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Finance 457
The Greek Letters
14
Chapter Fourteen
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Finance 457
Executive Summary
This chapter covers the way in which traders working
for financial institutions and market makers on the
floor of an exchange hedge a portfolio of
derivatives.
The software, DerivaGem for Excel, can be used to
chart the relationships between any of the Greek
letters and variables such as S0, K, r, s, and T.
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Finance 457
Chapter Outline
14.1 Illustration
14.2 Naked and Covered Positions
14.3 A Stop-Loss Strategy
14.4 Delta Hedging
14.5 Theta
14.6 Gamma
14.7 Relationship Between Delta, Theta, and Gamma
14.8 Vega
14.9 Rho
14.10 Hedging in Practice
14.11 Scenario Analysis
14.12 Portfolio Insurance
14.13 Stock Market Volatility
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14.1 Illustration
Finance 457
• A bank has sold for $300,000 a European call option on
100,000 shares of a non-dividend paying stock
• S0 = $49, K = $50, r = 5%, s = 20%,
T = 20 weeks, m = 13%
• The Black-Scholes value of the option is around $240,000
• How does the bank hedge its risk to lock in a $60,000
profit?
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Finance 457
14.2 Naked and Covered Positions
Naked position
Take no action:
wait for expiry and “hope for the best”
Covered position
Buy 100,000 shares today
this amounts to a covered call position
Both strategies leave the bank exposed to significant
risk.
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Finance 457
14.2 Naked and Covered Positions
• Neither a naked position nor a covered position
provides a satisfactory hedge.
Put-call parity shows that the exposure from writing a
covered call is the same as the exposure from
writing a naked put.
• For a perfect hedge the standard deviation of the
cost of writing and hedging the option is zero.
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14.3 A 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 in practice:
Purchases and subsequent sales cannot be made at K.
Transactions costs could easily eat the option premium
and then some.
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Finance 457
14.4 Delta Hedging
• Most traders use more sophisticated hedging
schemes.
• These involve calculating measures such as delta,
gamma, and vega.
• Delta was introduced in Chapter 10
• Delta is very closely related to the idea of the
replicating portfolio intuition.
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Delta (See Figure 14.2, page 302)
Finance 457
• Delta (D) is the rate of change of the option price with
respect to the underlying security
• If you have a pricing equation, just take a derivative
with respect to S
Option
price
Slope = D
B
A
Stock price
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Delta Hedging
Finance 457
• 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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Finance 457
Variation of Delta with Stock Price
D of call
D of put
+1
-1
K Stock price
K Stock price
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Finance 457
Delta Hedging
• The hedge position must be frequently rebalanced
• Delta hedging a written option involves a “buy high,
sell low” trading rule
• See Tables 14.2 (page 307) and 14.3 (page 308) for
examples of delta hedging
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Finance 457
Using Futures for Delta Hedging
• The delta of a futures contract 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
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Finance 457
Transactions costs
• Maintaining a delta-neutral position in a single
option and the underlying assets is likely to be
prohibitively expensive due to transactions costs.
• However, for a large portfolio of options, delta
hedging is much more feasible:
– Only 1 trade in the underlying assets is necessary to zero
out delta for the whole portfolio.
– The transactions costs of delta hedging could then be
spread over many different trades.
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Finance 457
14.5 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
• See Figure 14.5 for the variation of Q with respect
to the stock price for a European call
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Finance 457
14.6 Gamma
• Gamma (G) is the rate of change of delta (D) with
respect to the price of the underlying asset
• See Figure 14.9 for the variation of G with respect
to the stock price for a call or put option
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Finance 457
Gamma Addresses Delta Hedging Errors Caused
By Curvature
(Figure 14.7, page 312)
Call
price
C’’
C’
C
Stock price
S
S
’
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Interpretation of Gamma
Finance 457
• For a delta neutral portfolio,
dP  Q dt + ½GdS 2
dP
dP
dS
dS
Positive Gamma
Negative Gamma
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Finance 457
14.7 Relationship Between D, Q, and G
• Delta (D) is the rate of
change of the option price
with respect to the
underlying security
• Gamma (G) is the rate of
change of delta (D) with
respect to the price of the
underlying asset
• Theta (Q) of a derivative
(or portfolio of derivatives)
is the rate of change of the
value with respect to the
passage of time
f
D
S
D  f
G
 2
S S
2
f
Q
t
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14.7 Relationship Between D, Q, and G
Finance 457
• The price of a single derivative dependent on a nondividend paying stock must satisfy the Black-ScholesMerton differential equation:
f
f 1 2 2  2 f
 rS
 s S
 rf
2
t
S 2
S
1 2 2
Q  rSD  s S G  rf
2
f
Q
t
f
D
S
D  f
G
 2
S S
2
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14.7 Relationship Between D, Q, and G
For a portfolio of derivatives on a stock
paying a continuous dividend yield at rate q
1 2 2
Q  (r  q ) SD  s S G  rP
2
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14.8 Vega
• Vega (n) is the rate of change of the value of a
derivatives portfolio with respect to volatility
• See Figure 14.11 for the variation of n with respect
to the stock price for a call or put option
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Managing Delta, Gamma, & Vega
 D can be changed by taking a position in the
underlying
• To adjust G & n it is necessary to take a position in an
option or other derivative
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14.9 Rho
• Rho is the rate of change of the value of a derivative
with respect to the interest rate
• For currency options there are 2 rhos
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14.10 Hedging in Practice
• Traders usually ensure that their portfolios are deltaneutral at least once a day
• Whenever the opportunity arises, they improve
gamma and vega
• As a portfolio becomes larger hedging becomes less
expensive
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Hedging vs. Creation of a Synthetic Option
• 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
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14.11 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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14.12 Portfolio Insurance
• It is essential to understand what portfolio insurance
is:
• Portfolio insurance involves creating a long
position in an option synthetically.
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Portfolio Insurance
Finance 457
• 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
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Portfolio Insurance
• 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
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Portfolio Insurance
Finance 457
The strategy did not work well on October 19,
1987...
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Finance 457
14.13 Stock Market Volatility
Trading itself is a cause of volatility.
Portfolio insurance schemes such as those just described have the potential
to increase volatility.
When the market declines, they cause portfolio managers to either sell stock
or to sell index futures contracts. Either action may accentuate the
decline.
Selling obviously carries the potential to drive down prices
The sale of index futures contracts is liable to drive down futures prices,
this creates selling pressure on stocks via the index arbitrage mechanism.
Whether the portfolio insurance schemes affect volatility depends on how
easily the market can absorb the trades that are generated by portfolio
insurance.
Widespread use of portfolio insurance could have a destabilizing effect on
the market—which would of course increase the necessity of portfolio
insurance.
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Finance 457
Summary
• Delta (D) is the rate of
change of the option price
with respect to the
underlying security
• Gamma (G) is the rate of
change of delta (D) with
respect to the price of the
underlying asset
• Theta (Q) of a derivative
(or portfolio of derivatives)
is the rate of change of the
value with respect to the
passage of time
f
D
S
D  f
G
 2
S S
2
f
Q
t
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Finance 457
Summary
• Vega (n) is the rate of
change of the option price
with respect to the
volatility of the underlying
security
• Rho (r) of a derivative (or
portfolio of derivatives) is
the rate of change of the
value with respect to the
interest rate
f
n
s
f
r
r