Mathematical Finance and Probability A Discrete Introduction
The objective of this book is to give a self-contained presentation to the theory underlying the valuation of derivative financial instruments, which is becoming a standard part of the toolbox of professionals in the financial industry. Although a complet
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The Black-Scholes Formula Unfortunately, the mathematical tools employed in the Black-Scholes and Merton articles are quite advanced and have tended to obscure the underlying economics. However, thanks to a remark by William Sharpe it is possible to derive the same results using only elementary mathematics. J.C. Cox, S.A. Ross and M. Rubinstein
An introduction to mathematical finance would not be complete without an exposition of its most famous result: the Black-Scholes formula for the price of European call and put options.
14.1
Limiting Behavior of a Cox-Ross-Rubinstein Economy
We consider the Cox-Ross-Rubinstein economy over the time interval [0, T] as parametrized in Section 12.2. For each N E N trading activity may take place at the dates
tt
By choosing as a unit of time trading takes place at the dates t = 0,1, ... ,N . The underlying sample space is
P. K. Medina et al., Mathematical Finance and Probability © Birkhäuser Verlag 2003
Chapter 14.
248
On
n the sequence
The Black-Scholes Formula
(ZN,s) of independent random variables is given by
for W = (WI, ... ,WN). Setting,
s=1
we have P(ZN,s = 1) = PN
P(ZN,s = 0) = 1 - PN ,
and
where PN is the probability that in one of the subperiods of the model the economic development will be "good". Given the risk-free rate rN and the stock yields Yg,N and Yb,N the money market account at time t is given by
and the price of the stock by (14.1)
Parametrization As in Section 12.2 we use the expression SN,N(W)
= SoeHN(w)
with H
( ) ~1
N W
-
SN,N(W) og So (w) .
Also in Section 12.2 we showed that, given R such that e- RT is the price of a zero-bond with face value 1 currency unit and maturity T, and given historical observations of
~ ~E
J-l- T
PN
[1
SN,N og So j
and
a
2
def
=
1
-VarPN[log-s j '
T
SN,N
0
one can parametrize the Cox-Ross-Rubinstein model for large N E N by defining rN , Yg,N , Yb,N and PN as rN
def
e fR -1
Yg,N
def
e fla - 1, e -fla N - 1,
Yb,N PN
def
def
,
~+~~fft· 2 2a N
14.1. Limiting Behavior of a Cox-Ross-Rubinstein Economy
249
We will need the following auxiliary results.
Lemma 14.1 The following statements hold:
HN EQN [HN] VarQN [HN]
DN,N log(1 + Yg,N)
= (2qN -1)av'NT , = qN(1 - qN )4a 2T .
+ (N -
DN,N) log(1
+ Yb,N)
,
Proof The expression for HN follows immediately from the definition (14.1) of SN,N. To compute the expectation of HN just note that
EQN[DN,N]log(1 + Yg,N) + EQN[N - DN,N]log(1 N qN log(1 + Yg,N) + N(1 - qN) log(1 + Yb,N)
+ Yb,N)
NqN{f;a - N(1- qN){f;a (2qN-1)av'NT. The variance of HN is obtained as follows: VarQN [DN,N][log(1 + Yg,N) -log(1 + Yb,N W NqN(1- qN)[log(1 + Yg,N) -log(1 + Yb,N)]2 2T NqN(1- qN)4a N
qN(1- qN)4a 2 T.
o Risk-neutral probability Since we are interested in looking at prices of contingent claims we need to consider the risk neutral probability. As shown in Theorem 12.7 it is given by (14.2)
Lemma 14.2 The following holds: . I1m qN
1
= -.
2 Proof Applying de I'Hopital's rule we easily see that for any a N--+oo
e82a
_
e- 8
lim----
(14.3)
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