Brownian Motion
Denition. A Brownian motion is a stochastic process B = cfw_Bt , t 0 satisfying
(i) B0 = 0
(ii) for any 0 t0 < t1 < < tk , the rvs Bti Bti1 , i = 1, . . . , k, (increments)
are independent,
(iii) Bt Bs N (t s), 2 (t s), where t > s, R, >
Basic Monte Carlo
Suppose we want to nd
= E [G(X1 , , Xd )] =
G(x1 , . . . , xd )f (x1 , . . . , xd )dx1 dxd
where f is the j.d.f. of X = (X1 , , Xd ).
We can approximate using the following Monte Carlo simulation method:
(i) draw N independent values x
Stochastic Integral and Diusion Processes
Let us consider a simple growth model
dSt
= (t)St .
dt
In practice the value of (t) may not be completely known. Thus, we would
like to be able to write the following equation
dX
= b(t, Xt ) + (t, Xt )
dt
Wt ,
(
Numerical Solutions of SDEs
Suppose we want to generate N trajectories from the process cfw_Xt
that solves the SDE
dXt = (t, Xt)dt + (t, Xt)dBt,
X0 = x,
(1)
over a time interval [0, T ].
Euler approximation:
1 Partition [0, T ] into M subintervals 0 = t
Multidimensional Models
(Chapter 13)
Suppose we have n risky traded securities
S (t)T = [S1(t), . . . , Sn(t)]
and a contingent claim of the form
= (S (T ),
which we want to price and hedge.
Assumption:
Under the objective probability measure P , the S
Martingale Approach to Arbitrage Theory
Chapters 10, 11 and 12
Suppose we have N risky traded securities
S (t)T = [S1(t), . . . , SN (t)]
and another security S0(t) whose price is assumed to be strictly
positive. The latter will serve as the numeraire pr
Conditional Expectations, Filtrations and
Martingales
Let X and Y be jointly distributed continuous random variables with joint probability density function fX,Y (x, y ). We dene the conditional probability density function
fX |Y (x|y ) for the random va
Black-Scholes Model
Chapters 6, 7, 8 and 9
Black-Scholes model of a risky asset assumes
dSt = Stdt + StdBt,
S0 = x,
where and are instantaneous expected value and volatility
(standard deviation) of the rate of return, respectively.
This equation can be so