The Greek Letters
University of Macedonia
E
Executive
i MBA
Financial Engineering
Textbook: Hull, J.C. “Options Futures and Other
Derivatives”, 6e, Prentice Hall, 2006
Achilleas Zapranis
Hedging the Risk of the Issuer
z
If an issuer can sell an option for more than
its worth and then hedge away all the risk for
the rest of the option’s
option s life,
life he has locked a
guaranteed, risk-free profit
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Hedging
g g the Risk of the Issuer
continued
z
z
z
z
A bank has sold for $300,000 a European call
option
p
on 100,000
,
shares of a non dividendpaying stock
S0 = 49,, K = 50,, r = 5%,, σ = 20%,,
T = 20 weeks, μ = 13%
The Black
Black-Scholes
Scholes value of the option is
$240,000
How does the bank hedge its risk to lock in a
$60,000 profit?
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Naked & Covered Positions
•
Naked position
Take no action
•
Covered
C
d position
iti
Buyy 100,000
,
shares todayy
Both strategies leave the bank exposed to
significant risk
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Naked & Covered Positions
Payoff Diagrams
Profit/Loss
K
Profit/Loss
ST
Naked position
K
ST
Covered position
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Stop-Loss Strategy
z
z
This involves:
Buying 100
100,000
000 shares as soon as
price reaches $50
S lli 100
Selling
100,000
000 shares
h
as soon as
price falls below $50
This deceptively simple hedging
strategy does not work well
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Stop-Loss
p
Strategy
gy
continued
Caveats:
•The cash flows to
the
h h
hedger
d
occur
at different times
and must be
discounted
•Purchases and
sales cannot be
made at exactly
th same price
the
i
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Delta
z
Delta (Δ) is the rate of change of the
option price with respect to the underlying
Option
price
Slope = Δ
B
A
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The Greek Letters
Stock price
8
Delta Hedging
z
z
z
z
z
Suppose a stock call option with Δ = 0.6
Every time that the stock price changes the
option price changes by 60%
Imagine an investor who has sold 20 call option
contracts – that is options to buy 2,000 shares
If the
th price
i off the
th option
ti is
i c = $10 h
he can h
hedge
d
his position by buying 0.6 x 2,000 = 1,200 shares
The gain/loss of the position would then tend to
be offset by the loss/gain on the stock position
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Delta Hedging
z
If the stock price goes up by 1 (producing a gain of
$1,200 on the shares purchased), the option price
will tend to go up by 0.6 x $1 = $0.60 (producing a
loss of $1,200 on the options written)
z
If the stock p
price g
goes down by
y 1 (p
(producing
g a loss
of $1,200 on the shares purchased), the option price
will tend to down by 0.6 x $1 = $0.60 (producing a
gain of $1,200 on the options written)
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Delta Hedging
z
z
z
This involves maintaining a delta neutral
portfolio
The delta of a European call on a stock
paying dividends at rate q is
N (d 1)e– qT
The delta of a European put is
e– qT [N (d 1) – 1]
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Delta Hedging
continued
z
The hedge
g p
position must be frequently
q
y
rebalanced
z
Delta hedging a written option involves
a “buy
“b hi
high,
h sellll llow”” trading
di rule
l
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Simulation of Delta hedging
g g
Option Closes ITM
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Using Futures for Delta
Hedging
z
The delta of a futures contract is e(r-q)T times
the delta of a spot contract
z
The position required in futures for delta
hedging is therefore e-(r-q)T times the position
required in the corresponding spot contract
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Delta of a Portfolio
z
It is the weighted average of the Δ of the individual
positions in the portfolio n
Δ = ∑ wi Δ i
i =1
z
Suppose a financial institution in the US has the
following 3 positions in GBP:
z
z
z
A long
gp
position in 100,000 call options
p
with K = 0.55 and
expiration date in 3 months. Δ = 0.533
A short position in 200.000 call options with K = 0.56 and
p
date in 5 months. Δ = 0.468
expiration
A short position in 50.000 put options with K = 0.56 and
expiration date in 2 months. Δ = -0.508
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Making the Portfolio Delta
Neutral
z
Portfolio Δ:
100,000x0.533 – 200,000x0.468 – 50,000x(-0.508)
= -14,900
z
This means that
Thi
th t the
th portfolio
tf li can be
b made
d delta
d lt
neutral with a long position of 14,900 GBP
z
Another alternative is to take a long position on a 6month futures for buying14,650
y g
GBP ((suppose
pp
that
the risk-free rate in US is 8% and in UK is 4%
14,900e-(0.08-0.04)x0.5=14,605
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Theta
z
z
Theta (Θ) of a derivative (or portfolio of
derivatives) is the rate of change of the value
with respect to the passage of time
The theta of a call or put is usually negative
negative.
This means that, if time passes with the price of
the underlying asset and its volatility remaining
the same, the value of the option declines
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Gamma
z
z
z
Gamma (Γ) is the rate of change of delta (Δ) with
respect to the price of the underlying asset
For a European call or put option on a non dividend
paying stock:
N ′(d1 )
Γ=
Sσ T
For a European
p
index call or p
put option
p
with a
dividend yield q:
Γ=
N ′(d1 )e − qT
Sσ T
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Gamma
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Gamma Addresses Delta Hedging
Errors Caused By Curvature
Call
price
C''
C'
C
C
Stock price
S
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S'
20
Interpretation of Gamma
z
F a delta
For
d lt neutral
t l portfolio,
tf li
ΔΠ ≈ Θ Δt + ½ Γ ΔS 2
ΔΠ
ΔΠ
ΔS
ΔS
Positive Gamma
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Negative Gamma
21
Making a Portfolio Gamma
Neutral
z
A position on the underlying or on a future
has zero Γ
z
The only way to change the portfolio Γ is by
taking a position on wT options with ΓΤ
z
The portfolio Γ then becomes:
wT ΓΤ + Γ
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Making a Portfolio Gamma
Neutral (continued)
z
z
z
Hence, the required position on the option in
order for the portfolio to become gamma neutral
is
-Γ/ ΓΤ
However, including the option in the portfolio
changes the portfolio’s Δ, so the position on the
underlying
d l i h
has to b
be changed
h
d to maintain
i i d
delta
l
neutrality
A time
As
ti
passes, the
th position
iti on the
th option
ti iis
being adjusted so that the portfolio remains
gamma neutral
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Making a Portfolio Gamma
Neutral (Example)
z
z
z
z
z
Suppose that portfolio is delta neutral with Γ = 3,000
Suppose a call option with Δ = 0.62 and Γ = 1.50
The portfolio will become gamma neutral if we
include a long position on 3,000/1.5 = 2000 call
options
However, the portfolio Δ will change from 0 to
2,000 x 0.62 = 1,240
A quantity 1,240 of the underlying asset must be
sold
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Relationship Between Delta,
Gamma, and Theta
For a portfolio of derivatives on a stock
paying
p
y g a continuous dividend yyield at
rate q
1 2 2
Θ + (r − q ) SΔ + σ S Γ = rΠ
2
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Vega
z
z
z
Vega (ν) is the rate of change of the value
of a derivatives portfolio with respect to
volatility
Vega tends to be greatest for options that
are close to the money
A position on the underlying asset or on a
future written on the underlying has zero
vega
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Vega (continued)
•If V is the vega of the portfolio and VT is the
vega of a traded option
option, a position of
of–V/V
V/VT in
the option makes the portfolio vega neutral
•Unfortunately a portfolio that is gamma
neutral will not in general be vega neutral,
and vise versa
•If a hedger requires a portfolio to be both
gamma and vega neutral, at least two traded
derivatives dependent on the underlying
must be used
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Vega (continued)
z
z
z
z
z
z
Consider a portfolio that is delta neutral, with Γ = 5,000 and V=-8,000
Suppose a traded option with Γ=0.5, V=2, Δ=0.6
The portfolio can be made vega neutral by
including a long position on 8,000/2 = 4,000 options
This would increase Δ from 0 to 4,000 x 0.6 =
2 400
2,400
It is required 2,400 units of the asset to be sold for
the portfolio to remain delta neutral
Γ will change from –5,000 to -5,000 + 0.5 x 4,000 =
-3,000
,
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Vega (continued)
z
z
z
To make the portfolio vega and gamma neutral two
options are needed
Suppose that there is a 2nd option with Γ=0.8,
V=1 2 Δ=0
V=1.2,
Δ=0.5
5
If w1 and w2 are the quantities of the two options
included in the portfolio,
portfolio then it is required that
-5,000
5 000 + 0
0.5
5 x w1 +0.8
+0 8 x w2 = 0
-8,000 + 2.0 x w1 +1.2 x w2 = 0
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Vega (continued)
z
The solution to these equation is
w1 = 400
w2 = 6,000
z
The Δ of the portfolio will change to
400 x 0.6 + 6,000 x 0.5 = 3,240
z
Hence, 3,240
H
3 240 units
it off the
th underlying
d l i assett mustt b
be
sold to maintain delta neutrality
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Managing Delta, Gamma, &
Vega
•
z
Δ can be changed
g by
y taking
gap
position in
the underlying
To adjust Γ & ν it is necessary to take a
position in an option or other derivative
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Rho
z
z
z
z
Rho is the rate of change of the value
of a derivative with respect
p
to the
interest rate
For currency options there are 2 rhos
For example, suppose a European put
index with Rho = -52.67
52 67
That means that for 1% change in the
risk-free
i kf
rate
t (e.g.,
(
f
from
8% to
t 9%) the
th
value of the option increases by 0.5267
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Hedging in Practice
z
z
z
Traders usually ensure that their portfolios
are delta-neutral
delta neutral at least once a day
Whenever the opportunity arises, they
improve gamma and vega
As portfolio becomes larger hedging
becomes less expensive
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Scenario Analysis
A scenario analysis involves testing the effect
on the value of a portfolio of different
assumptions concerning asset prices and
their volatilities
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Hedging vs. Creation of an
Option Synthetically
z
z
When we are hedging we take
positions that offset Δ, Γ, ν, etc.
When we create an option
synthetically we take positions that
match
t h Δ,
Δ Γ
Γ, & ν
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Portfolio Insurance
z
z
In October of 1987 many portfolio
managers
g
attempted
p
to create a p
put option
p
on a portfolio synthetically
This involves initially selling enough of the
portfolio (or of index futures) to match the Δ
of the put option
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Portfolio Insurance
continued
z
z
As the value of the portfolio increases, the Δ
of the put becomes less negative and some
of the original portfolio is repurchased
As the value of the portfolio decreases
decreases, the Δ
of the put becomes more negative and more
off the
th portfolio
tf li mustt be
b sold
ld
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Portfolio Insurance
continued
The strategy
Th
t t
did nott workk wellll on O
October
t b 19
19,
1987...
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Assignment Questions
Hull, Chapter 15, p. 371
Try to answer assignment questions:
15.25
Optional
This presentation is an augmented version of the presentation accompanying Chapter 15 of J. Hull’s textbook “Option’s
Futures and other Derivatives”, 6e, Prentice Hall, 2006, downloaded from http://www.rotman.utoronto.ca/~hull
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