1. CF-5 Bank Hapoalim Jul-2001 Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Computational Finance

2. CF-5 Bank Hapoalim Jul-2001 Following T. Bjork, ch. 15 Arbitrage Theory in Continuous Time Bonds and Interest Rates

3. Zvi WienerCF5 slide 3 Bonds and Interest Rates Zero coupon bond = pure discount bond T-bond, denote its price at time t by p(t,T). principal = face value, coupon bond - equidistant payments as a % of the face value, fixed or floating coupons.

4. Zvi WienerCF5 slide 4 Assumptions There exists a frictionless market for T- bonds for every T > 0 p(t, t) =1 for every t for every t the price p(t, T) is differentiable with respect to T.

5. Zvi WienerCF5 slide 5 Interest Rates Let t < S < T, what is IR for [S, T]? at time t sell one S-bond, get p(t, S) buy p(t, S)/p(t,T) units of T-bond cashflow at t is 0 cashflow at S is -$1 cashflow at T is p(t, S)/p(t,T) the forward rate can be calculated...

6. Zvi WienerCF5 slide 6 The simple forward rate LIBOR - L is the solution of: The continuously compounded forward rate R is the solution of:

7. Zvi WienerCF5 slide 7 Definition 15.2 The simple forward rate for [S,T] contracted at t (LIBOR forward rate) is The simple spot rate for [S,T] LIBOR spot rate is (t=S):

8. Zvi WienerCF5 slide 8 Definition 15.2 The continuously compounded forward rate for [S,T] contracted at t is The continuously compounded spot rate for [S,T] is (t=S)

9. Zvi WienerCF5 slide 9 Definition 15.2 The instantaneous forward rate with maturity T contracted at t is The instantaneous short rate at time t is

10. Zvi WienerCF5 slide 10 Definition 15.3 The money market account process is Note that here t means some time moment in the future. This means

11. Zvi WienerCF5 slide 11 Lemma 15.4 For t  s  T we have And in particular

12. Zvi WienerCF5 slide 12 Models of Bond Market Specify the dynamic of short rate Specify the dynamic of bond prices Specify the dynamic of forward rates

13. Zvi WienerCF5 slide 13 Important Relations Short rate dynamics dr(t)= a(t)dt + b(t)dW(t)(15.1) Bond Price dynamics(15.2) dp(t,T)=p(t,T)m(t,T)dt+p(t,T)v(t,T)dW(t) Forward rate dynamics df(t,T)=  (t,T)dt +  (t,T)dW(t) (15.3) W is vector valued

14. Zvi WienerCF5 slide 14 Proposition 15.5 We do NOT assume that there is no arbitrage! If p(t,T) satisfies (15.2), then for the forward rate dynamics

15. Zvi WienerCF5 slide 15 Proposition 15.5 We do NOT assume that there is no arbitrage! If f(t,T) satisfies (15.3), then the short rate dynamics

16. Zvi WienerCF5 slide 16 Proposition 15.5 If f(t,T) satisfies (15.3), then the bond price dynamics

17. Zvi WienerCF5 slide 17 Proof of Proposition 15.5 Left as an exercise …

18. Zvi WienerCF5 slide 18 Fixed Coupon Bonds

19. Zvi WienerCF5 slide 19 Floating Rate Bonds L(T i-1,T i ) is known at T i-1 but the coupon is delivered at time T i. Assume that K =1 and payment dates are equally spaced. Now it is t<T 0. By definition of L we have

20. Zvi WienerCF5 slide 20 Floating Rate Bonds implies

21. Zvi WienerCF5 slide 21 This coupon will be paid at T i. The value of -1 at time t is -p(t, T i ). The value of the first term is p(t, T i-1 ). Thus the present value of each coupon is The present value of the principal is p(t,T n ).

22. Zvi WienerCF5 slide 22 The value of a floater is Or after a simplification

23. Zvi WienerCF5 slide 23 Forward Swap Settled in Arrears K - principal, R - swap rate, rates are set at dates T 0, T 1, … T n-1 and paid at dates T 1, … T n. T 0 T 1 T n-1 T n

24. Zvi WienerCF5 slide 24 Forward Swap Settled in Arrears If you swap a fixed rate for a floating rate (LIBOR), then at time T i, you will receive where c i is a coupon of a floater. And at T i you will pay the amount Net cashflow

25. Zvi WienerCF5 slide 25 Forward Swap Settled in Arrears At t < T 0 the value of this payment is The total value of the swap at time t is then

26. Zvi WienerCF5 slide 26 Proposition 15.7 At time t=0, the swap rate is given by

27. Zvi WienerCF5 slide 27 Zero Coupon Yield The continuously compounded zero coupon yield y(t,T) is given by For a fixed t the function y(t,T) is called the zero coupon yield curve.

28. Zvi WienerCF5 slide 28 The Yield to Maturity The yield to maturity of a fixed coupon bond y is given by

29. Zvi WienerCF5 slide 29 Macaulay Duration Definition of duration, assuming t=0.

30. Zvi WienerCF5 slide 30 Macaulay Duration What is the duration of a zero coupon bond? A weighted sum of times to maturities of each coupon.

31. Zvi WienerCF5 slide 31 Meaning of Duration r $

32. Zvi WienerCF5 slide 32 Proposition 15.12 TS of IR With a term structure of IR (note y i ), the duration can be expressed as:

33. Zvi WienerCF5 slide 33 Convexity r $

34. Zvi WienerCF5 slide 34 FRA Forward Rate Agreement A contract entered at t=0, where the parties (a lender and a borrower) agree to let a certain interest rate R*, act on a prespecified principal, K, over some future time period [S,T]. Assuming continuous compounding we have at time S:-K at time T: Ke R*(T-S) Calculate the FRA rate R* which makes PV=0 hint: it is equal to forward rate

35. Zvi WienerCF5 slide 35 Exercise 15.7 Consider a consol bond, i.e. a bond which will forever pay one unit of cash at t=1,2,… Suppose that the market yield is y - flat. Calculate the price of consol. Find its duration. Find an analytical formula for duration. Compute the convexity of the consol.

36. CF-5 Bank Hapoalim Jul-2001 Following T. Bjork, ch. 19 Arbitrage Theory in Continuous Time Change of Numeraire

37. Zvi WienerCF5 slide 37 Change of Numeraire P - the objective probability measure, Q - the risk-neutral martingale measure, We will introduce a new class of measures such that Q is a member of this class.

38. Zvi WienerCF5 slide 38 Intuitive explanation Assuming that X and r are independent under Q, we get In all realistic cases that X and r are not independent under Q. However there exists a measure T (forward neutral) such that

39. Zvi WienerCF5 slide 39 Risk Neutral Measure Is such a measure Q that for every choice of price process  (t) of a traded asset the following quotient is a Q-martingale. Note that we have divided the asset price  (t) by a numeraire B(t).

40. Zvi WienerCF5 slide 40 Conjecture 19.1.1 For a given financial market and any asset price process S 0 (t) there exists a probability measure Q 0 such that for any other asset  (t)/S 0 (t) is a Q 0 -martingale. For example one can take p(t,T) (fixed T) as S 0 (t) then there exists a probability measure Q T such that for any other asset  (t)/p(t,T) is a Q T -martingale.

41. Zvi WienerCF5 slide 41 Using p(T,T)=1 we get Using a derivative asset as  (t,X) we get

42. Zvi WienerCF5 slide 42 Assumption 19.2.1 Denote an observable k+1 dimensional process X=(X 1, …, X k, X k+1 ) where X k+1 (t)=r(t) (short term IR) Denote by Q a fixed martingale measure under which the dynamics is: dX i (t)=  i (t,X(t))dt +  i (t,X(t))dW(t), i=1,…,k+1 A risk free asset (money market account): dB(t)=r(t)B(t)dt

43. Zvi WienerCF5 slide 43 Proposition 19.1 The price process for a given simple claim Y=  (X(T)) is given by  (t,Y)=F(t,X(t)), where F is defined by

44. CF-5 Bank Hapoalim Jul-2001 Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Practical Numeraire Approach

45. Zvi WienerCF5 slide 45 Options with uncertain strike Stock option with strike fixed in foreign currency. How it can be priced? Margarbe 78 or Numeraire approach 1. Price it using this currency as a numeraire. foreign interest rate foreign current price foreign volatility! 2. Translate the resulting price into SHEKELS using the current exchange rate.

46. Zvi WienerCF5 slide 46 Options with uncertain strike Endowment warrants strike is increasing with short term IR. strike is decreasing when a dividend is paid What is an appropriate numeraire? A closed Money Market account. Result – price by standard BS but with 0 dividends and 0 IR.

47. Zvi WienerCF5 slide 47 Options with uncertain strike An option to choose by some date between dollar and CPI indexing (may be with some interest). Margrabe can be used or one can price a simple CPI option in terms of an American investor and then translate it to SHEKELS.

48. Zvi WienerCF5 slide 48 Convertible Bonds A convertible bond typically includes an option to convert it into some amount of ordinary shares. This can be seen as a package of a regular bond and an option to exchange this regular bond to shares of the company. If the company does not have traded debt there is a problem of pricing this option.

49. Zvi WienerCF5 slide 49 Convertible Bonds This is an option to exchange one asset to another and can be priced with Margrabe approach. However in order to use this approach one need to know the correlation between the two assets (stock and regular bond). When there is no market for regular bonds this might be a problem.

50. Zvi WienerCF5 slide 50 Convertible Bonds An alternative approach is with a numeraire. Denote by S t stock price at time t, B t price at time t of a regular bond (may be not observable). CB t price of a convertible bond. C - value of the conversion option, so that CB = C(B) + B at any time

51. Zvi WienerCF5 slide 51 Convertible Bonds Note that C is a decreasing function of B (the higher the strike price, the lower is the option’s value). This means that as soon as CB t < S t = C(B=0) the right hand side of the following equation (B - an unknown) CB t = C(B t ) + B t has a unique solution.

52. Zvi WienerCF5 slide 52 Convertible Bonds The left hand side is a known constant, the right hand side is a sum of two variables. The first one is decreasing in B, but its derivative is strictly less than one and approaches zero for large B. The second one is linear with slope one. This means that as soon as CB>C(B=0)+0=S there exists a unique solution.

53. Zvi WienerCF5 slide 53 Uniqueness of a solution B CB S

54. Zvi WienerCF5 slide 54 Pricing with known volatility Let’s use B t as a numeraire, then the stochastic variable is S t /B t. Assume that S t /B t has a constant volatility . Then this option has a fixed strike (in terms of B) and is equivalent to a standard option, which can be priced with BS equation. Call(S t /B t, T, 1, , r) (in terms of B t ), the dollar value is then B t Call(S t /B t, T, 1, , r).

55. Zvi WienerCF5 slide 55 Pricing with known volatility This means that when  is known the option can be priced easily and consequently the straight bond. However  that we need can not be observed. The solution is in the following procedure.

56. Zvi WienerCF5 slide 56 Pricing with known volatility Assume that  is stable but unknown. For any value of  we can easily price the option at any date, and hence we can also derive the value of B t. Take a sequence of historical data (meaning S t and CB t ). For any value of  we can construct the implied B t (  ). Then using these sequence of observations we can check whether the volatility of S t /B t is indeed . If our guess of  was correct this is true.

57. Zvi WienerCF5 slide 57 Pricing with known volatility However there is no reason why some value of  will give the same implied historical volatility. This means that we have to solve for  such that the implied volatility is equal . Numerically this can be done easily. Why there exists a unique solution??? Check monotonicity!!

58. Zvi WienerCF5 slide 58 Solution for  Implied volatitity

59. Zvi WienerCF5 slide 59 MMA implementation FindRoot[CB == B + bsCallFX[s, ttm, B, sg, 0, 0], {B,CB}] ConvertibleBondHistorical[StockHistory_, CBHistory_, ttm_] := Module[{sg, len, ff, BusinessDaysYear = 250, sgg, t1, t2}, len = Length[StockHistory]; ff[sg_] := Log[StockHistory/ MapThread[StraightBond[#1, #2, ttm, sg] &, {CBHistory, StockHistory}]]; FindRoot[sg == StandardDeviation[Rest[ff[sg]] - Drop[ff[sg], -1]]* Sqrt[BusinessDaysYear], {sg, 0.001, 1}][[1, 2]] ];

60. Zvi WienerCF5 slide 60 Example 1 CB S B

61. Zvi WienerCF5 slide 61 Example 2 CB S B

62. CF-5 Bank Hapoalim Jul-2001 Zvi Wiener The Hebrew University of Jerusalem mswiener@mscc.huji.ac.il Value of Value-at-Risk

63. Zvi WienerCF5 slide 63 P&L VaR 1 day 1% probability 1d 1 week 1% probability 1w

64. Zvi WienerCF5 slide 64 Model Bank’s choice of an optimal system Depends on the available capital Current and potential capital needs Queuing model as a base

65. Zvi WienerCF5 slide 65 Required Capital Let A be total assets C – capital of a bank  - percentage of qualified assets k – capital required for traded assets

66. Zvi WienerCF5 slide 66 Maximal Risk (Assets) The coefficient k varies among systems, but a better (more expensive) system provides more precise risk measurement, thus lower k. Cost of a system is p, paid as a rent (pdt during dt). A max is a function of C and p.

67. Zvi WienerCF5 slide 67 Risky Projects Deposits arrive and are withdrawn randomly. All deposits are of the same size. Invested according to bank’s policy. Can not be used if capital requirements are not satisfied.

68. Zvi WienerCF5 slide 68 Arrival of Risky Projects We assume that risky projects arrive randomly (as a Poisson process with density ). This means that there is a probability dt that during dt one new project arrives.

69. Zvi WienerCF5 slide 69 Arrival of Risky Projects A new project is undertaken if the bank has enough capital (according to the existing risk measuring system). We assume that one can NOT raise capital or change systems quickly.

70. Zvi WienerCF5 slide 70 Termination of Risky Projects We assume that each risky project disappears randomly (as a Poisson process with density  ).

71. Zvi WienerCF5 slide 71 Termination of Risky Projects We assume that each risky project disappears randomly (as a Poisson process with density  ). This means that there is a probability n  dt that during dt one out of n existing projects terminates. With probability (1-n  dt) all existing projects will be active after dt.

72. Zvi WienerCF5 slide 72 Profit We assume that each existing risky project gives a profit of  dt during dt. Thus when there are n active projects the bank has instantaneous profit (  n-p)dt.

73. Zvi WienerCF5 slide 73 States After C and p are chosen, the maximal number of active projects is given by s=A max (C,p).  0  1  2  s-1  s 0 1 2 s-1 s  2  s 

74. Zvi WienerCF5 slide 74 States  0  1  2  s-1  s 0 1 2 s-1 s  2  s  Stable distribution:  0 =  1   1 =  2 2  …  s-1 =  s s 

75. Zvi WienerCF5 slide 75 Probabilities Probability of losing a new project due to capital requirements is equal to the probability of being in state s, i. e.  s. Termination of projects does not have to be Poissonian, only mean and variance matter.

76. Zvi WienerCF5 slide 76 Expected Profit An optimal p (risk measurement system) can be found by maximizing the expected profit stream.

77. Zvi WienerCF5 slide 77 Example Capital requirement as a function of p (price) and q (scaling factor), varies between 1.5% and 8%.

78. Zvi WienerCF5 slide 78 Example q=0.5 q=1 q=3 p A max

79. Zvi WienerCF5 slide 79 Example of a bank Capital $200M Average project is $20K On average 200 new projects arrive each day Average life of a project is 2 years 15% of assets are traded and q=1 spread  =1.25%

80. Zvi WienerCF5 slide 80 Bank’s profit as a function of cost p. C=$200M, arrival rate 200/d, size $20K, average life 2 yr., spread 1.25%, q=1, 15% of assets are traded. rent p Expected profit

81. Zvi WienerCF5 slide 81 rent p Expected profit Bank’s profit as a function of cost p. C=$200M, arrival rate 200/d, size $20K, average life 2 yr., spread 1%, q=1, 5% of assets are traded.

82. Zvi WienerCF5 slide 82 Conclusion Expensive systems are appropriate for banks with low capitalization operating in an unstable environment Cheaper methods (like the standard approach) should be appropriate for banks with high capitalization small trading book operating in a stable environment many small uncorrelated, long living projects

83. Zvi WienerCF5 slide 83 A simple intuitive and flexible model of optimal choice of risk measuring method.

84. CF-5 Bank Hapoalim Jul-2001 Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html DAC

85. Zvi WienerCF5 slide 85 Life Insurance yearly contribution 10,000 NIS yearly risk premium 2,000 NIS first year agent’s commission 3,000 NIS promised accumulation rate 8,000 NIS/yr After the first payment there is a problem of insufficient funds. 8,000 NIS are promised (with all profits) and only 5,000 NIS arrived.

86. Zvi WienerCF5 slide 86 10,000 NIS Risk 2,000 NIS Client’s 8,000 NIS Agent 3,000 NIS insufficient funds if the client leaves insufficient profits

87. Zvi WienerCF5 slide 87 Risk measurement The reason to enter this transaction is because of the expected future profits. Assume that the program is for 15 years and the probability of leaving such a program is . Fees are 0.6% of the portfolio value each year 15% real profit participation

88. Zvi WienerCF5 slide 88 Obligations The most important question is what are the obligations? The Ministry of Finance should decide Transparent to a client Accounted as a loan

89. Zvi WienerCF5 slide 89 One year example Assume that the program is for one year only and there is no possibility to stop payments before the end. Initial payment P 0, fees lost L 0, fixed fee a% of the final value P 1, participation fee b% of real profits (we ignore real). Investment policy TA-25 (MAOF).

90. Zvi WienerCF5 slide 90 Liabilities (no actual loan) Assets (no actual loan)

91. Zvi WienerCF5 slide 91 Total=Assets-Liabilities Fair value

92. Zvi WienerCF5 slide 92 Liabilities (actual loan) Assets (actual loan)

93. Zvi WienerCF5 slide 93 Total=Assets-Liabilities (loan)

94. Zvi WienerCF5 slide 94 2 years liabilities (no actual loan) 2 years assets (no actual loan) In reality the situation is even better for the insurer, since profit participation fees once taken are never returned (path dependence).

95. Zvi WienerCF5 slide 95 2 years fair value, no loan

96. Zvi WienerCF5 slide 96 2 years liabilities (with a loan) 2 years assets (with a loan)

97. Zvi WienerCF5 slide 97 Stock index Profit No loan With a loan 10 years, L 0 =7%

98. Zvi WienerCF5 slide 98 Partial loan - portion q Theoretically q can be negative.

99. Zvi WienerCF5 slide 99 Mixed portfolio When the investment portfolio is a mix one should analyze it in a similar manner. Important: an option on a portfolio is less valuable than a portfolio of options. Another risk factor - leaving rate should be accounted for by taking actuarial tables as leaving rate.

100. Zvi WienerCF5 slide 100 Conclusions It is a reasonable risk management policy not to take a loan against DAC. Up to some optimal point it creates a useful hedge to other assets (call options and shares) of the firm. Intuitively DAC is good when the stock market performs badly and profit participation is valueless. DAC performs bad when the market performs well.