1. Zvi WienerContTimeFin - 1 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049

2. Zvi WienerContTimeFin - 1 slide 2 Main Books F Shimko D. Finance in Continuous Time, A Primer. Kolb Publishing Company, 1992, ISBN 1-878975-07-2 F Wilmott P., S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, A Student Introduction, Cambridge University Press, 1996, ISBN 0-521-49789-2

3. Zvi WienerContTimeFin - 1 slide 3 Useful Books F Duffie D., Dynamic Asset Pricing Theory. F Duffie D., Security Markets, Stochastic Models. F Neftci S., An Introduction to the Mathematics of Financial Derivatives. F Steele M., Invitation to Stochastic Differential Equations and Financial Applications. F Karatzas I., and S. Shreve, BM and Stochastic Calculus.

4. Zvi WienerContTimeFin - 1 slide 4 Primary Asset Valuation Discrete-time random walk: W(t+1) = W(t) + e(t+1); W(0) = W 0 ; e~i.i.d. N(0,1)

5. Zvi WienerContTimeFin - 1 slide 5 Primary Asset Valuation N(0,1)

6. Zvi WienerContTimeFin - 1 slide 6 Primary Asset Valuation Discrete-time random walk refinement: W(t+  ) = W(t) + e(t+  ); W(0) = W 0 ; e~i.i.d. N(0,  ) This process has the same expected drift and variance over n periods as the initial process had in one period.

7. Zvi WienerContTimeFin - 1 slide 7 Primary Asset Valuation N(0,1) t = 00.250.50.751

8. Zvi WienerContTimeFin - 1 slide 8 Primary Asset Valuation Set   dt W(t+ dt) = W(t) + e(t+ dt); W(0) = W 0 ; e~i.i.d. N(0, dt) Define dW(t) = W(t+dt) - W(t) white noise, (dt) a = 0 for any a > 1

9. Zvi WienerContTimeFin - 1 slide 9 Primary Asset Valuation Needs["Statistics`NormalDistribution`"] nor[mu_,sig_]:=Random[NormalDistribution[mu,sig]]; tt=NestList[ (#+nor[0, 0.1])&, 0, 300]; ListPlot[tt,PlotJoined->True,PlotLabel->"Random Walk"];

10. Zvi WienerContTimeFin - 1 slide 10 Primary Asset Valuation

11. Zvi WienerContTimeFin - 1 slide 11 Main Properties 1.E[dW(t)] = 0 2.E[dW(t) dt] = E[dW(t)] dt = 0 3.E[dW(t) 2 ] = dt

12. Zvi WienerContTimeFin - 1 slide 12 Main Properties (cont.) 4.Var[dW(t) 2 ] = E[dW(t) 4 ] - E 2 [dW(t) 2 ] = 3 dt 2 - dt 2 = 0 5.E[(dW(t)dt) 2 ] = E [dW(t) 2 ] (dt) 2 = 0 6.Var[dW(t)dt] = E[(dW(t)dt) 4 ] - E 2 [dW(t)dt] = 0

13. Zvi WienerContTimeFin - 1 slide 13 E[dW(t)] = 0 By definition the mean of the normally distributed variable is zero.

14. Zvi WienerContTimeFin - 1 slide 14 E[dW(t) dt] = E[dW(t)] dt = 0 The expectation of the product of a random variable (dW) and a constant (dt) equals the constant times the expected value of the random variable.

15. Zvi WienerContTimeFin - 1 slide 15 E[dW(t) 2 ] = dt For any distribution with zero mean the expected value of the squared random variable is the same as its variance.

16. Zvi WienerContTimeFin - 1 slide 16 Var[dW(t) 2 ] = E[dW(t) 4 ] - E 2 [dW(t) 2 ] = 3 dt 2 - dt 2 = 0 The fourth central moment of the standard normal distribution is 3, and (dt) 2 = 0.

17. Zvi WienerContTimeFin - 1 slide 17 E[(dW(t)dt) 2 ] = E [dW(t) 2 ] (dt) 2 = 0 Follows from properties 2 and 3.

18. Zvi WienerContTimeFin - 1 slide 18 Var[dW(t)dt] = E[(dW(t)dt) 4 ] - E 2 [dW(t)dt] = 0 Follows from properties 2 and 5.

19. Zvi WienerContTimeFin - 1 slide 19 Important Property if Var[f(dW)] = 0 then E[f(dW)] = f(dW)

20. Zvi WienerContTimeFin - 1 slide 20 Multiplication Rules Rule 1.(dW(t)) 2 = dt Rule 2.(dW(t)) dt = 0 Rule 3.dt 2 = 0

21. Zvi WienerContTimeFin - 1 slide 21 W(t) is called a standard Wiener process, or a Brownian motion.

22. Zvi WienerContTimeFin - 1 slide 22 Major Properties of W 1. W(t) is continuous in t. 2. W(t) is nowhere differentiable. 3. W(t) is a process of unbounded variation. 4. W(t) is a process of bounded quadratic variation.

23. Zvi WienerContTimeFin - 1 slide 23 Major Properties of W 5. The conditional distribution of W(u) given W(t), for u > t, is normal with mean W(t) and variance (u-t). 6. The variance of a forecast W(u) increases indefinitely as u .

24. Zvi WienerContTimeFin - 1 slide 24 Brownian Motion - BM The standard BM is useful since many general stochastic processes can be written in terms of W. X(t+1) = X(t) +  (X(t),t) +  (X(t),t) e(t+1) X(0) = X 0,e~i.i.d. N(0,1) generalized driftheteroscedastisity (changing variance)

25. Zvi WienerContTimeFin - 1 slide 25 Brownian Motion - BM Choose a shorter time interval  : X(t+  ) = X(t) +  (X(t),t)  +  (X(t),t) e(t+  ) X(0) = X 0,e~i.i.d. N(0,  ) generalized driftheteroscedastisity (changing variance)

26. Zvi WienerContTimeFin - 1 slide 26 Brownian Motion - BM As we let  dt we see that dX(t) =  (X(t),t) dt +  (X(t),t) dW(t) X(0) = X 0 generalized univariate Wiener process, (diffusion). dX =  (X,t) dt +  (X,t) dW, X(0) = X 0

27. Zvi WienerContTimeFin - 1 slide 27 Interpretation How can we interpret the fact that dX =  dt +  dW the random variable dX has local mean  dt and local variance  2 dt. A discrete analogy is X =  +  z.

28. Zvi WienerContTimeFin - 1 slide 28 Arithmetic BM dX =  dt +  dW Let  (X,t) = , and  (X,t) =  two constants. Then X follows an arithmetic Brownian Motion with drift  and volatility . This is an appropriate specification for a process that grows at a linear rate and exhibits an increasing uncertainty.

29. Zvi WienerContTimeFin - 1 slide 29 Arithmetic BM dX =  dt +  dW 1. X may be positive or negative. 2. If u > t, then X u is a future value of the process relative to time t. The distribution of X u given X t is normal with mean X t +  (u-t) and standard deviation  (u-t) 1/2. 3. The variance of a forecast X u tends to infinity as u does (for fixed t and X t ).

30. Zvi WienerContTimeFin - 1 slide 30 Arithmetic BM dX =  dt +  dW   time X

31. Zvi WienerContTimeFin - 1 slide 31 Arithmetic BM dX =  dt +  dW   time X

32. Zvi WienerContTimeFin - 1 slide 32 F Is appropriate for variables that can be positive and negative, have normally distributed forecast errors, and have forecast variance increasing linearly in time. F Example: net cash flows. F Is inappropriate for stock price. Arithmetic BM dX =  dt +  dW

33. Zvi WienerContTimeFin - 1 slide 33 Geometric BM dX =  Xdt +  XdW Let  (X,t) =  X, and  (X,t) =  X. Then X follows an geometric Brownian Motion.

34. Zvi WienerContTimeFin - 1 slide 34 Geometric BM dX =  Xdt +  XdW This is an appropriate specification for a process that F grows exponentially at an average rate of  F has volatility proportional to the level of the variable. F It also exhibits an increasing uncertainty.

35. Zvi WienerContTimeFin - 1 slide 35 Geometric BM dX =  Xdt +  XdW 1. If X(0) > 0, it will always be positive. 2. X has an absorbing barrier at X = 0. Thus if X hits zero (a zero probability event) it will remain there forever.

36. Zvi WienerContTimeFin - 1 slide 36 Geometric BM dX =  Xdt +  XdW 3. The conditional distribution of X u given X t is lognormal. The conditional mean of ln(X t ) is ln(X t ) +  (u-t) - 0.5  2 (u-t) and conditional standard deviation of ln(X t ) is  (u-t) 1/2. ln(X t ) is normally distributed.

37. Zvi WienerContTimeFin - 1 slide 37 Geometric BM dX =  Xdt +  XdW 4. The conditional expected value of X u is E t [X u ] = X t exp[  (u-t)] 5. The variance of a forecast X u tends to infinity as u does (for fixed t and X t ).

38. Zvi WienerContTimeFin - 1 slide 38 Geometric BM dX =  Xdt +  XdW time X

39. Zvi WienerContTimeFin - 1 slide 39 Geometric BM dX =  Xdt +  XdW time X

40. Zvi WienerContTimeFin - 1 slide 40 F Is often used to model security values, since the proportional changes in security price are independent and identically normally distributed (sometimes). F Example: currency price, stocks. F Is inappropriate for dividends, interest rates. Geometric BM dX =  Xdt +  XdW

41. Zvi WienerContTimeFin - 1 slide 41 Mean Reverting Process dX =  (  -X)dt +  X  dW Ornstein-Uhlenbeck when  = 1 Let  (X,t) =  (  -X), and  (X,t) =  X , where  0 - speed of adjustment  - long run mean  - volatility

42. Zvi WienerContTimeFin - 1 slide 42 Mean Reverting Process dX =  (  -X)dt +  X  dW This is an appropriate specification for a process that has a long run value but may be beset by short-term disturbances. We assume that , , and  are positive for simplicity.

43. Zvi WienerContTimeFin - 1 slide 43 Mean Reverting Process dX =  (  -X)dt +  X  dW 1. If X(0) > 0, it will always be positive. 2. As X approaches zero, the drift is positive and volatility vanishes. 3. As u becomes infinite, the variance of a forecast X u is finite.

44. Zvi WienerContTimeFin - 1 slide 44 Mean Reverting Process dX =  (  -X)dt +  X  dW 4. If  = 0.5, the distribution of X u given X t for u > t is non-central chi-squared, the mean of the distribution is: (X t -  ) exp[-  (u-t)] +  the variance of the distribution is (CIR 85):

45. Zvi WienerContTimeFin - 1 slide 45 Mean Reverting Process dX =  (  -X)dt +  X  dW time X 

46. Zvi WienerContTimeFin - 1 slide 46 Mean Reverting Process dX =  (  -X)dt +  X  dW time X 

47. Zvi WienerContTimeFin - 1 slide 47 Mean Reverting Process dX =  (  -X)dt +  X  dW SeedRandom[2] tt=NestList[ (#+0.3(1-#)+0.1*#*nor[0,0.1])&, 1.01,130]; ListPlot[tt,PlotJoined->True, Axes->False];

48. Zvi WienerContTimeFin - 1 slide 48 F Is often used to model economic variables and do not represent traded assets. F Example: interest rates, volatility. Mean Reverting Process dX =  (  -X)dt +  X  dW

49. Zvi WienerContTimeFin - 1 slide 49 Ito’s lemma Consider a real valued function f(X): R  R. Taylor series expansion:

50. Zvi WienerContTimeFin - 1 slide 50 Ito’s lemma If X is a “standard” variable, then  2 is o(  )

51. Zvi WienerContTimeFin - 1 slide 51 Ito’s lemma If X is a stochastic variable (following diffusion) then the term dX 2 does NOT vanish.

52. Zvi WienerContTimeFin - 1 slide 52 Ito’s lemma

53. Zvi WienerContTimeFin - 1 slide 53 Ito’s lemma If f = f(X,t) and dX =  dt +  dW, then

54. Zvi WienerContTimeFin - 1 slide 54 Financial Applications A Suppose that a security with value V guarantees $1dt every instant of time forever. This is the continuous time equivalent of a risk-free perpetuity of $1. If the risk-free interest rate is constant r, what is the (discounted) value of the security?

55. Zvi WienerContTimeFin - 1 slide 55 Financial Applications A 1. V = V(t), there are NO stochastic variables. dV = V t dt 2. The expected capital gain on V is ECG = E[dV] = V t dt 3. The expected cash flows to V is ECF = 1 dt 4. The total return on V is ECG + ECF = (V t +1)dt

56. Zvi WienerContTimeFin - 1 slide 56 Financial Applications A 5. Since there is no risk, the total return must be equal to the risk-free return on V, or rVdt. (V t +1) dt = r V dt 6. Divide both sides by dt: V t = rV - 1

57. Zvi WienerContTimeFin - 1 slide 57 Financial Applications A V t = rV - 1 DSolve[ V'[t]==r*V[t]-1, V[t], t ] V(t) = c Exp[r t] + 1/r given V(0) one can find c

58. Zvi WienerContTimeFin - 1 slide 58 Financial Applications B Suppose that X follows a geometric Brownian motion with drift  and volatility . A security with value V collects Xdt continuously forever. V represents a perpetuity that grows at an average exponential rate of , but whose risks in cash flow variations are considered diversificable. The economy is risk-neutral, and the risk-free interest rate is constant at r. What is the value of this security?

59. Zvi WienerContTimeFin - 1 slide 59 Financial Applications B 1. V = V(X), since V is a perpetual claim, its price does not depend on time. dV = V x dX + 0.5 V xx dX 2, dX =  Xdt +  XdW, dX 2 =  2 X 2 dt dV = [  XV x +0.5  2 X 2 V xx ]dt +  XV x dW

60. Zvi WienerContTimeFin - 1 slide 60 Financial Applications B 2. The expected capital gain: ECG = E[dV] = [  XV x +0.5  2 X 2 V xx ]dt since E[dW] = 0 3. The Expected cash flow: ECF = X dt

61. Zvi WienerContTimeFin - 1 slide 61 Financial Applications B 4. Total return: TR = ECG + ECF = [  XV x +X+0.5  2 X 2 V xx ]dt 5. But the return must be equal to the risk free return on the same investment V. rVdt = [  XV x +X+0.5  2 X 2 V xx ]dt 6. Thus the PDE: rV =  XV x +X+0.5  2 X 2 V xx

62. Zvi WienerContTimeFin - 1 slide 62 Financial Applications B rV =  XV x +X+0.5  2 X 2 V xx there are several ways to solve it. One can guess that doubling X will double the price V. If V is proportional to X, then V =  X, V x = , and V xx =0, then the equation becomes r  X=  X  +X  = 1/(r-  ) V(X) = X/(r-  )