(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

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