Financial Stochastic Models
Financial Stochastic Models
Yves Guo
The Chinese University of Hong Kong (SZ)
2020
Yves Guo
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Financial Stochastic Models
Table of Contents
1
Elements of Probability
2
Elements of Stochastic Calculus
3
Risk-Neutral Modelling
4
Black-Scholes-Merton Model
5
Discussion on Hedging
6
Some Numerical Methods for Option Pricing
7
American Options
8
Foreign Exchange Modeling and Composite/Quanto Options
9
Change of Numeraire and Vanilla Interest Rate Option
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Financial Stochastic Models
Elements of Probability
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Financial Stochastic Models
Basic Concepts
A sample space Ω is an nonempty set of all the possible outcomes for a random experiment.
For example, the coin toss of n times: Ω = {ω : ω = ω1 ω2 ...ωn } where ωi is H (head) or T
(tail), ∀i = 1, ..., n.
A subset of Ω, A ⊆ Ω, is called an event.
A partition of Ω is defined as a collection of nonempty sets of Ω known as blocks of the partition,
{A1 , A2 , ...An }, which satisfy the following:
• Ai are disjoint: ∀i 6= j, Ai ∩ Aj = ∅
• A1 ∪ A2 ∪ ... ∪ An = Ω
For A ∈ Ω, the function
Yves Guo
1A (ω) =
(
1,
ω ∈A
0,
ω∈
/A
is called indicator function for A.
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Financial Stochastic Models
σ-algebra
Let Ω be a nonempty set and F be a collection of subsets of Ω. F is called a σ-algebra if it
satisfies:
• F is closed under complements: A ∈ F ⇒ Ac ∈ F
• F is closed under countable union: ∀A1 , A2 , ... ∈ F ,
S∞
i=1
Ai ∈ F
• the empty set ∅ is in F : ∅ ∈ F (hence Ω ∈ F as Ω = ∅c .)
It can be shown that a σ-algebra is closed under intersections and set differences:
∀A, B ∈ F ⇒ A ∩ B ∈ F and A\B ∈ F (A\B is sometimes noted as A − B)
A partition is a special case of σ-algebra.
In probability theory, σ-algebra is used to represent a set of events.
The smallest σ-algebra generated by intervals in R is called Borel σ-algebra.
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Financial Stochastic Models
Probability Space
Let F be a σ-algebra of subsets of sample space Ω. A probability measure P is a function such
that:
• 0 ≤ P (A) ≤ 1, ∀A ∈ F
• P (Ω) = 1
S
• Countable additivity: P ( ∞
i=1 Ai ) = Σi=1 P (Ai ), for disjoint sets Ai ∈ F , i = 1, 2, ...
(i.e. Ai ∩ Aj = ∅, ∀i 6= j)
The triple (Ω, F , P ) is called a probability space.
If P (A) = 1, then, we say that the event A occurs P − almost surely (or simple, almost surely).
Example: Let Ω = 1, 2, ... be the set of all natural numbers. A1 is the set of all even numbers in
Ω and A2 is the set of all odd numbers in Ω. F = {∅, Ω, A1 , A2 } is a σ-algebra. Defining
P (A1 ) =
1
2
and P (A2 ) =
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1
,
2
then, the triple (Ω, F , P ) is a probability space.
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Financial Stochastic Models
Some Properties
• P (∅) = 0, P (Ω) = 1
• P (Ac ) = 1 − P (A)
• P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
• if A and B are disjoint, then P (A ∪ B) = P (A) + P (B)
• if A ⊆ B, then P (B\A) = P (B) − P (A)
• if A ⊆ B, then P (A) ≤ P (B)
Theorem of Total Probability: Let {B1 , B2 , ..., Bn } be a partition of Ω. Then, ∀A ∈ F ,
P (A) =
n
X
P (A ∩ Bi )
i=1
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Financial Stochastic Models
Independence
Given (Ω, F , P ), the events A ∈ F and B ∈ F are independent if
P (A ∩ B) = P (A)P (B)
The collection of events {A1 , ..., An } are independent if, for every subcollection of at least two
events {Ai1 , Ai2 , ..., Aik }, k = 2, ..., n,
P (Ai1 ∩ Ai2 ∩ ... ∩ Aik ) = P (Ai1 )P (Ai2 )...P (Aik )
Example:
In a two coin-toss experiment, let H1 be the event that the first coin is ”Head” with P (H1 ) =
and H2 be the event that the second coin is ”Head” with P (H2 ) =
1
2
. The two coin toss
experiments are independent, we have
P (H1 ∩ H2 ) = P (H1 )P (H2 ) =
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1
1
1
× =
2
2
4
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1
2
Financial Stochastic Models
Conditional Probability
Let (Ω, F , P ) be a probability space. The conditional probability of event A ∈ F given event
B ∈ F with P (B) > 0 is defined as
P (A | B) =
P (A ∩ B)
P (B)
.
We have
• 0 ≤ P (A | B) ≤ 1
• P (Ω | B) = 1
• P (Ac | B) = 1 − P (A | B)
• P (Ai ∪ Aj | B) = P (Ai | B) + P (Aj | B) − P (Ai ∩ Aj | B)
• if Ai and Aj are disjoint, then P (Ai ∪ Aj | B) = P (Ai | B) + P (Aj | B)
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Financial Stochastic Models
Conditional Probability
Example: Sophie has 3 coins in her pocket, 2 Euro coins and 1 Pound coin. She takes out
randomly one coin and makes a coin toss.
Let
• A be the event that the coin is Euro
• B the event that the coin toss outcome is Head
Then,
• Probability for getting a Euro coin: P (A) =
2
.
3
• Probability of having a Euro coin Head: P (A ∩ B) =
2
.
6
• Conditional probability for getting Euro coin Head if the coin is Euro:
2
1
P (A ∩ B)
P (B|A) =
= 62 =
P (A)
2
3
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Financial Stochastic Models
Conditional Probability
Let {B1 , B2 , ..., Bn } be a partition of Ω and P (Bi ) > 0, i = 1, ..., n.
Theorem on Total Probability: ∀A ∈ F ,
P (A) =
n
X
P (A | Bi )P (Bi )
i=1
Bayes’ Rule: ∀k = 1, .., n
P (A | Bk )P (Bk )
P (Bk | A) = Pn
i=1 P (A | Bi )P (Bi )
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Financial Stochastic Models
Random Variable
Let (Ω, F , P ) be a probability space and B an interval in R. A (real) random variable is a
real-valued function X: Ω → R such that
{X(ω) ∈ B} ∈ F
where {X(ω) ∈ B} is the notation for {ω ∈ Ω; X(ω) ∈ B}.
It is customary to denote
• P {X = x} (i.e. P {ω ∈ Ω; X = x}) by P (X = x) or P [X = x]
• P {X ∈ B} (i.e. P {ω ∈ Ω; X ∈ B}) by P (X ∈ B) or P [X ∈ B]
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Financial Stochastic Models
Distribution
The function F (x) = P {X ≤ x}, x ∈ R is called the cumulative distribution function (c.d.f.) of
X. It has the following properties:
• F (x) is nondecreasing of x
• 0 ≤ F (x) ≤ 1: limx→−∞ F (x) = 0, limx→∞ F (x) = 1
• F is right continuous: limy↓x F (y) = F (x)
For continuous random variable:P {a ≤ X ≤ b} = F (b) − F (a), ∀ a ≤ b
For discrete random variable: P {X = x} = F (x) − F (x− )
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Financial Stochastic Models
Probability Density Function
Let F be the distribution function of a random variables X. The integrable function f is called
probability density function (p.d.f.) of X if
Z
a
f (x)dx, ∀a ∈ R
F (a) =
−∞
Z
b
We have P {a ≤ X ≤ b} =
f (x)dx.
a
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Financial Stochastic Models
Normal Distribution
Standard Normal Distribution X ∼ N (0, 1):
x2
1
• Probability Density Function (p.d.f.): f (x) = √ e− 2
2π Z
a
x2
1
• Cumulative Distribution Function (c.d.f.): F (a) =
√ e− 2 dx
2π
−∞
Figure: Standard Normal Distribution
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Financial Stochastic Models
Normal Distribution
General Normal Distribution with mean µ and variance σ 2 , X ∼ N (µ, σ 2 ):
• Probability Density Function (p.d.f.): f (x) = √
1
2πσ
Z
• Cumulative Distribution Function (c.d.f.): F (a) =
−
e
(x−µ)2
2σ 2
a
√
−∞
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1
2πσ
−
e
(x−µ)2
2σ 2
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Financial Stochastic Models
Binomial Distribution
If the discrete random X is said to follow Binomial Distribution of (n, p), 0 < p < 1, if its
distribution is
P {X = k} =
n
pk (1 − p)n−k , k = 0, 1, ..., n
k
.
We note X ∼ B(n, p).
Example:
Consider a experiment with n trials. The probability of having A is p for a trial. X denotes the
number of of times of having A. Then,
P {X = k} =
n
pk (1 − p)n−k , k = 0, 1, ..., n
k
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Financial Stochastic Models
Expectation
The expectation of a random variable is the probability weighted average.
For a discrete random variable X ∈ {x1 , x2 , ..., xn } on the probability space (Ω, F , P ), it is
defined as
E[X] =
n
X
xi P {X = xi } or E[X] =
i=1
X
X(ω)P (ω)
ω∈Ω
For a continuous random variable in R with density f (x), the expectation is defined as
Z
+∞
E[X] =
xf (x)dx
−∞
The general notation for the expectation of a random variable X on (Ω, F , P ) is denoted by the
Lebesgue integration:
Z
E[X] =
X(ω)dP (ω)
Ω
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Financial Stochastic Models
Lebesgue Integration Illustration
Z
Lebesgue Integration
ydµ is the limiting value of
X
yi µ{yi } with partition on Y -axis, where
i
µ{yi } is the measure of yi .
For example:
Z
ydx ≈ y1 (x3 − x2 ) + (x5 − x4 ) + y2 (x2 − x1 ) + (x4 − x3 )
R
Figure: Lower Lebesgue Sum
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Financial Stochastic Models
Variance
The variance of the random variable X is defined as
V ar(X) = E[(X − E[X])2 ]
σ=
p
V ar(X) is called the standard deviation of X.
Properties of Variance:
• V ar(c) = 0, if c is a constant
• V ar(cX) = c2 V ar(X), if c is a constant
• if X, Y are two independent random variables, then V ar(X + Y ) = V ar(X) + V ar(Y )
The i-th moment of the random variable X is defined as E[X i ], i = 1, 2, ..., if it exists.
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Financial Stochastic Models
Bivariate Distribution
Let (X1 , X2 ) ∈ R2 be a vector of two random variables.
The joint distribution function of (X1 , X2 ) is defined as
F (a, b) = P {X1 ≤ a, X2 ≤ b}
If (X1 , X2 ) is a continuous random vector, then, the joint probability density function is
f (x1 , x2 ) the function such that
Z
b
Z
a
F (a, b) =
f (x1 , x2 )dx1 dx2
−∞
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−∞
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Financial Stochastic Models
Marginal Distribution
For (X1 , X2 ) with density f (x1 , x2 ), the marginal cumulative distribution function of X1 is
defined as
a
Z
FX1 (a) =
∞
Z
where fX1 (x1 ) =
fX1 (x1 )dx1
∞
f (x1 , x2 )dx2 is called marginal density.
−∞
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Financial Stochastic Models
Bivariate Normal Distribution
Bivariate Normal Distribution: (X1 , X2 ) ∼ N (µ1 , µ2 ; σ1 , σ2 ; ρ)
Probability Density Function (p.d.f.) f (x, y) equals to
2πσ1 σ2
1
p
exp
1 − ρ2
−
(x1 − µ1 )2
1
(x1 − µ1 )(x2 − µ2 )
(x2 − µ2 )2 − 2ρ
+
2(1 − ρ2 )
σ12
σ1 σ2
σ22
Cumulative Distribution Function (c.d.f.):
Z
b
Z
a
f (x1 , x2 )dx1 dx2
F (a, b) =
−∞
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Financial Stochastic Models
Multi-variate Distribution
Let (X1 , X2 , ..., Xn ) be a vector of random variables.
The joint distribution function of (X1 , X2 , ..., Xn ) is defined as
F (a1 , a2 , ..., an ) = P {X ≤ a1 , X2 ≤ a2 , ..., Xn ≤ an }
If (X1 , X2 , ..., Xn ) is a continuous random vector, then, the joint probability density function is
the function f (x1 , x2 , ..., xn ) such that
Z
an
Z
an−1
−∞
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Z
a1
f (x1 , x2 , ..., xn )dx1 dx2 ...dxn
...
F (a1 , a2 , ..., an ) =
−∞
−∞
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Financial Stochastic Models
Multi-variate Normal Distribution
Probability Density Function (p.d.f.) of multivariate Normal distribution:
1
1
exp − (x − µ)> Σ−1 (x − µ)
f (x1 , x2 , ..., xn ) = p
2
(2π)n |Σ|
where x = (x1 , x2 , ..., xn )> is a real n-dimensional column vector and |Σ| is the determinant of
Σ, the n × n covariance matrix which is symmetric and positive definite.
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Financial Stochastic Models
Covariance, Correlation
The covariance of two random variables X and Y is defined as
Cov(X, Y ) = E (X − E[X])(Y − E[Y ])
The correlation between X and Y is defined as
ρ= p
Cov(X, Y )
V ar(X)V ar(Y )
or ρ =
Cov(X, Y )
σX σY
V ar(X + Y ) = V ar(X) + V ar(Y ) + 2Cov(X, Y )
Properties of covariance:
• Cov(X, Y ) = E(XY ) − E(X)E(Y )
• Cov(X, Y ) = Cov(Y, X)
• Cov(aX, bY ) = abCov(X, Y )
• Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z)
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Financial Stochastic Models
Characteristic Function
The characteristic function of a random variable X is defined as the below, if it exists:
ΦX (t) = E[eitX ], t ∈ R
where i2 = −1
For a vector of random variables (X1 , X2 , ..., Xn ), the characteristic function is defined as the
below, if it exists:
Φ(X1 ,X2 ,...,Xn ) (t1 , t2 , ..., tn ) = E[ei
Pn
k=1 tk Xk
]
where tk ∈ R, k = 1, ..., n
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Financial Stochastic Models
Moment-Generating Function
The moment-generating function of a random variable X is defined as the below, if it exists:
MX (t) = E[etX ], t ∈ R
For a vector of random variables (X1 , X2 , ..., Xn ), the moment-generating function is defined as
the below, if it exists:
Pn
M(X1 ,X2 ,...,Xn ) (t1 , t2 , ..., tn ) = E[e
k=1 tk Xk
]
where tk ∈ R, k = 1, ..., n
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Financial Stochastic Models
Moment-Generating Function
The moment-generating function can be used to calculate the moments:
(1)
(2)
(k)
E[X] = MX (0), E[X 2 ] = MX (0), ..., E[X k ] = MX (0)
where
(k)
means k times derivative w.r.t. (with respect to) t.
Example of the Normal Distribution N (0, σ 2 ):
1 2 σ2
MX (t) = E[etX ] = e 2 t
Applying the above method, we obtain
(1)
1 2 σ2
MX = σ 2 te 2 t
1 2 σ2
(2)
MX = (σ 2 + σ 4 t2 )e 2 t
1 2 σ2
(3)
MX = (3σ 4 t + σ 6 t3 )e 2 t
1 2 σ2
(4)
MX = (3σ 4 + 6σ 6 t2 + σ 8 t4 )e 2 t
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(1)
−−−→
E[X] = MX (0) = 0
−−−→
E[X 2 ] = MX (0) = σ 2
−−−→
E[X 3 ] = MX (0) = 0
−−−→
E[X 4 ] = MX (0) = 3σ 4
(2)
(3)
(4)
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Financial Stochastic Models
Limit Theorems
Law of Large Numbers
Let X1 , X2 , ... be an infinite sequence of i.i.d. (independent and identically distributed) Lebesgue
integrable random variables with expected value E(X1 ) = E(X2 ) = ... = µ. Then,
1
(X1 + X2 + ... + Xn ) −−−→ µ
n→+∞
n
Central Limit Theorem
Let X1 , X2 , ... be an infinite sequence of i.i.d. (independent and identically distributed) random
variables with finite mean µ and variance σ 2 . Denote
Pn
Sn =
Xi − nµ
√
nσ
i=1
Then, Sn converges to a standard Normal random variable:
Sn ∼ N (0, 1), if n → +∞
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Financial Stochastic Models
Confidence Interval
Denote X̄ =
P
n
1X
Xi . The confidence interval of µ is obtained from normal distribution:
n i=1
X̄ − µ
< zα/2
√
σ/ n
=1−α
or,
σ
σ
P X̄ − √ zα/2 < µ < X̄ + √ zα/2 = 1 − α.
n
n
When n → ∞, σ ≈ s, where s2 =
n
1 X
(Xi − X̄)2 .
n − 1 i=1
Example
For 1 − α = 95%, we get zα/2 = 1.96. So the 95% confidence interval for µ (i.e. E[X]) is
s
s
X̄ − 1.96 √ , X̄ + 1.96 √
.
n
n
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Financial Stochastic Models
Independent Random Variables
The random variables X1 , X2 , ..., Xn are said to be independent if
P {X1 ∈ B1 , X2 ∈ B2 , ..., Xn ∈ Bn } = P {X1 ∈ B1 }P {X2 ∈ B2 }...P {Xn ∈ Bn }
For independent random variables X1 , X2 , ..., Xn , the following properties hold
• Joint cumulative distribution: FX1 ,X2 ,...,Xn (a1 , a2 , ..., an ) = FX1 (a1 )FX2 (a2 )...FXn (an ),
where FXi (ai ) represents the cumulative distribution function of Xi
• Joint cumulative distribution: if X1 , X2 , ..., Xn are continuous random variables with
densities fX1 , fX2 , ..., fXn and joint density fX1 ,X2 ,...,Xn (x1 , x2 , ..., xn ), then,
fX1 ,X2 ,...,Xn (x1 , x2 , ..., xn ) = fX1 (x1 )fX2 (x2 )...fXn (xn )
• Expectation: E[X1 X2 ...Xn ] = E[X1 ]E[X2 ]...E[Xn ] are continuous random variables
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Financial Stochastic Models
Conditional Probability Distribution
The conditional probability distribution of continuous random variable X given y, F (x|y) is
defined by
F (a|y) = P {X ≤ a|Y = y} =
lim
|Y −y|→0
P {X ≤ a|Y ∈ ∆y}
Z
a
f (x|y)dx, then, f (x|y) is called the conditional
If f (x|y) is positive function and F (a|y) =
−∞
probability density of X given Y = y.
We have
f (x, y) = f (x|y)f (y)
where f (y) is the marginal distribution of Y .
The conditional expectation given Y = y is defined as
Z
E[X|Y = y] =
xf (x|y)dx
Ω
For discrete variable X, Y , the conditional expectation of X given Y = y is defines as
E[X|Y = y] =
X
xi P {X = xi |Y = y}
i
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Financial Stochastic Models
Elements of Stochastic Calculus
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Financial Stochastic Models
Stochastic Process
A stochastic process is a collection of random variables indexed by a totally ordered set T (e.g.
”time”).
{Xt : t ∈ T }
The σ-algebra generated by a random variable Xt , denoted σ(Xt ), is the σ-algebra by the
collection of {Xt ∈ B} (i.e. {ω : Xt (ω) ∈ B}), where B is any interval in R.
Let G be a σ-algebra on Ω. X is said to be G-measurable if σ(X) ⊆ G.
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Financial Stochastic Models
Filtration
Given a probability space (Ω, F , P ), a filtration is a increasing collection of σ-algebras on Ω,
Ft , t ∈ T , indexed by the totally ordered set T i.e. for s, t ∈ T, s < t,
Fs ⊆ F t
A stochastic process X on T is said to be adapted to the filtration if, for every t ∈ T , Xt is
Ft -measurable.
Given a stochastic process X = {Xt : t ∈ T }, the natural filtration for this process is the
filtration where Ft is generated by all values of Xs up to s = t.
A stochastic process is always adapted to its natural filtration.
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Financial Stochastic Models
Asset Price Example
Assume a simplistic multi period asset price model where the price has only two possible movements for each
period driven by a single economic factor with 2 status:
• ”y” drives the price up by a certain amount with probability p, or
• ”n” drives the price down by a certain amount with probability q (q = 1 − p)
Consider a 2-period stock price model which can be represented by the following tree, where
p = 0.5, q = 1 − p = 0.5:
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Financial Stochastic Models
Filtration - Example
Sample space of S2 : Ω = {yy, yn, ny, nn}
The events on the tree:
• Ayy = {yy}, Ayn = {yn}, Any = {ny}, Ann = {nn} (these are called ”atom events”)
• Ay = {yy, yn}, An = {ny, nn}
We have Ay = Acn , An = Acy , Ay = Ayy ∪ Ayn , An = Any ∪ Ann , Ω = Ay ∪ An .
Probabilities for the events:
• P (Ayy ) = 0.25, P (Ayn ) = 0.25, P (Any ) = 0.25, P (Ann ) = 0.25
• P (Ay ) = 0.5, P (An ) = 0.5
• P (Ω) = 1
Filtration for each period:
• F0 = {∅, Ω}
• F1 = {∅, Ω, Ay , An }
• F2 = {∅, Ω, Ay , An , Ayy , Ayn , Any , Ann , Acyy , Acyn , Acny , Acnn , Ayy ∪ Any , Ayy ∪ Ann , Ayn ∪
Any , Ayn ∪ Ann }
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Financial Stochastic Models
Conditional Expectation Given an Event
Let (Ω, F , P ) be a probability space and B = {ω1 , ω2 , ...} ∈ F an event.
The conditional expectation of random variable X given event B is defined as
E[X|B] =
1
P (B)
Z
X(ω)dP (ω)
B
If X is a discrete random variable, the above expression can be also written as:
E[X|B] =
1 X
xi P ({X = xi } ∩ B)
P (B) i
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Financial Stochastic Models
Conditional Expectation Given a σ-algebra
The conditional expectation given σ-algebra G, G ⊂ F , is the G-measurable random variable Z,
denoted E[X|G], which satisfies
Z
Z
X(ω)dP (ω), ∀B ∈ G
Z(ω)dP (ω) =
B
B
Note that E[X|G] is not a single value in general. It is a variable representing the ”partial
averaging” for each B ∈ G. It is a generalization of the single valued conditional expectation with
single event.
For illustrating the concept, we use the following less general definition of conditional expectation.
Under probability space (Ω, F , P ), let {B1 , B2 , ...}, ∀i, Bi ∈ F , be a countable partition of Ω and
B = σ(Bi , i = 1, 2, ...) the σ-algebra generated by all Bi . The conditional expectation of random
variable X conditional by B is defined as
E[X|B] =
X
E[X|Bi ]1Bi
i
Yves Guo
The Chinese University of Hong Kong (SZ)
40 / 154
Financial Stochastic Models
Conditional Expectation - Examples
In the 2-period asset price example, we have the following conditional expectation representations on the 3
σ-algebras F0 , F1 , F2 .
E[S2 |F2 ]
=
81Ayy + 41Ayn + 51Any + 21Ann
E[S2 |F1 ]
=
8P (Ayy ∩ Ay ) + 4P (Ayn ∩ Ay ) + 5P (Any ∩ Ay ) + 2P (Ann ∩ Ay )
1Ay +
P (Ay )
8P (Ayy ∩ An ) + 4P (Ayn ∩ An ) + 5P (Any ∩ An ) + 2P (Ann ∩ An )
1An
P (An )
=
=
E[S2 |F0 ]
5P (Any ) + 2P (Ann )
8P (Ayy ) + 4P (Ayn )
1Ay +
1An
P (Ay )
P (An )
61Ay + 3.51An
=
8P (Ayy ∩ Ω) + 4P (Ayn ∩ Ω) + 5P (Any ∩ Ω) + 2P (Ann ∩ Ω)
1Ω
P (Ω)
8P (Ayy ) + 4P (Ayn ) + 5P (Any ) + 2P (Ann ) 1Ω
=
E[S2 ]1Ω = E[S2 ] = 4.75
=
Yves Guo
The Chinese University of Hong Kong (SZ)
41 / 154
Financial Stochastic Models
Conditional Expectation - Properties
Let G, H be two σ-algebras and G ⊂ H. Then, we have the following properties:
(1) Independence:
If X is independent of G, then E[X|G] = E[X]
(2) Law of Total Expectation:
E[E[X|G]] = E[X]
(3) Taking out Known Factors:
If X is G-measurable, then E[XY |G] = XE[Y |G]
(4) Tower Property (also known as Iterated Conditioning):
a. E[E[X|H]|G] = E[X|G]
b. E[E[X|G]|H] = E[X|G]
Yves Guo
The Chinese University of Hong Kong (SZ)
42 / 154
Financial Stochastic Models
Conditional Expectation - Illustration Examples
Illustration with Asset Price Example for (1) ”Independence”:
Let G be a σ-algebra which is independent of S2 . Assuming that {B1 , B2 } is a partition of G (hence
1 B1 + 1 B2
= 1), then
E[S2 |G] = E[S2 |B1 ]1B1 + E[S2 |B2 ]1B2
The first term of E[S2 |G] is
E[S2 |B1 ]
=
8P (Ayy ∩ B1 ) + 4P (Ayn ∩ B1 ) + 5P (Any ∩ B1 ) + 2P (Ann ∩ B1 )
P (B1 )
=
8P (Ayy )P (B1 ) + 4P (Ayn )P (B1 ) + 5P (Any )P (B1 ) + 2P (Ann )P (B1 )
P (B1 )
=
8P (Ayy ) + 4P (Ayn ) + 5P (Any ) + 2P (Ann )
=
E[S2 ]
We obtain similar result for the second term of E[S2 |G]. Hence,
E[S2 |G]
=
E[S2 ]1B1 + E[S2 ]1B2
=
E[S2 ](1B1 + 1B2 )
=
E[S2 ]
Yves Guo
The Chinese University of Hong Kong (SZ)
43 / 154
Financial Stochastic Models
Conditional Expectation - Illustration Examples
Illustration with Asset Price Example for (2) ”Law of Total Expectation”:
E[S2 |F1 ] =
E[E[S2 |F1 ]]
8P (Ayy ) + 4P (Ayn )
5P (Any ) + 2P (Ann )
1Ay +
1An
P (Ay )
P (An )
=
5P (Any ) + 2P (Ann )
8P (Ayy ) + 4P (Ayn )
E[1Ay ] +
E[1An ]
P (Ay )
P (An )
=
8P (Ayy ) + 4P (Ayn )
5P (Any ) + 2P (Ann )
P (Ay ) +
P (An )
P (Ay )
P (An )
=
8P (Ayy ) + 4P (Ayn ) + 5P (Any ) + 2P (Ann )
=
E[S2 ]
Yves Guo
The Chinese University of Hong Kong (SZ)
44 / 154
Financial Stochastic Models
Conditional Expectation - Illustration Examples
Illustration with Asset Price Example for (3) ”Taking out Known Factors”:
Let’s take the example of E[S1 S2 |F1 ] where S1 is F1 -measurable and S1 = 71Ay + 31An .
E[S1 S2 |F1 ]
=
3 × 5P (Any ) + 3 × 2P (Ann )
7 × 8P (Ayy ) + 7 × 4P (Ayn )
1Ay +
1An
P (Ay )
P (An )
=
5P (Any ) + 2P (Ann )
8P (Ayy ) + 4P (Ayn )
(71Ay + 31An )
1Ay +
1An
P (Ay )
P (An )
=
(71Ay + 31An )E[S2 |F1 ]
=
S1 E[S2 |F1 ]
Yves Guo
The Chinese University of Hong Kong (SZ)
45 / 154
Financial Stochastic Models
Conditional Expectation - Illustration Examples
Illustration with Asset Price Example for (4.a) ”Tower Property”:
E[E[S2 |F2 ]|F1 ]
=
E[81Ayy + 41Ayn + 51Any + 21Ann |F1 ]
=
8E[1Ayy |F1 ] + 4E[1Ayn |F1 ] + 5E[1Any |F1 ] + 2E[1Ann |F1 ]
The first term is
8E[1Ayy |F1 ] = 8
P (Ayy ∩ An )
P (Ayy )
P (Ayy ∩ Ay )
1Ay + 8
1An = 8
1Ay
P (Ay )
P (An )
P (Ay )
We obtain similar results for the other terms. Hence,
E[E[S2 |F2 ]|F1 ]
Yves Guo
P (Ayn )
P (Any )
P (Ann )
P (Ayy )
1Ay + 4
1Ay + 5
1An + 2
1A n
P (Ay )
P (Ay )
P (An )
P (An )
=
8
=
8P (Ayy ) + 4P (Ayn )
5P (Any ) + 2P (Ann )
1Ay +
1An
P (Ay )
P (An )
=
61Ay + 3.51An
=
E[S2 |F1 ]
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Financial Stochastic Models
Conditional Expectation - Illustration Examples
Illustration with Asset Price Example for (4.b) ”Tower Property”:
E[E[S2 |F1 ]|F2 ] = E[61Ay + 3.51An |F2 ] = 6E[1Ay |F2 ] + 3.5E[1An |F2 ]
The first term is
6E[1Ay |F2 ]
=
P (Ay ∩ Ayn )
P (Ay ∩ Any )
P (Ay ∩ Ann )
P (Ay ∩ Ayy )
1Ayy +
1Ayn +
1Any +
1Ann
6
P (Ayy )
P (Ayn )
P (Any )
P (Ann )
=
6
P (Ayn )
P (Ayy )
1Ayy +
1Ayn
P (Ayy )
P (Ayn )
= 61Ay
Similarly, we obtain
3.5E[1An |F2 ] = 3.51An
Hence,
E[E[S2 |F1 ]|F2 ] = 61Ay + 3.51An = E[S2 |F1 ]
Yves Guo
The Chinese University of Hong Kong (SZ)
47 / 154
Financial Stochastic Models
Martingale
Let (Ω, F , P ) be a probability space with the filtration Ft . An Ft -adapted stochastic process Mt
is said to be a
• martingale, if E[Mt |Fs ] = Ms
• supermartingale, if E[Mt |Fs ] ≤ Ms
• submartingale, if E[Mt |Fs ] ≥ Ms
∀0 ≤ s ≤ t
Yves Guo
The Chinese University of Hong Kong (SZ)
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Financial Stochastic Models
Change of Probability
Let (Ω, F , P) be a probability space, Z > 0, P-a.s. and E[Z] = 1. Defining a new probability as
Z
e
P(A)
=
ZdP,
∀A ∈ F , (noted as Z =
A
de
P
)
dP
then, we have
e
E[X]
= E[ZX]
and
e X = E[X],
E
Z
e with respect to P.
where Z is called the Radon-Nikodym derivative of P
Radon-Nikodym derivative process is defined as
Zt = E[Z|Ft ] (noted as Z =
de
P
| )
dP Ft
It can be shown that Zt is a martingale.
Note: ”Radon-Nikodym Theorem” proves the existence of Z.
Yves Guo
The Chinese University of Hong Kong (SZ)
49 / 154
Financial Stochastic Models
Bayes’ Formula
Propositon 2.1 (Bayes’ Formula)
Let 0 ≤ s ≤ t, Y be an Ft -measurable random variable, Zt =
derivative process, then
de
P
dP F
t
be a Radon-Nikodym
e |Fs ] = 1 E[Y Zt |Fs ].
E[Y
Zs
Proof
We check the “partial averaging” property of conditional expectation. For any A ∈ Fs (hence, 1A is
Fs -measurable),
Z
1
e 1A 1 E[Y Zt |Fs ] = E Z 1A 1 E[Y Zt |Fs ]
E[Y Zt |Fs ]de
P = E
Zs
Zs
A Zs
E[Z|Fs ]
1
= E E[Z 1A
E[Y Zt |Fs ]]|Fs = E
E[1A Y Zt |Fs ]
Zs
Zs
= E E[1A Y Zt |Fs ] = E[1A Y Zt ] = E[1A Y E[Z|Ft ]]
=
=
=
E[E[1A Y Z|Ft ]] = E[1A Y Z]
e 1A Y ]
E[
Z
Y de
P.
A
Yves Guo
The Chinese University of Hong Kong (SZ)
50 / 154
Financial Stochastic Models
Quadratic Variation
The quadratic variation of a function of t, Xt , is defined as
hX, Xit =
lim
X
kP k→0
(Xsi+1 − Xsi )2 , if it exists
i
where P = {s0 , s1 , ..., sn } represents a partition of the interval [0, t] and kP k is the maximum
step size.
It is also denoted as hXit .
Note:
If Xt has continuous derivative Xt0 and 0t |Xs0 |2 ds is finite, then hX, Xit = 0 which is illustrated in the below:
R
hX, Xit
=
=
≤
lim
X
kP k→0
lim
X
kP k→0
lim
i
0
2
∗
(Xs∗ (si+1 − si )) , si ∈ [si , si+1 ]
i
kP k
kP k→0
Yves Guo
2
(Xsi+1 − Xsi )
i
X
i
0
2
|Xs∗ | (si+1 − si ) =
i
(by Mean Value Theorem)
Z
lim
t
kP k
kP k→0
0
0 2
|Xs | ds = 0
The Chinese University of Hong Kong (SZ)
(as 0t |Xs0 |2 ds is finite)
R
51 / 154
Financial Stochastic Models
Cross Variation
The cross variation of two processes, Xt and Yt , is defined as the
hX, Y it =
lim
kP k→0
X
(Xsi+1 − Xsi )(Ysi+1 − Ysi ), if it exists
i
where P represents a partition of interval [0, t] and kP k is the maximum step size.
Yves Guo
The Chinese University of Hong Kong (SZ)
52 / 154
Financial Stochastic Models
Brownian Motion
A continuous process Wt starting from 0 (W0 = 0) is said to be a (Standard) Brownian Motion
if
(1) Wt has stationary and independent increments
(2) every increment Wt − Ws , 0 ≤ s < t, follows the Normal distribution N (0, t − s)
We have
• E[Wt − Ws ] = 0
• V ar(Wt − Ws ) = t − s
√
• Wt = g t where g ∼ N (0, 1) (because Wt = Wt − W0 )
Figure: Brownian Motion
Yves Guo
The Chinese University of Hong Kong (SZ)
53 / 154
Financial Stochastic Models
Properties of Brownian Motion
Independence of Brownian increments implies Wt − Ws (s < t) is independent of
Fs = σ(Wu , u ≤ s).
Stationarity of Brownian increments implies Wt − Ws (s < t) follows the same distribution as
Wt−s (=Wt−s − W0 ).
An Ft -adapted Brownian Motion is called Ft -Brownian Motion. It has the following properties:
1) Wt is Ft -martingale
2) Wt2 − t is Ft -martingale
1
3) e− 2 σ
2
t+σWt
is Ft -martingale
Yves Guo
The Chinese University of Hong Kong (SZ)
54 / 154
Financial Stochastic Models
Properties of Brownian Motion
Proof:
1)
E[Wt − Ws |Fs ] = E[Wt − Ws ]. Hence E[Wt |Fs ] = E[Ws |Fs ] = Ws (as Ws is Fs -measurable)
2)
E[Wt2 − Ws2 |Fs ] = E[(Wt − Ws )2 + 2Ws (Wt − Ws )|Fs ] = E[(Wt − Ws )2 |Fs ] = t − s.
Hence E[Wt2 − t|Fs ] = E[Ws2 |Fs ] − s = Ws2 − s (because Ws2 is Fs -measurable)
3)
1
E[e− 2 σ
2 t+σW
t |F
s]
1
= e− 2 σ
2 t+σW
s E[eσ(Wt −Ws ) |F ]
s
The second term is
Z +∞
√
√
g2
gσ t−s 1
−
= E[eσ(Wt −Ws ) ] = E[egσ t−s ] =
e
√ e 2 dg
2π
−∞
√
Z +∞
(g−σ t−s)2
1 2
1
−
2
= e 2 σ (t−s)
dg
√ e
2π
Z−∞
2
+∞
1 2
√
1
−x
= e 2 σ (t−s)
√ e 2 dx (x = g − σ t − s)
2π
−∞
1
= e2σ
2 (t−s)
1
Hence, we obtain E[e− 2 σ
Yves Guo
2 t+σW
t |F ]
s
1
= e− 2 σ
2 t+σW
1 2
s e 2 σ (t−s)
1
= e− 2 σ
2 s+σW
s
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Financial Stochastic Models
Quadratic Variation of Brownian Motion
Theorem Let Wt be a Brownian motion, then hW, W it = t, almost surely.
In differential
dWt dWt
dWt dt
dtdt
form, we have the following calculation rules
= dt,
= 0,
(because E[∆Wt ∆t] = ∆tE[∆Wt ] = 0)
=0
Yves Guo
The Chinese University of Hong Kong (SZ)
56 / 154
Financial Stochastic Models
Quadratic Variation of Brownian Motion
Proof:
Let P = {s0 , s1 , ..., sn } be a partition of [0, t] with kP k → 0. We want to prove that E[hW, W it ] = t and
V ar(hW, W it ) = 0.
E hW, W it = E
lim
X
2
(Wsi+1 − Wsi ) =
kP k→0
i
X
lim
kP k→0
V ar(∆Wsi ) =
i
lim
kP k→0
X
∆si = t
i
where ∆Wsi = Wsi+1 − Wsi and ∆si = si+1 − si .
V ar
X
2
(Wsi+1 − Wsi )
=
V ar
i
X
2 i
∆Ws
i
=
X
=
X
=
X
=
X
2 i
V ar ∆Ws
i
X
2
2
2 2
2
E ∆Ws − E[∆Ws ] =
E ∆Ws − ∆si
i
i
i
i
i
4 2
2
E ∆Ws − 2E[∆Ws ]∆si + ∆si
i
i
i
2
2
2
3∆si − 2∆si + ∆si =
i
Hence, V ar hW, W it = V ar
lim kP k
kP k→0
X
2∆si =
i
Yves Guo
lim
kP k→0
X
2
(Wsi+1 − Wsi ) =
i
X
2
2∆si
i
lim
kP k→0
X
2
2∆si ≤
i
lim 2kP kt = 0
kP k→0
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Financial Stochastic Models
Stochastic Integral
Let P = {t0 , t1 , ..., tn } be a partition of the interval [0, T ], ft be an Ft -adapted process and
R
E 0T ft2 dt < +∞, the (Ito) stochastic integral is defined as
T
Z
ft dWt =
0
lim
kP k→0
n
X
fti−1 (Wti − Wti−1 )
i=1
Note that the value of ft is known at the beginning of the sub interval. This is a crucial property
for the application of stochastic integration in finance. The integral can be applied to represent
the cumulative trading P/L where ft is determined by the trading strategy before each time
period.
t
Z
Properties of stochastic integral It =
fu dWu :
0
1) Martingale: It is Ft -martingale
Z t
2) Ito-Isometry: EIt2 = E[
fu2 du]
0
t
Z
3) Quadratic Variation: hIt , It i =
fu2 du
0
Yves Guo
The Chinese University of Hong Kong (SZ)
58 / 154
Financial Stochastic Models
Stochastic Integral
Remark:
1) may be understood from fact that E
T
Z
ft dWt |Fs = E
s
Z
ft dWt |Fs + E
Z
T
ft dWt |Fs . The second term
s
0
0
is actually zero because the expected value of any future Brownian increment is zero.
For 2), we notice that
E
n
X
fti−1 ∆Wti
2 n X
n
X
=
E fti−1 ftj−1 (Wti − Wti−1 )(Wtj − Wtj−1 )
i=1 j=1
i=1
For each term of the form i < j:
E fti−1 ftj−1 ∆Wti−1 ∆Wtj−1
= E E[fti−1 ftj−1 ∆Wti−1 ∆Wtj−1 |Fj−1 ]
E fti−1 ftj−1 ∆Wti−1 E[∆Wtj−1 |Fj−1 ] = 0 ( as E[∆Wtj−1 |Fj−1 ] = 0)
=
For terms with i = j:
2
2
2
E ft
∆Wt
i−1
i−1
2
2
∆Wt
|Fj−1 ]
i−1
i−1
= E E[ft
2
2
E ft
E[∆Wt
|Fj−1 ] = E ft
∆ti−1
i−1
i−1
i−1
=
Hence, E
Z
T
ft dWt
2 0
Yves Guo
=E
lim
kP k→0
n
X
fti−1 ∆Wti
i=1
2 =
lim
n
X
kP k→0
i=1
2
∆ti−1
i−1
E ft
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=E
Z
T
0
2
ft dt
59 / 154
Financial Stochastic Models
Stochastic Calculus - Itô-Doeblin Formula
Propositon 2.2 (Itô-Doeblin Formula)
∂f
∂f
∂2f
(t, x),
(t, x) and
(t, x) are defined and continuous,
∂t
∂x
∂x2
Let f (t, x) be a function for which
Xt be an Itô process:
t
Z
Xt = X0 +
t
Z
θs ds +
0
φs dWs
0
Then, for T > 0,
f (T, XT ) =
f (T, XT ) =
Z
∂f
dt +
∂t
Z
T
∂f
dt +
∂t
Z
0
Z
f (0, X0 ) +
0
Yves Guo
T
f (0, X0 ) +
T
∂f
1
dXt +
∂x
2
Z
T
∂f
θt dt +
∂x
T
0
0
Z
0
0
T
∂2f
dhX, Xit
∂x2
∂f
1
φt dWt +
∂x
2
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T
Z
0
∂2f 2
φ dt
∂x2 t
60 / 154
Financial Stochastic Models
Stochastic Calculus - Itô-Doeblin Formula
The differential form of Itô process is
dXt = θt dt + φt dWt
Itô-Doeblin Formula is written as follows in differential form:
df (t, Xt ) =
∂f
1 ∂2f
∂f
dt +
dXt +
dhX, Xit
∂t
∂x
2 ∂x2
df (t, X ) =
t
∂f
∂f
∂f
1 ∂2f 2
dt +
θt dt +
φt dWt +
φ dt
∂t
∂x
∂x
2 ∂x2 t
Yves Guo
The Chinese University of Hong Kong (SZ)
61 / 154
Financial Stochastic Models
Stochastic Calculus - Itô-Doeblin Formula
Itô-Doeblin Formula can be understood from the Taylor series of f (t, Xt ) :
df (t, Xt ) =
∂f
1 ∂2f
∂f
dt +
dXt +
dXt dXt + [other terms]
∂t
∂x
2 ∂x2
Applying the calculation rules of Brownian Motion:
dWt dWt = dt, dWt dt = 0, dtdt = 0,
the value of [other terms] is zero.
From
dXt = θt dt + φt dWt
we obtain
2
2
2
dXt dXt = φt dWt dWt + φt θt dWt dt + θt φt dtdWt + θt dtdt = φt dt
Hence, replacing dXt and dXt dXt in the Taylor series, we obtain
df (t, Xt ) =
∂f
∂f
1 ∂2f 2
∂f
dt +
θt dt +
φt dWt +
φ dt
∂t
∂x
∂x
2 ∂x2 t
Yves Guo
The Chinese University of Hong Kong (SZ)
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Financial Stochastic Models
Multivariable Stochastic Calculus
Correlated Brownian Motions
Two Brownian motions B, W are said to be Correlated if
hB, W it = ρt,
−1 ≤ ρ ≤ 1.
We have the following:
p
ct ,
Bt = ρWt + 1 − ρ2 W
ct are independent Brownian motions.
where Wt and W
Itô’s Product Rule
Let Xt and Yt be Itô processes, then
d(Xt Yt ) = Xt dYt + Yt dXt + dhX, Y it
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Financial Stochastic Models
Girsanov Theorem
Propositon 2.3 (Girsanov Theorem)
Let Wt be a Brownian motion on (Ω, F , P) and θt be an adapted process. Denote
Zt = exp{−
Rt
0
θs dWs −
1
2
Rt
0
θs2 ds} which is an exponential martingale.
Taking Zt as the Radon-Nikodym derivative process
Z
ft = Wt +
W
de
P
,
dP F
t
and define
t
θs ds,
0
then
e
ft is a Brownian motion under the probability measure P,
W
assuming that E
RT
0
θs2 Zs2 ds < ∞.
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The Chinese University of Hong Kong (SZ)
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Financial Stochastic Models
Predictable Martingale Representation
Propositon 2.4 (Predictable Martingale Representation Theorem)
Let Wt be a Brownian with Ft as its natural filtration and Mt be an Ft -measurable martingale,
then, there is a unique predictable process φs , such that
Z t
Mt = M0 +
φs dWs .
0
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The Chinese University of Hong Kong (SZ)
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Financial Stochastic Models
Stochastic Differential Equations
Stochastic Differential Equation (SDE)
A stochastic differential equation takes the form:
dXt = β(t, Xt )dt + γ(t, Xt )dWt ,
where
• β(t, x): the drift;
• γ(t, x): the diffusion;
• Xt = x: the initial condition at time t ≥ 0, x ∈ R.
Example (Geometric Brownian Motion)
For dSt = αSt dt + σSt dWt , it can be shown that the unique solution is
1
ST = St e(α− 2 σ
2
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)(T −t)+σ(WT −Wt )
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Financial Stochastic Models
Markov Property
The Markov Property
An Ft -measurable process Xt is said to be a Markov Process if, for every Borel-measurable
function h, there is another Borel-measurable function f , such that,
E[h(Xt )|Fs ] = E[h(Xt )|Xs ] = f (s, Xs ),
∀t > s.
Propositon 2.5 (Markov Property of SDE Solutions)
Let Xu , u ≥ 0 be a solution to the SDE with initial condition given at time 0. Then
E[h(XT )|Ft ] = Et,Xt h(XT ).
Remark
We can imagine that the value of XT should depend on
• Initial value Xt which is Ft -measurable;
• The increments of Xu , t ≤ u ≤ T , which, in turn, are determined by ∆Wu ;
• As ∆Wu is independent of Ft , we can drop the conditioning of Ft in the expectation.
Yves Guo
The Chinese University of Hong Kong (SZ)
67 / 154
Financial Stochastic Models
Risk-Neutral Modelling
Yves Guo
The Chinese University of Hong Kong (SZ)
68 / 154
Financial Stochastic Models
Interest Rate Conventions
In money market, the calculation of investment return is usually based on one of the below
conventions, according to the financial instrument:
• yield basis: for N0 at time 0, the investor receives Nt = N0 (1 + rm t) at maturity (e.g. bank deposit)
• discount basis: for N0 = Nt (1 − rd t) at time 0, the investor receives Nt at maturity (e.g. T-Bill)
The continuous rate r is not directly observable in the market. It assumes dNt = Nt rdt, leading
to Nt = N0 ert . r can be derived through the below relationships:
• yield v.s. continuous rate: 1 + rm Tm = erT
• discount rate v.s. continuous rate:
1
rT
=e
1 − rd Td
where Tm is the time fraction used by the instrument (e.g. 30E/360) and T usually follows
ACT/365 or ACT/ACT.
Note: in a 30/360 type of convention, the time fraction between two dates follows the below
principal Tm =
360×(Y2 −Y1 )+30×(M2 −M1 )+(D2 −D1 )
360
Yves Guo
The Chinese University of Hong Kong (SZ)
69 / 154
Financial Stochastic Models
Numeraire, Discount Factor
Money Market Numeraire
Let rt be an adapted interest rate process. The Money Market Numeraire Mt is define as
Mt = e
Rt
0
ru du
The numeraire can be considered as a risk-free ”deposit” growing at the rate rt continuously over
time.
Equivalently, we can define Dt =
Rt
1
= e− 0 ru du as the Discount Factor. If rt is a constant r,
Mt
we have Mt = ert and Dt = e−rt
Yves Guo
The Chinese University of Hong Kong (SZ)
70 / 154
Financial Stochastic Models
No-Arbitrage Conditions
Cash-and-Carry
The spot-forward no-arbitrage relationship for a stock is
K = S0 1 + (r − rrepo )Tm − div
where S0 , K, r, rrepo and div represent, respectively, the spot price, the forward price at T , the
money market interest rate, the Repo rate of the stock and the time-T value of the dividends
paid during the period (0, T ]. If rrepo and div are both zero, it becomes
K = S0 (1 + rTm ) or, in continuous-time modeling, K = S0 erT
Cash-and-Carry strategy consists of hedging the short position of a forward contract by holding
the stock until maturity through a financing in the money market.
Yves Guo
The Chinese University of Hong Kong (SZ)
71 / 154
Financial Stochastic Models
No-Arbitrage Conditions
Put-Call Parity is an arbitrage relationship stating that the followings are equivalent:
1
long vanilla call and short vanilla put, both at the same strike K;
2
long forward contract with strike K.
Notes:
• This arbitrage forces the vanilla call and put options with the same strike level to have the
same implied volatility value;
• It illustrates an interesting case that the combination of two options is no more an option.
Yves Guo
The Chinese University of Hong Kong (SZ)
72 / 154
Financial Stochastic Models
Risk-Neutral Probability Measure
Risk-Neutral Probability
e is defined as the one under which the expectation of the discounted
A Risk-Neutral Probability P,
e St |F0 = S0
value of any asset’s future price equals to the current price of the asset, i.e. E
Mt
M0
(M0 = 1).
Propositon 3.1 (Risk-Neutral Probability for 1-Factor Models)
Assume that St follows the stochastic differential equation:
dSt = µt St dt + σt St dWt , under P
e (”risk-neutral” measure) under which
Then, there exists a probability measure P
St
Mt
is a
martingale and
d
St
St f
= σt
dWt
Mt
Mt
Yves Guo
ft ).
(or, equivalently, dSt = rSt dt + σSt dW
The Chinese University of Hong Kong (SZ)
73 / 154
Financial Stochastic Models
Risk-Neutral Probability Measure
Proof
Itô’s product rule implies
d
St
St
St
St
= (µt − rt )
dt + σt
dWt = σt
Mt
Mt
Mt
Mt
Applying Girsanov Theorem with Zt =
µt − rt
dWt +
dt .
σt
Z t
Z t
µt − rt
de
P
1
2
θs ds where θt =
|F = exp −
θs dWs −
,
dP t
2
σt
0
0
ft = dWt + θt dt.
we obtain dW
Hence, d
St
St f
= σt
dWt , which implies that
Mt
Mt
Yves Guo
St
Mt
is a e
P-martingale.
The Chinese University of Hong Kong (SZ)
74 / 154
Financial Stochastic Models
Self-Financing Portfolio (with One Stock Example)
A portfolio with stocks and cash is represented as Xt = ∆t St + ζt Mt (∆t , ζt may be negative)
where ∆t , ζt represent respectively the number of stocks and the units in money market account.
A Self-Financing Portfolio is defined as the one satisfying dXt = ∆t dSt + ζt dMt .
Propositon 3.2 (Self-Financing Portfolio)
Xt = ∆t St + ζt Mt is self-financing portfolio if and only if
Z t
Xt
X0
Su
=
+
∆u d
Mt
M0
Mu
0
Hence, if
St
Mt
e
is a P-martingale,
Su
Comment: ∆u d M
=
u
Xt
Mt
1
∆ (dSu
Mu u
e
e Xt |Fs =
is also a P-martingale:
E
M
t
Xs
Ms
− ru Su du) is the present value of the P/L in holding ∆u
stocks financed by borrow, if ∆u ≥ 0. Otherwise, it is the P/L of the short position plus the
interest from lending the shorting proceed to money market.
Yves Guo
The Chinese University of Hong Kong (SZ)
75 / 154
Financial Stochastic Models
Complete Market, Hedging
Complete Market means the modeling under which every derivative security can be hedged
according to a ”complete market model”.
Hedging a contingent claim on tradable assets means the replication of the contingent claim by a
self-financing portfolio of the assets.
e the discounted value of the portfolio is a martingale;
• Under the risk-neutral probability P,
e is the necessary and sufficient condition for the predictable martingale
• The uniqueness of P
representation property which implies the existence of the self-financing portfolio process for
replicating the contingent claim. Actually, the integral part in the martingale representation
constitutes the dynamic hedging process.
Yves Guo
The Chinese University of Hong Kong (SZ)
76 / 154
Financial Stochastic Models
Hedging in Complete Market
Propositon 3.3 (Self-financing Hedging Process in Complete Market)
In complete market, the self-financing hedging process for a short position of a contingent claim
paying out VT at T is
Z t
Vt
V0
Su
=
+
∆u d
Mt
M0
Mu
0
(0 ≤ t ≤ T, M0 = 1)
Su
Mu
where ∆u is a unique predictable process and
e
is P-martingale.
Proof
We replicate (hedge) the contingent claim by a self-financing portfolio with terminal value XT = VT . In
X
complete market, e
P is unique and MT is e
P-martingale. Hence, by the predictable martingale representation
T
theorem, there exists a unique predictable process φu , s.t.
Z t
X0
Xt
fu , 0 ≤ t ≤ T
φu dW
=
+
Mt
M0
0
Defining the unique predictable process ∆u as
φu
, we obtain
S -
σu Mu u
Xt
X0
=
+
Mt
M0
t
Z
∆u σ u
0
Su f
X0
d Wu =
+
Mu
M0
t
Z
∆u d
0
Su
.
Mu
By Proposition 3.2, Xt is a self-financing portfolio. Setting Vt = Xt , ∀ 0 ≤ t ≤ T , we prove the proposition.
Yves Guo
The Chinese University of Hong Kong (SZ)
77 / 154
Financial Stochastic Models
Pricing in Complete Market
From the above analysis, we obtained the pricing for a derivative of European style paying out VT
at maturity by taking conditional expectation:
Fair Price at Inception
e
V0 = M 0 E
VT
|F0
MT
If Mt = ert , then M0 = 1. Hence
h
i
e e−rT VT |F0
V0 = E
Mark-To-Market Price at time t
e
Vt = M t E
VT
|Ft
MT
Yves Guo
The Chinese University of Hong Kong (SZ)
78 / 154
Financial Stochastic Models
Black-Scholes-Merton Model
Yves Guo
The Chinese University of Hong Kong (SZ)
79 / 154
Financial Stochastic Models
Black-Scholes-Merton Model
Black-Scholes-Merton assumes that the asset follow the following stochastic process:
dSt = µSt dt + σSt dWt ,
where
σ: constant volatility;
r: constant money market rate;
µ: the drift of the asset in its real probability P.
Yves Guo
The Chinese University of Hong Kong (SZ)
80 / 154
Financial Stochastic Models
Black-Scholes-Merton Model
Risk-Neutral Probability
According to Proposition 3.1, we have
d
St f
St
=σ
dWt .
Mt
Mt
where Zt =
de
P
|
dP Ft
Z
= exp −
t
θdWu −
0
1
2
Z
0
t
µ−r
θ2 du and θ =
.
σ
e St is a martingale.
Under P,
Mt
Stochastic Differential Equation for BSM Model
Applying Ito’s rule to the above result for
St
,
Mt
we obtain the SDE for the asset price process
ft .
dSt = rSt dt + σSt dW
1
It can be shown that the solution for the SDE is St = S0 e(r− 2 σ
Yves Guo
2
ft
)t+σ W
.
The Chinese University of Hong Kong (SZ)
81 / 154
Financial Stochastic Models
Black-Scholes-Merton Model
Black-Scholes PDE
Let Vt be the mark-to-market value of a derivative paying out h(ST ) which is a function of the
underlying’s price at maturity T .
The differential of
d
Vt
Mt
=
=
Vt
Mt
is
∂V
∂V
1 ∂2V
dt +
dS +
dhS, Sit
2
∂t
∂S
2 ∂S
∂V
1
∂V
∂2V
∂V
ft .
+ rS
+ σ2 S 2
dt + e−rt
σSdW
e−rt −rV +
2
∂t
∂S
2
∂S
∂S
e−rt
−rV dt +
Vt
is a martingale. Therefore, the dt term (drift) should be zero
Mt
which leads to the Black-Scholes PDE:
According to Proposition 3.2,
∂V
∂V
1
∂2V
+ rS
+ σ2 S 2
− rV = 0.
∂t
∂S
2
∂S 2
with terminal condition VT = h(ST )
Yves Guo
The Chinese University of Hong Kong (SZ)
82 / 154
Financial Stochastic Models
Black-Scholes-Merton Model
Pricing
The pricing of a European option paying out h(ST ) is
e −r(T −t) h(ST )|Ft ]
Vt = E[e
Hedging
St
ft , we have
From the above analysis and using d M
= e−rt σSt dW
t
d
Vt
∂Vt
ft = ∂Vt d St .
= e−rt
σSt dW
Mt
∂St
∂St Mt
It’s integral form is
V0
VT
=
+
MT
M0
T
Z
∆t d
0
St
Mt
where
∆t =
∂Vt
.
∂St
Yves Guo
The Chinese University of Hong Kong (SZ)
83 / 154
Financial Stochastic Models
Closed-form Solution for Call option
The pricing of a European Call option paying CT = (ST − K)+ at maturity T :
c0
=
e
E
CT
|F0
MT
=
+
e (ST − K) |F0
E
erT
h
i
1 2
f
−rT
e
E e
(S0 e(r− 2 σ )T +σWT − K)+ |F0
h
i
e e−rT (ST − K)+
f0 = 0 and W
fT − W
f0 is independent of F0 )
E
(W
h
i
e e−rT ST 1{S >K} − e−rT K 1{S >K}
E
T
T
h
i
−rT
−rT
e
e
E e
ST 1{ST >K} ] − E[e
K 1{ST >K}
=
A − B.
=
=
=
=
Yves Guo
The Chinese University of Hong Kong (SZ)
84 / 154
Financial Stochastic Models
h
fT > v, with v = ln
The condition ST > K is equivalent to W
h
fT
e e− 21 σ2 T +σW
A = S0 E
1{W
f
i
T >v}
h
e ZT 1 f
= S0 E
{W
1
2
fT
T +σ W
eliminated by applying Girsanov Proposition with Zt =
A
=
e T1 f
S0 E[Z
{W
=
e Q [1 Q
S0 E
f +σT >v} ]
{W
T >v}
i
T >v}
The first term inside the expectation ZT = e− 2 σ
Then
K
S0
i
− (r − 12 σ 2 )T /σ and hence
.
is an exponential martingale. It can be
dQ
de
P Ft
for defining a new probability Q.
]
(by Bayes’ formula and Girsanov Theorem)
σT
f Q > v − σT } = S0 Q g > v −
√
S0 Q{W
(g is standard Gaussian)
T
T
v − σT
S0 Q g < − √
(symmetry of Gaussian distribution)
T
!
ln SK0 + (r + 12 σ 2 )T
σT − v
√
√
S0 N
= S0 N
= S0 N (d1 ).
T
σ T
T
=
=
=
Yves Guo
The Chinese University of Hong Kong (SZ)
85 / 154
Financial Stochastic Models
B
=
=
e −rT K 1{S >K} ] = e−rT K E[
e1 f
E[e
]
T
{WT >v}
√
−rT
−rT
e
e
f
e
K P{WT > v} = e
K P{g T > v}
−rT
√
e
T < −v} = e−rT KN
K P{g
=
e
=
e−rT KN (d2 ).
ln
S0
K
+ (r − 12 σ 2 )T
√
σ T
!
Hence c0 = A − B = S0 N (d1 ) − e−rT KN (d2 ), where
d1 =
ln
S0
K
+ (r + 12 σ 2 )T
√
σ T
and
√
d2 = d1 − σ T .
In the same way, we can obtain ct = St N (d1 ) − e−r(T −t) KN (d2 ), where
d1 =
ln
St
K
+ (r + 21 σ 2 )(T − t)
√
σ T −t
Yves Guo
and
√
d2 = d1 − σ T − t.
The Chinese University of Hong Kong (SZ)
86 / 154
Financial Stochastic Models
Stock Dividends
Dividend Modeling
• Continuous dividend: dSt = µt St dt + σt St dWt − At St dt; It is evidently a theoretical
assumption for modeling;
• Proportional dividend: dividend amount of At St are paid at discrete time t1 , t2 , t3 , . . .;
• Cash dividend: dividends are paid as fixed cash amount at discrete times. This is the most
common dividend form.
Notes. Fixed cash dividend for long term is not a viable dividend modeling (a stock of $100
paying $3 dividend will not still pay $3 even when its price drops to $10).
In practice, people often use:
• cash dividends for the short term (e.g. ≤ 2 years);
• proportional dividends for long term (e.g. ≥ 5 years);
• mixture of cash dividends and proportional dividends in between.
Definition.
• “Price Return” of a stock is the return without adding back the already paid dividends
• “Total Return” reflects the return by reinvesting all the paid dividends in the same stock
Yves Guo
The Chinese University of Hong Kong (SZ)
87 / 154
Financial Stochastic Models
Risk-Neutral SDE with Continuous Dividend
For simplifying the illustration but without loss of generality, we assume that the dividend yield,
interest rate and the drift of asset price are all constants A, r and µ. The asset price model is
dSt = µSt dt + σt St dWt − ASt dt,
As St eAt is the total return of the asset, its discounted value
St eAt
Mt
should be a martingale under
the risk-neutral probability:
d
St eAt
Mt
=
d e−rt eAt St = d e(A−r)t St
=
(A − r)e(A−r)t St dt + e(A−r)t dSt
=
e(A−r)t St [(A − r)dt + (µ − A)dt + σt dWt ]
µ−r
dt + dWt
e(A−r)t σt St
σt
ft ,
e(A−r)t σt St dW
=
=
ft =
where dW
µ−r
dt
σt
+ dWt . Equating the second and last lines, we obtain the risk-neutral SDE,
ft .
dSt = (r − A)St dt + σt St dW
Yves Guo
The Chinese University of Hong Kong (SZ)
88 / 154
Financial Stochastic Models
Discussion on Hedging
Yves Guo
The Chinese University of Hong Kong (SZ)
89 / 154
Financial Stochastic Models
Sensitivities (Greeks)
In Black-Scholes-Merton model, the price of a portfolio Πt depends on the stock price, time to
maturity, volatility and risk-free interest rate: Πt = Π(T − t, St , σ, r). We denote these
sensitivities by greek letters:
• Theta: Θt = ∂Π (t);
∂t
• Vega: Vt = ∂Π (t).
• Delta: ∆t = ∂Π (t);
∂S
2
• Gamma: Γt = ∂ Π
2 (t);
Θt + rS∆t +
∂Π
(t);
∂r
∂σ
∂S
We can write the BSM PDE
• Rho: ρt =
∂Π
(t)
∂t
+ rS ∂Π
(t) +
∂S
1 ∂2Π
(t)σ 2 St2
2 ∂S 2
= rΠt as
1
Γt σ 2 St2 = rΠt .
2
If Π is delta-neutral (∆t = 0), then Θt + 21 Γt σ 2 St2 = rΠt . It shows that Gamma and Theta
should have opposite signs if their values are large, as Πt tends to be small for delta-hedged
portfolio.
Yves Guo
The Chinese University of Hong Kong (SZ)
90 / 154
Financial Stochastic Models
Yves Guo
The Chinese University of Hong Kong (SZ)
91 / 154
Financial Stochastic Models
Yves Guo
The Chinese University of Hong Kong (SZ)
92 / 154
Financial Stochastic Models
Yves Guo
The Chinese University of Hong Kong (SZ)
93 / 154
Financial Stochastic Models
Yves Guo
The Chinese University of Hong Kong (SZ)
94 / 154
Financial Stochastic Models
Some Practical Hedging Issues
The assumptions in Black-Scholes-Merton model are over simplistic. One of the issue is discrete
hedging.
Discrete Hedging
Delta hedging can only be done at discrete times (a few times a day at most). The negative
Gamma position will make the derivative under-hedged.
• The negative Gamma position is compensated by a positive Theta.
• Conversely, the positive Gamma position may get benefit from large market moves. However,
this benefit is counterbalanced by the negative Theta.
The Gamma-Theta ratio is an important factor for managing a trading book.
Yves Guo
The Chinese University of Hong Kong (SZ)
95 / 154
Financial Stochastic Models
Yves Guo
The Chinese University of Hong Kong (SZ)
96 / 154
Financial Stochastic Models
Some Numerical Methods for Option Pricing
Yves Guo
The Chinese University of Hong Kong (SZ)
97 / 154
Financial Stochastic Models
Numerical Methods for Option Pricing
There are several methods used in practice for option pricing :
• Closed-form solutions: Closed-form solutions (e.g. Black Scholes formula) are accurate and
fast to calculate. However, only a very few options have had the closed-forme solutions
found under simple models like Black-Scholes-Merton.
• Numerical Integration: It is also an efficient method. However, it is applicable only when
the asset’s probability of transition is known, which is not the case in general.
• Tree methods: Assume that, over a small time step, the asset price has only two possible
moves: ”up” by a proportion of u probability p, or ”down”by a proportion of d with
probability 1 − p. A Binomial-tree can be constructed with time steps until the maturity of
the considered option. The option price can then be calculated on the tree. Similarly, people
have also developed Trinomial-tree method. The tree methods are limited to options
involving less than two random variable in general.
Yves Guo
The Chinese University of Hong Kong (SZ)
98 / 154
Financial Stochastic Models
Numerical Methods for Option Pricing
• PDE method: there is a link between SDE (Stochastic Differential Equation) and PDE
(Partial Differential Equation). Instead of resolving the SDE for option pricing, one can
choose to resolve the relevant PDE. PDEs are well studied in mathematics and physics. Like
the tree methods, PDE method is limited to options involving less than two random variable
in general.
• Monte Carlo simulation: it is the simplest but the universal method for pricing all types of
options (excluding American options which requires special Monte Carlo methods). It
consists of simulating the random paths and then calculating the averaged discounted payoff
value as the option price.
In this course, we only present Binomial Tree, PDE and Monte Carlo method.
Yves Guo
The Chinese University of Hong Kong (SZ)
99 / 154
Financial Stochastic Models
Binomial Tree
In Binomial Tree method, it is assumed that, over a small time step ∆t, the asset price has only
two possible moves:
• up by a proportion of u with risk-neutral probability p, or
• down by a proportion of d with risk-neutral probability 1 − p.
A binomial tree will be constructed with time steps 0, ∆t, 2∆t, 3∆t, . . . , which represents a
diffusion of the asset prices with the transition probabilities.
Yves Guo
The Chinese University of Hong Kong (SZ)
100 / 154
Financial Stochastic Models
Binomial Tree
Determination of p, u and d
Parameters are chosen so that the tree gives correct values for the mean and variance of the asset
price changes under risk-neutral probability. A further condition on u and d is imposed in order to
get a third equation for solving 3 unknowns u, d and p:
Mean: pu + (1 − p)d = er∆t ,
2
Variance: pu2 + (1 − p)d2 = e2r∆t+σ ∆t ,
1
u= .
d
(1)
(2)
(3)
When terms of higher order than ∆t are ignored, the solution to (1), (2) and (3) is:
p=
er∆t − d
,
u−d
u = eσ
√
∆t
and
d = e−σ
√
∆t
.
To obtain the option price at each node, we have
V (t, S) = e−r∆t [pV (t + ∆t, Su) + (1 − p)V (t + ∆t, Sd)].
Yves Guo
The Chinese University of Hong Kong (SZ)
101 / 154
Financial Stochastic Models
Binomial Tree
e St+∆t |F (t)] = St , i.e.,
Derivation of (1): Under risk-neutral probability, we have E[
Mt+∆t
Mt
e
−r(t+∆t)
St [pu + (1 − p)d]
⇒
pu + (1 − p)d
=
=
e
−rt
St ,
e
r∆t
.
Derivation of (2):
From the property of log normal distribution, we have V ar(St+∆t ) = St2 e2r∆t (eσ
2 ∆t
− 1).
2
2
From V ar(St+∆t ) = E[St+∆t
] − (E[St+∆t ])2 , E[St+∆t
] = p(uSt )2 + (1 − p)(dSt )2 and
(E[S∆t ])2 = (puSt + (1 − p)dSt )2 , we obtain equation (2):
2
2
pu + (1 − p)d = e
Yves Guo
2r∆t+σ 2 ∆t
.
The Chinese University of Hong Kong (SZ)
102 / 154
Financial Stochastic Models
Binomial Tree
Yves Guo
The Chinese University of Hong Kong (SZ)
103 / 154
Financial Stochastic Models
Finite Difference Methods
We take the example of BSM model (i.e. dSt /St = rdt + σdWt ) for a European option with
payoff function h(ST ).
From previous analysis, the European option price Vt = E[e−r(T −t) h(ST )] satisfies the PDE
2
∂V + rS ∂V + 1 σ 2 S 2 ∂ V − rV = 0
∂t
∂S
2
∂S 2
VT = h(ST ).
Conversely, if St follows BSM model and Vt satisfies the above PDE, then
V0 = E[e−rT h(ST )].
Actually, applying Itô-Doeblin formula to e−rT VT , we have
e−rT VT = V0 +
T
Z
0
e−rt [
∂V
∂V
1
∂2V
+ rS
+ σ2 S 2
− rV ]dt +
∂t
∂S
2
∂S 2
Z
T
0
e−rt
∂Vt
St σdWt
∂St
As the PDE implies that the term before dt is zero, we obtain V0 = E[e−rT VT ] = E[e−rT h(ST )]
by taking expectation on both side.
The finite difference is the most straightforward numerical method. We illustrate the finite
difference implementation for BSM model, which is a 1-dimensional PDE.
Yves Guo
The Chinese University of Hong Kong (SZ)
104 / 154
Financial Stochastic Models
Finite Difference Methods
Finite Difference Methods for Derivatives Calculation
From the Taylor expansions:
V (x̄ + δx)
=
V (x̄ − δx)
=
∂V
(x̄)δx +
∂x
∂V
V (x̄) −
(x̄)δx +
∂x
V (x̄) +
1 ∂2V
1 ∂3V
(x̄)δx2 +
(x̄)δx3 + O(δx4 ),
2
2 ∂x
3! ∂x3
1 ∂2V
1 ∂3V
(x̄)δx2 −
(x̄)δx3 + O(δx4 ),
2 ∂x2
3! ∂x3
we obtain the different methods for calculating the derivatives:
• Forward difference (with precision O(δx)):
V (x̄ + δx) − V (x̄)
∂V
(x̄) =
;
∂x
δx
• Centered difference (with precision O(δx2 )):
∂V
(x̄)
∂x
2
∂ V
(x̄)
∂x2
=
=
V (x̄ + δx) − V (x̄ − δx)
;
2δx
V (x̄ + δx) + V (x̄ − δx) − 2V (x̄)
.
δx2
• ...
Yves Guo
The Chinese University of Hong Kong (SZ)
105 / 154
Financial Stochastic Models
Finite Difference Methods
Grid Construction
Because Black–Scholes–Merton is a log-normal model, it is more efficient to work with x = ln S:
1 ∂2V
∂V
+ σ2
+
∂t
2
∂x2
r−
σ2
2
∂V
− rV (x, t) = 0,
∂x
(4)
with European-type payoff V (x, T ) for −∞ < x < ∞.
We truncate the spatial domain as x ∈ [L1 , L2 ] and replace the boundary conditions V (−∞, t)
by V (L1 , t) and V (∞, t) by V (L2 , t).
Next we partition the solution domain [L1 , L2 ] × [0, T ] by grid lines:
• [L1 , L2 ] into n equal sub-intervals, each of length δx, and
• [0, T ] into m equal sub-intervals, each of length δt.
Denote xi = L1 + iδx and tj = jδt for 0 ≤ i ≤ n and 0 ≤ j ≤ m.
Yves Guo
The Chinese University of Hong Kong (SZ)
106 / 154
Financial Stochastic Models
Finite Difference Methods
Grid Construction
xO
V (L2 , t)
L2−
V (xi , tj )
•
•
•
•
L1 + 2δx−
•
•
•
•
L1 + δx −
•
•
•
L1
δt 2δt
V (x, T )
•
L1 + iδx −
jδt
T
/t
T
V (L1 , t)
Yves Guo
The Chinese University of Hong Kong (SZ)
107 / 154
Financial Stochastic Models
Finite Difference Methods
Terminal and Boundary Conditions
Denote V i (t) for V (xi , t) for i = 1, 2, . . . , n − 1.
• The terminal condition for V (x, T ) can be obtain according to the payoff of the option; (in
the terminology for differential equations, it is called Initial Value Problem as one can choose
to work with reversed time t0T − t instead of t);
• The boundary conditions can determined according to the payoff of the option, e.g. for a
European call option V (x0 , t) = V (L1 , t) ≈ 0 and V (xn , t) = V (L2 , t) ≈ eL2 − Ke−r(T −t) .
Instead of option specific the boundary conditions as the above example, a general boundary
condition may be applied to all options which is known as ”zero gamma boundary condition”
(∂xx V = 0) at L1 and L2 , i.e., V (x0 , t) = 2V 1 (t) − V 2 (t) and
V (xn , t) = 2V n−1 (t) − V n−2 (t).
Yves Guo
The Chinese University of Hong Kong (SZ)
108 / 154
Financial Stochastic Models
Finite Difference Methods
Equation System
Now, apply centered difference:
1 V i+1 (t) − 2V i (t) + V i−1 (t)
∂V i (t)
+ σ2
+
∂t
2
δx2
r−
σ2
2
V i+1 (t) − V i−1 (t)
− rV i (t) = 0
2δx
which leads to the following system of equations:
b
∂t V 2 (t) a
..
=
.
∂t V n−2 (t)
∂t V 1 (t)
∂t V n−1 (t)
where a =
c
b
..
.
c
..
a
.
..
.
b
a
V 1 (t)
V 2 (t)
..
+
.
c V n−2 (t)
b
V n−1 (t)
aV (x0 , t)
0
..
.
0
cV (xn , t)
r − σ 2 /2
σ2
σ2
r − σ 2 /2
σ2
−
, b=r+
,c=−
−
.
2
2
2δx
2(δx)
(δx)
2δx
2(δx)2
Yves Guo
The Chinese University of Hong Kong (SZ)
109 / 154
Financial Stochastic Models
Finite Difference Methods
Equation System
The above system of equations can be expressed as
∂v(t)
= Av(t) + f (t),
∂t
(5)
Recall that ”zero gamma boundary condition” implies:
• V (x0 , t) = 2V 1 (t) − V 2 (t)
• V (xn , t) = 2V n−1 (t) − V n−2 (t)
Hence, with ”zero gamma boundary condition”, f (t) will be a zero vector and we obtain:
∂v(t)
= Dv(t)
∂t
The choice of the finite difference for
(6)
∂v(t)
∂t
on the grid leads to different numerical methods. We
present the most usual ones in the following sections.
Yves Guo
The Chinese University of Hong Kong (SZ)
110 / 154
Financial Stochastic Models
Finite Difference Methods
Explicit Scheme
The explicit scheme is equivalent to, for j = 0, 1, . . . , m,
(vj − vj−1 )/δt = Dvj ,
or,
vj−1 = (I − δtD)vj ,
where I is an identity matrix of size n − 1.
Since vm = [V (x1 , tm ), . . . , V (xn−1 , tm )]| are known values at maturity, a time-marching
operation is performed to obtain vm−1 , vm−2 , and after m times, v0 at time 0.
The accuracy of explicit scheme is O(δt) and O(δx2 ).
Yves Guo
The Chinese University of Hong Kong (SZ)
111 / 154
Financial Stochastic Models
Finite Difference Methods
Implicit Scheme
The implicit scheme is equivalent to, for j = 0, 1, . . . , m,
(vj − vj−1 )/δt = Dvj−1 ,
or,
(I + δtD)vj−1 = vj .
The time-marching process starts from the known vm and a tridiagonal linear system needs to be
solved at each time step.
The accuracy of implicit scheme is O(δt) and O(δx2 ). Compared to the explicit method, the
implicit method is more robust.
Yves Guo
The Chinese University of Hong Kong (SZ)
112 / 154
Financial Stochastic Models
Finite Difference Methods
Crank-Nicolson Scheme
The Crank–Nicolson scheme combines the implicit and explicit schemes by introducing a half
time-step j −
I+
1
2
between j − 1 and j:
δt
D vj−1 = vj− 1
2
2
and
vj− 1 =
2
δt
I − D vj .
2
That is
I+
δt
δt
D vj−1 = I − D vj .
2
2
For the resolution, a tridiagonal system is solved at each time-step during time-marching.
The scheme is second order accurate in both time and space (O(δt2 ) and O(δx2 )).
Yves Guo
The Chinese University of Hong Kong (SZ)
113 / 154
Financial Stochastic Models
Finite Difference Methods
Graphical illustration of some of the finite difference methods
Yves Guo
The Chinese University of Hong Kong (SZ)
114 / 154
Financial Stochastic Models
Finite Difference Methods
Accuracy analysis for the finite difference methods
Denote the terms in the above partial differential equation as :
L(x̃, t̃) =
∂V
∂2V
∂V
(x̃, t̃) + H(x̃, t̃), where H(x̃, t̃) = a 2 (x̃, t̃) + b
(x̃, t̃) + cV (x̃, t̃).
∂t
∂x
∂x
The finite difference approximation of H(xi , t) is
e i , t) = a V (x
H(x
i+1 , t)
+ V (xi−1 , t) − 2V (xi , t)
V (xi+1 , t) − V (xi−1 , t)
+b
+ cV (xi , t).
2
δx
2δx
The error of the finite difference H̄(xi , t) is
e i , t) − H(xi , t)
H(x
=
a
2 ∂4V i
1 ∂3V i
(x , t)δx2 + O(δx4 ) + b
(x , t)δx2 + O(δx4 )
4! ∂x4
3! ∂x3
e i , t) − H(xi , t) = O(δx2 )
Hence, H(x
Yves Guo
The Chinese University of Hong Kong (SZ)
115 / 154
Financial Stochastic Models
Finite Difference Methods
The explicit finite difference approximation of L(xi , tj ) is
e ex (xi , tj )
L
V (xi , tj ) − V (xi , tj−1 )
e i , tj )
+ H(x
δt
1 ∂2V i j
∂V i j
e i , tj )
(x , t ) +
(x , t )δt + O(δt2 ) + H(x
∂t
2 ∂t2
=
=
e i , t) − H(xi , t) = O(δx2 ), the error of the explicit finite difference is
As H(x
2
e ex (xi , tj ) − L(xi , tj ) = 1 ∂ V (xi , tj )δt + O(δt2 ) + O(δx2 )
L
2 ∂t2
Hence, the order of accuracy for the explicit scheme is O(δt, δx2 ).
Similarly, the implicit finite difference approximation of L(xi , tj−1 ) is
e im (xi , tj−1 )
L
=
V (xi , tj ) − V (xi , tj−1 )
e i , tj−1 ),
+ H(x
δt
and the order of accuracy for the implicit scheme is O(δt, δx2 ).
Yves Guo
The Chinese University of Hong Kong (SZ)
116 / 154
Financial Stochastic Models
Finite Difference Methods
By the Taylor expansion of V (·, t̄ + ) and V (·, t̄ − ) where t̄ =
the finite difference approximations of
e
L
ex
j
(·, t )
im
(·, t
j−1
)
and
1 j−1
(t
2
+ tj ), =
δt
,
2
we obtain
L(·, tj−1 ):
V (·, tj ) − V (·, t̄)
e tj )
+ H(·,
∂V
1 ∂2V
2
e tj )
(·, t̄) +
(·, t̄) + O( ) + H(·,
∂t
2 ∂t2
=
=
e
L
L(·, tj )
=
=
V (·, t̄) − V (·, tj−1 )
e tj−1 )
+ H(·,
1 ∂2V
∂V
2
e tj−1 ).
(·, t̄) −
(·, t̄) + O( ) + H(·,
∂t
2 ∂t2
On the other hand, the Taylor expansion of
∂V
∂t
(·, tj ) and
∂V
∂t
(·, tj−1 ) with t̄ and gives
∂V
∂V
∂2V
∂V
∂V
∂2V
j
2
j−1
2
(·, t ) =
(·, t̄) +
(·, t̄) + O( ) and
(·, t
)=
(·, t̄) −
(·, t̄) + O( ).
∂t
∂t
∂t2
∂t
∂t
∂t2
Hence, we obtain
1 ∂V
∂V
∂V
(·, tj ) +
(·, tj−1 ) =
(·, t̄) + O(2 ).
2 ∂t
∂t
∂t
Yves Guo
The Chinese University of Hong Kong (SZ)
117 / 154
Financial Stochastic Models
Finite Difference Methods
Crank-Nicolson method consists of approximating the below
LC−N (xi , tj , tj−1 ) =
1
L(xi , tj ) + L(xi , tj−1 )
2
with
e C−N (xi , tj , tj−1 ) = 1 L
e ex (xi , tj ) + L
e im (xi , tj−1 )
L
2
By the results on the previous slide, we obtain the approximation error
e C−N (xi , tj , tj−1 ) − LC−N (xi , tj , tj−1 ) = O(2 ) + O(δx2 )
L
As =
δt
,
2
the order of accuracy for Crank Nicolson method is O(δt2 , δx2 ).
The equation for Crank-Nicolson method is
i j
i j−1 )
1 e i j
e C−N (xi , tj , tj−1 ) : V (x , t ) − V (x , t
e i , tj−1 ) = 0.
L
+
H(x , t ) + H(x
δt
2
Yves Guo
The Chinese University of Hong Kong (SZ)
118 / 154
Financial Stochastic Models
Monte Carlo Method
Monte Carlo simulation is the most simple and powerful method for pricing European options. It
consists of calculating the expectation based on the Law of Large Numbers.
Law of Large Numbers
Let Xk be i.i.d. (independent and identically distributed) random variables, and
E[|f (X)|] < +∞, then
lim
n→+∞
n
1 X
f (Xk ) = E[f (X)]
n k=1
Yves Guo
a.s.
The Chinese University of Hong Kong (SZ)
119 / 154
Financial Stochastic Models
Monte Carlo Method
Generating Random Variables
Denote U [0, 1] for the uniform distribution over [0, 1]. Let N (x) be the cumulative Normal
distribution function. Consider a random variable g generated as follows:
1
generate u from U [0, 1];
2
calculate g = N −1 (u).
Then, g follows Normal distribution: P{g < x} = N (x)
Yves Guo
The Chinese University of Hong Kong (SZ)
120 / 154
Financial Stochastic Models
Monte Carlo Method
Generating One Sample Path
Consider the stochastic process:
dSt = β(t, St )dt + γ(t, St )dWt
a sample path can be generated as described below:
• Discretize the final option maturity T into m time intervals t0 < t1 < t2 < · · · < tm . The
intervals depend on the option payoff and the model parameters. It is not necessary to have
equally spaced intervals;
• Generate ui from U [0, 1], i = 1, 2, . . . , m;
• Calculate the gi = N −1 (ui );
• Calculate the sample path with Euler scheme, i.e.,
Sti = Sti−1 + β(ti−1 , Sti−1 )∆ti + γ(ti−1 , Sti−1 )∆Wi ,
√
where ∆ti = ti − ti−1 and ∆Wi = gi ∆ti .
Yves Guo
The Chinese University of Hong Kong (SZ)
121 / 154
Financial Stochastic Models
Monte Carlo Method
Option Pricing with Monte Carlo Simulation
The Monte Carlo simulation for pricing a European option paying out f (ST ) is as follows:
1
Time discretization: decompose T into m time intervals: t = t0 < t1 < t2 < · · · < tm ;
2
Path generation: calculate k paths Sti , 1 ≤ k ≤ n, 1 ≤ i ≤ m;
3
Path payoff calculation: f (ST ), 1 ≤ k ≤ n;
4
(k)
(k)
Averaging and discounting (i.e. dividing by the numeraire): set option price as
n
1 X
(k)
e−rT
f (ST ).
n k=1
Yves Guo
The Chinese University of Hong Kong (SZ)
122 / 154
Financial Stochastic Models
Monte Carlo Method
Example: European Call Option under BSM Model
Payoff: max{0, ST − K}
2
Time discretization: one time step of T is sufficient because
• European call option is not path-dependent, and
• the parameters are constant;
fT , we have
Path generation: from ln ST = ln S0 + r − 12 σ 2 T + σ W
√
1
(k)
ln ST = ln S0 + r − σ 2 T + σgk T , gk ∼ N (0, 1),
2
n
o
(k)
(k)
and hence ST = exp ln ST
;
3
Path payoff calculation: V (k) = max{0, ST
1
4
(k)
− K}, 1 ≤ k ≤ n;
Averaging and discounting (i.e. dividing by the numeraire): set option price as
n
1 X (k)
V = e−rT
V
.
n k=1
Yves Guo
The Chinese University of Hong Kong (SZ)
123 / 154
Financial Stochastic Models
Monte Carlo Method
Generating Multivariate Normals
For multi-asset products or multi-factor models, correlated random normals can be generated by means of
Linear Transformation Property: X ∼ N (µ, Σ) ⇒ AX ∼ N (Aµ, AΣAT ), where
σ11
σ12
···
σ1n
σ21
Σ=
..
.
σ22
···
.
.
.
σ2n
..
σn1
σn2
···
. ,
.
.
.
σij = ρij σi σj , i, j ∈ [1, ..., n]
σnn
is the covariance matrix which is symmetric and positive semidefinite.
Let G ∼ N (0, I) where I is an n × n identity matrix and Σ be the covariance matrix. Now, we do the
following for X:
• Find A such that AAT = Σ, and
• Set X = AG.
Then, X ∼ N (0, Σ) (from linear transformation property, X = AG ∼ N (A0, AIAT ) ⇔ N (0, Σ))
Two methods are commonly used for finding A: Cholesky Factorization and Eigenvector Factorization.
Yves Guo
The Chinese University of Hong Kong (SZ)
124 / 154
Financial Stochastic Models
Monte Carlo Method
Cholesky Factorization
A representation of Σ (positive definite) as AAT with A lower triangular is a Cholesky factorization of Σ:
a11
0
a21
A=
.
.
.
a22
.
.
.
an1
an2
···
..
.
..
.
0
.
.
.
,
0
···
ann
and
T
Σ = AA
a11
0
a21
=
.
.
.
a22
.
.
.
an1
an2
Yves Guo
···
..
.
..
.
···
0
.
.
.
0
ann
a11
a21
···
an1
0
.
.
.
a22
···
..
.
..
an2
. .
.
.
0
···
0
.
ann
The Chinese University of Hong Kong (SZ)
125 / 154
Financial Stochastic Models
Monte Carlo Method
Cholesky Factorization
Given Σ, we can solve A recursively through the following system of equations:
a211 = σ11 ,
a11 a21 = σ12 ,
.
.
.
an1 a11 = σ1n ,
a221 + a222 = σ22 ,
.
.
.
a2n1 + a2n2 + · · · + a2nn = σnn .
In particular, when n = 2, σi = 1 (i = 1, 2) and Σ is the correlation matrix of normal variates, we have
2
a11 = 1
a11 a21 = ρ
2
a21 + a222 = 1
p
1 p 0
1 − ρ2 , or, A =
and
2
ρ
1−ρ
√ 1 p 0
B1
W1
g √
g1 √t
p
=A 1
t=
=
W2
g2
ρ
1 − ρ2 g2 t
ρB1 + 1 − ρ2 B2
√
√
(g1 , g2 : independent Normal variables; B1 = g1 t, B2 = g2 t).
implying a11 = 1, a21 = ρ, a22 =
Hence B1 , B2 are independent Brownians and W1 , W2 are two correlated Brownians.
Yves Guo
The Chinese University of Hong Kong (SZ)
126 / 154
Financial Stochastic Models
American Options
Yves Guo
The Chinese University of Hong Kong (SZ)
127 / 154
Financial Stochastic Models
American Option
An American Option gives the option holder the right to early terminate the option before its
maturity. Upon the option exercise, the option holder will be paid the intrinsic value or other
predefined payout.
An American option with discrete exercise dates is called Bermudan Option.
The modeling of American Option involves advanced mathematics. No closed-form formula has
been found. However, the pricing methods with Tree or Finite Difference are simple and intuitive
which are applicable to one underlying based options. For options with more than one underlying
assets, the ”Least-Square Method” is widely used in the financial industry. It is a special Monte
Carlo method proposed by Longstaff and Schwartz.
In this course, we only present the Tree method in detail.
Yves Guo
The Chinese University of Hong Kong (SZ)
128 / 154
Financial Stochastic Models
American Option Pricing with Binomial-Tree under BSM
The tree is constructed in the same way as the European option. But the option calculation is
different. With the backward induction, we calculate, at each node:
• the Continuation Value: the option value;
• the Exercise Value: the payout of the option with early exercise;
• the (conditional) option price at the node will be
max{Exercise Value, Continuation Value}.
We illustrate it through an example in the Excel spread sheet.
Yves Guo
The Chinese University of Hong Kong (SZ)
129 / 154
Financial Stochastic Models
Yves Guo
The Chinese University of Hong Kong (SZ)
130 / 154
Financial Stochastic Models
American Option Pricing with Finite Difference Methods
For American options, the option prices satisfy ”partial differential inequalities” with terminal
condition as well as boundary conditions for the early exercise. These systems can be solved with
finite difference methods.
The explicit finite difference method of American options is similar to the binomial tree method.
It consists of taking V (x, t) = max{Exercise Value, Continuation Value} as the American
option value on the grid where Continuation Value is calculated with the method for European
options at each time step t from the American option values of the precedent step.
For implicit finite difference method, special care needs to be taken. We can find the explication
in more detailed references, such as the book of Paul Wilmott, ”Derivatives: the theory and
practice of financial engineering”, John Wiley & Sons.
Yves Guo
The Chinese University of Hong Kong (SZ)
131 / 154
Financial Stochastic Models
Propositon 7.1
The price of an American call option is equal to the price of the corresponding European call if
• there is no dividend for the underlying stock, and
• the interest rate r is nonnegative.
Proof
Denote Ct (S0 , K, T ) and ct (S0 , K, T ) for, respectively, American and European Call option prices at time t
(∀t < T ). Evidently, Ct (S0 , K, T ) ≥ ct (S0 , K, T ). We show that
ct (S0 , K, T ) ≥ M ax(0, St − e
−r(T −t)
K)
Define two portfolios at date t:
• Portfolio A: Long ct (S0 , K, T ) + a bond with the value of e−r(T −t) K
• Portfolio B: Long one share St
At maturity T , we have
• Portfolio A: M ax(0, ST − K) + K = M ax(0, ST , K)
• Portfolio B: ST (no dividend payment)
Yves Guo
The Chinese University of Hong Kong (SZ)
132 / 154
Financial Stochastic Models
Proof (continued)
Hence, we should have A ≥ B, for any time until maturity, i.e.
ct (S0 , K, T ) + e
−r(T −t)
K > St
or
−r(T −t)
ct (S0 , K, T ) ≥ M ax(0, St − e
K)
(because ct (S0 , K, T ) ≥ 0)
Consequently, we have the following inequalities
Ct (S0 , K, T ) ≥ ct (S0 , K, T ) ≥ M ax(0, St − e
−r(T −t)
K) ≥ M ax(0, St − K)
(as r ≥ 0)
Because the early exercise price is M ax(0, St − K), it is not optimal for the option holder to exercise the
option before the maturity (the option holder should sell the option instead of exercising it). Therefore,
Ct (S0 , K, T ) = ct (S0 , K, T )
Yves Guo
The Chinese University of Hong Kong (SZ)
133 / 154
Financial Stochastic Models
Foreign Exchange Modeling
and
Composite/Quanto Options
Yves Guo
The Chinese University of Hong Kong (SZ)
134 / 154
Financial Stochastic Models
FX Market Terminology
Notes on terms used in FX market:
• we sometimes use the term Ccy1 for foreign currency and Ccy2 for domestic currency;
• Ccy1/Ccy2 denotes the exchange rate (units of Ccy2 per unit of Ccy1); Example: USD/JPY;
• markets usually fix the quotation order for the pairs; Example, EUR/USD, GBP/USD,
USD/JPY, USD/HKD...
• the Call option on Ccy1/Ccy2 is actually the Put option on Ccy2/Ccy1. As a consequence,
people often use the terms like USD call - JPY put for describing the Call option on
USD/JPY.
Yves Guo
The Chinese University of Hong Kong (SZ)
135 / 154
Financial Stochastic Models
Foreign Exchange Modeling
Basic arbitrage relationship for foreign exchange rate and risk-neutral SDE:
Ft = X0 e(rd −rf )t ,
where
• Ft : forward price of foreign exchange rate (units of domestic ccy per foreign ccy unit);
• X0 : current foreign exchange rate (i.e. the spot rate);
• rd : interest rate of the domestic currency;
• rf : interest rate of the foreign currency.
Yves Guo
The Chinese University of Hong Kong (SZ)
136 / 154
Financial Stochastic Models
Basic Model
From the domestic investor’s point of view, foreign currency can be considered as an asset paying
continuous dividend. If we assume that the FX rate follows geometric Brownian motion, its
risk-neutral SDE will be
ft
dXt = (rd − rf )Xt dt + σX Xt dW
(known as Garman-Kohlhagen Model).
Call/Put Options.
The closed-form solution can be derived in the same way as the BSM formula for stocks paying
continuous dividends:
CallFX
=
e−rf T X0 N (d1 ) − e−rd T KN (d2 ),
PutFX
=
e−rd T KN (−d2 ) − e−rf T X0 N (−d1 ),
where
d1 =
ln
X0
K
2 T
+ rd − rf + 21 σX
√
σX T
Yves Guo
and
d2 = d1 − σX
√
T.
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Financial Stochastic Models
Domestic Risk-Neutral Probability for a Foreign Asset
Let S t be a foreign asset process and Xt be the exchange rate process (units of domestic
currency per unit foreign currency). Assume the following SDEs:
ef :
• under the foreign risk-neutral probability P
f f (t)
dS t = rf S t dt + σS S t dW
S
(7)
ed :
• under the domestic risk-neutral probability P
f d (t).
dXt = (rd − rf )Xt dt + σX Xt dW
X
(8)
Propositon 8.1
ed :
The SDE for the foreign asset under the domestic risk-neutral probability P
f d (t)
dSt = (rf − ρσS σX )S t dt + σS St dW
S
and
f d (t)
dYt = rd Yt dt + σY Yt dW
Y
f d (t)dW
f d (t) and σY =
where Yt = Xt St , ρdt = dW
X
S
Yves Guo
q
2 + σ 2 + 2ρσ σ
σX
X S
S
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Financial Stochastic Models
Proof
f f (t) = dW
f d (t) − θdt
Girsanov proposition implies dW
S
S
We note that Yt = Xt St is actually a domestic asset. So, under e
Pd , we should have
f d (t)]
dYt = Yt [rd dt + σY dW
Y
(9)
We determine θ and σY in the follows:
dYt
=
S t dXt + Xt dS t + dhX, Sit
=
f (t) + rf dt + σS dW
f (t) + ρσS σX dt]
S t Xt [(rd − rf )dt + σX dW
X
S
=
f d (t) + σS dW
f d (t)]
Yt [(rd + ρσS σX − θσS )dt + σX dW
X
S
q
2 + σ 2 + 2ρσ σ dW
f d (t)]
Yt [(rd + ρσS σX − θσS )dt + σX
X
S
Y
S
=
d
f
Compare the above with (9) we obtain
θ = ρσX
and
σY =
q
2 + σ 2 + 2ρσ σ
σX
X S
S
f f (t) = dW
f d (t) − ρσX dt in (7), we prove the proposition.
Applying dW
S
S
Yves Guo
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Financial Stochastic Models
Composite Option.
Definition of a Composite Call Option: CallCompo (T ) = (XT ST − K)+ .
For pricing composite options, it suffices to use the SDE
f d (t),
dYt = rd Yt dt + σY Yt dW
Y
with Yt = Xt S t and σY =
q
2 + σ 2 + 2ρσ σ .
σX
X S
S
For example, Composite Call Option:
ed
CallCompo (0) = E
(XT ST − K)+
|F 0 = X0 S0 N (d1 ) − e−rd T KN (d2 ),
MT
where
d1 =
ln
X0 S0
K
2 )T
+ (rd + 12 σY
√
σY T
Yves Guo
and
d2 = d1 − σY
√
T.
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Financial Stochastic Models
Quanto Option.
Definition of a Quanto Call Option: CallQuanto (T ) = (ST − K)+ paying out in domestic
currency;
ed , we get
Pricing under P
+
+
e d (ST − K)
e d [e−rd T (ST − K)+ ]
e d (ST − K) |F 0 = E
=E
CallQuanto (0) = E
MT
MT
=
2
fdT
T +σS W
e d [e−rd T (S0 e(rf −ρσX σS )T − 12 σS
S
E
− K)+ ]
=
2
f d (T )
T +σS W
e d [e(−rd +rf −ρσX σS )T e−(rf −ρσX σS )T (S0 e(rf −ρσX σS )T − 12 σS
S
E
− K)+ ]
=
e(−rd +rf −ρσX σS )T [S0 N (d1 ) − e−(rf −ρσS σX )T KN (d2 )]
=
e−(rd −rf )T (e−ρσS σX T S0 N (d1 ) − e−rf T KN (d2 )),
where
d1 =
ln
S0
K
2 )T
+ (rf − ρσX σS + 12 σS
√
σS T
Yves Guo
and
√
d2 = d1 − σS T .
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Financial Stochastic Models
Hedging
Because of the presence of FX risk, the hedging of a short position of a derivative V is realized through the
following self-financing portfolio Π(t, S t , Xt ):
• holding ∆S
t units of the asset (stock) S t ;
• holding ∆X
t units of the foreign currency;
• financing the position with domestic interest rate rd ;
• investing the portfolio value Π(t, S t , Xt ) in money market with rate rd .
S
X
In the below, we simplify the writing of Π(t, S t , Xt ), X t , S t , ∆S
t , ∆t and V (t, S t , Xt ) with Π, X, S, ∆ ,
∆X and V .
Yves Guo
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Financial Stochastic Models
Hedging - continued
b =
Denote Mt = erd t , Π
Π
Mt
= e−rd t Π and X ∗ = erf t X.
b is a martingale under e
The discounted portfolio Π
Pd , which implies
S
b = ∆ d(
dΠ
XS
X∗
X∗
b = ∆S d(e−rd t XS) + e−rf t ∆X d(e−(rd −rf )t X).
)+∆
d(
) = dΠ
Mt
Mt
where ∆X = erf t ∆X
∗
Or, equivalently,
dΠ
=
rd Πdt + e
rd t
S
∆ d(e
−rd t
XS) + e
S
(rd −rf )t
X
∆ d(e
−(rd −rf )t
X)
X
=
rd Πdt + ∆ [d(XS) − rd XSdt] + ∆ [dX − (rd − rf )Xdt]
=
rd Πdt + ∆ d(XS) + ∆ dX + rf ∆ Xdt − rd (∆ XS + ∆ X)dt,
S
X
X
S
X
where the fourth term is the carry return from the holding of foreign ccy and the last term represents the
financing in domestic ccy.
The value of the derivative paying out V (T, ST , XT ) can be delta-hedged by setting
V (t, S t , Xt ) = Π(t, S t , Xt ).
Yves Guo
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Financial Stochastic Models
Hedging - continued
b (t, S t , Xt ) = e−rd t V (t, S t , Xt ), we get
Applying Itô-Doeblin formula to V
b =
dV
b
b
∂V
∂V
∂V
∂V
−r t
dS +
dX + [. . .]dt = e d
dS +
dX + [. . .]dt.
∂S
∂X
∂S
∂X
(10)
From the previous slide, we have
b
dΠ
=
e
−rd t
∆ [d(XS) − rd XSdt] + e
=
e
−rd t
∆ (XdS + SdX) + e
S
S
−rd t
−rd t
X
∆ [dX − (rd − rf )Xdt]
X
∆ dX + [. . .]dt,
or,
b =e
dΠ
−rd t
S
[∆ XdS + (∆
X
S
+ ∆ S)dX] + [. . .]dt.
(11)
Equating the dS and dX terms in (10) and (11), we get:
∆
S
=
1 ∂V
X ∂S
and
Yves Guo
∆
X
=
∂V
S
− ∆ S.
∂X
The Chinese University of Hong Kong (SZ)
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Financial Stochastic Models
Change of Numeraire
and
Vanilla Interest Rate Option
(Optional)
Yves Guo
The Chinese University of Hong Kong (SZ)
145 / 154
Financial Stochastic Models
Numeraires
Numeraire and Risk Neutral Modelling
A numeraire is the unit of account in which other assets are denominated.
Rt
In the previous modeling with martingales, the money market account Mt = e 0 ru du has been
used as numeraire.
We note that
e all non-dividend paying asset price
• under the money market risk-neutral probability P,
R
St
− 0t ru du
St is a martingale;
denominated in units of Mt M = e
t
• any strictly positive, non-dividend-paying asset can be used as numeraire.
Yves Guo
The Chinese University of Hong Kong (SZ)
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Financial Stochastic Models
Numeraires
Tool Kit for Change of Numeraire
e(M ) be the probability measure under which all non-dividend
• Let M be a numeraire and P
paying assets denominated in the unites of M (i.e.
St
)
Mt
is a martingale.
• Let U be a strictly positive non-dividend paying asset. Hence,
• Define Zt =
Ut /U0
.
Mt /M0
e(U ) with
• We define P
Ut
Mt
e (M ) [ Zt |Fs ]
E
Zs
is a martingale.
We have
e (M ) [Zt ]
E
de
P(U )
de
P(M ) Ft
= Zt as Radon-Nikodym derivative process.
= 1 and
= 1.
The useful formulas for the change of numeraire are as follows:
e (M ) [ Y Zt | Fs ] ,
e (U ) [Y |Fs ] = 1 E
E
Zs
f (M ) it
f (U ) = dW
f (M ) − dhZ, W
dW
,
t
t
Zt
(Bayes’ formula)
(General form of Girsanov proposition)
for all 0 ≤ s ≤ t and any Ft -measurable variable Y .
Yves Guo
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Financial Stochastic Models
Usual Numeraires and Pricing Frameworks
(1) Money Market Account Mt = e
Rt
0
ru du
.
The pricing of a European derivative paying VT at T is
e VT |Ft ].
Vt = Mt E[
MT
(2) Zero Coupon Bond
A zero coupon bond paying 1 at maturity T is noted as Pt (T ) for its value at time t. The
probability measure with zero coupon bond as numeriare is called T -Forward Measure. The
pricing framework under T -Forward Measure admits a simple form which does not contain
discounting term inside the expectation operator.
e T [VT |Ft ].
Proposition Vt = Pt (T )E
Proof
Vt
=
=
e
Mt E
VT
Ft
MT
T
e [VT |Ft ]
Pt (T )E
Yves Guo
e
= Mt E
PT (T )VT
Ft
MT
(because PT (T ) = 1)
numeraire change with ZT =
PT (T )/Pt (T )
MT /Mt
.
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Financial Stochastic Models
Usual Numeraires and Pricing Frameworks
Denote Lt (Ti ) for the LIBOR rate seen at t for the period δi = Ti+1 − Ti . Then
Lt (Ti ) =
Pt (Ti ) − Pt (Ti+1 )
δi Pt (Ti+1 )
(arbitrage relationship).
eTi+1 .
Proposition Lt (Ti ) is a martingale under P
Proof Let 0 < s < t. We have
eTi+1 [Lt (Ti )|Fs ] = E
eTi+1
E
=
Ps (Ti ) − Ps (Ti+1 )
δi Ps (Ti+1 )
=
Ls (Ti ).
Yves Guo
Pt (Ti ) − Pt (Ti+1 )
Fs
δi Pt (Ti+1 )
Ps (Ti ) − Ps (Ti+1 )
is a portfolio ofnon-dividend paying assets
δi
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Financial Stochastic Models
Pricing of Cap/Floor Options
An interest rate Cap contract is a strip of call options (each option is called a Caplet) on LIBOR
rate paying δi (LTi (Ti ) − K)+ at time Ti+1 .
eTi+1 , Lt (Ti ) is a positive martingale and we have
If Lt (Ti ) is always positive, then, under P
Ti+1
f
dLt (Ti ) = Lt (Ti )σi (t)dW
t
.
The pricing of a Caplet is
e Ti+1 [δi (LT (Ti ) − K)+ |Ft ],
Vi (t) = Pt (Ti+1 )E
i
which can be solved by Black’s formula as in the swaption case.
Similar results can be obtained for the interest rate Floor contract which is a strip of put options
on LIBOR rate.
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Financial Stochastic Models
Usual Numeraires and Pricing Frameworks
(3) Annuity Factor
Annuity factor is defined as At =
n
X
δi−1 Pt (Ti ) where Ti ’s represent the LIBOR fixing/payment
i=1
e
dates. It is a P-martingale
and can be used as numeraire.
The receiver of a standard swap contract will:
• receive the fixed rate leg: P VF ixd = st At ;
• pay the floating rate leg: P VF loating = Pt (T0 ) − Pt (Tn )
n−1
X
actually P VF loating =
δi Lt (Ti )Pt (Ti+1 ) = Pt (T0 ) − Pt (Tn ) .
i=0
The market fair price of the swap rate is
st =
Pt (T1 ) − Pt (Tn )
At
(by equating P VF ixed and P VF loating ).
(Note: the discounting in the Swap valuation depends on the collateral situation with the counterparty. OIS curve is often used for
Swap between banks.)
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Financial Stochastic Models
Pricing of Swaptions
e A [(sT − K)+ |Ft ].
Proposition The pricing of a swaption is Vt = At E
Proof Let T be the start date of the underlying swap. Then, for t < T ,
"
Vt
=
=
#
AT (sT − K)+
Ft
MT
AT /At
A
e [(sT − K)+ |Ft ]
.
At E
numeraire change with ZT =
MT /Mt
e
Mt E
eA .
Proposition The swap rate is a martingale under P
Proof
eA [st |Fu ] = E
eA
E
=
Pu (T1 ) − Pu (Tn )
Au
=
su .
Yves Guo
Pt (T1 ) − Pt (Tn )
Fu
At
(Pu (T1 ) − Pu (Tn ) is a portfolio of non-dividend paying assets)
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Financial Stochastic Models
Pricing of Swaptions
eA , s is a positive martingale and we have
If s is positive, then, under P
fA .
dst = st σt dW
t
The price of a swaption (European call option on a swap starting at T ) is
e A [(sT − K)+ |Ft ],
Vt = A t E
e A [(sT − K)+ |Ft ] can be obtained by Black’s formula:
where E
st N (d1 ) − KN (d2 )
with
d1 =
ln
st
K
+1
qR 2
T
t
RT
t
2 du
σu
2 du
σu
Yves Guo
and
d2 =
ln
st
K
−1
qR 2
T
t
RT
t
2 du
σu
.
2 du
σu
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Financial Stochastic Models
Pricing of Vanilla Interest Rate Options
Note on Negative Interest Rate Situation
As we have already experienced negative interest rates for certain currencies, the modelling
requirement for non-negative interest rate is no more valid.
For Vanilla interest rate options (Cap/Floor and Swaption), some practitioners use Bachelier
Model (i.e. dSt = µdt + σdWt ) which assumes Normal distribution (instead of Log-Normal
distribution) for the underlying.
Yves Guo
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154 / 154
