108156.pdf

198 Mathematics for Finance

we should design a portfolio with both delta and vega equal to zero (delta-
vega neutral). Adelta-gamma neutralportfolio will be immune against larger
changes of the stock price. Examples of such hedging portfolios will be exam-
ined below.
The Black–Scholes formula allows us to compute the derivatives explicitly
for a single option. For a European call we have

deltaCE=N(d 1 ),

gammaCE=^1

Sσ

√

2 πT

e−

d^21

(^2) ,
thetaCE=− Sσ
2

√

2 πT

e−

d^21

(^2) −rXe−rTN(d 2 ),
vegaCE=

S

√

√T

2 π

e−

d^21

(^2) ,
rhoCE=TXe−rTN(d 2 ).
(The Greek parameters are computed at timet=0.)

Remark 9.1

It is easy to see from the above that

thetaCE+rSdeltaCE+^1

2

σ^2 S^2 gammaCE=rCE.

In general, the priceDof any European derivative security can be shown to
satisfy theBlack–Scholes equation

∂D

∂t

+rS

∂D

∂S

+

1

2

σ^2 S^2

∂^2 D

∂S^2

=rD.

Exercise 9.5

Show that a delta neutral portfolio with initial value zero hedging a

single call option will gain in value with time if the stock price, volatility

and risk-free rate remain unchanged.

Exercise 9.6

Derive formulae for the Greek parameters of a put option.