Chapter 10
Applications of Stochastic
Calculus in Finance
Since Black and Scholes (1973) and Merton (1973), the idea of using stochastic
calculus for modelling prices of risky assets (e.g., share prices of stock, stock
indices such as Dow Jones, foreign exchange rates, interest rates, etc.) has been
generally accepted.
This led to a new branch of applied probability theory,
the field of mathematical finance.
It is a symbiosis of stochastic modelling,
economic reasoning and practical financial engineering.
In this chapter, we consider the BlackScholes model for pricing options
(in particular, a European option), using two different approaches (with em
phasis on the latter):
(i) PDE approach;
(ii) probabilistic approach
The basic
financial concepts
needed are:
bond, stock, option, portfo
lio, volatility, trading strategy, hedging, maturity of a contract, selffinancing,
arbitrage
. On the other hand, some terminology in
stochastic calculus
will
be encountered such as:
Brownian motion (BM), Geometric BM, Ito’s inte
gral, Stochastic differential equivation (SDE), martingale, equivalent martin
gale measure, etc.
The key tools include
•
Ito’s Lemma,
•
Girsanov Theorem (change of measure theorem),
•
Martingale representation theorem,
•
FeymannKac theorem
1
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10.1
Introduction to some basic concepts
We are given a filtered probability space (Ω
,
F
,
{F
t
, t
∈
[0
, T
]
}
, P
).
1.
Risky asset: stock
The price
S
t
of a
risky asset
(called
stock
) is supposed to follow a
geometric Brownian motion
(BM):
dS
t
S
t
=
μdt
+
σdB
t
(
P
)
⇐⇒
S
t
=
S
0
e
(
μ

0
.
5
σ
2
)
t
+
σB
t
(
P
)
(1.1)
Remark
10.1
(a) The measure
P
is also called the market measure.
Formally, we
have
E
ˆ
dS
t
S
t
!
=
μdt,
and
V ar
ˆ
dS
t
S
t
!
=
σ
2
V ar
(
dB
t
) =
σ
2
dt.
So
μ
and
σ
are the mean and standard deviation (or
volatility
) of
rate of return (or relative return, or log return) per unit of time.
(b) In (6.19), the rates of return are assumed to be: (i) constant per
unit of time; (ii) independent for different time intervals.
How
ever, empirical evidence indicates that, although the rates of return
are uncorrelated for different time periods, they are not independent
since the absolute values or the squares of rates of returns are corre
lated). Therefore, the geometric BM is only a reasonable, but crude,
first approximation to a real price process.
Incidentally, in time series, GARCH models are introduced to take
care of the correlated volatilities.
(c) If
σ
2
= 0
, then
S
t
=
S
0
e
μt
. Namely, taking away the noise, stock
markets grow exponentially.
People in economics believe in expo
nential growth, and empirical evidence generally supports this belief
as well.
2.
Riskless asset: bond
Suppose that there is also a
bond
(
riskless asset
) market with fixed
interest rate,
r
,
dβ
t
β
t
=
rdt
⇐⇒
β
t
=
β
0
e
rt
.
For simplicity, we might assume that the initial capital
β
0
= 1.
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 Spring '11
 D
 Calculus, Sets, Mathematical finance, Option style

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