Quantitative Finance for Physicists

(so-called term structure). In a different approach, the lognormal

stock price distribution is substituted with another statistical distri-

bution. Also, the jump-diffusion stochastic processes are sometimes

used instead of the geometric Brownian motion.

Other directions for expanding BST address the market imperfec-

tions, such as transaction costs and finite liquidity. Finally, the option

price in the current option pricing theory depends on time and price

of the underlying asset. This seemingly trivial assumption was ques-

tioned in [9]. Namely, it was shown that the option price might

depend also on the number of shares of the underlying asset in the

arbitrage-free portfolio. Discussion of this paradox is given in the

Appendix section of this chapter.

9.5 REFERENCES FOR FURTHER READING

Hull’s book is the classical reference for the first reading on finan-

cial derivatives [1]. A good introduction to mathematics behind the

option theory can be found in [4]. Detailed presentation of the option

theory, including exotic options and extensions to BST, is given in

[2, 3].

9.6 APPENDIX: THE INVARIANT

OF THE ARBITRAGE-FREE PORTFOLIO

As we discussed in Section 9.4, the option price F(S, t) in BST is a

function of the stock price and time. The arbitrage-free portfolio in

BST consists of one share and of a number of options (M

0

) that hedge

this share [5]. BST can also be derived with the arbitrage-free port-

folio consisting of one option and of a number of shares M

1

0

(see,

e.g., [1]). However, if the portfolio with an arbitrary number of shares

N is considered, and N is treated as an independent variable, that is,

F ¼ F(S, t, N) (9:6:1)

then a non-zero derivative, @F=@N, can be recovered within the

arbitrage-free paradigm [9]. Since options are traded independently

from their underlying assets, the relation (9.6.1) may look senseless to

the practitioner. How could this dependence ever come to mind?

Option Pricing 105

Recall the notion of liquidity discussed in Section 2.1. If a market

order exceeds supply of an asset at current ‘‘best’’ price, then the

order is executed within a price range rather than at a single price. In

this case within continuous presentation,

S ¼ S(t, N) (9:6:2)

and the expense of buying N shares at time t equals

ð

N

0

S(t, x)dx (9:6:3)

The liquidity effect in pricing derivatives has been addressed in [10,

11] without proposing (9.6.1). Yet, simply for mathematical general-

ity, one could assume that (9.6.1) may hold if (9.6.2) is valid. Surpris-

ingly, the dependence (9.6.1) holds even for infinite liquidity. Indeed,

consider the arbitrage-free portfolio P with an arbitrary number of

shares N at price S and M options at price F:

P(S, t, N) ¼ NS(t) þ MF(S, t, N) (9:6:4)

Let us assume that N is an independent variable and M is a parameter

to be defined from the arbitrage-free condition, similar to M

0

in BST.

As in BST, the asset price S ¼ S(t) is described with the geometric

Brownian process

dS ¼ mSdt þ sSdW: (9:6:5)

In (9.6.5), m and s are the price drift and volatility, and W is the

standard Wiener process. According to the Ito’s Lemma,

dF ¼

@F

@t

dt þ

@F

@S

dS þ

s

2

2

S

2

@

2

F

@S

2

dt þ

@F

@N

dN (9:6:6)

It follows from (9.6.4) that the portfolio dynamic is

dP ¼ MdF þ NdS þ SdN (9:6:7)

Substituting equation (9.6.6) into equation (9.6.7) yields

dP ¼ [M

@F

@S

þ N]dS þ [M

@F

@N

þ S]dN þ M

@F

@t

þ

s

2

2

S

2

@

2

F

@S

2



dt

(9:6:8)

106 Option Pricing

As within BST, the arbitrage-free portfolio grows with the risk-free

interest rate, r

dP ¼ rPdt (9:6:9)

Then the combination of equation (9.6.8) and equation (9.6.9)

yields

[M

@F

@S

þ N]dS þ [M

@F

@t

þ

s

2

2

MS

2

@

2

F

@S

2

 rMF  rNS]dtþ

[M

@F

@N

þ S]dN ¼ 0

(9:6:10)

Since equation (9.6.10) must be valid for arbitrary values of dS, dt

and dN, it can be split into three equations

M

@F

@S

þ N ¼ 0(9:6:11)

M

@F

@t

þ

s

2

2

S

2

@

2

F

@S

2

 rF



 rNS ¼ 0(9:6:12)

M

@F

@N

þ S ¼ 0(9:6:13)

Let us present F(S, t, N) in the form

F(S, t, N) ¼ F

0

(S, t)Z(N) (9:6:14)

where Z(N) satisfies the condition

Z(1) ¼ 1(9:6:15)

Then it follows from equation (9.6.11) that

M ¼N= Z

@F

0

@S



: (9:6:16)

This transforms equation (9.6.15) and equation (9.6.16), respectively,

to

@F

0

@t

þ rS

@F

0

@S

þ

s

2

2

S

2

@

2

F

0

@S

2

 rF

0

¼ 0(9:6:17)

dZ

dN

¼ (S=F

0

)

@F

0

@S

(Z=N), (9:6:18)

Option Pricing 107