An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu - - PowerPoint PPT Presentation

An Introduction to Stochastic Calculus

Haijun Li

lih@math.wsu.edu 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 claim payable at time t, then its value at time t ≤ T is given by Vt = EQ

• e−

T

t

ruduVT|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 claim payable at time t, then its value at time t ≤ T is given by Vt = EQ

• e−

T

t

ruduVT|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 ∂t + µ(t, x)∂u ∂x + 1 2σ2(t, x)∂2u ∂x2 = r(x)u(t, x), x ≥ 0, t ∈ [0, T] 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 ∂t + µ(t, x)∂u ∂x + 1 2σ2(t, x)∂2u ∂x2 = r(x)u(t, x), x ≥ 0, t ∈ [0, T] 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 ∂t + µ(t, x)∂u ∂x + 1 2σ2(t, x)∂2u ∂x2 = r(x)u(t, x), x ≥ 0, t ∈ [0, T] 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 ∂t + µ(t, x)∂u ∂x + 1 2σ2(t, x)∂2u ∂x2 = r(x)u(t, x), x ≥ 0, t ∈ [0, T] 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−

T

t

r(Xs)dsh(XT)|Xt = x

• .

Haijun Li An Introduction to Stochastic Calculus Week 12 6 / 18

A PDE Connection

Consider a parabolic partial differential equation ∂u ∂t + µ(t, x)∂u ∂x + 1 2σ2(t, x)∂2u ∂x2 = r(x)u(t, x), x ≥ 0, t ∈ [0, T] 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−

T

t

r(Xs)dsh(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. Consider Mt = exp{ t

0 bsσ(s, Xs)dBs − 1 2

t

0 b2 sσ2(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. Consider Mt = exp{ t

0 bsσ(s, Xs)dBs − 1 2

t

0 b2 sσ2(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 satisfying Novikov’s condition E(exp{1

2

T

0 b2 sσ2(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. Consider Mt = exp{ t

0 bsσ(s, Xs)dBs − 1 2

t

0 b2 sσ2(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 satisfying Novikov’s condition E(exp{1

2

T

0 b2 sσ2(s, Xs)ds}) < ∞.

Example: In Girsanov’s Theorem, Mt = exp{−qBt − 1

2q2t} 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. If µ = 0, Xt = X0 + t

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 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. If µ = 0, Xt = X0 + t

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(M2

t ) < ∞ for

any t ≥ 0, then there exists a unique Ft-adapted stochastic process σM(t) with E(σ2

M(t)) < ∞ (called the volatility process), such that

Mt = M0 + t

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. If µ = 0, Xt = X0 + t

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(M2

t ) < ∞ for

any t ≥ 0, then there exists a unique Ft-adapted stochastic process σM(t) with E(σ2

M(t)) < ∞ (called the volatility process), such that

Mt = M0 + t

0 σM(s)dBs.

Example: Let X be a random variable on the probability space (Ω, FT, P) with EX 2 < ∞. Then X = E(X|FT) = E(X) + T

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

• f 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

• f 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

• f 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. Example: In Girsanov’s Theorem, dQ

dP = exp{−qBt − 1 2q2t} 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. Define a new probability measure Q(A) =

• A MTdP, where

Mt = exp{ t

0 bsdBs − 1 2

t

0 b2 sds}, 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. Define a new probability measure Q(A) =

• A MTdP, where

Mt = exp{ t

0 bsdBs − 1 2

t

0 b2 sds}, t ∈ [0, T], is an exponential

martingale with respect to Ft. Clearly, Q and P are equivalent. The stochastic process ˜ Bt = − t

0 bsds + 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. Define a new probability measure Q(A) =

• A MTdP, where

Mt = exp{ t

0 bsdBs − 1 2

t

0 b2 sds}, t ∈ [0, T], is an exponential

martingale with respect to Ft. Clearly, Q and P are equivalent. The stochastic process ˜ Bt = − t

0 bsds + Bt, t ∈ [0, T], is standard

Brownian motion under the probability measure Q. Note that − t

0 bsds + Bt represents a stochastic process with a

predetermined drift − t

0 bsds 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. Define a new probability measure Q(A) =

• A MTdP, where

Mt = exp{ t

0 bsdBs − 1 2

t

0 b2 sds}, t ∈ [0, T], is an exponential

martingale with respect to Ft. Clearly, Q and P are equivalent. The stochastic process ˜ Bt = − t

0 bsds + Bt, t ∈ [0, T], is standard

Brownian motion under the probability measure Q. Note that − t

0 bsds + Bt represents a stochastic process with a

predetermined drift − t

0 bsds 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 µ′(t, x) be a continuous function such that µ′(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 µ′(t, x) be a continuous function such that µ′(t, x) − µ(t, x) σ(t, x) satisfies Novikov’s Condition. Construct a new probability measure Q with the Radon-Nikodym derivative dQ dP = exp T bsdBs − 1 2 T b2

sds

• , bt = µ′(t, Xt) − µ(t, Xt)

σ(t, Xt) .

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 µ′(t, x) be a continuous function such that µ′(t, x) − µ(t, x) σ(t, x) satisfies Novikov’s Condition. Construct a new probability measure Q with the Radon-Nikodym derivative dQ dP = exp T bsdBs − 1 2 T b2

sds

• , bt = µ′(t, Xt) − µ(t, Xt)

σ(t, Xt) . Under Q, X is a solution of the SDE dXt = µ′(t, Xt)dt + σ(t, Xt)d ˜ Bt, where ˜ Bt 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 determined at time t, is defined as f(t, s) = −∂ ln P(t,s)

∂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 determined at time t, is defined as f(t, s) = −∂ ln P(t,s)

∂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 determined at time t, is defined as f(t, s) = −∂ ln P(t,s)

∂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 V(t, s) = e−

t

0 ruduP(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 determined at time t, is defined as f(t, s) = −∂ ln P(t,s)

∂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 V(t, s) = e−

t

0 ruduP(t, s), 0 ≤ t ≤ s,

is a martingale with respect to Pt. Thus V(t, s) = EQ(V(s, s)|Pt) which leads to P(t, s) = EQ

• e−

s

t rudu|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 rt = r0e−κt + κ t

0 e−κ(t−u)θ(u)du + σ

t

0 e−κ(t−u)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 rt = r0e−κt + κ t

0 e−κ(t−u)θ(u)du + σ

t

0 e−κ(t−u)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 rt = r0e−κt + κ t

0 e−κ(t−u)θ(u)du + σ

t

0 e−κ(t−u)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 time t is given by C(t) = EQ

• e−

s

t ruduC(s)|Ft

• .

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 rt = r0e−κt + κ t

0 e−κ(t−u)θ(u)du + σ

t

0 e−κ(t−u)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 time t is given by C(t) = EQ

• e−

s

t ruduC(s)|Ft

• .

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 risk-neutral measure, we have µ(t, s) = σ(t, s) 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 risk-neutral measure, we have µ(t, s) = σ(t, s) s

t σ(t, y)dy. So the

term structures are uniquely determined. Using Itô Lemma, the bond price satisfies the SDE dP(t, s) = rtP(t, s)dt − s

t

σ(t, y)dy

• P(t, s)dBt

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 risk-neutral measure, we have µ(t, s) = σ(t, s) s

t σ(t, y)dy. So the

term structures are uniquely determined. Using Itô Lemma, the bond price satisfies the SDE dP(t, s) = rtP(t, s)dt − s

t

σ(t, y)dy

• P(t, s)dBt

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, with µ(t, s) = σ(t, s) 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, with µ(t, s) = σ(t, s) 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, with µ(t, s) = σ(t, s) 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 sometime difficult because the joint distribution of e−

s

t rudu 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, with µ(t, s) = σ(t, s) 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 sometime difficult because the joint distribution of e−

s

t rudu and

C(s) under Q needs to be identified. Rewrite: df(t, s) = σ(t, s) 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, with µ(t, s) = σ(t, s) 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 sometime difficult because the joint distribution of e−

s

t rudu and

C(s) under Q needs to be identified. Rewrite: df(t, s) = σ(t, s) s

t σ(t, y)dydt + dBt

• , 0 ≤ t ≤ s.

Consider ˜ Bs

t =

t

0 b(u, s)du + Bt, where b(u, s) = −

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 ˜ Bs

t , 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 ˜ Bs

t , 0 ≤ t ≤ s, is standard Brownian motion under Qs.

Under Qs, df(t, s) = σ(t, s)d ˜ Bs

t , 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 ˜ Bs

t , 0 ≤ t ≤ s, is standard Brownian motion under Qs.

Under Qs, df(t, s) = σ(t, s)d ˜ Bs

t , 0 ≤ t ≤ s, becomes a martingale.

For any Ft-contingent claim C(s) payable at time s, its discounted price e−

t

0 ruduC(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 ˜ Bs

t , 0 ≤ t ≤ s, is standard Brownian motion under Qs.

Under Qs, df(t, s) = σ(t, s)d ˜ Bs

t , 0 ≤ t ≤ s, becomes a martingale.

For any Ft-contingent claim C(s) payable at time s, its discounted price e−

t

0 ruduC(t) is a martingale under Q.

It follows from the Martingale Representation Theorem that d

• e−

t

0 ruduC(t)

• = σC(t)dBt for some volatility process σC(t).

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 ˜ Bs

t , 0 ≤ t ≤ s, is standard Brownian motion under Qs.

Under Qs, df(t, s) = σ(t, s)d ˜ Bs

t , 0 ≤ t ≤ s, becomes a martingale.

For any Ft-contingent claim C(s) payable at time s, its discounted price e−

t

0 ruduC(t) is a martingale under Q.

It follows from the Martingale Representation Theorem that d

• e−

t

0 ruduC(t)

• = σC(t)dBt for some volatility process σC(t).

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 ˜ Bs

t , 0 ≤ t ≤ s, is standard Brownian motion under Qs.

Under Qs, df(t, s) = σ(t, s)d ˜ Bs

t , 0 ≤ t ≤ s, becomes a martingale.

For any Ft-contingent claim C(s) payable at time s, its discounted price e−

t

0 ruduC(t) is a martingale under Q.

It follows from the Martingale Representation Theorem that d

• e−

t

0 ruduC(t)

• = σC(t)dBt for some volatility process σC(t).

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 e−

s

t rudu is separated from the contingent claim payoff 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 ˜ Bs

t , 0 ≤ t ≤ s, is standard Brownian motion under Qs.

Under Qs, df(t, s) = σ(t, s)d ˜ Bs

t , 0 ≤ t ≤ s, becomes a martingale.

For any Ft-contingent claim C(s) payable at time s, its discounted price e−

t

0 ruduC(t) is a martingale under Q.

It follows from the Martingale Representation Theorem that d

• e−

t

0 ruduC(t)

• = σC(t)dBt for some volatility process σC(t).

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 e−

s

t rudu is separated from the contingent claim payoff under Qs.

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 d P(t, T) P(t, s)

• = −P(t, T)

P(t, s) T

s

σ(t, y)dy

• d ˜

Bs

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)+. 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 d P(t, T) P(t, s)

• = −P(t, T)

P(t, s) T

s

σ(t, y)dy

• d ˜

Bs

t .

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 d P(t, T) P(t, s)

• = −P(t, T)

P(t, s) T

s

σ(t, y)dy

• d ˜

Bs

t .

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 d P(t, T) P(t, s)

• = −P(t, T)

P(t, s) T

s

σ(t, y)dy

• d ˜

Bs

t .

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