The Greek Letters
Chapter 17
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
1
Greek Letters



Greek letters are the partial derivatives with
respect to the model parameters that are liable
to change
Usually traders use the Black-Scholes-Merton
model when calculating partial derivatives
The volatility parameter in BSM is set equal to
the implied volatility when Greek letters are
calculated. This is referred to as using the
“practitioner Black-Scholes” model
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
2
Delta Hedging




This involves maintaining a delta neutral
portfolio
For each long position that is owned the
trader can sell 1/Delta calls
For example of the Delta is ½ and the trader
owns one share then they can sell two calls
The delta of a European call on a nondividend-paying stock is N (d 1)
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
3
Delta of a Stock Option (K=50, r=0, s =
25%, T=2, Figure 17.3, page 365)
Call
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
4
Variation of Delta with Time to
Maturity(S0=50, r=0, s=25%, Figure 17.4, page 366)
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
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
The theta of a call or put is usually
negative. This 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
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
6
Theta for Call Option: S0=K=50,
s = 25%, r = 5%, T = 1
Fundamentals of Futures and Options Markets, 8th Ed, Ch 17, Copyright © John C. Hull 2013
7
Gamma


Gamma (G) is the rate of change of
delta (D) with respect to the price of
the underlying asset
Gamma is greatest for options that
are close to the money
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
8
Gamma for Call or Put Option:
(K=50, s = 25%, r = 0%, T = 2, Figure 17.9, page 375)
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
9
Vega

Vega (n) is the rate of change of the
value of a derivative with respect to
implied volatility
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
10
Vega for Call or Put Option (K=50,
s = 25%, r = 0, T = 2)
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
11
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
12
Elasticity
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
13
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, 9th Ed, Ch 17, Copyright © John C. Hull 2016
14
Rho


Rho is the rate of change of the
value of a derivative with respect
to the interest rate
As interest rates increase calls are
worth more and puts are worth
less.
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
15
Call
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
16
Call
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
17
Put
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
18
Hedging in Practice



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, 9th Ed, Ch 17, Copyright © John C. Hull 2016
19
Portfolio Insurance



In October of 1987 many portfolio
managers attempted to create a put
option on a portfolio synthetically
As the value of the portfolio decreases,
more of the portfolio must be sold
The strategy did not work well on
October 19, 1987...
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
20
Fundamentals of Futures and Options Markets, 9th Ed, Ch 17, Copyright © John C. Hull 2016
21