(Informal) Introduction to Stochastic Calculus
Paola Mosconi
Banca IMI
Bocconi University, 15/02/2016
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Lecture 1-2
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Disclaimer
The opinion expressed here are solely those of the author and do not represent in
any way those of her employers
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Lecture 1-2
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Main References
D. Brigo and F. Mercurio
Interest Rate Models – Theory and Practice. With Smile, Inflation and Credit
Springer (2006)
Appendix C
S. Shreve
Stochastic Calculus for Finance II
Springer (2004)
Chapter 6
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Outline
1
From Deterministic to Stochastic Differential Equations
2
Ito’s formula
3
Examples
4
Change of Measure
5
No-Arbitrage Pricing
6
Exercises
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Probability Space
Probability Space I
A probability space (Ω, F , (Ft )t , P) can be interpreted as an experiment, where:
ω ∈ Ω represents a generic result of the experiment
Ω is the set of all possible outcomes of the random experiment
a subset A ⊂ Ω represents an event
the σ-field F is a collection of subsets of Ω
the σ-field Ft represents the information up to time t.
We have Ft ⊆ Fu ⊆ F for all t ≤ u, meaning that “the information increases in
time”, never exceeding the whole set of events
the family of σ-fields (Ft )t≥0 is called filtration
P is a probability measure
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Probability Space
Probability Space II
If:
ω ∈ A ∈ F we say that the event A has occurred
ω ∈ A ∈ Ft we say that the event A has occurred at a time smaller or equal
to t
Expectations
The symbol E denote expectation and E[·|F ] denotes expectation conditional
on the information contained in F .
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Deterministic Differential Equations
Deterministic Differential Equations (DDE)
EXAMPLE: Population Growth Model
Let x(t) = xt ∈ R, xt ≥ 0, denote the population at time t, and assume for
simplicity a constant (proportional) population growth rate, so that the change
in the population at t is given by the deterministic differential equation:
dxt = K xt dt, x0
where K is a real constant.
Assume now that x0 is a random variable X0 (ω) and that the population
growth is modeled by the following differential equation:
dXt (ω) = K Xt (ω)dt, X0 (ω)
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Deterministic Differential Equations
From Deterministic to Stochastic Differential Equations
The solution to this equation is:
Xt (ω) = X0 (ω) e Kt
where all the randomness comes from the initial condition X0 (ω).
For each experiment result ω, the map t 7→ Xt (ω) is called the path of X
associated to ω.
As a further step, suppose that even K is not known for certain, but that also
our knowledge of K is perturbed by some randomness, which we model as the
increment of a stochastic process {Wt (ω)}, t ≥ 0, so that
dXt (ω) = (K dt + dWt (ω)) Xt (ω), X0 (ω), K ≥ 0
(1)
where dWt (ω) represents a noise process that adds randomness to K .
Eq. (1) represents an example of stochastic differential equation (SDE).
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Stochastic Differential Equations
Stochastic Differential Equations (SDE)
More generally, a SDE is written as
dXt (ω) = ft (Xt (ω)) dt + σt (Xt (ω)) dWt (ω), X0 (ω)
(2)
The function f corresponds to the deterministic part of the SDE and is called
the drift. The function σt is called the diffusion coefficient.
The randomness enters the SDE from two sources:
the noise term dWt (ω)
the initial condition X0 (ω)
The solution X of the SDE is also called a diffusion process.
In general the corresponding paths t 7→ Xt (ω) are continuous.
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Brownian Motion
Brownian Motion
The process whose increments dWt (ω) are candidates to represent the noise
process in the SDE given by Eq. (2) is the Brownian motion.
It has two important properties:
it has stationary and independent Gaussian increments dWt (ω).
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Brownian Motion
Brownian Motion: Definition and Properties
Definition
Given a probability space (ω, F, (Ft )t P), for each ω ∈ Ω there is a continuous function
Wt , t ≥ 0 such that it depends on ω and W0 = 0. Then, Wt is a Brownian motion if
and only if for any 0 < s < t < u and any h > 0 it has:
Independent increments: Wu (ω) − Wt (ω) independent of Wt (ω) − Ws (ω)
Stationary increments: Wt+h (ω) − Ws+h (ω) ∼ Wt (ω) − Ws (ω)
Gaussian increments: Wt (ω) − Ws (ω) ∼ N (0, t − s)
Although the paths are continuous, they are (almost surely) nowhere differentiable, i.e.
Ẇt (ω) =
d
Wt (ω)
dt
does not exist.
Go to Brownian Motion: Exercises
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Stochastic Integrals
Stochastic Integrals I
The integral form of the general SDE, given by Eq. (2), is:
Xt (ω) = X0 (ω) +
Z t
fs (Xs (ω)) ds +
0
Z t
σs (Xs (ω)) dWs (ω)
(3)
0
where:
Rt
f (Xs (ω)) ds is a Riemann-Stieltjes integral.
0 s
Rt
0 σs (Xs (ω)) dWs (ω) is a Stieltjes-like integral, since it has unbounded variation.
The resulting integral depends on the chosen points of the sub-partitions used
in the limit that defines the integral.
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Stochastic Integrals
Stochastic Integrals II
Take the interval [0, T ] and consider the following partitions of this interval:
i
i = 0, 1, . . . , ∞
Tin = min T , n
2
where n is an integer.
For all i > 2n T all terms collapse to T , i.e. Tin = T .
For each n we have such a partition, and when n increases the partition contains
more elements, giving a better discrete approximation of the continuous interval
[0, T ].
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Stochastic Integrals
Ito and Stratonovich Integrals
Then define the integral as:
Z T
φs (ω)dWs (ω) = lim
0
n→∞
∞
X
i =0
h
i
φtin (ω) WTin+1 (ω) − WTin (ω)
where tin is any point in the interval Tin , Tin+1 . By choosing:
tni := Tni (initial point), we have the Ito integral;
tni :=
Tni +Tni+1
2
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(middle point), we have the Stratonovich integral.
Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Stochastic Integrals
Ito Integral vs Stratonovich Integral
Ito =⇒
Z t
Ws (ω)dWs (ω) =
0
Stratonovich =⇒
Z t
Wt (ω)2
1
− t
2
2
Ws (ω)dWs (ω) =
0
Wt (ω)2
2
Ito
Stratonovich
Martingale property
No martingale property
No standard chain rule
Standard chain rule
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Martingales
Martingales I
Consider a process X satisfying the following conditions:
Measurability:
Ft includes all the information on X up to time t, i.e. (Xt )t is adapted to
(Ft )t .
Integrability:
The relevant expected values exist.
A martingale is a process satisfying these two conditions and such that:
E[XT |Ft ] = Xt
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for each t ≤ T
Lecture 1-2
(4)
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From Deterministic to Stochastic Differential Equations
Martingales
Martingales II
In other words...
if t is the present time, the expected value at a future time T , given the
current information, is equal to the current value
a martingale represents a picture of a fair game, where it is not possible to
lose or gain on average
the martingale property is suited to model the absence of arbitrage, i.e.
there is no safe way to make money from nothing (no free lunch)
Go to No-Arbitrage Pricing
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Martingales
Martingales and SDEs
Consider a SDE admitting a unique solution. This solution is a martingale if the
equation has zero drift, i.e. ft (·) = 0 for all t in eq. (2):
dXt (ω) = σt (Xt (ω)) dWt (ω)
(5)
Martingale means a driftless diffusion process.
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Martingales
Submartingales, Supermartingales and Semimartingales
A submartingale is a similar process X satisfying:
E[XT |Ft ] ≥ Xt
for each t ≤ T
i.e. the expected value of the process grows in time.
A supermartingale satisfies:
E[XT |Ft ] ≤ Xt
for each t ≤ T
i.e. the expected value of the process decreases in time.
A process X that is either a submartingale or a supermartingale is termed a
semimartingale.
Go to Martingales: Exercises
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From Deterministic to Stochastic Differential Equations
Variations/Covariantions
Quadratic Variation I
The quadratic variation of a stochastic process Yt , with continuous paths, is
defined as follows:
∞ h
i2
X
hY iT = lim
YTin (ω) − YTin−1 (ω)
n→∞
i =1
or in form of a second order integral:
hY iT = “
Z T
2
[dYs (ω)] ”
0
or in differential form:
“dhY it = dYt (ω) dYt (ω)”
A process whose paths are differentiable for almost all ω satisfies hY it = 0.
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Variations/Covariantions
Quadratic Variation II
If Y is a Brownian motion W , it can be proved that:
hW iT = T
for each T
or, equivalently:
dWt (ω) dWt (ω) = dt
This comes from the fact that the Brownian motion moves so quickly that second order
effects are not negligible. Instead, a process with differentiable trajectories cannot move
so quickly and therefore second order effects do not contribute.
If Y is the deterministic process t 7→ t, then dYt = 0 and
dt dt = 0
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Variations/Covariantions
Quadratic Covariation I
The quadratic covariation of two stochastic processes Yt and Zt , with continuous
paths, is defined as follows:
hY , Z iT = lim
n→∞
∞ h
X
i =1
ih
i
YTin (ω) − YTin−1 (ω) ZTin (ω) − ZTin−1 (ω)
or in form of a second order integral:
hY , Z iT = “
Z T
dYs (ω)dZs (ω)”
0
or in differential form:
“dhY , Z it = dYt (ω) dZt (ω)”
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Variations/Covariantions
Quadratic Covariation II
If Y is a Brownian motion W and Z a deterministic process t 7→ t it follows:
hW , tiT = 0
for each T
or, equivalently:
dWt (ω) dt = 0
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Solutions to SDEs
Solution to a General SDE
Let us consider a general SDE:
dXt (ω) = ft (t, Xt (ω)) dt + σt (t, Xt (ω)) dWt (ω), X0 (ω)
Existence and uniqueness of the solution are guaranteed by:
Lipschitz continuity:
|f (t, x) − f (t, y )| ≤ C |x − y | and |σ(t, x) − σ(t, y )| ≤ C |x − y | x, y ∈ Rd
Linear growth bound:
|f (t, x)| ≤ D(1 + |x|) and |σ(t, x)| ≤ D(1 + |x|) x ∈ Rd
(See Øksendal (1992) for the details.)
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Solutions to SDEs
Interpretation of the Coefficients: DDE Case
For a deterministic differential equation
dxt = f (xt )dt
with f a smooth function, we have:
xt+h − xt
= f (y )
h→0
h
xt =y
lim
(xt+h − xt )2
=0
h→0
h
xt =y
lim
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Lecture 1-2
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From Deterministic to Stochastic Differential Equations
Solutions to SDEs
Interpretation of the Coefficients: SDE Case
For a stochastic differential equation
dXt (ω) = f (Xt (ω))dt + σ(Xt (ω))dWt (ω)
functions f and σ can be interpreted as:
Xt+h (ω) − Xt (ω)
lim E
Xt = y = f (y )
h→0
h
lim E
h→0
[Xt+h (ω) − Xt (ω)]2
Xt = y
h
= σ 2 (y )
The second limit is non-zero because of the infinite velocity of the Brownian motion.
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Lecture 1-2
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Ito’s formula
Outline
1
From Deterministic to Stochastic Differential Equations
2
Ito’s formula
3
Examples
4
Change of Measure
5
No-Arbitrage Pricing
6
Exercises
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Lecture 1-2
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Ito’s formula
Chain Rule
Deterministic Case
For a deterministic differential equation such as
dxt = f (xt )dt
given a smooth transformation φ(t, x), we can write the evolution of φ(t, xt ) via
the standard chain rule:
dφ(t, xt ) =
Paola Mosconi
∂φ
∂φ
(t, xt )dt +
(t, xt )dxt
∂t
∂x
Lecture 1-2
(6)
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Ito’s formula
Chain Rule
Stochastic Case: Ito’s Formula I
Let φ(t, x) be a smooth function and Xt (ω) the unique solution to the SDE (2).
The chain rule – Ito’s formula reads as:
dφ(t, Xt (ω)) =
∂φ
∂φ
(t, Xt (ω))dt +
(t, Xt (ω))dXt (ω)
∂t
∂x
1 ∂2φ
+
(t, Xt (ω))dXt (ω)dXt (ω)
2 ∂x 2
(7)
or, in compact notation:
dφ(t, Xt ) =
Paola Mosconi
∂φ
∂φ
1 ∂2φ
(t, Xt )dt +
(t, Xt )dXt +
(t, Xt )dhX it
∂t
∂x
2 ∂x 2
Lecture 1-2
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Ito’s formula
Chain Rule
Stochastic Case: Ito’s Formula II
The term dXt (ω)dXt (ω) can be developed by recalling the rules on quadratic variation and covariation:
dWt (ω)dWt (ω) = dt,
dWt (ω)dt = 0,
dtdt = 0
thus giving:
dφ(t, Xt (ω)) =
∂φ
∂φ
(t, Xt (ω)) +
(t, Xt (ω))f (Xt (ω))
∂t
∂x
1 ∂2φ
2
+
(t, Xt (ω))σ (Xt (ω)) dt
2 ∂x 2
∂φ
+
(t, Xt (ω))σ(Xt (ω))dWt (ω)
∂x
Go to Ito’s Formula: Exercises
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Lecture 1-2
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Ito’s formula
Leibniz Rule
Leibniz Rule
It applies to differentiation of a product of functions.
Deterministic Leibniz rule
For deterministic and differentiable functions x and y :
d(xt yt ) = xt dyt + yt dxt
Stochastic Leibniz rule
For two diffusion processes Xt (ω) and Yt (ω):
d(Xt (ω) Yt (ω)) = Xt (ω) dYt (ω) + Yt (ω) dXt (ω) + dXt (ω)dYt (ω)
or, in compact notation:
d(Xt Yt ) = Xt dYt + Yt dXt + dhX , Y it
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Lecture 1-2
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Examples
Outline
1
From Deterministic to Stochastic Differential Equations
2
Ito’s formula
3
Examples
4
Change of Measure
5
No-Arbitrage Pricing
6
Exercises
Paola Mosconi
Lecture 1-2
32 / 57
Examples
Linear SDE
Linear SDE with Deterministic Diffusion Coefficient I
A SDE is linear if both its drift and diffusion coefficients are first order polynomials
in the state variable.
Consider the particular case:
dXt (ω) = (αt + βt Xt (ω)) dt + vt dWt (ω),
X0 (ω) = x0
(8)
where α, β, v are deterministic functions of time, regular enough to ensure existence
and uniqueness of the solution.
The solution is:
Xt (ω) = e
Rt
0
= x0 e
βs ds
Rt
0
x0 +
βs ds
Z t
e
−
Rs
0
βu du
αs ds +
0
+
Z t
e
Rt
s
βu du
αs ds +
0
Paola Mosconi
Z t
Rs
Z t
e
e
βu du
0
Rt
s
−
0
βu du
vs dWs (ω)
(9)
vs dWs (ω)
0
Lecture 1-2
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Examples
Linear SDE
Linear SDE with Deterministic Diffusion Coefficient II
The distribution of the solution Xt (ω) is normal at each time t:
R
Z t R
Z t R
t
t
t
β
ds
β
du
2
β
du
2
s
u
u
Xt ∼ N x0 e 0
+
e s
αs ds,
e s
vs ds
0
0
Major examples: Vasicek SDE (1978) and Hull and White SDE (1990).
Go to Vasicek Model
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Lecture 1-2
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Examples
Lognormal Linear SDE
Lognormal Linear SDE
The lognormal SDE can be obtained as an exponential of a linear equation with
deterministic diffusion coefficient.
Let us take Yt = exp(Xt ), where Xt evolves according to (34), i.e.:
d ln Yt (ω) = (αt + βt ln Yt (ω)) dt + vt dWt (ω),
Y0 (ω) = exp(x0 )
Equivalently, by Ito’s formula we can write:
1
dYt (ω) = de Xt (ω) = e Xt (ω) dXt (ω) + e Xt (ω) dXt (ω)dXt (ω)
2
1 2
= αt + βt ln Yt (ω) + vt dt + vt Yt (ω)dWt (ω)
2
The process Y has a lognormal marginal density. Major examples: Black Karasinski model (1991) and Geometric Brownian Motion.
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Lecture 1-2
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Examples
Lognormal Linear SDE
Geometric Brownian Motion I
The GBM is a particular case of lognormal linear process.
Its evolution is defined by:
dXt (ω) = µ Xt (ω)dt + σ Xt (ω) dWt (ω),
X0 (ω) = X0
where µ and σ are positive constants.
By Ito’s formula, one can solve the SDE, by computing d ln Xt :
1 2
Xt (ω) = X0 exp
µ − σ t + σ Wt (ω)
2
From the work of Black and Scholes (1973) on, processes of this type are frequently
used in option pricing theory to model the asset price dynamics.
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Lecture 1-2
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Examples
Lognormal Linear SDE
Geometric Brownian Motion II
The GBM process is a submartingale:
E[XT |Ft ] = e µ(T −t) Xt ≥ Xt
The process Yt (ω) = e −µ t Xt (ω) is a martingale, since we obtain:
dYt (ω) = σ Yt (ω) dWt (ω)
i.e. the drift of the process is zero.
Go to Geometric Brownian Motion: Excercise
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Lecture 1-2
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Examples
Square Root Process
Square Root Process
It is characterized by a non-linear SDE:
dXt (ω) = (αt + βt Xt (ω)) dt + vt
p
Xt (ω) dWt (ω),
X0 (ω) = X0
Square root processes are naturally linked to non-central ξ-square distributions.
Major examples: the Cox Ingersoll and Ross (CIR) model (1985) and a particular
case of the constant-elasticity variance (CEV) model for stock prices:
p
dXt (ω) = µ Xt (ω)dt + σ Xt (ω) dWt (ω),
X0 (ω) = X0
Go to Cox Ingersoll Ross Model
Go to SDE: Excercise
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Lecture 1-2
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Change of Measure
Outline
1
From Deterministic to Stochastic Differential Equations
2
Ito’s formula
3
Examples
4
Change of Measure
5
No-Arbitrage Pricing
6
Exercises
Paola Mosconi
Lecture 1-2
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Change of Measure
Change of Measure
The way a change in the underlying probability measure affects a SDE is
defined by the Girsanov theorem
The theorem is based on the following facts:
the SDE drift depends on the particular probability measure P
if we change the probability measure in a “regular”way, the drift of the
equation changes while the diffusion coefficient remains the same.
The Girsanov theorem can be useful when we want to modify the drift coefficient
of a SDE.
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Lecture 1-2
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Change of Measure
Radon-Nikodym Derivative
Two measures P∗ and P on the space (Ω, F , (Ft )t ) are said to be equivalent, i.e.
P∗ ∼ P, if they share the same sets of null probability.
When two measures are equivalent, it is possible to express the first in terms of
the second through the Radon-Nikodym derivative.
There exists a martingale ρt on (Ω, F, (Ft )t , P) such that
Z
P∗ =
ρt (ω) dP(ω), A ∈ Ft
A
which can be written as:
dP∗
= ρt
dP Ft
The process ρt is called the Radon-Nikodym derivative of P∗ with respect to P
restricted to Ft .
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Lecture 1-2
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Change of Measure
Expected Values
When in need of computing the expected value of an integrable random variable
X , it may be useful to switch from one measure to another equivalent one.
Expectations
Z
Z
dP∗
dP∗
∗
∗
E [X ] =
X (ω) dP (ω) =
X (ω)
(ω) dP(ω) = E X
dP
dP
Ω
Ω
Conditional expectations
E∗ [X | Ft ] =
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Lecture 1-2
h
i
∗
E X dP
dP
ρt
42 / 57
Change of Measure
Girsanov Theorem
Consider SDE, with Lipschitz coefficients, under dP:
dXt (ω) = f (Xt (ω))dt + σ(Xt (ω))dWt (ω),
x0
Let be given a new drift f ∗ (x) and assume (f ∗ (x) − f (x))/σ(x) to be bounded.
Define the measure P∗ through the Radon-Nikodym derivative:
dP∗
(ω)
= exp
dP
Ft
(
1
−
2
)
Z t ∗
Z t ∗
f (Xs (ω)) − f (Xs (ω)) 2
f (Xs (ω)) − f (Xs (ω))
ds +
dWs (ω)
σ(Xs (ω))
σ(Xs (ω))
0
0
Then P∗ is equivalent to P and the process W ∗ defined by:
∗
f (Xt (ω)) − f (Xt (ω))
dWt∗ (ω) = −
dt + dWt (ω)
σ(Xt (ω))
is a Brownian motion under P∗ and
dXt (ω) = f ∗ (Xt (ω))dt + σ(Xt (ω))dWt∗(ω),
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Lecture 1-2
x0
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Change of Measure
Example: from P to Q I
A classical example involves moving from the real world asset price dynamics P
to the risk neutral one, Q, i.e. from
dSt = µ St dt + σSt dWtP
under P
(10)
dSt = r St dt + σSt dWtQ
under Q
(11)
to
The risk neutral measure Q is used in pricing problems while the real-world (or
historical) measure P is used in risk management.
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Lecture 1-2
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Change of Measure
Example: from P to Q II
Start from the asset dynamics under the real-world measure P, eq. (10):
dSt = µ St dt + σSt dWtP
Consider the discounted asset price process S̃t = St e −rt .
This process satisfies the following SDE:
d S̃t = (µ − r ) S̃t dt + σ S̃t dWtP
(12)
The goal is to find a measure Q, equivalent to P, such that the discounted
asset price process is a martingale under the new measure, i.e.
d S̃t = σ S̃t dWtQ
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Lecture 1-2
(13)
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Change of Measure
Example: from P to Q III
To this purpose, rewrite eq. (12) as follows:
µ−r
P
P
dt + dWt σ S̃t
d S̃t = (µ − r ) dt + σ dWt S̃t =
σ
and define, according to Girsanov theorem, a new Brownian process:
dWt∗ ≡
µ−r
dt + dWtP .
σ
Therefore,
d S̃t = σ S̃t dWt∗
(14)
i.e. S̃t is a martingale under the equivalent measure P∗ .
Comparing (13) with (14), we obtain P∗ ≡ Q.
Going back to the asset price process St = S̃t e rt , we finally get eq. (11):
dSt = r St dt + σSt dWtQ
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Lecture 1-2
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No-Arbitrage Pricing
Outline
1
From Deterministic to Stochastic Differential Equations
2
Ito’s formula
3
Examples
4
Change of Measure
5
No-Arbitrage Pricing
6
Exercises
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Lecture 1-2
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No-Arbitrage Pricing
No-Arbitrage Pricing
We refer to Brigo and Mercurio, Chapter 2.
As already mentioned, absence of arbitrage is equivalent to the impossibility to
invest zero today and receive tomorrow a non-negative amount that is positive with
positive probability. In other words, two portfolios having the same payoff at a given
future date must have the same price today.
Historically, Black and Scholes (1973) showed that, by constructing a suitable
portfolio having the same instantaneous return as that of a risk-less investment, the
portfolio instantaneous return was indeed equal to the instantaneous risk-free rate,
which led to their celebrated option-pricing formula.
Paola Mosconi
Lecture 1-2
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No-Arbitrage Pricing
Harrison and Pliska Result (1983)
A financial market is arbitrage free and complete if and only if there exists a
unique equivalent (risk-neutral or risk-adjusted) martingale measure.
Stylized characterization of no-arbitrage theory via martingales:
The market is free of arbitrage if (and only if) there exists a martingale measure
The market is complete if and only if the martingale measure is unique
In an arbitrage-free market, not necessarily complete, the price πt of any attainable
claim is uniquely given, either by the value of the associated replicating strategy, or
by the risk neutral expectation of the discounted claim payoff under any of the
equivalent (risk-neutral) martingale measures:
πt = E[D(t, T )ΠT |Ft ]
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Paola Mosconi
Lecture 1-2
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Exercises
Outline
1
From Deterministic to Stochastic Differential Equations
2
Ito’s formula
3
Examples
4
Change of Measure
5
No-Arbitrage Pricing
6
Exercises
Paola Mosconi
Lecture 1-2
50 / 57
Exercises
Brownian Motion: Exercises
1
Time reversal
Prove that the continuous time stochastic process defined by:
Bt = WT − WT −t ,
t ∈ [0, T ]
is a standard Brownian motion.
2
Brownian scaling
Let Wt be a standard Brownian motion. Given a constant c > 0, show that
the stochastic process Xt defined by:
1
Xt = √ Wct ,
c
t>0
is a standard Brownian motion.
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Lecture 1-2
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Exercises
Martingales: Exercises
1
Let X1 , X2 , . . . be a sequence of independent random variables in L1 such that
E[Xn ] = 0 for all n. If we set:
S0 = 0, Sn = X1 + X2 + ... + Xn , for n ≥ 1
F0 = {∅, Ω} and Fn = σ(X1 , X2 , ..., Xn ), for n ≥ 1
prove that the process (Sn )n is an (Fn )n -martingale.
2
Consider a filtration (Fn )n and an Fn -adapted stochastic process (Xn )n such that
X0 = 0 and E[|Xn |] ≤ ∞ for all n ≥ 0. Also, let (cn )n be a sequence of constants.
Define M0 = 0 and
Mn = cn Xn −
n
X
j=1
cj E[Xj − Xj−1 |Fj−1 ] −
n
X
j=1
(cj − cj−1 )Xj−1 ,
for n ≥ 1
Prove that (Mn )n is an Fn -martingale.
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Lecture 1-2
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Exercises
Ito’s formula: Exercises
1
Consider a standard one-dimensional Brownian motion Wt . Use Ito’s formula to
calculate:
Z t
Wt2 = t + 2
Ws dWs
0
and
Wt27 = 351
Z t
Ws25 ds + 27
0
2
Z t
Ws26 dWs
0
Consider a standard one-dimensional Brownian motion Wt . Given k ≥ 2 and t ≥ 0,
use Ito’s formula to prove that:
Z t
1
E[Wtk ] = k(k − 1)
E[Wuk−2 ]du
2
0
Use this expression to calculate E[Wt4 ] and E[Wt6 ].
Hint: stochastic integrals are martingales, so their expectation is zero.
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Lecture 1-2
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Exercises
Vasicek Model
In the Vasicek model (1977) for interest rates, the dynamics of the short rate process rt
is given by the following SDE:
drt = k(θ − rt ) dt + σdWt
with k, θ and σ strictly positive constants, and initial condition r0 .
1
Show that the solution to the above SDE is given by:
Z t
rt = θ + (r0 − θ) e −kt + σ
e k(s−t) dWs
0
Hint: Consider the Ito processes Xt and Yt defined by Xt = e kt and Yt = rt and
integrate by parts.
2
Calculate the mean E[rt ] and the variance var(rt ) of the random variable rt .
Hint: Use Ito’s isometry and assume that all stochastic integrals are martingales, so
they have zero expectation.
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Lecture 1-2
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Exercises
Geometric Brownian Motion: Exercise
Consider the Geometric Brownian motion SDE:
dSt = µt St dt + σt St dWt
with deterministic time-dependent coefficients, µt and σt , and initial condition S0 .
Prove that its solution is given by:
Z t Z t
σ2
σu dWu
St = S0 exp
µu − u du +
2
0
0
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Lecture 1-2
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Exercises
Cox Ingersoll Ross Model
In the Cox Ingersoll Ross model (1985) for interest rates, the dynamics of the short rate
process rt is given by the following SDE:
√
drt = k(θ − rt ) dt + σ rt dWt
with k, θ and σ strictly positive constants, and initial condition r0 .
1
Show that the solution to the above SDE is given by:
Z t
√
rt = θ + (r0 − θ) e −kt + σ
e k(s−t) rs dWs
0
Hint: Consider the Ito processes Xt and Yt defined by Xt = e kt and Yt = rt and
integrate by parts.
2
Calculate the mean E[rt ] and the variance var(rt ) of the random variable rt .
Hint: Use Ito’s isometry and assume that all stochastic integrals are martingales, so
they have zero expectation.
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Lecture 1-2
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Exercises
SDE: Exercise
Consider the following integral SDE:
Z t
Z t
Zt = −
Zu du +
e −u dWu
0
0
Prove that:
Zt = e −t Wt
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Lecture 1-2
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