1. Zvi WienerContTimeFin - 5 slide 1 Financial Engineering Continuous Time Finance Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049

2. Zvi WienerContTimeFin - 5 slide 2 F Futures Contracts F Mark to market F Convergence property F Spot-futures parity F Cost-of-carry F Martingale F Risk-neutral Measure F Forwards and Futures F Girsanov’s Theorem and its counterpart F Feynman-Kac Formula F Stochastic optimization F The Maximum Principle

3. Zvi WienerContTimeFin - 5 slide 3 Futures Markets Futures and forward contracts are similar to options in that they specify purchase or sale of some underlying security at some future date. However a future contract means an obligation of both sides. It is a commitment rather than an investment.

4. Zvi WienerContTimeFin - 5 slide 4 Basics of Futures Contracts Delivery of a commodity at a specified place, price, quantity and quality. Example: no. 2 hard winter wheat or no. 1 soft red wheat delivered at an approved warehouse by December 31, 1997.

5. Zvi WienerContTimeFin - 5 slide 5 Basics of Futures Contracts Long position – commits to purchase the commodity. Short position – commits to deliver. At maturity: Profit to long = Spot pr. at maturity – Original futures pr. Profit to short = Original futures pr. –Spot pr. at maturity it is a zero sum game

6. Zvi WienerContTimeFin - 5 slide 6 Futures Markets The initial investment is zero however some margin is required. The later cash flow is mark-to-market for a future contract and is concentrated in one point for the forward contract. Futures are standardized and not specify the counterside.

7. Zvi WienerContTimeFin - 5 slide 7 Futures Markets F Currencies – all major currencies, including cross rate F Agricultural – corn, wheat, meat, coffee, sugar, lumber, rice F Metals and Energy – copper, gold, silver, oil, gas, aluminum F Interest Rates Futures – eurodollars, T-bonds, LIBOR, Municipal, Fed funds F Equity Futures – S&P 500, NYSE index, OTC, FT-SE, Toronto

8. Zvi WienerContTimeFin - 5 slide 8 Mechanism of Trading Long Short commodity money Clearinghouse Long Short

9. Zvi WienerContTimeFin - 5 slide 9 Marking to Market Example: initial margin on corn is 10%. 1 contract is for 5,000 bushels, price of one bushel is 2.2775, so you have to post the initial margin = $1,138.75 = 0.1*2.2755*5000 If the futures price goes from 2.2775 to 2.2975 the clearinghouse credits the margin account of the long position for 5000 bushels x 2 cents or $100 per contract.

10. Zvi WienerContTimeFin - 5 slide 10 Marking to Market Your balance time Maint. margin margin call Initial margin

11. Zvi WienerContTimeFin - 5 slide 11 Marking to Market and Margin The current futures price for silver delivered in five days is $5.10 (per ounce). One futures contract is for 5,000 ounces

12. Zvi WienerContTimeFin - 5 slide 12 Marking to Market and Margin DayFutures Price 0 (today)$5.10 1$5.20 2$5.25 3$5.18 4$5.18 5$5.21

13. Zvi WienerContTimeFin - 5 slide 13 Marking to Market and Margin DayFuturesP&L/oz.Margin 1$5.20 5.20-5.10= 0.10 500 2$5.25 5.25-5.20= 0.05 250 3$5.18 5.18-5.25=-0.07-350 4$5.18 5.18-5.18= 0.00 0 5$5.21 5.21-5.18= 0.03 150 Total:$550 Compare the total to forward: (5.21–5.10)5000

14. Zvi WienerContTimeFin - 5 slide 14 Convergence Property The futures price and the spot price must converge at maturity. Otherwise there will be an arbitrage based on actual delivery. Sometimes delivery is costly!

15. Zvi WienerContTimeFin - 5 slide 15 Futures Markets Cash delivery: sometimes is allowed, sometimes is the only way to deliver. The question of quality is resolved with a conversion factor. The cheapest to deliver option.

16. Zvi WienerContTimeFin - 5 slide 16 Futures Markets The S&P 500 futures calls for delivery of $500 times the value of the index. If at maturity the index is at 475, then $500x475=$237,500 cash is the delivery value. If the contract was written on the futures price 470 (some time ago), who will pay money? Short side will pay to the long side.

17. Zvi WienerContTimeFin - 5 slide 17 Futures Markets Strategies Hedging and Speculation – efficient tool for hedging and speculation. A significant leverage effect.

18. Zvi WienerContTimeFin - 5 slide 18 Basis Risk and Hedging The basis is the difference between the futures price and the spot price. (At maturity it approaches zero). This risk is important if the futures position is not held till maturity and is liquidated in advance. Spread position is when an investor is long a futures with one ttm and short with another.

19. Zvi WienerContTimeFin - 5 slide 19 Spot-Futures Parity Theorem Create a riskless position involving a futures contract and the spot position. Buy one stock for S and take a short futures position in it. The only difference is from dividends. Thus F + D – S is riskless. The amount of money invested is S.

20. Zvi WienerContTimeFin - 5 slide 20 Spot-Futures Parity Theorem Create a riskless position involving a futures contract and the spot position. Buy one stock for S and take a short futures position in it. The only difference is from dividends. Thus F + D – S is riskless. The amount of money invested is S.

21. Zvi WienerContTimeFin - 5 slide 21 Spot-Futures Parity Theorem Cost-of-carry relationship

22. Zvi WienerContTimeFin - 5 slide 22 Spot-Futures Parity Theorem Cost-of-carry relationship For contract maturing in T periods

23. Zvi WienerContTimeFin - 5 slide 23 Relationship for Spreads This is a rough approximation based on an assumption that there is a single source of risk and all contracts are perfectly correlated.

24. Zvi WienerContTimeFin - 5 slide 24 Martingale X - a stochastic time dependent variable. E t - expectation based on information available at time t. X t is a martingale if for any s > t E t (X s ) = X t

25. Zvi WienerContTimeFin - 5 slide 25 Martingale Most financial variables are not martingales because of the drift component (inflation, interest rates, cost of storage, etc.) However one can change a numeraire so that the new financial variable becomes a martingale. What can be chosen for an ABM, GBM?

26. Zvi WienerContTimeFin - 5 slide 26 Martingale dX =  dt +  dZABM E t (X s ) = X t +  (s-t) set Y t = X t -  t then dY t =  dt +  dZ -  dt =  dZ hence E t (Y s ) = Y t

27. Zvi WienerContTimeFin - 5 slide 27 Martingale dX =  Xdt +  XdZGBM What is E t (X s )? set Y t = X t e -  t dY = e -  t dX -  e -  t Xdt = e -  t  Xdt + e -  t  XdZ -  e -  t Xdt = (  -  )Ydt +  YdZ. What is E t (Y s )?

28. Zvi WienerContTimeFin - 5 slide 28 Martingale dY = (  -  )Ydt +  YdZ then d(lnY) = (  -  - 0.5  2 ) dt +  dZ lnY t = lnY 0 + (  -  - 0.5  2 ) t +  Z if a~N( ,  ), then E(e a ) =exp(  +0.5  2 ) lnY t ~N(lnY 0 + (  -  - 0.5  2 ) t,  t) Then E 0 (Y t ) = Y 0 exp((  -  - 0.5  2 )t+0.5  2 t).

29. Zvi WienerContTimeFin - 5 slide 29 Martingale E 0 (Y t ) = Y 0 exp((  -  - 0.5  2 )t+0.5  2 t). Set  =  E 0 (Y t ) = Y 0 E t (Y s ) = Y t - martingale! What is the economic meaning of Y?

30. Zvi WienerContTimeFin - 5 slide 30 Equivalent Martingale Measure F Harrison and Kreps F Harrison and Pliska There exists a risk neutral probability measure. There exists an equivalent martingale measure. For a detailed explanation, see Duffie. Extension to a stochastic volatility, see Grundy, Wiener.

31. Zvi WienerContTimeFin - 5 slide 31 Forward Contract if W and r are independent F t =E t Q (W)

32. Zvi WienerContTimeFin - 5 slide 32 Futures Contract Mark-to-market procedure equates the instantaneous price to zero.

33. Zvi WienerContTimeFin - 5 slide 33 Girsanov’s Theorem Let dX =  (X,t)dt +  (X,t)dZ. If there exist and , such that  =  -, then there exists a new probability measure equivalent to the original one, such that relative to the new measure the original process X becomes: dX = (X,t)dt +  (X,t)dZ*

34. Zvi WienerContTimeFin - 5 slide 34 by a change of the probability measure (note B*), if there exists a process such that. Girsanov’s Theorem can be transformed to *

35. Zvi WienerContTimeFin - 5 slide 35 change of variables can be transformed to (Girsanov) * 1. (Theorem 1) 2. (Theorem 1’) 3.

36. Zvi WienerContTimeFin - 5 slide 36 Monotonic change of variables preserves order x y 2 y 2 x 1 1 y x

37. Zvi WienerContTimeFin - 5 slide 37 Monotonic change of variables preserves order x y 1 x 1

38. Zvi WienerContTimeFin - 5 slide 38 change of variables: Constant volatility case: leads to Example

39. Zvi WienerContTimeFin - 5 slide 39 Theorem 1. The diffusion process is transformed by the following change of variables into a process with a deterministic diffusion parameter Free parameters: a(t) – defines the resulting diffusion parameter A(t) – defines zero level of the new variable

40. Zvi WienerContTimeFin - 5 slide 40 Feynman-Kac Formula 0.5  2 f xx +  f x + f t - rf + h=0 f(X,T) = g(X) The solution is given by: the discount factor

41. Zvi WienerContTimeFin - 5 slide 41 Stochastic Optimization In many cases financial assets involve decisions. In some cases we should assume that decision makers are rational and try to use an optimal decision, in some cases we assume not rational behavior.

42. Zvi WienerContTimeFin - 5 slide 42 A Time-Homogeneous Problem Values do not depend on time explicitly. A financial asset V, which depends on a set of variables X, and time t. Control variable .

43. Zvi WienerContTimeFin - 5 slide 43 A Time-Homogeneous Problem Sometimes the control variable  is a constant, sometimes it is a function of time and state. The expected cash flow is: ECF = u(X,  )ds The capital gain is: CG = dV = V x dX+0.5V xx (dX) 2 The expected capital gain is: ECG = (  V x +0.5  2 V xx )dt

44. Zvi WienerContTimeFin - 5 slide 44 A Time-Homogeneous Problem The value of V does not depend on time. The optimally managed total return per unit of time is given by: ETR = max(ECF+ECG)= max  [u(X,  )+  (X,  )V x +0.5  2 (X,  )V xx ] It must be equal the risk free return: rV= max  [u(X,  )+  (X,  )V x +0.5  2 (X,  )V xx ]

45. Zvi WienerContTimeFin - 5 slide 45 The Maximum Principle X follows an ABM with parameters  and . An asset pays continuous cash flow at the rate Xdt. There is no limited liability option. A manager can influence the growth rate of X. Suppose that for any  one has to pay  2 dt to managers. What is the optimal strategy?

46. Zvi WienerContTimeFin - 5 slide 46 The Maximum Principle

47. Zvi WienerContTimeFin - 5 slide 47 The Maximum Principle Note that Shimko assumes that one can not replace a manager, thus  opt is constant and hence V xx =0.

48. Zvi WienerContTimeFin - 5 slide 48 The Maximum Principle With this assumption we get V=2X  opt + C

49. Zvi WienerContTimeFin - 5 slide 49 The Maximum Principle Assuming one-time decision we can value the security as a sum of linearly growing perpetuity (ABM) minus a level perpetuity (constant payment of  2 forever. Optimizing with respect to  we obtain:

50. Zvi WienerContTimeFin - 5 slide 50 The Maximum Principle Without this assumption we get: A non-linear ODE, must be solved numerically. What are the appropriate boundary conditions?

51. Zvi WienerContTimeFin - 5 slide 51 Multiple State Variables Consider a perpetually lived value-maximizing monopolist who produces output at a rate of qdt, but faces a stochastically varying demand. Assume that the demand is linear p = a - bq, where p is the price of the good, and a, b are given by:

52. Zvi WienerContTimeFin - 5 slide 52 Multiple State Variables The initial conditions are a(0)=a 0, b(0)=b 0. Assume that the cost of production is zero. The value of the firm is V, such that:

53. Zvi WienerContTimeFin - 5 slide 53 Multiple State Variables The expected cash flow is: (a-bq)qdt The capital gain component is: dV = V a da+V b db+0.5V aa (da) 2 +V ab dadb+0.5V bb (db) 2 The expected capital gain is: ECG=E[dV]=fV a + gV b +0.5  2 V aa +  V ab +0.5 2 V bb

54. Zvi WienerContTimeFin - 5 slide 54 Multiple State Variables The maximum total return is: max(TR) = max(ECF+ECG) = rV Therefore The first order condition is:

55. Zvi WienerContTimeFin - 5 slide 55 Multiple State Variables Assume that f(a,b,q) = af 0 g(a,b,q) = bg 0  (a,b,q) = a  0 (a,b,q) = b 0 The value of the firm is:

56. Zvi WienerContTimeFin - 5 slide 56 Optimal Asset Allocation Merton 1971. Utility function:U= r - 0.5A  2 Here r is the expected rate of return and  - its standard deviation. A - is the individual’s coefficient of risk aversion.

57. Zvi WienerContTimeFin - 5 slide 57 Optimal Asset Allocation Denote by  - proportion invested in risky assets. Then

58. Zvi WienerContTimeFin - 5 slide 58 Optimal Asset Allocation Maximizing utility with respect to , we get:

59. Zvi WienerContTimeFin - 5 slide 59 Dynamic Asset Allocation How one can apply the Girsanov’s theorem? Perfect markets, no taxes, costs, restrictions. The budget equation:

60. Zvi WienerContTimeFin - 5 slide 60 Dynamic Asset Allocation The objective function is to maximize the expected lifetime discounted utility.

61. Zvi WienerContTimeFin - 5 slide 61 Problem 4.3 The height of a tree at time t is given by X t, where X t follows an ABM. We must decide when to cut the tree. The tree is worth $1 per unit of height, and if the tree is cut down at time  at height Y, then its value today is: V = e -r  Y.

62. Zvi WienerContTimeFin - 5 slide 62 Problem 4.3 a. What PDE must the value of the tree satisfy? b. What are the boundary conditions? c. Value the tree, assuming that the value is zero when the tree’s height is - . d. What is the optimal cutting policy?