - September 2009 Futures Contract on S&P500 Index
(Futurespros.com, 2009)
Figure 2 Historical Prices - September 2009 Futures Contract on S&P500 Index
(Futurespros.com, 2009)
6
Chapter 2
METHODOLOGY
Basic Replication Strategy
The price movements of the stock and its options tend to fluctuate in the same direction.
But the variation in option price may be smaller than the variation in the underlying
stock. In other words, a dollar change in the stock price resembles to less than a dollar
change in the price of its option. The conditions for replicating a call option on S&P 500
index futures contracts are: a) the initial investment must provide the same absolute
dollar return, as a call for small changes in the underlying asset price; b) it must be equal
to the value of the call to give the same rate of return; c) and it must not cost anything
more than the initial investment. (Rubinstein and Leland, 1995).
Hedging the Delta
The basic replication strategy is the hedge with delta. The basic requirement for this
option replication strategy is an accurate delta for the option from the pricing model. The
delta measures the change in the price of option at a specific instance, with the change in
the price of the underlying asset. So, after the hedge, there needs to be the same
7
movement in the position that is equivalent to the realized change in the option price.
This can be understood with the help of following example.
The Black-Scholes option pricing model determined the delta for S&P500 Index Futures
call option on September 2009 contract, with strike price $1025, spot price $756, time to
expire 206 days, interest rate 1.08151%, dividend yield 3.42% (Standard and Poor's,
2009) and implied volatility 30.3108%2, as 9.812%. This means the option price changes
9.812% as compared to the price of the underlying asset. If the price of S&P500 index
future contract for September 2009 (denoted as SPU9, on a Bloomberg Terminal), goes
up by $1, then the call option price will go up by $0.09812. In order to reflect this payoff for the call option, holding 9.812% position in the underlying will serve the purpose
and so a $1 rise in the underlying lead us to realize 9.812% of $1 = $0.09812 pay-off.
Similarly, if the price change in the underlying is $2, then the net change realized, by
holding 9.812% of the underlying asset, should be $0.19624. In this case, looking at the
pay-off, anyone will not be able to know whether the performance is a result of
replication, which is holding the underlying asset, or holding a position in the associated
option.
1
For calculation purposes a constant interest rate from 3-month Eurodollar futures taken as of May 29,
2009 through the entire project.
2
Volatility used in the project is the implied volatility calculated from the Black-Scholes option pricing
model using the actual market price.
8
The hedging process is described in figure.3. The target call option is shown as the
function of the price of the underlying asset, which is the SPU9 contract here. The slope
of this curve at current price is given by its delta.
Figure 3 Delta Hedge: Tangents to Historical Price Curve of Target Call Option on
September ’09 S&P 500 Index Futures Contract
Historical Price - Payoff curvatures
x-axis:
Underlying
Price
30.00
y-axis : Option
price
25.00
Hedging
Option
20.00
15.00
Asset 2
Asset 1
Target
Option
10.00
5.00
Historical
Price
805
798.2
788.8
779.6
773.5
772.7
763
755.9
751.5
745.9
728.2
710.9
699.9
682.5
670.7
0.00
Expon.
(Historical
Price)
Historical price series is the actual price of the SPU9 contract call option. The Expon
(Historical Price) is the target option trend line. Introducing a certain specific
proportion security such that the pay-off trend traces the target option curve then we will
be very close to replicating actual payoff of the call option. The Asset (1) and Asset (2)
9
are the tangents to the trend line at two different prices. These tangents imply the
percentage of the underlying positions we need to hold to get the payoff. For instance,
holding 50%position in a stock will give 50% of profits and the payoff will have a line
with slope of 0.5.
Figure 3 illustrates the relation in target option price and the delta which used to replicate
it. For instance, when the price of September ’09 contract on S&P500 price or SPU9C
(Historical Prices, 2009) is $763.0, the call option with strike price $1025 is $8.00 with a
delta of 10.5172%. The asset (1) line is the tangent to the target option price curve at
underlying price $763 with a slope of 0.1051. At this point it can generate a payoff
similar to the target option pay off by buying 10.5172% of September ’09 S&P 500 index
futures contract (SPU9) (Historical Prices, 2009). It is this payoff that the replicating
strategy must produce. This is in fact matching the slope which in turn is delta for the
target call option. Because of the price volatility in the underlying futures contract, there
is a continuous significant price movement in the associated target call option, and hence
there is a continuous change in delta, which is represented by gamma for the call option.
This price volatility will fail to replicate the pay-off if the proportion of position in the
underlying contract is not changed with the change in delta. As the price of the contract
increases from $763.3 to $788.8, the option price is likely to increase and the slope of the
tangent increases to 11.9986%, which is the delta for the call option on this contract at
strike price $1025. The change in the delta indicates the change in the position of the
underlying asset from 10.5172% to 11.9986% to receive the same pay-off as the trend
10
line for the target option. Higher frequency of adjusting the hedge, by maintaining the
position as per the delta of target option, will lead to a better replication of the target
option pay-off. Consequently, multiple changes the position for the underlying contract
also increases the transaction cost, and thereby increases cost to the replication.
In this case, the curvature of the target option is expected because the option has different
delta with the change in the price of the underlying contract. To hedge this change in
delta of the target option with price of underlying contract, and generate equivalent payoff as of target option, a gamma neutral hedge should be added to the delta-neutral hedge
in the replication strategy. In practice, it is not realistic to change positions continuously
for the underlying holdings after every point change in the price as this can lead to very
high trade activity and transaction costs, and thereby higher cost for replication.
As the gamma hedge requires the hedging on a curve of target pay-off, an instrument
with a curved payoff can provide a better hedge. Another call option with shorter
duration to expiration than the target call option can be used to hedge the gamma effect in
replication. The hedge is possible, because the option with shorter duration to expiration
tend to have a greater gamma than the gamma of an option that has higher expiration
period (Bookstaber, 1991). Thus, a combination of a specific portion of higher gamma
option with specific portion zero gamma underlying assets in the hedge will match both
the slope and the curvature of the target option pay-off.
11
Figure 4 Trend Lines showing Curvature Pay-offs for September and June 2009 S&P
500 Index Futures Contract Call Options
140.00
120.00
Historical price for SPU9 and SPM9
contract call options with strike price
$1025
x-axis :
underlying
contract
prices
100.00
80.00
y-axis: option
prices
60.00
SPM9
SPU9
40.00
20.00
0.00
880.00 900.00 920.00 940.00 960.00 980.00 1000.001020.001040.00
Poly.
(SPM9)
Poly.
(SPU9)
-20.00
Figure 4 shows SPU9 – S&P index futures September’09 contract and SPM9 – S&P
index futures June’09 contract. The dotted lines show the trend lines for historical prices
for the call options on individual contracts.
Combined Hedge for Delta and Gamma
The theory for delta-gamma hedge says that the change in the price of the underlying as
well as passage of time leads to inaccuracies in hedging (Bookstaber, 1991). It can be a
single option or a portfolio of options. Therefore, it is very important to know the factors
that are contributing to the change in the option price. The major contributors are the
price of the underlying asset, the interest rates, and the volatility. The pricing model
12
(Black-Scholes Option Pricing Model) that is being used to calculate the prices must
provide appropriate considerations to all prior mentioned factors when determining the
option price or implied volatility.
Expiry date for S&P 500 index futures June 2009 call option (SPM9C) with $1025 strike
is June 15, 2009 while that for S&P 500 index futures September 2009 call option
(SPU9C) with strike price $1025 is September 18, 2009. The delta for associated with
the per cent change relationship is shown in the table below and presented in the graph.
It also shows that the gamma for the SPM9 contract is higher as the time to expiration for
the SPM9 call option is lower than the SPU9 call option at strike price 1025.
Table 1 Change in Delta for Sep ’09 and June ’09 Call Options Prices
Date
Change in
Delta(Sep'09
Call option)
Change in
Underlying
Asset
Price(SPU9)
Change in
Delta(June'09
Call option)
28-Nov-08
-21.9%
-9.0%
36.3%
16-Jan-09
-16.4%
-5.1%
21.2%
9-Feb-09
-20.0%
-4.5%
18.0%
27-Feb-09
-16.4%
-3.9%
2.2%
13-Jan-09
-11.2%
-3.4%
24.1%
28-Jan-09
-15.2%
-3.3%
20.0%
6-Apr-09
-14.9%
-2.0%
8.3%
12-Dec-08
-4.5%
-1.5%
35.1%
24-Feb-09
-6.7%
-0.9%
4.2%
24-Dec-08
0.7%
0.5%
26.5%
7-Apr-09
4.1%
1.0%
6.1%
15-Apr-09
6.9%
1.5%
7.7%
29-Dec-08
6.9%
2.0%
26.6%
4-Dec-08
8.3%
3.0%
31.5%
20-Jan-09
9.7%
4.0%
16.9%
13
15-Dec-08
11.4%
4.6%
33.3%
9-Mar-09
29.5%
6.0%
0.9%
20-Mar-09
44.2%
7.0%
4.4%
Table 1 shows some of the calculation change in the delta for Sep ’09 and June ’09 call
options as they relate to the per cent change in the underlying contract for Sep ’09 S&P
500 index futures.
Figure 5 Chart showing Change in Delta for Sep’09 and June’09 Call Options for a
Specific Period
52.0%
x-axis: Date
42.0%
y-axis: %
Change in
delta
32.0%
22.0%
12.0%
2.0%
Change in
Delta Sep '09
Call
Option
-8.0%
-18.0%
-28.0%
8-Nov-08
18-Nov-08
28-Nov-08
8-Dec-08
18-Dec-08
28-Dec-08
7-Jan-09
17-Jan-09
27-Jan-09
6-Feb-09
16-Feb-09
26-Feb-09
8-Mar-09
18-Mar-09
28-Mar-09
7-Apr-09
17-Apr-09
27-Apr-09
7-May-09
17-May-09
27-May-09
6-Jun-09
16-Jun-09
26-Jun-09
-38.0%
Change in
Delta June '09
Call
Option
14
Figure 6 Percent change in option delta to percent change in underlying index prices
50.0%
x-axis : Change in the
underlying price
40.0%
30.0%
y-axis: Change in delta
20.0%
Change in Delta
for Sep'09 Call
option
10.0%
0.0%
Change in Delta
for June '09 Call
option
-10.0%
-20.0%
Poly. (Change in
Delta for Sep'09
Call option)
-30.0%
-100.0%
-3.9%
-2.4%
-2.1%
-1.7%
-1.4%
-1.2%
-0.8%
-0.2%
0.0%
0.1%
0.3%
0.6%
0.8%
1.1%
1.4%
2.0%
2.7%
3.2%
4.0%
-40.0%
Poly. (Change in
Delta for June
'09 Call option)
Figure 6 shows a chart of the change in the delta values for Sep ’09 (SPU9C) and June
’09 (SPM9C) call options versus the change in the underlying asset prices. It is apparent
that the series for the SPM9C call option is having higher changes in delta with the
change in underlying price as compared with the call for SPU9 contract, except when the
change in the underlying contract price rises above 1.4%. Higher change in delta shows
higher gamma for the option. The data from the table above the graph is used to draw
this relationship. The replication strategy that uses the higher gamma option to neutralize
the curve for the target option is shown in the below graph.
15
Figure 7 Delta and Gamma Hedge on Target Option Pay-off Curvature
Historical Price - Payoff curvatures
30.00
x-axis:
Underlying
Price
25.00
y-axis : Option
price
Hedging
Option
20.00
15.00
Asset 2
Asset 1
Target
Option
10.00
5.00
Historical
Price
805
798.2
788.8
779.6
773.5
772.7
763
755.9
751.5
745.9
728.2
710.9
699.9
682.5
670.7
0.00
Expon.
(Historical
Price)
Based on the analysis from figure 5 and figure 6, it is apparent that the June ’09 S&P 500
index futures call option can be used to replicate the September ’09 S&P 500 index
futures call option pay-off. The basic replication strategy to be applied is shown in figure
7. In order to hedge the gamma for the target call option, it requires a specific proportion
of the investment to be invested in the call option with higher gamma, in this case the
June ’09 S&P 500 index futures call option or SPM9C. This multiplier to unit call option
investment in the portfolio for replication is the gamma hedge ratio.
16
Gamma Hedge Ratio in the Replication
The gamma hedge ratio is the proportion of option used to replicate the target option in
order to develop a gamma neutral hedge. This ratio is the multiplier to unit call option;
and can be calculated as follows:
1. Calculate the delta and gamma for the target call option and the call option to be
used in replication, which is having higher gamma than the target call option.(In
this case, June’09 call option is with higher gamma and Sep’09 call option is the
target call option.)
2. Calculate the fraction that converts the gamma of June’09 call option equal to
gamma of September ’09 call option.
3. This fraction is the gamma hedge ratio for the June’09 call option investment.
When the call option with higher gamma (June ’09) is invested in a proportion equal to
the gamma hedge ratio found from previous description, the net Greeks (delta, gamma,
rho and kappa) can be calculated. The net Greeks are the difference of target call option
Greeks and the Greeks multiplied with the gamma hedge ratio. For instance, net delta is
the delta for target call option (September ’09 call option) minus product of gamma
hedge ratio and delta of higher gamma call option (June ’09 call option). The net gamma
turns out to be zero (gamma neutral hedge), and there appears to be values for net kappa
and net rho. The net delta value can now be used as the delta for the replication and delta
17
hedge can be executed. To be more specific, the net delta is the proportion of investment
to be made in the underlying asset (S&P 500 index futures contracts), so as to create a
delta-neutral hedge after having previous gamma neutral hedge.
18
Chapter 3
IMPLEMENTATION OF METHODOLOGY
Example of Replicating September S&P500 Index Futures Call Option using the
Greeks Hedging Strategy
Delta-neutral hedging and delta-gamma neutral hedging are the two replication
strategies that are used to replicate the September 2009 call option, on S&P 500 index
futures contract, with strike price of $1025. The aim is to reduce the error during
replication and generating the pay-off close to the original option.
Hedging the Error Related to Delta for the Target Call Option
As discussed in methodology, in order to replicate a September 2009 call option on
S&P 500 index futures contract, the delta for the call option is calculated for
associated spot price of underlying asset (S&P 500 index futures contract for
September 2009) using Black-Scholes option pricing model. For ease of explanation
three specific pair of dates, April 9th, 2009, January 15th, 2009 and January 27th, 2009,
are taken as examples of single day pay-off replication. These dates can provide a
better explanation as the underlying S&P 500 index futures contract for September
19
’09 price change on these dates is highly diverse. Note: The term “September’09 call
option” in the tables is September 2009 call option on S&P 500 index futures
contract and the term “June’09 call option” means the June 2009 call option on S&P
500 index futures.
Table 2 shows September ’09 call option prices as well as underlying prices for April
9, 2009, January 15, 2009, and January 27, 2009. Adjacent to them are the next
business day values, so that the values of underlying price, delta, gamma, rho and
theta can be compared. It is apparent, from the table 3, that the change in the
underlying prices is diverse. The replication error can be calculated for three
different scenarios, in this case, small underlying asset price change of up to $2,
underlying price change up to $10, and high price change observed in the underlying
asset price.
20
Table 2 Delta-Neutral Hedge on September ’09 S&P500 Index Futures Call Option for
April 9, 2009, January 15, 2009 and January 27, 2009
Sep '09 Call
Option
9-Apr-09 13-Apr-09 15-Jan-09 16-Jan-09 27-Jan-09
28-Jan-09
Underlying
contract price
September
09 call option
price
Strike price
Interest Rate
Dividend
Yield
Implied
volatility
Time to
expire(days)
Delta
Gamma
Rho
Theta
Change in
underlying
price
$849.10
$841.10
$832.70
$865.00
$19.00
$18.80
$33.90
$35.50
$1,025.00 $1,025.00 $1,025.00 $1,025.00
1.0850% 1.0850% 1.0850% 1.0850%
$26.40
$1,025.00
1.0850%
$32.80
$1,025.00
1.0850%
3.4200%
3.4200%
32.1692% 32.2341% 35.6672% 35.3623% 32.8105%
32.0699%
163
159
247
246
235
0.20388 0.20344 0.26210
0.27121
0.23223
0.00154 0.00155 0.00132
0.00134
0.00138
68.82500 67.18462 124.62747 129.81784 107.50820
53.27246 54.14368 52.62253 53.62200 46.66398
235
0.27129
0.00148
129.41694
50.94521
3.4200%
$850.50
3.4200%
$1.40
$832.00
3.4200%
3.4200%
$9.10
$32.30
Change in the
option price
(pay-off)
-$0.20
$1.60
$6.40
0.2039*84 0.2039*85 0.2621*83 0.2621*84 0.2322*832
9.1=
0.5=
2=
1.1=
.7= 0.2322*865=
Delta neutral
hedge
$173.1174 $173.4028 $218.0659 $220.4510 $193.3809
$200.8820
Delta neutral
hedge pay-off
$0.2854
$2.3851
$7.5011
Delta neutral
hedge error
0.29-(0.20)=
2.39(1.60)=
7.50(6.40)=
$0.4854
$0.7851
$1.1011
Delta-neutral hedge is shown in the calculations of table 2. The three vertical
sections calculate the delta-neutral hedge pay-offs for dates April 9, 2009, January 15,
21
2009, and January 27, 2009. The change in the underlying from April 9th to April
13th is $1.4 (small underlying asset price change), from January 15th to January 16th is
$9.1(underlying asset price change around $10) and from January 27th to January 28th
is $32.3(high change in underlying asset price). The delta for Sep ’09 call option on
April 9th is 0.20388, so as per the methodology discussed in chapter 2, the delta
hedge pay-off equals to difference in delta times the underlying prices, which is
(0.20388*850.5 – 0.20388*849.1=) $0.285. In case the investment was done in the
call option, the realized pay-off would be ($18.8-$19=) -$0.2. This shows the
realized replication error with delta-neutral hedge strategy with an underlying asset
price change at -$0.485. Similarly, the replication error for January 15th with a $9.1
change in underlying asset price, as shown in table 2, is $0.7851 and for January 27th,
with a $32.3 change in the underlying asset price, the replication error is $1.1011.
22
Figure 8 Comparison of Change in Option Price with Delta-Neutral Hedge Payoff (Daily Hedge)
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
Change
in option
price ($)
8-Jun-09
25-May-09
11-May-09
27-Apr-09
13-Apr-09
30-Mar-09
16-Mar-09
2-Mar-09
16-Feb-09
2-Feb-09
19-Jan-09
5-Jan-09
22-Dec-08
8-Dec-08
24-Nov-08
Delta
neutral
hedge
pay-off
($)
Figure 8 shows comparison of the September 2009 call option price pay-off with
delta-neutral hedge pay-off for the period November 24, 2008 to June 15, 2009. The
dotted line is the September ’09 call option pay-off and the continuous line shows the
delta-neutral hedge pay-off. The mismatch in the curves shows the error in
replication. This hedge is a daily basis hedge, which means for every single day if the
underlying asset price is changed, the replication hedge needs to be adjusted to
receive target call option pay-off.
23
Hedging for Delta and Gamma
The second most important factor to be considered, when using the delta as a hedging
base, is the gamma of the target call option. For every dollar change in the
underlying price, the change observed in delta of the option, on that underlying asset,
is given by the gamma for that option ( Option Trading Tips, 2009). The table 3
shows the values for the delta, gamma, rho and theta for the June 2009 call option.
Table 3 Values of Greeks for June 2009 Call Option
June '09 Call
Option
Underlying
contract price
June '09 call
option price
Strike price
Interest Rate
9-Apr-09 13-Apr-09
$852.60
15-Jan-09
$854.00
$835.80
$5.30
$4.70
$1,025.00 $1,025.00
$19.90
$1,025.00
1.085%
1.085%
1.085%
16-Jan-09 27-Jan-09 28-Jan-09
$844.90
$835.70
$868.00
$20.70
$12.60
$16.80
$1,025.00 $1,025.00 $1,025.00
1.085%
1.085%
1.085%
Dividend Yield
3.42%
3.42%
3.42%
3.42%
3.42%
3.42%
Implied
volatility
32.3396% 32.0908% 35.5896% 34.9898% 31.6391% 30.5918%
Time to
expire(days)
68
64
152
151
140
139
Delta
0.09952
0.09261
0.20432
0.21204
0.15905
0.19970
Gamma
0.00147
0.00144
0.00147
0.00152
0.00147
0.00170
Rho
14.81989 13.04295 62.82797 65.55013 46.14975 59.61317
Theta
53.72233 52.29450 60.77288 61.80816 48.26203 55.63571
24
In order to find the proportion of June’09 call option for the gamma neutral
hedge, the gamma values of the September’09 call option from the table 2 and
gamma values of June’09 call option from table 3 are used.
Table 4 Calculations: Multiplier for June’09 Call Option for Gamma Neutral
Hedge
9-Apr-09 13-Apr-09 15-Jan-09 16-Jan-09 27-Jan-09 28-Jan-09
Price
change in
underlying
Sep 09
contract
Price
change in
Sep '09
call option
Gamma
hedge ratio
for June
'09 call
option
Gamma
hedge ratio
* June'09
call option
price
Modified
delta
Modified
gamma
Modified
rho
Modified
theta
$1.40
$9.10
$32.30
-$0.20
$1.60
$6.40
1.05113
0.89816
0.93592
$5.57
$4.94
$17.87
$18.59
June'09 call option
$11.79
$15.72
0.10460
0.09734
0.18351
0.19044
0.14886
0.18690
0.00154
0.00152
0.00132
0.00136
0.00138
0.00159
15.57762 13.70983 56.42957 58.87450 43.19237 55.79303
56.46913 54.96830 54.58376 55.51361 45.16930 52.07046
25
For April 9, 2009, the gamma value for September ’09 call option is 0.001542 and
for June ’09 call option, the gamma value is 0.001443. As a result, the gamma
hedge ratio for June’09 call option is, (0.001542/0.001443=)1.05113. This
means, in order to create a gamma neutral hedge, 1.05113 of June’09 call option
should be bought in the portfolio. The net exposure to Greeks should now be
modified by 1.05113 times, which means the delta for 1.05113 of June ’09 call
option is (0.09952*1.05113=) 0.104604. Similarly, now the modified rho is
15.57762 and modified theta value is 56.46913. The modified gamma value is
0.001542, which justifies the gamma hedge ratio is used to match the gamma
exposure of September’09 call option and create a gamma neutral hedge.
The replication can be hedged for delta by using the delta-neutral hedge strategy.
The proportion of required investment in underlying asset S&P500 index futures
September 2009 contract, in this case, must be the value of net delta 0.09928,
which is the difference of delta for September’09 call option(0.20388) and
modified delta for 1.05113 of June’09 call option(0.104604). Similarly, net rho
and net theta values should be 53.24737 and -3.19667, respectively. The net
gamma value is 0, which shows a gamma neutral hedge in the replication process.
As described in the methodology in chapter 2, and after finding out the gamma
hedge ratio for gamma hedge and proportion of investment in underlying asset,
the delta and gamma combined neutral hedge can be applied to replication by
26
adding 1.05113 of June’09 call option and 0.09928 of underlying asset (S&P 500
index futures September 2009 contract) to the portfolio.
The pay-offs for this delta-gamma neutral hedge is shown in table 5.
Table 5 Calculations: Delta and Gamma Neutral Hedge Pay-offs and Errors in
Replication
Net delta
Net gamma
Net rho
Net theta
Delta neutral
hedge
Delta and gamma
neutral hedge
Realized price
change in June'09
call option position
Realized price
change with
position in
underlying Sep '09
contract
Delta gamma
neutral hedge payoff
Error in replication
9-Apr-09
0.09928
0.00000
53.24737
-3.19667
15-Jan-09
0.07859
0.00000
68.19790
-1.96123
27-Jan-09
0.08337
0.00000
64.31583
1.49468
0.0993 of
0.0786 of
0.0834 of
underlying asset
underlying asset
underlying asset
Pay-offs
1.0511 of June'09 0.8982 of June'09 0.9359 of June'09
call option +
call option +
call option +
0.0993 of
0.0786 of
0.0834 of
underlying asset
underlying asset
underlying asset
-$0.6307
$0.7185
$3.9309
$0.1390
$0.7152
$2.6930
-$0.4917
$1.4337
$6.6238
$0.2917
$0.1663
$0.2238
The realized price change in 1.05113 of June’09 call option position is 1.05113 *
(4.7-5.3) = -$0.6307. Similarly, the realized price change, with the delta-neutral
27
hedge, in the underlying September 2009 S&P 500 index futures contract is
0.09928 * (850.5 – 849.1) = $0.1390. The net delta-gamma neutral hedge pay-off
will be -$0.4917, which shows replication error of $0.2917 ( = absolute value of $0.4917-(-$0.2)), when compared to the actual pay-off realized with the
September 2009 call option price change.
8-Jun-09
25-May-09
11-May-09
27-Apr-09
13-Apr-09
30-Mar-09
16-Mar-09
2-Mar-09
16-Feb-09
2-Feb-09
19-Jan-09
5-Jan-09
22-Dec-08
8-Dec-08
24-Nov-08
Figure 9 Comparison of Change in Option Price with Delta and Gamma Neutral
Hedge Pay-off (Daily Hedge)
15
13
11
9
Change
7
in option
5
price ($)
3
1
-1
-3
Delta
-5
gamma
-7
hedge
-9
pay-off
-11
($)
-13
-15
Figure 9 shows the comparison for September 2009 call option price pay-off, the
dotted line, on daily basis with the delta-gamma hedge pay-off, the continuous line.
28
Comparing the replication error from table 2 to table 5 shows that error is reduced by
$0.1937489, for April 9, 2009 Sep’09 call option replication, when using deltagamma hedge as compared to delta hedge. Similarly, the replication errors for
September ’09 call option on date January 15, 2009 and January 27, 2009 can be
calculated. The values for replication error for delta-gamma hedge, from table 5, are
$0.1663 and $0.2238, respectively. There is an apparent $0.6188(=0.7851-0.1663)
and $0.87730 (=1.1011-0.2238) benefit in replication with delta-gamma hedge as
compared to replication with delta hedge.
8-Jun-09
25-May-09
11-May-09
27-Apr-09
13-Apr-09
30-Mar-09
16-Mar-09
2-Mar-09
16-Feb-09
2-Feb-09
19-Jan-09
5-Jan-09
22-Dec-08
8-Dec-08
24-Nov-08
Figure 10 Comparison of Delta Hedge Error with Delta-Gamma Hedge Error
7
6.5
6
5.5
Delta
hedge
5
error ($)
4.5
4
3.5
3
2.5
2
1.5
Delta
gamma
1
hedge
0.5
error ($)
0
Figure 10 shows the comparison of replication errors due to delta hedge and deltagamma combined hedge. This graph considers values on daily basis replication.
29
Ideally, for any replication this error line should be a horizontal line crossing 0 on xaxis. It can be observed that the delta-gamma combined hedge provides a better
replication as compared to delta hedge.
30
Chapter 4
HEDGING WITH OTHER GREEKS AND VOLATILITY CONSIDERATIONS
Role of Carrying Cost on Replication of a Target Option
Carrying cost, which is the difference of the risk-free interest rate and the dividend
yield (Investopedia, 2009) considered in the Black-Scholes option pricing model, has
a significant role on replication when using delta and gamma hedge strategy on the
target option. An example showing the effect of change in carrying cost on
replication, of a September 2009 S&P 500 index futures call option, can illustrate the
concept in a better way.
For calculation purposes, it is assumed, in this case, that changes the carrying cost
change is only due to changes in interest rate and the dividend yield is constant for the
specific period considered in the example. Also, for calculating the call option prices
for June’09 and September’09, with changed carrying cost, the implied volatility is
assumed to be the same as compared implied volatility when there are no carrying
cost alterations. Table 6 calculates the delta-gamma hedge replication error when the
carrying cost drops by 100 basis points. This is a result of risk-free interest rate drop
from 1.085% to 0.085%.
31
Table 6 Replication without Rho Hedging
Underlying contract price
Call option price
Strike price
Interest Rate
Dividend Yield
Implied volatility
Time to expiry (days)
Sep'09 call option
27-Jan-09 28-Jan-09
$832.70
$865.00
$26.40 $31.5243
$1,025.00 $1,025.00
1.0850% 0.0850%
3.4200% 3.4200%
32.8105% 32.0699%
235
234
Delta
Gamma
Rho
Theta
0.23223
0.00138
107.50820
46.66398
Price change in underlying asset
Price change for call option
Gamma hedge ratio
0.9359 of June'09 call option price
June '09 call option
Modified delta
Modified gamma
Modified rho
Modified theta
June'09 call option
27-Jan-09 28-Jan-09
$835.70
$868.00
$12.60 $16.2121
$1,025.00 $1,025.00
1.0850% 0.0850%
3.4200% 3.4200%
31.6391% 30.5918%
140
139
0.15905
0.00147
46.14975
48.26203
$32.30
$5.12
0.93592
-
$32.30
$3.61
=0.00138/0.00147
$11.79
$15.17
0.148860
0.001379
43.192470
45.169399
Net delta 0.083374
Net gamma 0.000000
Net rho 64.315728
Net theta 1.494580
Delta neutral hedge
0.08337 of underlying asset
Pay-offs
Delta gamma neutral hedge
Realized price change in June'09 call
option position
Realized price change with position in
underlying Sep '09 contract
Delta gamma neutral hedge pay-off
Error in replication (without rho hedge)
0.93592 of June'09 call option + 0.08337 of
underlying asset
$3.3807
$2.6930
$6.0736
$0.9493
32
Because of the change in the risk-free interest rate, the single day price change for
June’09 and September’09 call options is different from the call prices shown in table
2 and table 3. This change is $5.12 for the target option (September ’09 call option)
and $3.61 for the June’09 call option which is $6.40(=$32.8-$26.4) for September ’09
call option and $4.2 (=$16.8-$12.6) for June’09 call option, on January 27, 2009, if
there is no change in the interest rate.(Please refer table 2 and table 3 for prices of
September’09 and June’09 call options respectively.)
The gamma hedge ratio stays the same 0.9352; and with this ratio the realized June
’09 call option price change is $3.3807. The delta hedge pay-off from table is
$2.6930, so there is a net delta-gamma hedge error of $0.9493 for $32.3 change in
underlying asset price. Comparing with the error value in table 5, this error is higher
by $0.7255 (=0.9493-0.2238).
Rho Hedging
A Eurodollar futures is an instrument that is specifically related to interest rates.
(Investopedia, 2009). This instrument assumes $1.00 change in future value for 100
basis points change in the risk-free interest rate. The values for delta and gamma are
zero for Eurodollar futures. Hence, introducing them in the portfolio can mitigate the
interest rate risk.
33
Eurodollar futures Intial Price Price after 100 bps drop
Price (may 29th)
98.915
99.915
Interest rate
1.085%
0.085%
The exposure for the Eurodollar futures is interest rate exposure and it has rho equal
to -0.01, which can be used to hedge the error due to carrying cost.
Table 7 Delta-Gamma Rho Hedging
Modified delta
Modified gamma
Modified rho
Modified theta
0.148860
0.001379
43.192470
45.169399
Net delta
Net gamma
Net rho
Net theta
0.083374
0.000000
64.315728
1.494580
Delta neutral hedge
0.08337 of underlying asset
Rho hedge
= rho(Eurodollar futures) * net rho
-0.01 * 64.31573 = -0.64316 of
Eurodollar futures
99.915 - 98.915 =$1
-$0.64
Price change for Eurodollar futures
Realized price change for rho hedge
Pay-offs
Delta gamma rho neutral hedge
Realized price change in June'09 call
option position
Realized price change with position in
underlying Sep '09 contract
Realized price change with position in
Eurodollar futures
Delta gamma neutral hedge pay-off
Error in replication (with rho hedging)
0.93592 of June'09 call option +
0.08337 of underlying asset+ -0.64316
of Eurodollar futures
$3.3807
$2.6930
-$0.6432
$5.4305
$0.3061
Table 7 shows the rho hedging added to delta-gamma hedging calculations from table
6. The rho hedge can be done by multiplying the rho of Eurodollars with net rho,
34
obtained after subtracting the modified rho (table 6) from the rho for target call option
(September’09 call option on January 27, 2009). The calculations in table 7 differ
from table 6 only for the part of rho hedging, which is shown in third and the last
section (pay-off section) of table 7.
The resulting replication error is calculated as $0.3061 which is $0.6432 (=$0.9493$0.3061) less than the replication error, $0.9493, from table 6 (replication without rho
hedging).
Hedging for Kappa
The kappa hedging, during replication with Greeks of the target option, is hedging the
volatility of the option by introducing instruments with different volatility, thereby
reducing the replication error as compared to hedging for the delta of the target
option. Apparently, there is a cost associated to introducing instruments with
different volatility in the portfolio, and it tends to increase the net cost of the
replication. There is no instrument available directly related to the volatility risk for a
predetermined cost over the life of the target option. Since, the Black-Scholes option
pricing model takes into consideration the factors that are highly contributive to the
volatility; it is not possible to determine the total volatility cost as it is a function of
expiration of the option, exercise price, and current volatility in underlying asset
price. And since the volatility hedge is uncertain for every instrument we use to
35
hedge the delta, gamma or rho, it actually happen to introduce more uncertainty per
instrument it can add to replicating portfolio, and more uncertainty in determining the
cost associated to it.
Hedging for Theta
Considering the net theta for the target option from the table 5, after introducing
gamma hedge to the replicating portfolio, it is apparent that the net theta is having a
considerably small value or less than zero.
Upon introducing the underlying asset in a proportion equal to the delta hedge ratio,
the net theta value for the portfolio is reduced to a negative value, which means that
there is no theta exposure in the replicating. The reason behind the automatic hedge
is that the theta exposure is closely related to gamma exposure. So application of
gamma hedge to the replication portfolio reduces the theta exposure to a small value
which can be hedged using the delta hedge.
Table 8 shows the calculations of modified theta as the product of delta hedge ratio
with the theta of target call option. The gamma hedge on the September’09 call
option, for January 27, 2009, leaves a positive 1.49 value for the theta exposure. But
by introducing, 0.08377 of underlying asset, S&P 500 index futures contract for
September 2009, the modified theta increases to a value of 3.89, which is higher by
2.395 units to the value of net theta, 1.49, after gamma hedge. This results in a net
36
negative value for the replicating portfolio, consisting of 0.03591 of June ’09 call
option and 0.08337 of underlying asset, for September ’09 target call option.
Table 8 Theta Hedge
(Refer table 5)
Sep'09 Call option
Delta
Gamma
Rho
Theta
Delta gamma hedge for Sep'09 call option
9-Apr-09
15-Jan-09
27-Jan-09
0.20388
0.00154
68.82500
53.27246
0.26210
0.00132
124.62747
52.62253
0.23223
0.00138
107.50820
46.66398
June'09 Call Option
Delta
Gamma
Rho
Theta
0.099516528
0.001466906
14.81988895
53.72232753
0.204318288
0.001468703
62.82797493
60.77287753
0.15905
0.00147
46.14975
48.26203
Gamma hedge ratio
1.05112967
0.89815994
0.935917871
0.09928
0.00000
53.24737
-3.19667
0.07859
0.00000
68.19790
-1.96123
0.08337
0.00000
64.31583
1.49468
0.099278574
0.07858796
0.08337
5.288813591
4.135497323
3.890563779
-8.48549
-6.09673
-2.39588
Net delta
Net gamma
Net rho
Net theta
Delta hedge ratio
modified theta (=delta
hedge ratio * theta of
target option
Effect on Net theta by
introducing underlying in
a proportion equal to
delta hedge ratio
37
Project Limitations
The project has a limited scope to the discussion of hedging with delta, gamma and
rho hedging. The discussion of hedging other Greeks such as: eta, lambda etc. for
the option is not covered in the project. The replication discussed and applied is good
for one day period replication. The hedge needs to be adjusted for associated Greek
hedging. For instance, gamma hedge needs to be adjusted with the change in gamma
value and rho hedge needs to be adjusted every time the interest rate changes. The
major factors behind those limitations of discussed replication strategy are:
1. Accuracy of Black-Scholes option pricing model consideration of volatility as it
predicts the target option price.
2. There is a change in the price as number of outstanding options, used in the
replication, changes. Also, the number of contracts available at an acceptable
price to provide the hedge changes overtime depending on trade activity (Asay and
Edelsburg, 1986).
3. A hedge for replication depends on the underlying asset price gap between two
consecutive price changes. There exists an uncertainty when aiming for an
anticipated minimum price gap to adjust the hedge for replication.
38
Chapter 5
CONCLUSION
Replication of an option can be done by hedging the Greeks: delta, gamma, rho,
kappa, and theta of the option. The hedging for delta exposure can be done by
investing a proportion, called as delta hedge ratio in the project, equal to delta of the
target option, in the underlying asset. The gamma exposure can be hedged by
investing a proportion in another option with higher gamma or more liquid option and
this proportion can be a fraction needed to neutralize the gamma of the target option
payoff curve. Investing in more than one option with shorter maturity periods as
compared to the target option helps reduce the gamma for replication or a portfolio of
options.
The empirical results from the project conclude that delta-gamma neutral hedging
replicates S&P500 Index Futures call option with comparatively less error as
compared with delta-neutral hedge. The values for replication error for delta-gamma
neutral hedge is less than $0.6187, $0.8773 and $0.1937 for September’09 call option
on dates January 15, 2009, January 27, 2009 and April 9, 2009 respectively, as
compared with delta-neutral hedge.
39
The importance of rho is towards changes in carrying cost, in this case, the interest
rates. Whenever there is a high change in price along with the change in the interest
rate, the replication is adjusted for risk, if it is hedged by a proper interest rate
addressing instruments like Eurodollar futures or other interest rate futures. The
volatility hedge during replication can lead to higher transaction costs and can also
induce uncertainty per instrument included depending on how the instrument is
correlated to other instruments in the same portfolio. This problem is associated with
kappa hedging, as there is no specific instrument that is having a specific volatility at
a predetermined price. Lastly, the accuracy of the option pricing model has high
resemblance to better hedging as it captures the factors affecting the payoff of the
target option.
40
APPENDIX
Rates from Eurodollar Futures versus US Treasury Bills
The risk-free rate derived from the mean of rates of months June’08-Oct ’09 can be
used. But there are related arguments for using those rates instead of 1.085% as the
risk-free rate. The investor sentiment does consider the historical rates even though it is
outlier but looking to higher rates, there are more capital flights into risk-free assets
which ultimately draw the rates down. It is logical to take the median over the mean of
the interest rates. The resulting risk-free rate comes out to be 1.65%.
The project uses Eurodollar futures rates instead of U.S. Treasury bill, for couple of
reasons:
First, the spread for adjacent quarterly Eurodollar futures which historically seem to be
small and during the period of trading it remains nearly constant as compared to U.S.
Treasury bills. Second, interest rates provided by Eurodollar futures prices can be well
predicted with US treasury yield curve and TED – (Eurodollar deposit rate and T Bill
rate spread) (Cretien, 2006).
41
Figure 11 Yield curves for July-Dec 2000, July’07, Oct ’07, July’08, Oct’08
6.5000
6.2500
6.0000
5.7500
5.5000
5.2500
5.0000
4.7500
4.5000
4.2500
4.0000
3.7500
3.5000
3.2500
3.0000
2.7500
2.5000
2.2500
2.0000
1.7500
1.5000
1.2500
1.0000
0.7500
0.5000
0.2500
-
Yield Curves
Mean of
July2000 to Dec
2000
Mean of July'07
Mean of Oct '07
Mean of July'08
Mean of Oct '08
1 mo3 mo6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr20 yr30 yr
This graph shows the yield curves for rates on t-bills with different maturities and
shows a comparison between October 2008, July 2008, October 2007, July 2007 and a
period of July 2000 to December 2000. Data is taken from the website of US
Department of Treasury, historical prices. Refer to the bibliography section.
42
Figure 12 TED spread larger period (Jan’00 – Oct ’09)
5.0000
Euro$ - Tbill spread (3 month contracts)
4.0000
3.0000
2.0000
1.0000
Jan-00
Apr-00
Jul-00
Oct-00
Jan-01
Apr-01
Jul-01
Oct-01
Jan-02
Apr-02
Jul-02
Oct-02
Jan-03
Apr-03
Jul-03
Oct-03
Jan-04
Apr-04
Jul-04
Oct-04
Jan-05
Apr-05
Jul-05
Oct-05
Jan-06
Apr-06
Jul-06
Oct-06
Jan-07
Apr-07
Jul-07
Oct-07
Jan-08
Apr-08
Jul-08
Oct-08
Jan-09
Apr-09
Jul-09
Oct-09
-
This graph shows the Eurodollar and T bill spread (TED) for the period January
2000 – October 2009.
Figure 13 TED spread for shorter period (Jan ’08 -Oct ’08)
3month Euro$ - 3 monthTbill spread
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
This graph shows the TED for a period January 2008 – October 2009.
Oct-09
Sep-09
Aug-09
Jul-09
Jun-09
May-09
Apr-09
Feb-09
Mar-09
Jan-09
Dec-08
Nov-08
Oct-08
Sep-08
Aug-08
Jul-08
Jun-08
May-08
Apr-08
Mar-08
Feb-08
Jan-08
3month Euro$ - Tbill spread
43
Under normal distribution major capital in the markets was distributed on both risky
and riskless assets. But as market started freezing, there was a capital flight to quality
and preservation and driving the demand of riskless assets high up and resulting in
lowering the returns (Broadle, Chernov and Johannes, 2009). The TED (Eurodollar to T
bill spread) shows the surge starting July’07 going up by 72 basis points as compared to
Oct ’07 and in 2008, it went up by 327 basis points starting from July’08 to October
’08. The increasing spread showed an increase in default risks. Under such
circumstances, the investors might have had sentiment to invest in riskless instruments.
Referring back to the chart of T-bill yield curves, the mean of July to December in 2000
was an inverted yield curve that showed intent of upcoming economic recession.
Mean of July ’07 T Bill yield curve was close to flat showing an increased demand in
riskless instruments and its high trading resulted in decreasing interest rates for
upcoming years. And certainly October ’07 and July ’08 yield curves show
resemblance with spiked TED spread for the same period. This shows a transition in
slope of yield curves have a relation to TED spread, and ultimately increases the
window of risk for less risky assets. As the slope of the yield curve changes, the TED
spread increases, which means investors are tending towards safer investments. But to
what extent this effect exists is beyond the scope of this project analysis.
44
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