UoM - Martingales with Applications to Finance: Part II

TOC

1. Continuous Time Martingales 1.1 Intro and Definition 1.2 Stopping time and Doob’s theorem 1.3 Doob-Meyer Decomposition Theorem 2. Discrete Time Random Models in Finance 2.1 Definition and Settings 2.2 The No-Arbitrage Conditions 2.3 Completeness of the market 2.4 Asset Pricing 2.5 The Cox-Ross-Rubinstein Model

1. Continuous Time Martingales

1.1 Intro and Definition

In Part I note, the time index we consider is . Almost all the results for discrete time martingales also hold for continuous time martingales. Let be a stochastic process, that is, a collection of random variables which describe systems that evolve randomly in time. Let be a family of increasing -fields, .

Definition: Continuous Martingale

is said to be a martingale (supermartingale, submartingale) w.r.t. if

is -measurable (determined)

is integrable, i.e.

for every :

Martingale:
supermartingale:
submartingale:

Definition: Brownian motion

is called a Brownian motion if:

is continuous in wiht .

has independent increments, i.e. for any

are independent random variables.

for every , . In particular, .

Example 1

Show a Brownian motion is a martingale w.r.t. .

Proof

field is generated by process , thus is -measurable.

As , is integrable.

let , we have

Thus a Brownian motion is a martingale w.r.t.

Example 2

Let be a Brownian motion. Show is a martingale w.r.t. .

Proof

is function of thus is -measurable.

Since , is integrable.

let , we have

Thus is a martingale w.r.t.

Definition: Poisson Process

is said to be Poisson process of rate if:

is right continuous with left limits in and .

has independent increments, i.e. for any

are independent random variables.

for every , , i.e.

Example 1

Let be a Poisson process of rate . Show is a martingale w.r.t. .

Proof

is a function of thus is -measurable.

Since has a Poisson distribution, is integrable.

let , we have

Thus is a martingale w.r.t. .

Example 2

Let be a Poisson process of rate . For , show is a martingale w.r.t.

Proof

is a funciton of and thus is -measurable.

As has a Poisson distribution, is integrable.

let , we have

Note that

Substitute this back to the above equation to get

1.2 Stopping time and Doob’s theorem

Stopping time: A non-negative random variable is a stopping time w.r.t. if for any .

Doob’s optional stopping theorem

Let be a martingale and and almost surely finite stopping time. In each of the flowing two cases, we have .

Case (i). is bounded, i.e. there is a constant such that .

Case (ii). The sequence is bounded by some integrable random variable , i.e.

for all .

Doob’s maximum inequality

Let be a martingale such that , for some . Then for every

and if ,

Martingale convergence theorem

Let be a Brownian motion with . Find the probability that the Brownian motino leaves the interval at the point where are two positive numbers.

Solution

is a martingale. Define

is the first time at which hits or . The is a stopping time w.r.t. . Since or , we have

Since for all , it follows from the Doob’s theorem that , which is

Solve (1) and (2), we get

Note that all the processes considered are assumed to be right continuous with left limits.

1.3 Doob-Meyer Decomposition Theorem

Definition: local martingale

is said to be a local martingale if there exists an increasing sequence of stopping times such that

almost surely as

for every , the stopped process is a martingale

Definition: bounded variation (function)

Let be a real-valued function on .

We say that is of bounded variation on the interval if

where the sup is taken over all the possible partitions of the interval .

Example 1

If is differentiable, say , then is of bouned variation on any finite interval .

Solution

Let be any partition of the interval . We have

which implies that

Example 2

If is an increasing function, then is of bounded variation on any finite interval .

Solution

Let be any partition of the interval . We have

Hence,

Definition: bounded variation (process)

We say that a process is a bounded variation process if for almost all , the function is of bounded variation.

Definition: semimartingale

A process is said to be a semimartingale if for some local martingale and bounded variation process .

Definition: quadratic covariation process

Let be two semimartingales. The quadratic covariation process of and , denoted by is defined as

where is a sequence of partitions of the interval such that as .

If , is also called the quadratic variation process of .

Example 1

If is a continuous process of bounded variation, then .

Solution

Let be a sequence of partitions of the interval such that as . We have

where is some constant because is of bounded variation. Since is continuous in and since , it follows that as . Hence we deduce that

Example 2

Let be a standard Brownian motion. Let be a sequence of partitions of the interval such that as . Prove

Hence, .

Proof

Since , we have

Hence,

If , by the independence we have

On the other hand,

Combining the above calculations together, we obtain

as .

Doob-Meyer Decomposition Theorem

Let be a supermatingale. Then has a decomposition , where is a local martingale and is a predictable, increasing process, and . Such a decomposition is unique.

2. Discrete Time Random Models in Finance

2.1 Definition and Settings

We work on a fixed probability space , where is finite, i.e. and . The probability space is equipped with a sequence of increasing -fields with . represents the collection of events (information) up to time . In the model, the initial time (current) is and the terminal date is . The time runs from to .

Definition: Asset Prices

A financial market contains traded financial assets, whose prices at time are denoted by , where are random variables representing the prices of risky assets, for example, stocks, houses, and is the price of a riskless asset (e.g. bank account). We assume that is -measurable (determined) meaning that is determined by what happened up to time on the market.

Definition: Portfolio

A portfolio at time is a division of the investor’s capital between different assets. It means that units are put in the bank account, units for the asset 1 (e.g. stock).

We assume that is -measurable (determined) meaning that is determined by what happened up to time . This is reasonable because the investor selects his time portfolio after observing what happened before time (up to time ). A portfolio is also called a trading strategy.

Definition: Value of Portfolio

The value of a portfolio at time is defined by

reflects the market value of the portfolio just after it has been established at time , while is the value jjust after time prices are observed, but before changes are made in the portfolio. Hence,

is the change in the market value due to changes in asset prices which occur between time and .

Definition: Gain Process

The gain (cumulative) process of a porfolio is given by

Definition: Discounted Price Process

is the price of the riskless asset, thus

is called the discounted prices process.

Definition: Discounted Value Process

The discounted value process of a portfolio is defined by

The discounted gain process is

Definition: Self-financing

A portfolio is called self-financing if by

The self-financing says that although the investor adjusts his strategy, the total wealth (value) remains the same as . This means that the investor does not bring in or consume any wealth.

Note that a trading strategy is self-financing w.r.t. iff it is self-financing w.r.t. the discounted price (i.e. ) because

That is

Proposition I: A trading strategy is self-financing iff

meaning that the discounted value at time is the sum of the initial value and the net gain.

Proof

i.e.

Conversly, suppose

for , let to get

which implies that

Let to get

which yields

Continuing the above procedure, we obtain

Hence is self-financing.

Proposition II: If is measurable and is measurable, there is a unique process such that is measurable and is a self-financing strategy with initial value .

Proof

We will determine according to the condition for self-financing. If is self-financing, then by proposition I

where . On the other hand

Comparing the above two equations, we obtain that

So is uniquely determined and self-financing.

2.2 The No-Arbitrage Conditions

Definition: Arbitrage Opportunity (Strategy)

A self-financing portfolio is called an arbitrage opportunity or arbitrage strategy if and the terminal wealth of satisfies

That is, using the arbitraget strategy , one starts with nothing , but end up with something . Arbitrage is “making something out of nothing”.

Definition: Arbitraget Free

We say that a market is arbitrage free if there are no arbitrage opportuities in the class of self-financing trading strategies.

Definition: Martingale Probability Measure

A probability measure on with is called a martingale probability measure if the discounted prices is -martingale, i.e.

Denote by the collection of all equivalent martingale probability measures.

Proposition: Let be an equivalent martingale probability measure and is any self-financing strategy. The the discounted value process is a -martingale, i.e.

where denotes the expectation computed under the new probability measure .

Proof

Since is self-financing, by Porsosition I: .

Hence,

Since is determined by and is -martingale. We have

Completing the proof

Theorem: Arbitrage-free Condition

A market is arbitrage-free iff there exists an equivalent martingale probability measure (i.e. is a martingale under ).

Sufficiency Proof

Assume that an equivalent martingale measure exists. For any portfolio with and , we have

On the other hand, is a martingale w.r.t. , thus

the equation is valid because measurement changes do not change the “zero property”.

This yields that , a.s. and hence . If one starts with nothing, one ends up still with nothing. So the market is free of arbitrage.

Example (*)

Consider a single time period model, where we have the current time and the terminal time . Suppose that the sample space consists of two outcomes. The market has two assets: a riskless bank account with price and a stock with price . Suppose the price at is given by

The prices at time are determined by the following rules. where is the interest rate.

Determin for which values of the market is free of arbitrage.

Solution:

According to the Arbitrage-free Condition, the market is arbitrage-free iff there exists a equivalent martigale measure . Suppose , . If is an equivalent martingale probability measure, then the discounted price process is a martingale. In particular

That is

On the other hand,

which yields

Rearranging the terms

thus

To make sure exists, we must have , which is equivalent to

2.3 Completeness of the market

Definition: Contingent Claim

A contingent claim (payoff) with maturity date is an arbitrary non-negative measurable random variable representing a cash flow.

Definition: attainable

A contingent claim is said to be attainable if there exists a self-financing portfolio is a replicating strategy of the claim .

Note, if is attainable, then one can find a trading strategy such that terminal wealth is equal to .

Definition: Market Complete

A market is complete if every contingent claim is attainable, i.e. there exists a replicating self-financing portfolio such that .

Theorem: An arbitrage-free market is complete iff there exists a unique equivalent martingale probability measure .

Necessity Proof

Suppose the market is complete. Then there is only one equivalent martingale probability measure. If be two martingale probability measure, we will show .

Since are both martingale meaure, thus

This yields

For any , choose and use above equation we have

Thus .

2.4 Asset Pricing

In an arbitrage-free complete market, the claim is attainable. Let be a replicating strategy for . Then . The value process is called the arbitrage-free price of the contingent claim at time .

Let be the equivalent martingale measure. Then the discounted value process is a martingale w.r.t. . Hence for

Therefore,

In particular, the arbitrage free pricea at time is given by

In summary, to find the arbitrage free price of a claim , one first determines a martingale probability measure and then compute the above equation.

In example (*):

Find the arbitrage-free price at time 0 for the claim if .

Determine a replicating strategy for the claim above.

Solution

The unique martingale probability measure is given by

If , then

The price is given by

The replicating strategy for the claim is determined by

Put and in the above equation to get

which yields . So the strategy is to do nothing for the bank account and buy one share of the stock in order to replicate .

European Call Option

Cosider an European call option with strike price and terminal date . They payoff of this option is

The price of the European option is

The contingent claim (payoff) of an European put option with strike price is

Example

Consider a single time period financial market, where we have the current time and the terminal time . Suppose that the sample space consists of two outcomes. The market has two assets: a riskless bank account with price and a stock with price . Suppose the price at is given by

The prices at time are determined by the following rules. , where is the interest rate

Determine for which values of the market is free of arbitrage.

Determine whether the arbitrage-free market is complete or not.

Consider an European put option (for the risky asset) at a strike price . Find the price for the European put option when .

Find a replicating strategy for the claim in 3

Solution

If is an equivalent martingale probability measure, then the dsicounted price procss is a martingale. In particular

That is

Since

thus

which yields

To make sure exists, we must have , which is equivalent to

The market is complete since the martingale probability measure is uniquely given by

The European put option has the payoff

If , then and . The price of the European put option is

The replicating strategy for the claim is determined by

thus

Solve the above equation to obtain that .

Suppose there are one riskless asset with interest rate and one risky asset whose price is determined by

Let

The sample space is the collection of T-tuples , where the family of -fields if taken as

is the probability meaure on such that are independent and identically distributed random variables with .

2.5 The Cox-Ross-Rubinstein Model

Theorem:

A martingale probability measure exists iff

If the above equation holds true, there is a unique such measure in characterized by

Proof

Since , we have

So the discounted price is a -martingale iff for

Since is independent of we have,

thus

In order that exists one must have

which is equivalent to and it’s easy to see that is uniquely determined.

Proposition:

If , then the Cox-Ross-Rubinstein model is arbitrage-free and complete and the arbitrage-free price of a contingent claim in the Cox-Ross-Rubinstein model is given by

where

Example:

A European call option with expiry and strike price wirtten on the stock . The arbitrage price is given by

where .

Proof

Since

The claim for the European call option is . So the price is given by

Note that the random variable takes the values with

Therefore, the equation is valid.

Heding

Since the Cox-Ross-Rubinstein model is complete, we can hedge every claim. To hedge the European call option, we set

Proposition: heding (replicating) strategy

The hedging (replicating) strategy of the European call option with time of expiry and strike price is given by

Proof

The discounted value process is a martingale, then the value process is given by

Therefore, we have

Since or , so

Subtract get:

thus

Use any of the equations in the above system to obtain