Differential Equations: An Introduction to Modern Methods and Applications
Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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Chapter 8.P2, Problem 4P

Variance Reduction by Antithetic Variates. A simple and widely used technique for increasing the efficiency and accuracy of Monte Carlo simulations in certain situations with little additional increase in computational complexity is the method of antithetic variates. For each k = 1 , , M , use the sequence { ε 1 ( k ) , ε N 1 ( k ) } in Eq. (4) to simulate a payoff f ( S N ( k + ) ) and also use the sequence { ε 1 ( k ) , ε N 1 ( k ) } in Eq. (4) to simulate an associated payoff f ( S N ( k ) ) . Thus, the payoffs are simulated in pairs { f ( S N ( k + ) ) , f ( S N ( k ) ) } . A modified Monte Carlo estimate is then computed by replacing each payoff f ( S N ( k ) ) in Eq. (6) by the average [ f ( S N ( k + ) ) + f ( S N ( k ) ) ] 2 ,

C ^ A V ( S ) = 1 M k = 1 M f ( S N ( k + ) ) + f ( S N ( k ) ) 2 ( 1 + r Δ t ) N . (iii)

Use the parameters specified in Problem 3 to compute several (say, 20 or so) option price estimates using Eq. (6) and an equivalent number of option price estimates using (iii). For each of the two methods, plot a histogram of the estimates and compute the mean and standard deviation of the estimates. Comment on the accuracies of the two methods.

The difference equation (4): S n + 1 ( k ) = S n ( k ) + r S n ( k ) Δ t + σ S n ( k ) n + 1 k Δ t , S 0 k = s

Use the differential equation (4) to generate an ensemble of stock prices S N ( k ) = S ( k ) ( N Δ t ) , k = 1 , , M (where T = N Δ t ) and then use formula (6) to compare a Monte Carlo estimate of the value of a five-month call option ( T = 5 12 years ) for the following parameter values: r = 0.06 , σ = 0.2 , and K = $ 50 . Find estimates corresponding to current stock prices of S ( 0 ) = s = $ 45 , $ 50 , and $ 55 . Use N = 200 time steps for each trajectory and M 10 , 000 sample trajectories for each Monte Carlo estimate. Check the accuracy of your results by comparing the Monte Carlo approximation with the value computed from the exact Black-Scholes formula

C ( s ) = s 2 erfc ( d 1 2 ) K 2 e r T erfc ( d 2 2 ) , (ii)

where

d 1 = 1 σ T [ ln ( s k ) + ( r + σ 2 2 ) T ] , d 2 = d 1 σ T

And erfc ( x ) is the complementary error function,

erfc ( x ) = 2 π x e t 2 d t .

The difference equation (4) is given below:

S n + 1 ( k ) = S n ( k ) + r S n ( k ) Δ t + σ S n ( k ) n + 1 k Δ t , S 0 k = s

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Chapter 8 Solutions

Differential Equations: An Introduction to Modern Methods and Applications

Ch. 8.1 - In each of Problems 11 through 14 , use Eular’s...Ch. 8.1 - In each of Problems 11 through 14 , use Eular’s...Ch. 8.1 - In each of Problems 11 through 14 , use Eular’s...Ch. 8.1 - In each of Problems 11 through 14 , use Eular’s...Ch. 8.1 - Consider the initial value problem...Ch. 8.1 - Consider the initial value problem Use Euler’s...Ch. 8.1 - Consider the initial value problem...Ch. 8.1 - Consider the initial value problem Where is a...Ch. 8.1 - Consider the initial value problem y=y2t2,y(0)=,...Ch. 8.2 - In each of Problem 1 through 6, find approximate...Ch. 8.2 - In each of Problem 1 through 6, find approximate...Ch. 8.2 - In each of Problem 1 through 6, find approximate...Ch. 8.2 - In each of Problem 1 through 6, find approximate...Ch. 8.2 - In each of Problem 1 through 6, find approximate...Ch. 8.2 - In each of Problem 1 through 6, find approximate...Ch. 8.2 - In each of Problem 7 through 12, find approximate...Ch. 8.2 - In each of Problem 7 through 12, find approximate...Ch. 8.2 - In each of Problem 7 through 12, find approximate...Ch. 8.2 - In each of Problem 7 through 12, find approximate...Ch. 8.2 - In each of Problem 7 through 12, find approximate...Ch. 8.2 - In each of Problem 7 through 12, find approximate...Ch. 8.2 - Complete the calculations leading to the entries...Ch. 8.2 - Using three terms in the Taylor series given in...Ch. 8.2 - In each of Problems 15 and 16, estimate the local...Ch. 8.2 - In each of Problems 15 and 16, estimate the local...Ch. 8.2 - In each of Problems 17 and 20, obtain a formula...Ch. 8.2 - In each of Problems 17 and 20, obtain a formula...Ch. 8.2 - In each of Problems 17 and 20, obtain a formula...Ch. 8.2 - In each of Problems 17 and 20, obtain a formula...Ch. 8.2 - Consider the initial value problem y=cos5t,y(0)=1....Ch. 8.2 - Using a step size h=0.05 and the Euler method,...Ch. 8.2 - The following problem illustrates a danger that...Ch. 8.2 - The distributive law a(bc)=abac does not hold, in...Ch. 8.2 - In this section we stated that the global...Ch. 8.3 - In each of Problem 1 through 6, find approximate...Ch. 8.3 - In each of Problem 1 through 6, find approximate...Ch. 8.3 - In each of Problem 1 through 6, find approximate...Ch. 8.3 - In each of Problem 1 through 6, find approximate...Ch. 8.3 - In each of Problem 1 through 6, find approximate...Ch. 8.3 - In each of Problem 1 through 6, find approximate...Ch. 8.3 - In each of Problem 7 through 12, find approximate...Ch. 8.3 - In each of Problem 7 through 12, find approximate...Ch. 8.3 - In each of Problem 7 through 12, find approximate...Ch. 8.3 - In each of Problem 7 through 12, find approximate...Ch. 8.3 - In each of Problem 7 through 12, find approximate...Ch. 8.3 - In each of Problem 7 through 12, find approximate...Ch. 8.3 - Complete the calculation leading to the entries in...Ch. 8.3 - Confirm the results in Table 8.3.2 by executing...Ch. 8.3 - Consider the initial value problem y=t2+y2,y(0)=1....Ch. 8.3 - Consider the initial value problem Draw a...Ch. 8.3 - In this problem, we establish that the local...Ch. 8.3 - Consider the improved Euler method for solving the...Ch. 8.3 - In each of Problems 19 and 20, use the actual...Ch. 8.3 - In each of Problems 19 and 20, use the actual...Ch. 8.3 - In each of Problems 21 through 24, carry out one...Ch. 8.3 - In each of Problems 21 through 24, carry out one...Ch. 8.3 - In each of Problems 21 through 24, carry out one...Ch. 8.3 - In each of Problems 21 through 24, carry out one...Ch. 8.4 - In each of Problems 1 through 6, determine...Ch. 8.4 - In each of Problems 1 through 6, determine...Ch. 8.4 - In each of Problems 1 through 6, determine...Ch. 8.4 - In each of Problems 1 through 6, determine...Ch. 8.4 - In each of Problems 1 through 6, determine...Ch. 8.4 - In each of Problems 1 through 6, determine...Ch. 8.4 - Consider the example problemwith the initial...Ch. 8.4 - Consider the initial value problem...Ch. 8.P1 - Assume that the shape of the dispensers are...Ch. 8.P1 - After viewing the results of her computer...Ch. 8.P2 - Show that Euler’s method applied to the...Ch. 8.P2 - Simulate five sample trajectories of Eq. (1) for...Ch. 8.P2 - Use the differential equation (4) to generate an...Ch. 8.P2 - Variance Reduction by Antithetic Variates. A...
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