AMS-511 Foundations of Quantitative Finance

Transcription

1 AMS-511 Foundations of Quantitative Finance Fall 013 Lecture 15 Monday 09 Dec 013 Final Examination Review Robert J. Frey, Research Professor Stony Brook University, Applied Mathematics and Statistics 1. Mathematica Exercises Write a single Mathematica statement to simulate a Weiner process W(t) for W(0) = 0, t = 0 to 1 and Dt = 0.01, Any of the solutions below would work. FoldListBPlus, 0, RandomRealBNormalDistributionB0, 0.01 F, 100FF 80, , , , , 0.006, 0.853, , , 0.814, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , <

2 ams cn-m-00.nb PrependBAccumulateBRandomRealBNormalDistributionB0, 0.01 F, 100FF, 0F 80, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , < JoinB80<, AccumulateBRandomRealBNormalDistributionB0, 0.01 F, 100FFF 80, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.031, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.196, , , , , 0.340, , , , , , , , , , , , , , , , < solve the equation 1 + x + x ã 1. SolveB 1 + x ã 1, xf + x 99x Ø H-1L 1ê3 =, 9x Ø -H-1L ê3 == c. produce a function which normalizes the elements of a numeric vector so that they sum to 1;

3 ams cn-m-00.nb 3 Any of the solutions below would work. Ò Total@ÒD : 1 10, 1 5, 3 10, 5 > Ò Plus üü Ò : 1 10, 1 5, 3 10, 5 > &@81,, 3, 4<D &@81,, 3, 4<D Function@x, x ê Total@xDD@81,, 3, 4<D : 1 10, 1 5, 3 10, 5 > d. produces a list of points suitable for plotting by ListPlot of 1 + x - 3 x, where x is sampled from - to in increments of 0.1, TableA9x, 1 + x - 3 x =, 8x, -,, 0.1<E 88-., -15.<, 8-1.9, <, 8-1.8, -1.3<, 8-1.7, <, 8-1.6, -9.88<, 8-1.5, -8.75<, 8-1.4, -7.68<, 8-1.3, -6.67<, 8-1., -5.7<, 8-1.1, -4.83<, 8-1., -4.<, 8-0.9, -3.3<, 8-0.8, -.5<, 8-0.7, -1.87<, 8-0.6, -1.8<, 8-0.5, -0.75<, 8-0.4, -0.8<, 8-0.3, 0.13<, 8-0., 0.48<, 8-0.1, 0.77<, 80., 1.<, 80.1, 1.17<, 80., 1.8<, 80.3, 1.33<, 80.4, 1.3<, 80.5, 1.5<, 80.6, 1.1<, 80.7, 0.93<, 80.8, 0.68<, 80.9, 0.37<, 81., 0.<, 81.1, -0.43<, 81., -0.9<, 81.3, -1.47<, 81.4, -.08<, 81.5, -.75<, 81.6, -3.48<, 81.7, -4.7<, 81.8, -5.1<, 81.9, -6.03<, 8., -7.<<

4 4 ams cn-m-00.nb e. for yd = Sin@x yd Cos@xD compute f x y, The two statements below are syntactically equivalent. x,y Sin@x yd Cos@xD Cos@x yd Sec@xD - x y Sec@xD Sin@x yd + x Cos@x yd Sec@xD Tan@xD D@Sin@x yd ê Cos@xD, x, yd Cos@x yd Sec@xD - x y Sec@xD Sin@x yd + x Cos@x yd Sec@xD Tan@xD f. estimate E@X Sin@XDD f or X ~ N@0.05, 0.D via Monte Carlo simulation using a sample size of 100, Mean@Ò Sin@ÒD & êü RandomReal@NormalDistribution@0.05, 0.D, 100DD g. and write a function which takes a numeric integer vector and replaces even components with a 0 and odd with 1. The EvenQ[] function has the attribute Listable and will distribute itself over the components of its arguments; either of the alternatives below will work. Note the nesting of pure functions.

5 ams cn-m-00.nb 5 If@EvenQ@ÒD, 0, 1D & êü Ò &@81,, 3, 7, 7,, 8<D 81, 0, 1, 1, 1, 0, 0< If@Ò, 0, 1D & êü EvenQ@ÒD &@81,, 3, 7, 7,, 8<D 81, 0, 1, 1, 1, 0, 0< Better, perhaps, would be to use the Mod[] function which is itself also Listable. Mod@Ò, D &@81,, 3, 7, 7,, 8<D 81, 0, 1, 1, 1, 0, 0<. Mathematica Exercises Write a single Mathematica statement to find the roots of the polynomial x - 5 x + 1 ã 0; SolveAx - 5 x + 1 ã 0, xe ::x Ø 1 I5-1 M>, :x Ø 1 I5 + 1 M>> produce a function which, given a real vector x, returns a vector containing only those elements of x which are greater that zero. Any of the solutions below would work. x = 81,, -3, 0, 4, -9, 0, 8< 81,, -3, 0, 4, -9, 0, 8< Select@x, Ò > 0 &D 81,, 4, 8< Select@x, PositiveD 81,, 4, 8< c. produces a table of numbers (as a vector of 3-vectors) of the values of x, Sin[x] and Cos[x] for x =

6 6 ams cn-m-00.nb produces a table of numbers (as a vector of 3-vectors) of the values of x, Sin[x] and Cos[x] for x = -p/ to p/, dividing the interval into 1 equal increments; Table@8x, Sin@xD, Cos@xD<, 8x, -p ê, p ê, p ê 0<D ::- p 9 p, -1, 0>, :- 0, -CosB p p p F, SinB F>, : , , 1 4 I M>, :- 7 p 0, -CosB 3 p 3 p 3 p F, SinB F>, : , 1 4 I-1-5 M, >, :- p 4, - 1, 1 >, :- p 5, , 1 4 I1 + 5 M>, :- 3 p 0, -SinB 3 p 0 F, CosB 3 p 0 F>, :- p 10, 1 4 I1-5 M, >, :- p 0, -SinB p p F, CosB 0 0 F>, 80, 0, 1<, : p 0, SinB p 0 F, CosB p 0 F>, : p 10, 1 4 I M, >, : 3 p 0, SinB 3 p 3 p F, CosB 0 0 F>, : p 5, , 1 4 I1 + 5 M>, : p 4, 1, 1 >, : 3 p 10, 1 4 I1 + 5 M, >, : 7 p 0, CosB 3 p 3 p F, SinB 0 0 F>, : p 5, , p I M>, : 0, CosB p p F, SinB 0 0 F>, : p, 1, 0>> d. for -y compute the definite integral of f for x from 0 to 1; The two statements below are syntactically equivalent. 1 -y y 0-1 +

7 ams cn-m-00.nb 7 Integrate@Exp@-yD, 8y, 0, 1<D -1 + e. estimate VarB Abs@XD F f or X ~ N@0, 1D via Monte Carlo simulation using a sample size of 50. Variance@Sqrt@Abs@ÒDD & êü RandomReal@NormalDistribution@D, 50DD Time Value of Money A individual wishes to take out a loan from a bank for $5,000 which will be paid back with a single payment in two years. The bank's standard interest charge for such a loan is 10% per year compounded continuously. However, the bank also charges a $100 origination and processing fee at the time the loan is made. What is the effective annual interest rate of the loan; i.e., what interest rate is the borrower actually paying for the loan net the fee? First, solve for the payment P r T M = P 0.10µ 5000 = P ï P = then for the effective rate given the proceeds to the borrower of M - f r eff T HM - f L = P r effµ H L = ï r eff = Compound Interest What is the effective rate for 10% compounded Semi-annually? The effective rate for semi-annual compounding is

8 8 ams cn-m-00.nb i.e., 10.5%. H êL - 1 = Continuously? The effective rate for continuous compounding is i.e., 10.5% = Discount Factor What is the discount factor for a cash payment occurring 10-years from now assuming an annual interest rate of 10% and a quarterly compounding frequency? d n = 1 = 1 = H1 + r êkl n k 4µ10 H ê4L 6. Time Value of Money Frank agrees to pay Marion $10,000 in two years and in return he receives $9, from her today. Assume continous compounding in what follows. As time goes on he realizes that he will be unable to pay her on time and requests that he pay her back in three years instead of two. How much should Marion demand after three years if she is to realize the same rate of interest? The original loan is: = r ï r = LogB Fì ï r = The value of the original $9, after three years is: i.e., $10, r =

9 ams cn-m-00.nb 9 He agrees to pay her $5,000 after two years but then wants to pay the balance off a year later. How much should Marion demand for that final payment in the third year if she is to realize the same rate of interest? Frank owes $10,000 after two years but only pays back $5,000, then he is in effect borrowing the remaining balance for another year. His payment in year three would be: i.e., $5, = Bond Valuation A bond with a face value of $100,000, a semi-annual coupon of $,500 and exactly 0 years to maturity has a market yield quoted at 4.8%. Assuming that there is no accured interest. What is the price of the bond? S = F + c H1 + y êkl k n y êk 1-1 = H1 + y êkl k n µ0 H êL 0.048ê 1-1 = µ0 H êL 8. Spot Rates Given the spot rate and payment data: Years Spot Payment What is the present value of the stated cash flows?

10 10 ams cn-m-00.nb B = 3 c i t=1 H1 + s i L i = = What is the current price of a zero coupon bond with a face value of $10,000 that matures in three years? If Z is the price of the zero coupon bond: F Z = = = H1 + s 3 L Forward Rates Given the spot rate curve data: Years Spot Assume annual compounding: What is the discount factor for a cash payment three years forward in time? d n = 1 H1 + s n L = 1 = n What is the forward rate for a sum borrowed two years hence and paid back three years hence? f i, j = I1 + s jm j H1 + s i L i 1êHj-iL - 1 = = 0.047

11 ams cn-m-00.nb Dividend Discount Model A stock has just paid an annual dividend of $.50/share. (a) Assuming an annual dividend growth rate of 3% and an interest rate of 4%, estimate the price of the stock using a dividend discount model. (b) All other factors held constant, what would the interest rate have to be in order to cut the price of the stock in half? (c) All other factors held constant, what would would the interest rate have to be in order for the price of the stock to double? (a) It is important to remember that D 1 = H1 + gl D 0 and you have to be clear whether the question is talking about D 0 or D 1. In this instance, D 1 = H L.50 =.575. S 0 = D 1 r - g = = (b) The interest rate that would result in the stock price being cut in half is (b) and to double the interest rate rate is µ57.50 =.575 =.575 r ï r = 0.05 r ï r = Project Selection A firm is considering a number of capital projects. It has a total investment budget of 650. You are given the following list of projects with their costs and present values: Project Cost PV Write out a mathematical program whose solution maximizes the present value of projects selected within the stated investment budget with the addditional condition that if Project 1 is selected then Project must also be selected (although Project can be selected without Project 1).

12 1 ams cn-m-00.nb Project must also be selected (although Project can be selected without Project 1). The program, with x i representing the decision to select (1) project i or not (0), is: maximize: 00 x x x x 4 subject to: 100 x x + 50 x x x 1 - x 0 x i œ {0,1} What constraint would be required if the additional condition was modified so that either both Project 1 and Project must be selected or neither Project 1 and Project must be selected. The program, with x i representing the decision to select (1) project i or not (0), is: maximize: 00 x x x x 4 subject to: 100 x x + 50 x x x 1 - x ã 0 x i œ {0,1} 1. Portfolio Optimization Two uncorrelated risk assets have mean returns of 10% and 1%, respectively, and standard deviations of return of 10% and 15%, respectively. The two risk assets are uncorrelated. The risk-free rate of return is %. Assume that you have one unit of capital to allocate: Assume that there no restrictions on short selling, what allocation represents the minimum variance portfolio? The general solution is for a portfolio on the efficient frontier: x = l Q -1 m l 1T Q -1 m Q -1 1, 1 T Q -1 1 with the minimum variance portfolio found by setting l = 0. Thus, 0 l x = Q T Q -1 1

13 ams cn-m-00.nb 13 Substituting the appropriate values: x 1 x = = Note that, even though we didn't explicitly deal with the no shorting restriction, it was not an issue in the solution. Assume that there are restrictions on short selling, what allocation represents the maximum expected return portfolio? Obviously, we just put all of the capital into the asset with the highest expected return. x 1 x = Portfolio Optimization Two uncorrelated risk assets have mean returns of 10% and 1%, respectively, and standard deviations of return of 10% and 15%, respectively. The two risk assets are uncorrelated. The risk-free rate of return is %. Assume that you have one unit of capital to allocate and that there is no restrictions on short selling: What allocation of the risk assets represent the tanget portfolio? The general solution for the tangent portfolio: x tang = Q-1 Hm -1 r f L 1 T Q -1 Im -1 r f M Solving to proportionality: then normalizing to unity gives: x 1 x µ = x 1 = x

14 14 ams cn-m-00.nb What is the mean and standard deviation of the tangent portfolio? The mean and standard deviation of the portfolio are: m P = x 1 x T m 1 m = T = s P = x 1 x T s s x 1 x = T = Options with Costs from the Underlying On p. 37 of Luenberger s text he describes the risk neutral pricing of a commodity where there are storage costs involved. Rewrite the Black-Scholes option pricing formula for the situation in which storage costs are present. Rewrite the Black-Scholes PDE originally presented on p. 35 of Luenberger s text to account for an option on a commodity with storage costs. (a) The price of the underlying follows the following Itô process SHtL = m SHtL t + s SHtL WHtL The actual portfolio, however, is the underlying net of holding costs. That follows a modified SDE S «HtL = Hm - cl S «HtL t + s S «HtL WHtL We are hedging the net underlying against the option; thus, we make the substitution Hm - cl Ø r fl m Ø r + c In terms of the original underlying the risk neutral SDE is SHtL = Hr + cl SHtL t + s SHtL W è HtL where W è represents Brownian motion under the risk neutral measure. Thus, we come to the conclusion that we must simply substitute (r + c) for r in the standard Black Scholes formul (b) As in the vanilla case in which there are no dividends or holding costs we form a delta hedged portfolio

15 ams cn-m-00.nb 15 PHtL = FHtL - F S SHtL PHtL = F t + 1 s SHtL If we assume that we can maintain the hedge from instant to instant and that there are no liquidity limitations or transaction costs, then a arbitrage free market demands that the riskless portfolio PHtL returns the risk-free rate plus the holding costs because the holder of the underlying pays them and the option holder does not. This suggests a modified condition on PHtL PHtL = Hr + cl PHtL t We continue the development of the Black Scholes PDE using the modified no-arbitrage relationship above F S t Hr + cl PHtL t = F t + 1 s SHtL F S t Hr + cl FHtL - F F SHtL t = S t + 1 s SHtL F S t F t + 1 s SHtL F F + Hr + cl SHtL - Hr + cl FHtL = 0 S S Thus, we come to the same conclusion as before and simply substitute (r + c) for r in the standard Black Scholes formul 15. Binomial Pricing We are given a single-step geometric binomial pricing lattice as the model for the price dynamics of a stock with current price S(0): S + HDL = u µ SH0L p + SH0L p - = 1 p + S - HDL = SH0Lêu Consider a case in which S(0) = 95, D = 0.5, u = 1.05, and the riskfree rate of return r =.5%, then...

16 16 ams cn-m-00.nb Prove that the market represented by the stock and cash is arbitrage free. The state price equations are 1 SH0L = r D r D u SH0L SH0Lê u If we divide the bottom row of the matrix above by S(0), then we see that the state prices for a geometric binomial lattice can be restated as: 1 1 = r D r D u 1êu 1 1 = 0.05µ ê µ0.5 The state prices are positive; hence, the market is arbitrage-free > 1.05 > 1 ê 1.05 True What is the risk neutral measure? y + y - y + y - y + ï y + = y - y q + = r D y + ï q + = q - y - q c. What is the price of a call option with a strike price K = 110 expiring at D? The price of the call option is then y + y - T Max@K - uµsh0l, 0D Max@K - SHDLêu, 0D = T Max@ µ95.00, 0D Max@ ê1.05, 0D = Itô's Lemma For an underlying whose price dynamic follows the following SDE

17 ams cn-m-00.nb 17 Let SHtL = m SHtL t + s SHtL WHtL XHtL = 1 Use Itô's lemma to determine the SDE which describes the price dynamics of X(t). Given the SDE and suitably differentiable function Itô's lemma states SHtL S(t) = a(s(t), t) t + b(s(t), t) W(t) XHtL = g S a + g X(t) = g(s(t), t) t + 1 g S b t + g S b W HtL For the problem presented we have a = m S(t), b = s S(t), and g = 1êSHtL; therefore, and we have substituting X(t) for 1ê SHtL 1 g S a = - m S g t = 0 g S b = s S g XHtL = Is - mm S b = - s S 1 SHtL t -s 1 SHtL WHtL XHtL = Is - mm XHtL t -s XHtL WHtL 17. Itô's Lemma For an underlying whose price dynamic follows the following SDE SHtL = m SHtL t + s SHtL WHtL Let XHtL = SHtL

18 18 ams cn-m-00.nb Use Itô's lemma to determine the SDE which describes the price dynamics of X(t). Given the SDE S(t) = a(s(t), t) t + b(s(t), t) W(t) and suitably differentiable function X(t) = g(s(t), t) Itô's lemma states XHtL = g S a + g t + 1 g S b t + g S b W HtL For the problem presented we have a = m S(t), b = s S(t), and g = SHtL ë; therefore, and we have g S a = m S g t = 0 1 g S b = s S g S b = s S XHtL = I m + s M SHtL t + s SHtL WHtL substituting X(t) for SHtL XHtL = I m + s M XHtL t + s XHtL WHtL 18. Risk Neutral Valuation Consider a "safe call" with strike price K and expiration date T which gives the holder the right to receive at expiration either SHTL - K or a fixed payment of M < K. Assume the risk free rate is r with continuous compounding. Let S(t) denote the price of the underyling. Let V(t) denote the price of the option. Adapt the Black-Scholes option pricing formula for European options to price the V-option. (Note: It is only necessary to express the result in terms of the prices of calls and puts with appropriately chosen parameters.) Under the risk neutral measure Q the value of the option at t T is the discounted expected value of the terminal value:

19 ams cn-m-00.nb 19 VHtL = e -rht-tl E - K, MD Note that max@s - k, md = max@s - Hk + ml, 0D + m, so we can rewrite the above as: VHtL = e -rht-tl I E - HK + ML, 0D + MDM This is simply the mean of the value of a call with a strike price of HK + ML and a cash position with a future value of M: VHtL = CHSHtL, t K + ML + e -rht-tl M 19. Risk Neutral Valuation Consider a "V-option" with strike price K and expiration date T which gives the holder the right to receive at expiration half the value of either SHTL - K or K - SHTL. Assume the risk free rate is r with continuous compounding. Let S(t) denote the price of the underyling. Let V(t) denote the price of the option. Adapt the Black-Scholes option pricing formula for European options to price the V-option. (Note: It is only necessary to express the result in terms of the prices of calls and puts with appropriately chosen parameters.) Under the risk neutral measure Q the value of the option at t T is the discounted expected value of the terminal value: VHtL = e -rht-tl E Q BMaxB SHTL - K, K - SHTL F VHtL = 1 e-rht-tl E - K, K - SHTLD Note that max[a-k, k-a] = max[a-k, 0] + max[k-a, 0], so we can rewrite the above as: VHtL = 1 e-rht-tl I E - K, 0D + E - SHTL, 0DM This is simply the mean of the value of a call and a put with the same parameters as the V-option. CHSHtL, t L + PHSHtL, t L VHtL =