Transcript The Greek Letters

The Greek Letters
Chapter 17
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Example (Page 359)




A bank has sold for $300,000 a European call
option on 100,000 shares of a non-dividendpaying stock
S0 = 49, K = 50, r = 5%, s = 20%,
T = 20 weeks, m = 13%
The Black-Scholes-Merton value of the option
is $240,000
How does the bank hedge its risk?
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Naked & Covered Positions
Naked position
Take no action
Covered position
Buy 100,000 shares today
Both strategies leave the bank
exposed to significant risk
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Stop-Loss Strategy


This involves:
Buying 100,000 shares as soon as
price reaches $50
Selling 100,000 shares as soon as
price falls below $50
This deceptively simple hedging
strategy does not work well
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Delta (See Figure 17.2, page 363)

Delta (D) is the rate of change of the
option price with respect to the underlying
Option
price
Slope = D
B
A
Stock price
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Delta Hedging



This involves maintaining a delta neutral
portfolio
The delta of a European call on a nondividend-paying stock is N (d 1)
The delta of a European put on the stock is
[N (d 1) – 1]
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Delta Hedging
continued


The hedge position must be frequently
rebalanced
Delta hedging a written option involves a
“buy high, sell low” trading rule
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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First Scenario for the Example:
Table 17.2 page 366
Week
Stock
price
Delta
Shares
purchased
Cost
(‘$000)
Cumulative
Cost ($000)
Interest
0
49.00
0.522
52,200
2,557.8
2,557.8
2.5
1
48.12
0.458
(6,400)
(308.0)
2,252.3
2.2
2
47.37
0.400
(5,800)
(274.7)
1,979.8
1.9
.......
.......
.......
.......
.......
.......
.......
19
55.87
1.000
1,000
55.9
5,258.2
5.1
20
57.25
1.000
0
0
5263.3
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Second Scenario for the Example
Table 17.3 page 367
Week
Stock
price
Delta
Shares
purchased
Cost
(‘$000)
Cumulative
Cost ($000)
Interest
0
49.00
0.522
52,200
2,557.8
2,557.8
2.5
1
49.75
0.568
4,600
228.9
2,789.2
2.7
2
52.00
0.705
13,700
712.4
3,504.3
3.4
.......
.......
.......
.......
.......
.......
.......
19
46.63
0.007
(17,600)
(820.7)
290.0
0.3
20
48.12
0.000
(700)
(33.7)
256.6
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Theta

Theta (Q) of a derivative (or portfolio of
derivatives) is the rate of change of the value
with respect to the passage of time
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Theta for Call Option: S0=K=50,
s = 25%, r = 5% T = 1
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Gamma

Gamma (G) is the rate of change of
delta (D) with respect to the price of the
underlying asset
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Gamma for Call or Put Option:
S0=K=50, s = 25%, r = 5% T = 1
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Gamma Addresses Delta Hedging
Errors Caused By Curvature
(Figure 17.7, page 371)
Call
price
C′′
C′
C
Stock price
S
S′
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Interpretation of Gamma

For a delta neutral portfolio,
DP  Q Dt + ½GDS 2
DP
DP
DS
DS
Positive Gamma
Negative Gamma
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Relationship Among Delta,
Gamma, and Theta
For a portfolio of derivatives on a nondividend-paying stock paying
1 2 2
Q  rS 0 D  s S 0 G  rP
2
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Vega

Vega (n) is the rate of change of the
value of a derivatives portfolio with
respect to volatility
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Vega for Call or Put Option:
S0=K=50, s = 25%, r = 5% T = 1
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Managing Delta, Gamma, &
Vega
Delta can be changed by taking a
position in the underlying asset
 To adjust gamma and vega it is
necessary to take a position in an
option or other derivative

Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Rho

Rho is the rate of change of the
value of a derivative with respect
to the interest rate
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Hedging in Practice
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
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Traders usually ensure that their portfolios
are delta-neutral at least once a day
Whenever the opportunity arises, they
improve gamma and vega
As portfolio becomes larger hedging
becomes less expensive
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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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
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Using Futures for Delta Hedging


The delta of a futures contract on an asset
paying a yield at rate q is e(r-q)T times the
delta of a spot contract
The position required in futures for delta
hedging is therefore e-(r-q)T times the
position required in the corresponding spot
contract
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Hedging vs Creation of an Option
Synthetically
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
When we are hedging we take
positions that offset D, G, n, etc.
When we create an option
synthetically we take positions
that match D, G, & n
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Portfolio Insurance


In October of 1987 many portfolio
managers attempted to create a put
option on a portfolio synthetically
This involves initially selling enough of
the portfolio (or of index futures) to
match the D of the put option
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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Portfolio Insurance
continued
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
As the value of the portfolio increases, the
D of the put becomes less negative and
some of the original portfolio is
repurchased
As the value of the portfolio decreases, the
D of the put becomes more negative and
more of the portfolio must be sold
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
26
Portfolio Insurance
continued
The strategy did not work well on October
19, 1987...
Fundamentals of Futures and Options Markets, 7th Ed, Ch 17, Copyright © John C. Hull 2010
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