Hedge with an EdgeMathematics of Finance

April 7th and 14th 2012Riaz Ahmed & Adnan Khan

Lahore University of Management Sciences

The Good• To blame organizational failures solely on

derivatives is to miss the point. A better answer lies in greater reliance on market forces to control derivative-related risk taking.

10 Myths About Financial Derivativesby Thomas F. Siems (CATO Institute)

The Bad• “……and it was all caused by derivatives. So,

everybody said we better regulate derivatives better, we oughta have more sunshine, we oughta have more transparency..”

JOSEPH STIGLITZInterview with ABC

And the Ugly……• What are derivatives?• How does one determine the price of such

instruments?• The Math behind it all…………..

The Ugly

From Random Walks to Brownian Motion

• Consider a random walk

• Move right or left based on a coin toss

Random Walk• Define

• The mean of Ri

• The Variance of Ri

Coin Tossing Game• Heads - wins you a rupee• Tails - loses you a rupee• After n tosses the earnings are given by the rv

• Starting with no money, expected earnings after n tosses

Coin Tossing Game• The variance of the earnings is given by

• So we have

Making it more Interesting

• Consider the quadratic variation

• Let’s start flipping the coin faster say n tosses in time t

• Q: What should the winnings be so that the quadratic variation is finite and non zero?

In terms of the random walk• If we take n steps in time t, how long should

each step be so that the variation remains finite and non zero?

The right scaling

• Let the winnings (or alternately the step size) be

• i.e.

• The quadratic variation then is

Brownian Motion

• Consider the random walk, with step size taken every time interval

• In the limit as this scaling keeps the random walk finite and non zero

Brownian Motion• The expectation is given by

• The variance is

as

• The limiting process is called Brownian Motion Bt or Weiner Process Wt

Diffusion Equation-who ordered that?• Consider a random walk • Suppose at time t one is at position x• Consider the probability distribution at the

next time step

Who ordered that…...

• Taylor expanding etc…

• Taking vanishingly small time steps and noting we scaled so that remained finite

Variations……• Consider a function • Want to know how much the function changes

over an interval , where • Summing over all intervals we define the n

variation as

• n=1 (absolute) and n=2 (quadratic) variation

Functions….Nice and otherwise

• Nice functions are those with total variation finite

• Can define the integral over an interval for such functions (our old friend the Riemann Steiltjes Integral)

• Bad functions may not have finite total variation, e.g. on

Weiner Process• A continuous time continuous space stochastic

process • • Sample paths are continuous• Increments are Normally distributed

• i.e. has pdf given by

Weiner Process• Increments are independent

are i.id• The covariance is given by

• In general

Martingales• A sequence of random variables (stochastic

process) where the expectation of the next value is equal to the present observed value

• This means knowledge of past events cannot help predict the future

• E.g. Ones position in a random walk• E.g. Earnings in the coin tossing game

Technical Mumbo Jumbo

• A stochastic process is a P-Martingale with respect to a filtration if

• is an information set for the process

Wiener Process

• Wiener Process is a Martingale

• Increments are mean zero normal and since we are taking expectations at time t, the process is determined up to time t

• Hence

Weiner Process is Markovian

• A stochastic process satisfies the Markov property if

• i.e. The process is memoryless

• The future depends on what is now irrespective of the past

Mean Square Convergence• Consider the function F(X). If then it is said that F(X)converges to in the

mean square sense

• For a Weiner process

• This ‘means’ that in the mean square sense

Simulating Weiner Processes

• Consider the discretization

• where and

• Also each increment is given by

Sample Paths for Weiner Process

Numerical Expectation and Variance

• Theoretically on the interval [0,t]

Taylor Series Ito’s Lemma• For a deterministic variable X

• However if we try the same for Brownian motion, the higher order terms cannot be dropped as (at least the expected value in the MSS).

• Actually as

Ito’s Lemma• For a stochastic variable X

• Q: If , what is

• A: Using Ito’s lemma• This is an example of an Ito Stochastic Differential

Equation (SDE)

Ito’s Lemma• Consider a function of a Weiner Process

• Using Taylor’s rule

• So we have• Ito SDE

Examples• Q: Obtain an SDE for• A: We have• Using Ito we get

• Q: Obtain an SDE for• A: Using Ito we have

Stochastic Integration

• Consider Ito’s lemma

• Integrating from 0 to t we have

Example• Q : Evaluate

• A: Noting

• And using the formula derived

Example

• Q: Evaluate

• A: Note that

• The using the formula derived we have

Ito Stochastic Integral• Let f(x(t),t) be a function of the Stochastic

Process X(t)• The Ito Stochastic Integral is defined if

• The integral is defined as

• where the limit is in the sense that given

means

Properties of Ito Stochastic Integral

• Linearity

• Zero Mean

• Ito Isometry

Diffusion Processes• Consider the SDE

• Here A(G(t),t) is called the drift• And B(G(t),t) is called the diffusion

• Q: Given A and B can we determine G? i.e. solve the SDE

‘Solving’ SDE

• We derived SDE given the process

• Usually are given SDE, want to ‘solve’ the SDE to find the process

• Need to extend Ito’s lemma to be able to do this

Extension of Ito’s Lemma

• Consider a stochastic process

• And a function of the process

• An extension of Ito’s lemma gives

Solving SDE using Extended Ito’s Lemma

• Geometric Brownian Motion

• Let

• Using Ito’s lemma we have

On to the Promised LandFinally some finance

After Lunch