Stochastic Calculus
Ilya Dergunov
Quantitative Finance
Higher School of Economics
Module 2, November 2023
Outline
Aim of the lecture
Plan for the lecture:
Brownian Motion and its properties
Stochastic Integral and Ito formula
Examples
Martingales and change of prob. measure
Fundamental PDE
1Essential Reading: Tomas Bjork Arbitrage Theory in
Continuous Time– Ch. 4 & 5
2Further reading: S. Shreve: Stochastic Calculus for Finance II,
Ch. 3 & 4.
Stochastic Calculus
Ilya Dergunov 1
Properties of logreturns
Variance of logreturns
Binomial model
logreturn between iand i+ 1: Li+1 = log(Sti+1 )−log(Sti)
logreturn between iand i+ 2:
log(Sti+2 )−log(Sti) = Li+1 +Li+2
logreturns Li+1 and Li+2 are independent ⇒
var [log(Sti+2 )−log(Sti)] = var[Li+1 +Li+2] = var(Li+1) + var(Li+2)
Variance of the logreturns is proportional to the number of
steps:
var log(Sti+k)−log(Sti)∼k
Stochastic Calculus
Ilya Dergunov 2
Properties of logreturns
Data analysis
Can we empirically confirm that the variance is prop. to time?
Data: closing prices of S&P500 3/01/1996 – 31/12/2022
st. deviation
daily logreturns 1.23%
weekly logreturns 2.48%
monthly logreturns 5.23%
Stochastic Calculus
Ilya Dergunov 3
Properties of logreturns
Variance of logreturns
The Black-Scholes (BS) model is the continuous time limit of the
binomial model.
Similar assumptions:
logreturn variance is proportional to time
logreturns on disjoint time intervals are independent
In addition:
In the BS model logreturns are assumed to be normally dis-
tributed
Stochastic Calculus
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Properties of logreturns
Data analysis
Histogram of S&P500 logreturns 3/01/1996 – 31/12/2022
Stochastic Calculus
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Properties of logreturns
Data analysis cont’d
Autocorrelation of S&P500 logreturns 3/01/1996 – 31/12/2022
Stochastic Calculus
Ilya Dergunov 6
Brownian Motion
Intuition
Let ϵt
iid
∼N(0,1)
We construct a stochastic process Xwith X0= 0 and changes
∆X≡Xt−Xt−∆tdriven by ϵ
∆X=ϵ√∆t
⇒Xk∆t=X0+
k
X
i=1
∆Xi∆t
=X0+
k
X
i=1
ϵi∆t√∆t
⇒Xk∆t∼N(0,k∆t)
Stochastic Calculus
Ilya Dergunov 7
Brownian Motion
Intuition
Properties of this process:
starting value 0
i.i.d. changes
mean of 0 for any k
variance of k∆t, i.e., variance grows linearly in time
Next step: construct continuous-time process with similar
characteristics
Stochastic Calculus
Ilya Dergunov 8
Brownian Motion
Brownian motion
A stochastic process {Wt}t∈[0,T]with continuous paths is
called a Brownian motion, if
the initial value is given by W0= 0
for any 0 ≤s<t≤u<v≤T, the increments
Wt−Wsand Wv−Wuare independent
for any 0 ≤t<u≤Tthe increment (Wu−Wt) is
normally distributed with
Wu−Wt∼ N(0,u−t)
Stochastic Calculus
Ilya Dergunov 9
Brownian Motion
BM trajectories
Stochastic Calculus
Ilya Dergunov 10
Brownian Motion
Properties of Brownian motion
Variance of increments is proportional to time period
var(Wt−Ws) = t−s
This is also called √dt-effect:
