Download The Greek Letters

The Greek Letters
Based on Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
Introduction
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Each of the Greek letters measures a different dimension
to the risk in an option position.
The aim of a trader is to manage the Greeks so that all
risks are acceptable.
A bank has sold for $300,000 a European call option on
100,000 shares of a non-dividend paying stock.
The points that will be made apply to other types of
options and derivatives.
S0 = 49, K = 50, r = 5%, s = 20%,
T = 20 weeks (0.3846 years), m = 13%
The Black-Scholes-Merton value of the option is
$240,000(i.e., $2.40 for an option to buy one share).
A good hedge would ensure that the cost is always equal
or at least close to $240,000.
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Naked and Covered Positions
Strategy A: Naked position
 Suppose that the bank takes a naked position against
risk: it takes no specific action to hedge risk.
 If the stock price at the end of the 20 weeks remains
below K=$50, then this strategy works well.
 A naked position does not work well if the call is
exercised because the bank then has to buy 100,000
shares at the market price prevailing in 20 weeks to
cover the call.
 The cost to the financial institution is 100,000 times
the amount by which the stock price exceeds the
strike price.
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Naked and Covered Positions
Strategy B: Covered position
 As an alternative to a naked position, the financial
institution can adopt a covered position.
 This involves buying 100,000 shares as soon as the
option has been sold.
 If the option is exercised, this strategy works well, but
in other circumstances it could lead to a significant
loss.
 For example, if the stock price drops to $40, the
financial institution loses $900,000 on its stock
position.
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Stop-loss Strategy
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Neither a naked nor a covered position provides a good
hedge on its own.
An alternative hedging procedure is the stop-loss strategy,
which is a combination of both naked and covered
strategies.
This involves:
Buying 100,000 shares as soon as the stock price rises
above $50.
Selling 100,000 shares as soon as price falls below $50.
The objective is to hold a naked position whenever the
stock price is less than K and a covered position
whenever the stock price is greater than K.
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Stop-loss Strategy
The stop-loss strategy involves buying the stock at time
t1, selling it at time t2, buying it at time t3, selling it at
time t4, buying it at time t5, and delivering it at time T.
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Stop-loss Strategy
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The procedure is designed to ensure that at time T
the institution owns the stock if the option closes in
the money and does not own it if the option closes
out of the money.
Options are referred to as in the money, at the
money, or out of the money.
A call option is in the money when S > K, at the
money when S = K, and out of the money when S <
K.
A put option is in the money when S < K, at the
money when S = K, and out of the money when S >
K.
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Stop-loss Strategy
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The cost of setting up the hedge initially is 𝑆0 if 𝑆0 > 𝐾and
zero otherwise. It seems as though the total cost, Q, of
writing and hedging the option is the option’s initial
intrinsic value:
𝑄 = max(𝑆0 − 𝐾, 0)
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The intrinsic value of an option is defined as the maximum
of zero and the value the option would have if it were
exercised immediately.
 For a call option, the intrinsic value is max 𝑆0 − 𝐾, 0
 For a put option, it is max⁡(𝐾 − 𝑆0 , 0)
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As we can see, all purchases and sales subsequent
to time 0 are made at price K.
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Stop-loss Strategy
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There are two key reasons why the cost equation is
incorrect.
The first is that the cash flows to the hedger occur at
different times and must be discounted.
The second is that purchases and sales cannot be made
at exactly the same price K.
If we assume a risk-neutral world with zero interest rates,
we can justify Q ignoring the time value of money.
But we cannot legitimately assume that both purchases
and sales are made at the same price.
The hedger cannot know whether the stock price equals
K, it will continue above or below K.
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Stop-loss Strategy
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In practice, purchases are made at a price K + e and sales
are made at a price K + e, for some small e>0.
Thus, every purchase and subsequent sale involves a cost
(apart from transaction costs) of 2e.
If the path of the stock price crosses the strike price level
many times, the procedure is quite expensive.
Assuming that stock prices change continuously, e can be
made arbitrarily small by monitoring the stock prices
closely.
But as e is made smaller, trades tend to occur more
frequently. Thus, the lower cost per trade is offset by the
increased frequency of trading.
As e→0, the expected number of trades →∞.
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Delta hedging
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The delta (Δ) of an option is defined as the rate of
change of the option price with respect to the price of the
underlying asset.
 It is the slope of the curve that relates the option price to
the underlying asset price.
 Suppose that the delta of a call option on a stock is 0.6.
This means that when the stock price changes by a small
amount, the option price changes by about 60% of that
amount. In general:
𝜕𝑐
Δ=
𝜕S
where c is the price of the call option and S is the stock
price.
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Delta hedging
The figure shows the relationship between a call price and the
underlying stock price. When the stock price corresponds to
point A, the option price corresponds to point B and is the
slope of the line indicated.
Call option
price
Slope = D = 0.6
B
A
Stock price
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Delta hedging
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Suppose that the stock price is $100 and the price of a
call option is $10.
An investor has sold 20 call option contracts -that is,
options on 2,000 shares.
The investor’s position could be hedged by buying:
0.6 x 2,000 = 1,200 shares
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The delta of the trader’s short position in 2,000 options
is:
0.6 x (-2,000) = -1,200 shares
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Delta hedging
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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).
If the stock price goes down by $1 (producing a loss of
$1,200 on the shares purchased), the option price will tend
to go down by $0.60 (producing a gain of $1,200 on the
options written).
The gain (loss) on the stock position would then tend to
offset the loss (gain) on the option position.
The delta of the stock position offsets the delta of the option
position.
A position with a delta of zero is referred to as delta neutral.
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Delta hedging
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The delta of an option does not remain constant.
Therefore, the trader’s position remains delta hedged (or
delta neutral) for only a relatively short period of time.
The hedge has to be adjusted periodically. This is known as
rebalancing.
Suppose that delta rises from 0.60 to 0.65.
An extra 0.05 x 2,000 = 100 shares would then have to be
purchased to maintain the hedge.
A procedure such as this, where the hedge is adjusted on a
regular basis, is referred to as dynamic hedging.
It can be contrasted with static hedging, where a hedge is
set up initially and never adjusted.
Static hedging is sometimes also referred to as hedge-andforget.
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Delta hedging
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For a European call option on a non-dividend-paying stock, it
can be shown that:
Δ(call) = N(d1)
where:
N(x) is the cumulative distribution function for a s.n. distribution.
d1 =
ln( S 0 / K )  ( r   2 / 2)T
 T
 The above formula gives the delta of a long position in one call
option.
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The delta of a European futures call option is defined as
the rate of change of the option price with respect to the
futures price (not the spot price). Δ(call) = 𝑒 −𝑟𝑇 N(d1)
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The delta of a short position in one call option is given by:
Δ(call) = - N(d1)
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Delta hedging
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For a European put option on a non-dividend-paying stock, it
can be shown that:
Δ(put) = N(d1) - 1
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Delta is negative, which means that a long position in a
put option should be hedged with a long position in the
underlying stock.
 A short position in a put option should be hedged with a
short position in the underlying stock.
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Delta hedging
Variation of delta with stock price for (a) a call option and (b)
a put option on a non-dividend-paying stock.
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Delta hedging: Exercise
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What is the delta of a short position in 1,000 European
call options on silver futures?
The options mature in eight months, and the futures
contract underlying the option matures in nine months.
The current nine-month futures price is $8 per ounce.
The exercise price of the options is $8.
The risk-free interest rate is 12% per annum.
The volatility of silver is 18% per annum.
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Theta
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The theta (Θ) of a portfolio of options is the rate of change
of the value of the portfolio with respect to the passage of
time with all else remaining the same.
Theta is sometimes referred to as the time decay of the
portfolio.
The theta of a call or put is usually negative.
A negative theta means that, if time passes with the price
of the underlying asset and its volatility remaining the
same, the value of a long call or put option declines.
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Gamma
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The Gamma (G) is the rate of change of delta (D) with respect
to the price of the underlying asset.
 Gamma is the second partial derivative of the portfolio with
respect to asset price:
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If gamma is small, delta changes slowly, and adjustments to
keep a portfolio delta neutral need to be made only relatively
infrequently.
 However, if gamma is highly negative or highly positive, delta is
very sensitive to the price of the underlying asset.
 It is then quite risky to leave a delta-neutral portfolio unchanged
for any length of time.
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Relationship Between Delta, Gamma, and
Theta
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For a portfolio of derivatives on a stock paying a
continuous dividend yield at rate q it follows from the
Black-Scholes-Merton differential equation that:
1 2 2
  rSD 
 S  = r
2
For a delta-neutral portfolio (Δ = 0), we obtain:
1 2 2
   S  = r
2
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This shows that, when Θ is large and positive, gamma of
a portfolio tends to be large and negative, and vice versa.
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