MSF 526: Mid-term Exam Solutions Prof. Matthew Dixon October 17th, 2020 This exam consists of three questions. You are required to answer all three questions in two and a half hours (150 minutes). Each question carries equal marks. Write your name clearly on each sheet of paper that you hand in and show all intermediate calculations. This is an open book exam. I 1 Question 1 A digital call option either pays a fixed amount if the underlying is above the strike at maturity or nothing so that the payoff takes the form h(ST ) = { S0 −K, ST > K 0, otherwise. For a digital call option expiring in 0.25 years and struck at 39 (K = 39), use antithetic sampling to estimate its spot price using a two-step Euler Scheme with 10 paths. You may assume that S0 = 40, µ = r = 0.01, σ = 0.2, the fixed payoff is S0 −K = 1 and = [ 0.44163989 1.0941102 0.18084294 −0.89863916 0.94760756 −0.19369481 −1.30991276 −0.21051674 1.43932212 −0.29027246 ] where each row corresponds to the time step and each column to the first five paths. You should generate the additional five ”mirror” paths from those given. [16] Estimate the standard deviation and standard error of the digital call price estimate. [4] Solution The mirror paths are created from = − 1 = [ −0.44163989 −1.0941102 −0.18084294 0.89863916 −0.94760756 0.19369481 1.30991276 0.21051674 −1.43932212 0.29027246 ] Using the Euler scheme along path j and time step i to simulate the prices Sji+1 = S j i (1 + r∆t+ σ √ ∆tj) gives the following simulated values S1 =  41.29914624 43.14461097 40.56150108 37.50826462 42.73023893 38.80085376 36.95538903 39.53849892 42.59173538 37.36976107  and S2 =  40.78512504 39.20227993 40.00841331 41.37257021 41.90659808 39.38078262 40.42456951 40.1764835 38.3101823 38.18350117  The digital option payoffs are: V2 = [ 1 1 1 1 1 1 1 1 0 0 ] and so the estimated mean spot price V0 = e −rTE[VT ] = $0.798 with a standard deviation of 0.42164 and standard error of 0.13333. 2 Question 2 Estimate the current price of a European call option expiring in two years and struck at $38 using a two-step binomial tree. The spot price of the underlying stock is $40 and it pays a dividend at an annualized rate of 2%. The risk-free rate is 5%. Assume that the upstate multiplicative factor u = 1.2 and the downstate multiplicative factor d = 1/1.2. Calculate the spot price of the call option. [15] Explain in detail how you would modify your methodology for American style exercise and recompute the price if it is different. [5] 2 Solution Beginning by plotting the binomial tree: $40 $33.333 $48 $27.778 $40 $57.6 (1− q) q q (1− q) q (1− q) Now setting up the notation and assigning terminal European Call option pay-off values to the tree. V0 V −1 V +1 V −−2 = 0 V +−2 = 2 V ++2 = 19.6 (1− q) q q (1− q) q (1− q) The risk neutral probability q is calculated using the following formula: q = e(r−δ)∆t − d u− d . where u and d represent the up and down multiplicative factors respectively and δ is the annual dividend yield. We first calculate the risk neutral proba- 3 bility q. Substituting the relevant values for the interest rate, dividend yield and interval length into the risk neutral probability equation q = 48× e(0.05−0.02) − 40 57.6− 40 = 0.5376 Since the value of the option at the previous node is equal to the discounted expected value of the option payoff under the risk neutral probability mea- sures we have: V +1 = e −r∆t(q × 19.6 + (1− q)× 2) = $10.9028 and V −1 = e −r∆t(q × 2 + (1− q)× 0) = $1.0227 Repeating these steps to find the spot price of the option we arrive at the answer V0 = e −r∆t(q × V +1 + (1− q)× V −1 ) = $6.0254. Answer can be given to two decimal places. Since the Call option pays dividends there may be an advantage to early exercise so the American Call may not be the same value as the European Call. To check this, let’s modify the methodology to price an American call by comparing the option value (intrinsic value) based on exercising at the time of the node with the discounted price of the option at the next node: V +1 = max($10.9028, $48− $38 = $10) = $10.9028 and V −1 = max($1.0227, $0) = $1.0227 and V0 = max($6.0254, $40− $38) = $6.0254. Answer can be given to two decimal places. 3 Question 3 Under an arithmetic brownian motion (ABM) model for the spread S be- tween two instruments, the price of a European spread option is given by the following partial differential equation (PDE). ∂V ∂t + rS ∂V ∂S + σ2 2 ∂2V ∂S2 = rV. 4 Note that this is not the usual Black-Scholes PDE which has a S2 term in the coefficient for the second order derivative. The spread may be treated as a single underlying instrument whose dynamics are prescribed by ABM. Using the following grid S = −50,−25, 0, 25, 50 t = 0, 0.25 estimate the spot value of a European Call Spread Option struck at 0 (K = 0) and expiring in 3 months with an explicit finite difference scheme. Use the parameters S0 = 0, r = 0.05, σ = 0.1 and ∆t = 0.25 [20]. Solution Beginning by calculating the formulating the explicit finite difference ap- proximation of the ABM partial differential equation: rVi,j+1 = Vi,j+1 − Vi,j ∆t +ri∆S[ Vi+1,j+1 − Vi−1,j+1 2∆S ]+ 1 2 σ2[ Vi+1,j+1 − 2Vi,j+1 + Vi−1,j+1 ∆S2 ] Simplifying this expression gives the formula for updating each inner-grid node: Vi,j = αiVi−1,j+1 + βiVi,j+1 + γiVi+1,j+1 where αi = ∆t 2∆S2 [σ2 − ri∆S2] (1) βi = 1− ∆t ∆S2 [σ2 + r∆S2] (2) γi = ∆t 2∆S2 [σ2 + ri∆S2] (3) with values at each inner grid node i αi βi γi -1 0.006252 0.987496 -0.006248 0 0.000002 0.987496 0.000002 1 -0.006248 0.987496 0.006252 The payoffs of the call spread option at maturity are: V1 = [ 0. 0. 0. 25. 50. ] 5 and using the update formula gives the call spread option values at time t0 V0 = [ 0 0 0.00005 25 49.37889002 ] Answer: V0 = $0.00005 since S0 = 0. 4 Question 4 Using the following grid S = 0, 25, 50, 75, 100 t = 0, 0.5, 1 estimate the spot value of the Call Option struck at 50 and expiring in 1 year with an implicit finite difference scheme. Use the parameters r = 0.05, q = 0.03, σ = 0.1. [20] . Give your answer in the form Ax = b and compute all elements of A and b for the last time step only (i.e. solve for V at t = 0.5). Note that you do not need to actually solve the linear system for this question. HINT: Discretize the standard Black-Scholes PDE and be sure to include q in the drift term. Question 3 Solution Begin by computing the coefficients from the discretized expression (see Lecture notes on using an implicit stencil). αi = ∆t 2 [(r − q)i− σ2i2] (4) βi = 1 + ∆t[σ 2i2 + r] (5) γi = ∆t 2 [(r − q)i+ σ2i2] (6) and evaluating on the inner grid only to give the table. Now the implicit linear system, corresponding to the inner grid, can be written as the following linear system to solve for the unknown option price Vj at time tj 6 i αi βi γi 1 0.0025 1.03 -0.0075 2 0 1.045 -0.02 3 -0.0075 1.07 -0.0375  β1 γ1 0α2 β2 γ2 0 α3 β3  V1,jV2,j V3,j  =  V1,j+1V2,j+1 V3,j+1 −  α1 × V0,j0 γ3 × V4,j  For a call option V0,j and V4,j equal the boundary conditions V0,j = 0 V4,j = exp{−r(N + 1− j)∆t}|Smax −K|+. Note that all values in the matrix are given in the table and do not depend on time. Since we consider the last time step, let us set j = 1 and set Vi,j+1 to the payoffs for i ∈ {1, 2, 3}. We have V1,j+1V2,j+1 V3,j+1  =  |25− 50|+ = 0|50− 50|+ = 0 |75− 50|+ = 25  The boundary nodes are V0,1 = 0 V4,1 = exp{−r∆t}|100− 50|+ = 48.765 The right hand side vector is therefore b =  00 25 + 0.0375× 48.765 = 26.829  So we have found A = M and b corresponding to the linear system Ax = b where x = [V1,j , V2,j , V3,j ] T . 7