An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 12
Haijun Li
An Introduction to Stochastic Calculus
Week 12
1 / 18
Outline
1
More on Change of Measure
Risk-Neutral Measure
Construction of Risk-Neutral and Distorted Measures
Continuous-Time Interest Rate Models
The Forward Risk Adjusted Measure and Bond Option Pricing
The World is Incomplete
Haijun Li
An Introduction to Stochastic Calculus
Week 12
2 / 18
Risk-Neutral Measure
A risk-neutral measure is a probability measure under which the
underlying risky asset has the same expected return as the
riskless bond (or money market account).
Haijun Li
An Introduction to Stochastic Calculus
Week 12
3 / 18
Risk-Neutral Measure
A risk-neutral measure is a probability measure under which the
underlying risky asset has the same expected return as the
riskless bond (or money market account).
We often demand more for bearing uncertainty. To price assets,
the calculated values need to be adjusted for the risk involved.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
3 / 18
Risk-Neutral Measure
A risk-neutral measure is a probability measure under which the
underlying risky asset has the same expected return as the
riskless bond (or money market account).
We often demand more for bearing uncertainty. To price assets,
the calculated values need to be adjusted for the risk involved.
One way of doing this is to first take the expectation under the
physical distribution and then adjust for risk.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
3 / 18
Risk-Neutral Measure
A risk-neutral measure is a probability measure under which the
underlying risky asset has the same expected return as the
riskless bond (or money market account).
We often demand more for bearing uncertainty. To price assets,
the calculated values need to be adjusted for the risk involved.
One way of doing this is to first take the expectation under the
physical distribution and then adjust for risk.
A better way is to first adjust the probabilities of future outcomes
by incorporating the effects of risk, and then take the expectation
under those adjusted, ‘virtual’ risk-neutral probabilities.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
3 / 18
Risk-Neutral Measure
A risk-neutral measure is a probability measure under which the
underlying risky asset has the same expected return as the
riskless bond (or money market account).
We often demand more for bearing uncertainty. To price assets,
the calculated values need to be adjusted for the risk involved.
One way of doing this is to first take the expectation under the
physical distribution and then adjust for risk.
A better way is to first adjust the probabilities of future outcomes
by incorporating the effects of risk, and then take the expectation
under those adjusted, ‘virtual’ risk-neutral probabilities.
Definition
A risk-neutral measure is a probability measure under which the
current value of all financial assets at time t is equal to the expected
future payoff of the asset discounted at the risk-free rate, given the
information structure available at time t.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
3 / 18
Complete Market
The existence of a risk-neutral measure involves absence of
arbitrage in a complete market.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
4 / 18
Complete Market
The existence of a risk-neutral measure involves absence of
arbitrage in a complete market.
A market is complete with respect to a trading strategy if all cash
flows for the trading strategy can be replicated by a similar
synthetic trading strategy.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
4 / 18
Complete Market
The existence of a risk-neutral measure involves absence of
arbitrage in a complete market.
A market is complete with respect to a trading strategy if all cash
flows for the trading strategy can be replicated by a similar
synthetic trading strategy.
For example, consider the put-call parity: A put is synthesized by
buying the call, investing the strike at the risk-free rate, and
shorting the stock.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
4 / 18
Complete Market
The existence of a risk-neutral measure involves absence of
arbitrage in a complete market.
A market is complete with respect to a trading strategy if all cash
flows for the trading strategy can be replicated by a similar
synthetic trading strategy.
For example, consider the put-call parity: A put is synthesized by
buying the call, investing the strike at the risk-free rate, and
shorting the stock.
If at some time before maturity, they differ, then someone else
could purchase the cheaper portfolio and immediately sell the
more expensive one to make risk-less profit (since they have the
same value at maturity).
Haijun Li
An Introduction to Stochastic Calculus
Week 12
4 / 18
Complete Market
The existence of a risk-neutral measure involves absence of
arbitrage in a complete market.
A market is complete with respect to a trading strategy if all cash
flows for the trading strategy can be replicated by a similar
synthetic trading strategy.
For example, consider the put-call parity: A put is synthesized by
buying the call, investing the strike at the risk-free rate, and
shorting the stock.
If at some time before maturity, they differ, then someone else
could purchase the cheaper portfolio and immediately sell the
more expensive one to make risk-less profit (since they have the
same value at maturity).
In insurance markets, a complete market models the situation that
agents can buy insurance contracts to protect themselves against
any future time and state-of-the-world.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
4 / 18
Fundamental Theorem of Arbitrage-Free Pricing
Consider a finite state market.
1
There is no arbitrage if and only if there exists a risk-neutral
measure that is equivalent to the physical probability measure.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
5 / 18
Fundamental Theorem of Arbitrage-Free Pricing
Consider a finite state market.
1
There is no arbitrage if and only if there exists a risk-neutral
measure that is equivalent to the physical probability measure.
2
In absence of arbitrage, a market is complete if and only if there is
a unique risk-neutral measure that is equivalent to the physical
probability measure.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
5 / 18
Fundamental Theorem of Arbitrage-Free Pricing
Consider a finite state market.
1
There is no arbitrage if and only if there exists a risk-neutral
measure that is equivalent to the physical probability measure.
2
In absence of arbitrage, a market is complete if and only if there is
a unique risk-neutral measure that is equivalent to the physical
probability measure.
Let B = (Bt , t ≥ 0) denote standard Brownian motion and Ft the
natural filtration generated by B. When risky asset price is driven by a
single Brownian motion, there is a unique risk-neutral measure Q.
Harrison-Pliska Theorem
If (rt , t ≥ 0) is the short rate process driven by Brownian motion, and
Vt is any Ft -adapted contingent claimR payable at time t, then its value
T
at time t ≤ T is given by Vt = EQ e− t ru du VT |Ft .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
5 / 18
Fundamental Theorem of Arbitrage-Free Pricing
Consider a finite state market.
1
There is no arbitrage if and only if there exists a risk-neutral
measure that is equivalent to the physical probability measure.
2
In absence of arbitrage, a market is complete if and only if there is
a unique risk-neutral measure that is equivalent to the physical
probability measure.
Let B = (Bt , t ≥ 0) denote standard Brownian motion and Ft the
natural filtration generated by B. When risky asset price is driven by a
single Brownian motion, there is a unique risk-neutral measure Q.
Harrison-Pliska Theorem
If (rt , t ≥ 0) is the short rate process driven by Brownian motion, and
Vt is any Ft -adapted contingent claimR payable at time t, then its value
T
at time t ≤ T is given by Vt = EQ e− t ru du VT |Ft .
The result can be extended to the case when the asset price is driven
by a semi-martingale (see Delbaen and Schachermayer 1994).
Haijun Li
An Introduction to Stochastic Calculus
Week 12
5 / 18
A PDE Connection
Consider a parabolic partial differential equation
∂u
∂u 1 2
∂2u
+ µ(t, x)
+ σ (t, x) 2 = r (x)u(t, x), x ≥ 0, t ∈ [0, T ]
∂t
∂x
2
∂x
subject to the terminal condition u(T , x) = h(x).
Haijun Li
An Introduction to Stochastic Calculus
Week 12
6 / 18
A PDE Connection
Consider a parabolic partial differential equation
∂u
∂u 1 2
∂2u
+ µ(t, x)
+ σ (t, x) 2 = r (x)u(t, x), x ≥ 0, t ∈ [0, T ]
∂t
∂x
2
∂x
subject to the terminal condition u(T , x) = h(x).
The functions µ, σ, h and r are known functions, and T is a
parameter.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
6 / 18
A PDE Connection
Consider a parabolic partial differential equation
∂u
∂u 1 2
∂2u
+ µ(t, x)
+ σ (t, x) 2 = r (x)u(t, x), x ≥ 0, t ∈ [0, T ]
∂t
∂x
2
∂x
subject to the terminal condition u(T , x) = h(x).
The functions µ, σ, h and r are known functions, and T is a
parameter.
It turns out that the solution can be expressed as a conditional
expectation with respect to an Itô process starting at x
dXt = µ(t, Xt )dt + σ(t, Xt )dBt , X0 = x.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
6 / 18
A PDE Connection
Consider a parabolic partial differential equation
∂u
∂u 1 2
∂2u
+ µ(t, x)
+ σ (t, x) 2 = r (x)u(t, x), x ≥ 0, t ∈ [0, T ]
∂t
∂x
2
∂x
subject to the terminal condition u(T , x) = h(x).
The functions µ, σ, h and r are known functions, and T is a
parameter.
It turns out that the solution can be expressed as a conditional
expectation with respect to an Itô process starting at x
dXt = µ(t, Xt )dt + σ(t, Xt )dBt , X0 = x.
The Feynman-Kac Formula
u(t, x) = E e−
Haijun Li
RT
t
r (Xs )ds
h(XT )|Xt = x .
An Introduction to Stochastic Calculus
Week 12
6 / 18
A PDE Connection
Consider a parabolic partial differential equation
∂u
∂u 1 2
∂2u
+ µ(t, x)
+ σ (t, x) 2 = r (x)u(t, x), x ≥ 0, t ∈ [0, T ]
∂t
∂x
2
∂x
subject to the terminal condition u(T , x) = h(x).
The functions µ, σ, h and r are known functions, and T is a
parameter.
It turns out that the solution can be expressed as a conditional
expectation with respect to an Itô process starting at x
dXt = µ(t, Xt )dt + σ(t, Xt )dBt , X0 = x.
The Feynman-Kac Formula
u(t, x) = E e−
RT
t
r (Xs )ds
h(XT )|Xt = x .
Example: For the Black-Scholes PDE of the European call option,
µ(t, x) = rx, σ(t, x) = σx, r (t, x) = r and h(x) = (x − K )+ .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
6 / 18
Exponential Martingales
Positive martingales play a central role in changing probability
measures. Since a necessary condition for an Itô process to be a
martingale is that its drift term vanishes, many continuous positive
martingales used in option pricing have an exponential form in
connection with Itô processes.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
7 / 18
Exponential Martingales
Positive martingales play a central role in changing probability
measures. Since a necessary condition for an Itô process to be a
martingale is that its drift term vanishes, many continuous positive
martingales used in option pricing have an exponential form in
connection with Itô processes.
As usual, let X denote a solution of an Itô SDE
dXt = µ(t, Xt )dt + σ(t, Xt )dBt .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
7 / 18
Exponential Martingales
Positive martingales play a central role in changing probability
measures. Since a necessary condition for an Itô process to be a
martingale is that its drift term vanishes, many continuous positive
martingales used in option pricing have an exponential form in
connection with Itô processes.
As usual, let X denote a solution of an Itô SDE
dXt = µ(t, Xt )dt + σ(t, Xt )dBt .
Rt
Rt
Consider Mt = exp{ 0 bs σ(s, Xs )dBs − 21 0 bs2 σ 2 (s, Xs )ds}, where
(bt , t ≥ 0) is an Ft -adapted stochastic process.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
7 / 18
Exponential Martingales
Positive martingales play a central role in changing probability
measures. Since a necessary condition for an Itô process to be a
martingale is that its drift term vanishes, many continuous positive
martingales used in option pricing have an exponential form in
connection with Itô processes.
As usual, let X denote a solution of an Itô SDE
dXt = µ(t, Xt )dt + σ(t, Xt )dBt .
Rt
Rt
Consider Mt = exp{ 0 bs σ(s, Xs )dBs − 21 0 bs2 σ 2 (s, Xs )ds}, where
(bt , t ≥ 0) is an Ft -adapted stochastic process.
Novikov’s Condition
The process Mt is a martingale with respect to Ft for any process bt
RT
satisfying Novikov’s condition E(exp{ 21 0 bs2 σ 2 (s, Xs )ds}) < ∞.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
7 / 18
Exponential Martingales
Positive martingales play a central role in changing probability
measures. Since a necessary condition for an Itô process to be a
martingale is that its drift term vanishes, many continuous positive
martingales used in option pricing have an exponential form in
connection with Itô processes.
As usual, let X denote a solution of an Itô SDE
dXt = µ(t, Xt )dt + σ(t, Xt )dBt .
Rt
Rt
Consider Mt = exp{ 0 bs σ(s, Xs )dBs − 21 0 bs2 σ 2 (s, Xs )ds}, where
(bt , t ≥ 0) is an Ft -adapted stochastic process.
Novikov’s Condition
The process Mt is a martingale with respect to Ft for any process bt
RT
satisfying Novikov’s condition E(exp{ 21 0 bs2 σ 2 (s, Xs )ds}) < ∞.
Example: In Girsanov’s Theorem, Mt = exp{−qBt − 12 q 2 t} is a
martingale.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
7 / 18
Itô Integral Representation
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Consider an Itô process dXt = µ(t, Xt )dt + σ(t, Xt )dBt .
Rt
If µ = 0, Xt = X0 + 0 σ(s, Xs )dBs becomes a martingale with
respect to Ft . Conversely, such an integral representation holds
for any square integrable martingale.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
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Itô Integral Representation
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Consider an Itô process dXt = µ(t, Xt )dt + σ(t, Xt )dBt .
Rt
If µ = 0, Xt = X0 + 0 σ(s, Xs )dBs becomes a martingale with
respect to Ft . Conversely, such an integral representation holds
for any square integrable martingale.
Martingale Representation Theorem
If a martingale (Mt , t ≥ 0) with respect to Ft satisfies E(Mt2 ) < ∞ for
any t ≥ 0, then there exists a unique Ft -adapted stochastic process
2 (t)) < ∞ (called the volatility process), such that
σM (t) with E(σM
Rt
Mt = M0 + 0 σM (s)dBs .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
8 / 18
Itô Integral Representation
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Consider an Itô process dXt = µ(t, Xt )dt + σ(t, Xt )dBt .
Rt
If µ = 0, Xt = X0 + 0 σ(s, Xs )dBs becomes a martingale with
respect to Ft . Conversely, such an integral representation holds
for any square integrable martingale.
Martingale Representation Theorem
If a martingale (Mt , t ≥ 0) with respect to Ft satisfies E(Mt2 ) < ∞ for
any t ≥ 0, then there exists a unique Ft -adapted stochastic process
2 (t)) < ∞ (called the volatility process), such that
σM (t) with E(σM
Rt
Mt = M0 + 0 σM (s)dBs .
Example: Let X be a random variable on the probability space
RT
(Ω, FT , P) with EX 2 < ∞. Then X = E(X |FT ) = E(X ) + 0 σX (s)dBs .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
8 / 18
Adjusted Measure: A Fundamental Idea of Distortion
We may want to price in uncertainty by adjusting the probability
measure under which our expectation is taken.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
9 / 18
Adjusted Measure: A Fundamental Idea of Distortion
We may want to price in uncertainty by adjusting the probability
measure under which our expectation is taken.
For a given Itô process, it means to adjust the probability of each
path of the process so that the Itô process under the new
probabilities has a specific drift.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
9 / 18
Adjusted Measure: A Fundamental Idea of Distortion
We may want to price in uncertainty by adjusting the probability
measure under which our expectation is taken.
For a given Itô process, it means to adjust the probability of each
path of the process so that the Itô process under the new
probabilities has a specific drift.
For pricing an option or a contingent claim, this often requires
finding a equivalent probability measure Q under which the
underlying asset price process has the same stochastic return as
that of the money market account (i.e., risk-neutral) or a process
of our choice (e.g., a long-term zero-coupon bond).
Haijun Li
An Introduction to Stochastic Calculus
Week 12
9 / 18
Adjusted Measure: A Fundamental Idea of Distortion
We may want to price in uncertainty by adjusting the probability
measure under which our expectation is taken.
For a given Itô process, it means to adjust the probability of each
path of the process so that the Itô process under the new
probabilities has a specific drift.
For pricing an option or a contingent claim, this often requires
finding a equivalent probability measure Q under which the
underlying asset price process has the same stochastic return as
that of the money market account (i.e., risk-neutral) or a process
of our choice (e.g., a long-term zero-coupon bond).
The Randon-Nikodym derivative dQ
dP of the adjusted measure Q
with respect to the physical measure P can be viewed as a
distortion factor for P that incorporates uncertainty. This distortion
factor often takes an exponential form.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
9 / 18
Adjusted Measure: A Fundamental Idea of Distortion
We may want to price in uncertainty by adjusting the probability
measure under which our expectation is taken.
For a given Itô process, it means to adjust the probability of each
path of the process so that the Itô process under the new
probabilities has a specific drift.
For pricing an option or a contingent claim, this often requires
finding a equivalent probability measure Q under which the
underlying asset price process has the same stochastic return as
that of the money market account (i.e., risk-neutral) or a process
of our choice (e.g., a long-term zero-coupon bond).
The Randon-Nikodym derivative dQ
dP of the adjusted measure Q
with respect to the physical measure P can be viewed as a
distortion factor for P that incorporates uncertainty. This distortion
factor often takes an exponential form.
1 2
Example: In Girsanov’s Theorem, dQ
dP = exp{−qBt − 2 q t} is a
distortion factor.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
9 / 18
An Extension of the Girsanov’s Theorem
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Let (bt , t ≥ 0) denote an Ft -adapted stochastic process, satisfying
Novikov’s condition.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
10 / 18
An Extension of the Girsanov’s Theorem
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Let (bt , t ≥ 0) denote an Ft -adapted stochastic process, satisfying
Novikov’s condition.
R
Define a new probability measure Q(A) = A MT dP, where
Rt
Rt
Mt = exp{ 0 bs dBs − 12 0 bs2 ds}, t ∈ [0, T ], is an exponential
martingale with respect to Ft . Clearly, Q and P are equivalent.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
10 / 18
An Extension of the Girsanov’s Theorem
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Let (bt , t ≥ 0) denote an Ft -adapted stochastic process, satisfying
Novikov’s condition.
R
Define a new probability measure Q(A) = A MT dP, where
Rt
Rt
Mt = exp{ 0 bs dBs − 12 0 bs2 ds}, t ∈ [0, T ], is an exponential
martingale with respect to Ft . Clearly,
Q and P are equivalent.
Rt
The stochastic process B̃t = − 0 bs ds + Bt , t ∈ [0, T ], is standard
Brownian motion under the probability measure Q.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
10 / 18
An Extension of the Girsanov’s Theorem
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Let (bt , t ≥ 0) denote an Ft -adapted stochastic process, satisfying
Novikov’s condition.
R
Define a new probability measure Q(A) = A MT dP, where
Rt
Rt
Mt = exp{ 0 bs dBs − 12 0 bs2 ds}, t ∈ [0, T ], is an exponential
martingale with respect to Ft . Clearly,
Q and P are equivalent.
Rt
The stochastic process B̃t = − 0 bs ds + Bt , t ∈ [0, T ], is standard
Brownian motion
under the probability measure Q.
Rt
Note that − 0 bs ds + Bt represents a stochastic process with a
Rt
predetermined drift − 0 bs ds under P. To make the drift
disappear, we adjust the probability of each path by multiplying a
distortion factor MT .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
10 / 18
An Extension of the Girsanov’s Theorem
Let B = (Bt , t ≥ 0) be standard Brownian motion on the probability
space (Ω, F, P), and Ft = σ(Bs , s ≤ t) the Brownian filtration.
Let (bt , t ≥ 0) denote an Ft -adapted stochastic process, satisfying
Novikov’s condition.
R
Define a new probability measure Q(A) = A MT dP, where
Rt
Rt
Mt = exp{ 0 bs dBs − 12 0 bs2 ds}, t ∈ [0, T ], is an exponential
martingale with respect to Ft . Clearly,
Q and P are equivalent.
Rt
The stochastic process B̃t = − 0 bs ds + Bt , t ∈ [0, T ], is standard
Brownian motion
under the probability measure Q.
Rt
Note that − 0 bs ds + Bt represents a stochastic process with a
Rt
predetermined drift − 0 bs ds under P. To make the drift
disappear, we adjust the probability of each path by multiplying a
distortion factor MT .
Consider an Itô SDE dXt = µ(t, Xt )dt + σ(t, Xt )dBt under the
probability measure P. It has a new drfit µ(t, Xt ) + σ(t, Xt )bt under
the distorted probability measure Q.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
10 / 18
Adjust for a Specified Drift
Pricing a contingent claim often requires us to find a probability
measure for which the underlying risky asset has a specified drfit.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
11 / 18
Adjust for a Specified Drift
Pricing a contingent claim often requires us to find a probability
measure for which the underlying risky asset has a specified drfit.
Consider an Itô SDE dXt = µ(t, Xt )dt + σ(t, Xt )dBt under the
probability measure P.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
11 / 18
Adjust for a Specified Drift
Pricing a contingent claim often requires us to find a probability
measure for which the underlying risky asset has a specified drfit.
Consider an Itô SDE dXt = µ(t, Xt )dt + σ(t, Xt )dBt under the
probability measure P.
Let µ0 (t, x) be a continuous function such that
µ0 (t, x) − µ(t, x)
σ(t, x)
satisfies Novikov’s Condition.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
11 / 18
Adjust for a Specified Drift
Pricing a contingent claim often requires us to find a probability
measure for which the underlying risky asset has a specified drfit.
Consider an Itô SDE dXt = µ(t, Xt )dt + σ(t, Xt )dBt under the
probability measure P.
Let µ0 (t, x) be a continuous function such that
µ0 (t, x) − µ(t, x)
σ(t, x)
satisfies Novikov’s Condition.
Construct a new probability measure Q with the Radon-Nikodym
derivative
)
(Z
Z
T
dQ
1 T 2
µ0 (t, Xt ) − µ(t, Xt )
= exp
bs dBs −
bs ds , bt =
.
dP
2 0
σ(t, Xt )
0
Haijun Li
An Introduction to Stochastic Calculus
Week 12
11 / 18
Adjust for a Specified Drift
Pricing a contingent claim often requires us to find a probability
measure for which the underlying risky asset has a specified drfit.
Consider an Itô SDE dXt = µ(t, Xt )dt + σ(t, Xt )dBt under the
probability measure P.
Let µ0 (t, x) be a continuous function such that
µ0 (t, x) − µ(t, x)
σ(t, x)
satisfies Novikov’s Condition.
Construct a new probability measure Q with the Radon-Nikodym
derivative
)
(Z
Z
T
dQ
1 T 2
µ0 (t, Xt ) − µ(t, Xt )
= exp
bs dBs −
bs ds , bt =
.
dP
2 0
σ(t, Xt )
0
Under Q, X is a solution of the SDE
dXt = µ0 (t, Xt )dt + σ(t, Xt )d B̃t , where B̃t is standard Brownian
motion under Q.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
11 / 18
Relation Between Bond Price and Short Rate
Consider a continuously trading bond market over [0, T ].
Let P(t, s), 0 ≤ t ≤ s ≤ T , be the price of a default-free zero
coupon bond at time t that pays one monetary unit at maturity s.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
12 / 18
Relation Between Bond Price and Short Rate
Consider a continuously trading bond market over [0, T ].
Let P(t, s), 0 ≤ t ≤ s ≤ T , be the price of a default-free zero
coupon bond at time t that pays one monetary unit at maturity s.
Let Pt be the σ-field generated by the bond prices P(t, s).
Haijun Li
An Introduction to Stochastic Calculus
Week 12
12 / 18
Relation Between Bond Price and Short Rate
Consider a continuously trading bond market over [0, T ].
Let P(t, s), 0 ≤ t ≤ s ≤ T , be the price of a default-free zero
coupon bond at time t that pays one monetary unit at maturity s.
Let Pt be the σ-field generated by the bond prices P(t, s).
The forward rate, compounded continuously for time s that is
P(t,s)
.
determined at time t, is defined as f (t, s) = − ∂ ln ∂s
Haijun Li
An Introduction to Stochastic Calculus
Week 12
12 / 18
Relation Between Bond Price and Short Rate
Consider a continuously trading bond market over [0, T ].
Let P(t, s), 0 ≤ t ≤ s ≤ T , be the price of a default-free zero
coupon bond at time t that pays one monetary unit at maturity s.
Let Pt be the σ-field generated by the bond prices P(t, s).
The forward rate, compounded continuously for time s that is
P(t,s)
.
determined at time t, is defined as f (t, s) = − ∂ ln ∂s
The short rate (i.e., instantaneous interest rate) at time t is defined
as rt = f (t, t).
Haijun Li
An Introduction to Stochastic Calculus
Week 12
12 / 18
Relation Between Bond Price and Short Rate
Consider a continuously trading bond market over [0, T ].
Let P(t, s), 0 ≤ t ≤ s ≤ T , be the price of a default-free zero
coupon bond at time t that pays one monetary unit at maturity s.
Let Pt be the σ-field generated by the bond prices P(t, s).
The forward rate, compounded continuously for time s that is
P(t,s)
.
determined at time t, is defined as f (t, s) = − ∂ ln ∂s
The short rate (i.e., instantaneous interest rate) at time t is defined
as rt = f (t, t).
To ensure no arbitrage for the the bond market, there exists a
risk-neutral measure Q such that for all s ≥ 0, the discounted
process
Rt
V (t, s) = e− 0 ru du P(t, s), 0 ≤ t ≤ s,
is a martingale with respect to Pt .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
12 / 18
Relation Between Bond Price and Short Rate
Consider a continuously trading bond market over [0, T ].
Let P(t, s), 0 ≤ t ≤ s ≤ T , be the price of a default-free zero
coupon bond at time t that pays one monetary unit at maturity s.
Let Pt be the σ-field generated by the bond prices P(t, s).
The forward rate, compounded continuously for time s that is
P(t,s)
.
determined at time t, is defined as f (t, s) = − ∂ ln ∂s
The short rate (i.e., instantaneous interest rate) at time t is defined
as rt = f (t, t).
To ensure no arbitrage for the the bond market, there exists a
risk-neutral measure Q such that for all s ≥ 0, the discounted
process
Rt
V (t, s) = e− 0 ru du P(t, s), 0 ≤ t ≤ s,
is a martingale with respect to Pt .
Thus V (t, s) = ERQ (V (s, s)|Pt ) which leads to
s
P(t, s) = EQ e− t ru du |Pt .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
12 / 18
Hull-White (Extended Vasicek) Interest Rate Model
Assume that the short rate rt follows the SDE
drt = κ(θ(t) − rt )dt + σdBt
under the risk-neutral measure Q, where the mean-reverting
intensity κ is a positive constant and the long-run average θ(t) is a
deterministic function.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
13 / 18
Hull-White (Extended Vasicek) Interest Rate Model
Assume that the short rate rt follows the SDE
drt = κ(θ(t) − rt )dt + σdBt
under the risk-neutral measure Q, where the mean-reverting
intensity κ is a positive constant and the long-run average θ(t) is a
deterministic function.
Solving it, the short
rate (Markov) process
is given by
R t −κ(t−u)
R t −κ(t−u)
−κt
rt = r0 e
+κ 0e
θ(u)du + σ 0 e
dBu .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
13 / 18
Hull-White (Extended Vasicek) Interest Rate Model
Assume that the short rate rt follows the SDE
drt = κ(θ(t) − rt )dt + σdBt
under the risk-neutral measure Q, where the mean-reverting
intensity κ is a positive constant and the long-run average θ(t) is a
deterministic function.
Solving it, the short
rate (Markov) process
is given by
R t −κ(t−u)
R t −κ(t−u)
−κt
rt = r0 e
+κ 0e
θ(u)du + σ 0 e
dBu .
Pt = Ft , the natural Brownian filtration.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
13 / 18
Hull-White (Extended Vasicek) Interest Rate Model
Assume that the short rate rt follows the SDE
drt = κ(θ(t) − rt )dt + σdBt
under the risk-neutral measure Q, where the mean-reverting
intensity κ is a positive constant and the long-run average θ(t) is a
deterministic function.
Solving it, the short
rate (Markov) process
is given by
R t −κ(t−u)
R t −κ(t−u)
−κt
rt = r0 e
+κ 0e
θ(u)du + σ 0 e
dBu .
Pt = Ft , the natural Brownian filtration.
The bond price P(t, s) can then be solved, and more generally, for
any Ft -contingent claim C(s) payable
at time s, its price C(t) at
R
− ts ru du
C(s)|Ft .
time t is given by C(t) = EQ e
Haijun Li
An Introduction to Stochastic Calculus
Week 12
13 / 18
Hull-White (Extended Vasicek) Interest Rate Model
Assume that the short rate rt follows the SDE
drt = κ(θ(t) − rt )dt + σdBt
under the risk-neutral measure Q, where the mean-reverting
intensity κ is a positive constant and the long-run average θ(t) is a
deterministic function.
Solving it, the short
rate (Markov) process
is given by
R t −κ(t−u)
R t −κ(t−u)
−κt
rt = r0 e
+κ 0e
θ(u)du + σ 0 e
dBu .
Pt = Ft , the natural Brownian filtration.
The bond price P(t, s) can then be solved, and more generally, for
any Ft -contingent claim C(s) payable
at time s, its price C(t) at
R
− ts ru du
C(s)|Ft .
time t is given by C(t) = EQ e
The closed form expression for P(t, s) is given by the so-called
affine form P(t, s) = eA(t,s)−B(s−t)rt , where A and B are explicit,
deterministic and independent of the short rate.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
13 / 18
The One-Factor Gaussian Forward Rate Model
Assume that under a risk-neutral probability measure Q, the
forward rate is governed by the SDE
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
where the deterministic function µ(t, s) is the term structure of the
forward rate drifts and the deterministic function σ(t, s) is the term
structure of the forward rate volatilities.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
14 / 18
The One-Factor Gaussian Forward Rate Model
Assume that under a risk-neutral probability measure Q, the
forward rate is governed by the SDE
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
where the deterministic function µ(t, s) is the term structure of the
forward rate drifts and the deterministic function σ(t, s) is the term
structure of the forward rate volatilities.
The forward rate processes are Gaussian.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
14 / 18
The One-Factor Gaussian Forward Rate Model
Assume that under a risk-neutral probability measure Q, the
forward rate is governed by the SDE
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
where the deterministic function µ(t, s) is the term structure of the
forward rate drifts and the deterministic function σ(t, s) is the term
structure of the forward rate volatilities.
The forward rate processes are Gaussian.
Since the discounted bond price V (t, s) is a martingale
under the
Rs
risk-neutral measure, we have µ(t, s) = σ(t, s) t σ(t, y )dy . So the
term structures are uniquely determined.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
14 / 18
The One-Factor Gaussian Forward Rate Model
Assume that under a risk-neutral probability measure Q, the
forward rate is governed by the SDE
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
where the deterministic function µ(t, s) is the term structure of the
forward rate drifts and the deterministic function σ(t, s) is the term
structure of the forward rate volatilities.
The forward rate processes are Gaussian.
Since the discounted bond price V (t, s) is a martingale
under the
Rs
risk-neutral measure, we have µ(t, s) = σ(t, s) t σ(t, y )dy . So the
term structures are uniquely determined.
Using Itô Lemma, the bond price satisfies the SDE
Z s
dP(t, s) = rt P(t, s)dt −
σ(t, y )dy P(t, s)dBt
t
Haijun Li
An Introduction to Stochastic Calculus
Week 12
14 / 18
The One-Factor Gaussian Forward Rate Model
Assume that under a risk-neutral probability measure Q, the
forward rate is governed by the SDE
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
where the deterministic function µ(t, s) is the term structure of the
forward rate drifts and the deterministic function σ(t, s) is the term
structure of the forward rate volatilities.
The forward rate processes are Gaussian.
Since the discounted bond price V (t, s) is a martingale
under the
Rs
risk-neutral measure, we have µ(t, s) = σ(t, s) t σ(t, y )dy . So the
term structures are uniquely determined.
Using Itô Lemma, the bond price satisfies the SDE
Z s
dP(t, s) = rt P(t, s)dt −
σ(t, y )dy P(t, s)dBt
t
This linear homogeneous equation with multiplicative noise can be
solved using the standard method.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
14 / 18
From Risk-Neutral to Forward Risk
Consider again the one-factor Gaussian forward rate model:
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
Rs
with µ(t, s) = σ(t, s) t σ(t, y )dy under the risk-neutral measure
Q.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
15 / 18
From Risk-Neutral to Forward Risk
Consider again the one-factor Gaussian forward rate model:
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
Rs
with µ(t, s) = σ(t, s) t σ(t, y )dy under the risk-neutral measure
Q.
For any Ft -adapted contingent claim C(s), payable at time s, its
price C(t), t ≤ s, can be obtained via expectation under Q.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
15 / 18
From Risk-Neutral to Forward Risk
Consider again the one-factor Gaussian forward rate model:
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
Rs
with µ(t, s) = σ(t, s) t σ(t, y )dy under the risk-neutral measure
Q.
For any Ft -adapted contingent claim C(s), payable at time s, its
price C(t), t ≤ s, can be obtained via expectation under Q.
The implementation of calculating the risk-neutral expectation
is
R
− ts ru du
sometime difficult because the joint distribution of e
and
C(s) under Q needs to be identified.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
15 / 18
From Risk-Neutral to Forward Risk
Consider again the one-factor Gaussian forward rate model:
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
Rs
with µ(t, s) = σ(t, s) t σ(t, y )dy under the risk-neutral measure
Q.
For any Ft -adapted contingent claim C(s), payable at time s, its
price C(t), t ≤ s, can be obtained via expectation under Q.
The implementation of calculating the risk-neutral expectation
is
R
− ts ru du
sometime difficult because the joint distribution of e
and
C(s) under Q needs to be identified.
R s
Rewrite: df (t, s) = σ(t, s) t σ(t, y )dydt + dBt , 0 ≤ t ≤ s.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
15 / 18
From Risk-Neutral to Forward Risk
Consider again the one-factor Gaussian forward rate model:
df (t, s) = µ(t, s)dt + σ(t, s)dBt , 0 ≤ t ≤ s,
Rs
with µ(t, s) = σ(t, s) t σ(t, y )dy under the risk-neutral measure
Q.
For any Ft -adapted contingent claim C(s), payable at time s, its
price C(t), t ≤ s, can be obtained via expectation under Q.
The implementation of calculating the risk-neutral expectation
is
R
− ts ru du
sometime difficult because the joint distribution of e
and
C(s) under Q needs to be identified.
R s
Rewrite: df (t, s) = σ(t, s) t σ(t, y )dydt + dBt , 0 ≤ t ≤ s.
Rt
Rs
Consider B̃ts = 0 b(u, s)du + Bt , where b(u, s) = − u σ(u, y )dy .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
15 / 18
Forward Risk Adjusted Measure
The Girsanov’s Theorem implies that there is a probability
measure Qs , called the forward risk adjusted measure, such that
B̃ts , 0 ≤ t ≤ s, is standard Brownian motion under Qs .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
16 / 18
Forward Risk Adjusted Measure
The Girsanov’s Theorem implies that there is a probability
measure Qs , called the forward risk adjusted measure, such that
B̃ts , 0 ≤ t ≤ s, is standard Brownian motion under Qs .
Under Qs , df (t, s) = σ(t, s)d B̃ts , 0 ≤ t ≤ s, becomes a martingale.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
16 / 18
Forward Risk Adjusted Measure
The Girsanov’s Theorem implies that there is a probability
measure Qs , called the forward risk adjusted measure, such that
B̃ts , 0 ≤ t ≤ s, is standard Brownian motion under Qs .
Under Qs , df (t, s) = σ(t, s)d B̃ts , 0 ≤ t ≤ s, becomes a martingale.
For any F
-contingent claim C(s) payable at time s, its discounted
Rt
− 0t ru du
price e
C(t) is a martingale under Q.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
16 / 18
Forward Risk Adjusted Measure
The Girsanov’s Theorem implies that there is a probability
measure Qs , called the forward risk adjusted measure, such that
B̃ts , 0 ≤ t ≤ s, is standard Brownian motion under Qs .
Under Qs , df (t, s) = σ(t, s)d B̃ts , 0 ≤ t ≤ s, becomes a martingale.
For any F
-contingent claim C(s) payable at time s, its discounted
Rt
− 0t ru du
price e
C(t) is a martingale under Q.
It follows
from the Martingale Representation Theorem that
R
− 0t ru du
C(t) = σC (t)dBt for some volatility process σC (t).
d e
Haijun Li
An Introduction to Stochastic Calculus
Week 12
16 / 18
Forward Risk Adjusted Measure
The Girsanov’s Theorem implies that there is a probability
measure Qs , called the forward risk adjusted measure, such that
B̃ts , 0 ≤ t ≤ s, is standard Brownian motion under Qs .
Under Qs , df (t, s) = σ(t, s)d B̃ts , 0 ≤ t ≤ s, becomes a martingale.
For any F
-contingent claim C(s) payable at time s, its discounted
Rt
− 0t ru du
price e
C(t) is a martingale under Q.
It follows
from the Martingale Representation Theorem that
R
− 0t ru du
C(t) = σC (t)dBt for some volatility process σC (t).
d e
This martingale representation, the SDE for P(t, s) and the Itô
Lemma imply that C(t)/P(t, s), 0 ≤ t ≤ s, is a martingale under
the forward risk adjusted measure Qs .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
16 / 18
Forward Risk Adjusted Measure
The Girsanov’s Theorem implies that there is a probability
measure Qs , called the forward risk adjusted measure, such that
B̃ts , 0 ≤ t ≤ s, is standard Brownian motion under Qs .
Under Qs , df (t, s) = σ(t, s)d B̃ts , 0 ≤ t ≤ s, becomes a martingale.
For any F
-contingent claim C(s) payable at time s, its discounted
Rt
− 0t ru du
price e
C(t) is a martingale under Q.
It follows
from the Martingale Representation Theorem that
R
− 0t ru du
C(t) = σC (t)dBt for some volatility process σC (t).
d e
This martingale representation, the SDE for P(t, s) and the Itô
Lemma imply that C(t)/P(t, s), 0 ≤ t ≤ s, is a martingale under
the forward risk adjusted measure Qs .
C(t)
= P(t, s)EQs (C(s)|Ft ). That is, the discounted process
R
− ts ru du
is separated from the contingent claim payoff under Qs .
e
Haijun Li
An Introduction to Stochastic Calculus
Week 12
16 / 18
Forward Risk Adjusted Measure
The Girsanov’s Theorem implies that there is a probability
measure Qs , called the forward risk adjusted measure, such that
B̃ts , 0 ≤ t ≤ s, is standard Brownian motion under Qs .
Under Qs , df (t, s) = σ(t, s)d B̃ts , 0 ≤ t ≤ s, becomes a martingale.
For any F
-contingent claim C(s) payable at time s, its discounted
Rt
− 0t ru du
price e
C(t) is a martingale under Q.
It follows
from the Martingale Representation Theorem that
R
− 0t ru du
C(t) = σC (t)dBt for some volatility process σC (t).
d e
This martingale representation, the SDE for P(t, s) and the Itô
Lemma imply that C(t)/P(t, s), 0 ≤ t ≤ s, is a martingale under
the forward risk adjusted measure Qs .
C(t)
= P(t, s)EQs (C(s)|Ft ). That is, the discounted process
R
− ts ru du
is separated from the contingent claim payoff under Qs .
e
This is useful for pension valuation for which one often needs to
evaluate the expected cash flow from a fixed income portfolio and
then discount it using a yield curve.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
16 / 18
Bond Option Pricing
Consider European call options on zero-coupon bond P(t, T ) with
strike price K and maturity s, t ≤ s ≤ T .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
17 / 18
Bond Option Pricing
Consider European call options on zero-coupon bond P(t, T ) with
strike price K and maturity s, t ≤ s ≤ T .
The payoff of the option is (P(s, T ) − K )+ .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
17 / 18
Bond Option Pricing
Consider European call options on zero-coupon bond P(t, T ) with
strike price K and maturity s, t ≤ s ≤ T .
The payoff of the option is (P(s, T ) − K )+ .
The forward rate f (t, s) follow the one-factor Gaussian model.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
17 / 18
Bond Option Pricing
Consider European call options on zero-coupon bond P(t, T ) with
strike price K and maturity s, t ≤ s ≤ T .
The payoff of the option is (P(s, T ) − K )+ .
The forward rate f (t, s) follow the one-factor Gaussian model.
The process P(t, T )/P(t, s) is a martingale under the forward risk
adjusted measure Qs , and satisfies
"Z
#
T
P(t, T )
P(t, T )
d
=−
σ(t, y )dy d B̃ts .
P(t, s)
P(t, s)
s
Haijun Li
An Introduction to Stochastic Calculus
Week 12
17 / 18
Bond Option Pricing
Consider European call options on zero-coupon bond P(t, T ) with
strike price K and maturity s, t ≤ s ≤ T .
The payoff of the option is (P(s, T ) − K )+ .
The forward rate f (t, s) follow the one-factor Gaussian model.
The process P(t, T )/P(t, s) is a martingale under the forward risk
adjusted measure Qs , and satisfies
"Z
#
T
P(t, T )
P(t, T )
d
=−
σ(t, y )dy d B̃ts .
P(t, s)
P(t, s)
s
Hence P(s, T ) = P(s, T )/P(s, s) has a lonnormal distribution
under Qs .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
17 / 18
Bond Option Pricing
Consider European call options on zero-coupon bond P(t, T ) with
strike price K and maturity s, t ≤ s ≤ T .
The payoff of the option is (P(s, T ) − K )+ .
The forward rate f (t, s) follow the one-factor Gaussian model.
The process P(t, T )/P(t, s) is a martingale under the forward risk
adjusted measure Qs , and satisfies
"Z
#
T
P(t, T )
P(t, T )
d
=−
σ(t, y )dy d B̃ts .
P(t, s)
P(t, s)
s
Hence P(s, T ) = P(s, T )/P(s, s) has a lonnormal distribution
under Qs .
The price of the call option can then be calculated using
φc (t) = P(t, s)EQs (P(s, T ) − K )+ .
Haijun Li
An Introduction to Stochastic Calculus
Week 12
17 / 18
Bond Option Pricing
Consider European call options on zero-coupon bond P(t, T ) with
strike price K and maturity s, t ≤ s ≤ T .
The payoff of the option is (P(s, T ) − K )+ .
The forward rate f (t, s) follow the one-factor Gaussian model.
The process P(t, T )/P(t, s) is a martingale under the forward risk
adjusted measure Qs , and satisfies
"Z
#
T
P(t, T )
P(t, T )
d
=−
σ(t, y )dy d B̃ts .
P(t, s)
P(t, s)
s
Hence P(s, T ) = P(s, T )/P(s, s) has a lonnormal distribution
under Qs .
The price of the call option can then be calculated using
φc (t) = P(t, s)EQs (P(s, T ) − K )+ .
The corresponding put price φp (t) = P(t, s)EQs (K − P(s, T ))+
may be obtained by the put-call parity.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
17 / 18
Market is Incomplete
If stock prices are modelled by Lévy processes, then a problem
arising from non-Gaussian option pricing is that the market is
incomplete.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
18 / 18
Market is Incomplete
If stock prices are modelled by Lévy processes, then a problem
arising from non-Gaussian option pricing is that the market is
incomplete.
That is, there may be more than one possible pricing formula. This
is clearly undesirable, and a number of selection principles, such
as entropy minimization, have been employed to overcome this
problem.
Haijun Li
An Introduction to Stochastic Calculus
Week 12
18 / 18