Hedge with an Edge

1. Hedge with an Edge Mathematics of Finance April 7th and 14th 2012 Riaz Ahmed & Adnan Khan Lahore University of Management Sciences

2. 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 Derivatives by Thomas F. Siems (CATO Institute)

3. 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 STIGLITZ Interview with ABC

4. And the Ugly…… • What are derivatives? • How does one determine the price of such instruments? • The Math behind it all…………..

5. The Ugly

6. From Random Walks to Brownian Motion • Consider a random walk • Move right or left based on a coin toss

7. Random Walk • Define • The mean of Ri • The Variance of Ri

8. 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

9. Coin Tossing Game • The variance of the earnings is given by • So we have

13. 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?

14. 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?

15. The right scaling • Let the winnings (or alternately the step size) be • i.e. • The quadratic variation then is

16. 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

17. Brownian Motion • The expectation is given by • The variance is as • The limiting process is called Brownian Motion Bt or Weiner Process Wt

18. 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

19. Who ordered that…... • Taylor expanding etc… • Taking vanishingly small time steps and noting we scaled so that remained finite

20. 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

21. 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

22. Weiner Process • A continuous time continuous space stochastic process • Sample paths are continuous • Increments are Normally distributed • i.e. has pdf given by

23. Weiner Process • Increments are independent are i.id • The covariance is given by • In general

24. 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

25. Technical Mumbo Jumbo • A stochastic process is a P-Martingale with respect to a filtration if • is an information set for the process

26. 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

27. 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

28. 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

29. Simulating Weiner Processes • Consider the discretization • where and • Also each increment is given by

30. Sample Paths for Weiner Process

31. Numerical Expectation and Variance • Theoretically on the interval [0,t]

32. 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

33. 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)

34. Ito’s Lemma • Consider a function of a Weiner Process • Using Taylor’s rule • So we have • Ito SDE

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

36. Stochastic Integration • Consider Ito’s lemma • Integrating from 0 to t we have

37. Example • Q : Evaluate • A: Noting • And using the formula derived

38. Example • Q: Evaluate • A: Note that • The using the formula derived we have

39. 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

40. Properties of Ito Stochastic Integral • Linearity • Zero Mean • Ito Isometry

41. 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

42. ‘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

43. Extension of Ito’s Lemma • Consider a stochastic process • And a function of the process • An extension of Ito’s lemma gives

44. Solving SDE using Extended Ito’s Lemma • Geometric Brownian Motion • Let • Using Ito’s lemma we have

46. On to the Promised Land Finally some finance After Lunch