Elementary Probability For Applications 1st Edition Rick Durrett - Solutions

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If1 is an increasing sequence of o-algebras, then U is merely an algebra.
In the notation of Exercise 1, prove that A T(T(A)) with set equality holding if T is one-to-one. Also, T(T(B))
Let 7:0, Q and A, Q, for 2e A. Show that TU4)=UT(4) but that T(4-A) need not equal T(A)-T(A), where T(A) = {T(w): w = A), T() {T(A): AE}. If T is-one-to-one and onto, then af a 6-algebra on Q, entails T() is a -algebra on Q.
Let (2) be measurable spaces and o() with es, 1 i n. If 9A, x Axx A: Ae, Isis n. then a(9)=xxx. on , xxx.
If is an algebra such that for every disjoint sequence (F,n 1) of sets in UFF, then is a 6-algebra.
Prove that of is a -algebra if and only if it is both a -class and a z-class. If a class 9, then o()=(D).
If {f, n0} is a sequence of real functions with falfo and g. fast for some constants - sa
If {f, then A, m+ m +(-1)-(m) (3) 0} is a sequence of real functions on with, fo and A = {w: f(w) > c). A., and lim A, = A0-1 =>
Apropos of (2), prove that if B., is the set of points belonging to exactly m(1 mn) of A, A, then (m + 18 = Sm m
Iff is a real function on 2, then f = fifff is an indicator of some subset of
If A, Q, n 1, show that An Imax I #21 Inmin L #21
= lim 1, whenever either side exists. ====
Prove that = lim and that im
Prove that (limA) = lim 4; and (lim. A) = lim also that lim B, = B implies lim B=B and lim A & B, = AA B.
Verify that = ', C=AAB iff A = BAC, BA
For any sequence of sets (A,, n 1), define B, A, B+BAA1. Prove that lim, B, exists iff lim A, exists and is empty.
If a is a sequence of real numbers, 0a, sx, prove that a)()]-[(ap [ (" + )] - [0. Sup (" + ")] U [o. a.) = [0. sup a.). =
7. Prove that the set S consisting of all infinite sequences with entries 0 or 1 is non- denumerable and conclude that the set of real numbers in [0, 1] or any nondegenerate interval is nondenumerable. Hint: If S were countable. i.e., S(s,. n 1) where S=(XX), then (1x11, 1-X22X) would be an
Demonstrate that if B is a countable set and B, ((b,...):b, e B for Isisn). then B, is countable, n
Prove that lim,(A, B) = lim, Alim, B, and lim, A, B, lim, 4, lim, B Moreover, lim A, A and lim B = B imply lim, (A, B) = AB and lim A,B,= AB. =>
10
Moreover, if (4,. n 1) is a sequence of disjoint sets, i.e., 4,4,0,ij. then lim UA,=
Verify that 4, = lim U 4, and 4 = lim -
3. Prove that lim A, lim, A. Specify lim 4, and lim 4, when A2, B. A-1 C.j=1.2.....
Show that [0, n/(n + 1)) = [0. 1). (0.1/n) =
Prove (i) if A, is countable, n 1, so is 4, (ii) if A is uncountable and B > A. then B is uncountable.
If (Q,, ) is a measure space, prove that lim 4) slim (4) for A, Analogously, if A) ii. Put (4)--((b)-fla)) for AU (a,b) e %, where 0sa,
Show that a finite measure on a class is (i) subtractive. i.c. A, ei= 1.2. A, A, and 4, A, e implies 4,-4)(A)-4,), and (ii) monotone. ie., (4) S4) with 4, as in (i). In view of (i), if there is one set in of with finite measure, the proviso (2)-0 is automatically satisfied.
1.2. In MATLAB/SCILAB write a script to use randn and cumsum to generate, say, five realizations of aWiener process—approximate it by taking n = 1000 time steps over 0 < t ≤ 1 . Plot the realizations. Introduce a drift μ and volatility σ and plot realizations of the resultant process; for
1.8. Quantitatively investigate convergence as time step h→0 at time t=1 for a range of realizations of exponential Brownian motion (use relative differences). Summarize your findings.
1.1. Get your friends and family to play this simple little game that illustrates a key aspect of stochastic mathematics in application to finance.1. Draw on a big sheet of paper a sequence of 30 squares and label them consecutively 0 (bankrupt), 1, 1, 2, 2, 3, 3, 4, . . . , 14, 14, 15
1.3. Estimate (albeit very crudely) the stock drift and stock volatility of the price of silver(US$/ounce) from the data in Table 1.2.Table 1.2. Annual closing prices of silver (US$/ounce).Year Close 2000 $4.595 2001 $4.650 2002 $4.790 2003 $5.960 2004 $6.845 2005 $8.910 Table 1.3. Annual closing
1.4. Estimate (albeit crudely) the stock drift and stock volatility of the value of the Japanese Nikkei Index (yen) from the data in Table 1.3. Plot the Index as a function of time and copy and paste the data into MATLAB/SCILAB or an equivalent program, and compute estimates of the stock drift and
1.5. Implement the Euler code of Example 1.6. Plot numerical solutions for α = 1 andβ = 2 over 0 ≤ t ≤ 1 . Now generate simultaneously m = 300 realizations of the solution of the SDE (for α = 1 and β = 2) and draw a histogram of their values at time t = 1 , say, using bins of width 0.25
3.1. Show that Var[X(t)]=X20e(2α+β2)t−e2αtfor X(t)=X0exp[(α−12β2)t+βW(t)] .Give an example showing that even when the expectation decays to zero, the variance may nonetheless grow in time.
1.6. Modify the Euler code of Example 1.6 to numerically solve and plot five different realizations of the (Ornstein–Uhlenbeck process) solutions of the SDE: dS =−Sdt+dW with initial conditions S0 = 0,1,2,3, and 4 . Perhaps integrate over 0 ≤ t ≤ 4 .
1.7. Modify the Euler code of Example 1.6 to numerically solve and plot five different realizations of the solutions of the following SDEs:1. dX =2X 1+t +(1+t)2dt+(1+t)2dW with X0 = 0 and X0 = 1 ;2. dX = dt+2√XdW with X0 = 1 ;3. dX=−12 e−2Xdt+e−XdW with X0 = 0 ;4. dX =12
1.9. What is the value of a forward contract on an asset when 1. its current value is $60, the agreed price after a year is $57, and the bond rate is 4% per year?2. its current value is $35, the agreed price after two years is also $35, and the bond rate is 5% per year?
2.8. Let an Ito process Y(t) = 1/(t+X(t)) in terms of the Ito process X(t) = W(t)2, where W(t) is a Wiener process. Use Ito’s formula to deduce an expression for dY in terms of only t, dt, Y, and dW.
3.2. Consider in turn each of the stochastic processes illustrated in Figures 3.14–3.17.Each figure shows ten realizations of some Ito process, a different Ito process for each figure. Assuming each Ito process is settling upon some steady state PDF p(x);sketch the PDF.
3.3. The realizations plotted in Exercise 3.2 all come from Ito processes you have met or will soon meet. Identify as best you can which Ito process shown in Exercise 3.4 corresponds to each of the four figures.
3.4. Use the Fokker–Planck equation to investigate the steady state PDFs for the following stochastic processes:1. dX = (2X−X2)dt+XdW for X(t) ≥ 0 ;2. dX = (3X−2X2)dt+2XdW for X(t) ≥ 0 ;3. dX = β1+X2dW ;4. dX=−Xdt+1+X2dW ;5. dX = Xdt+(1+X2)dW ;6. dX=−Xdt+√2(1+X2)1/4dW .
3.5. Use the Fokker–Planck equation to find the structure of the steady state PDF of solutions X(t) to the SDE dX=−2Xdt+1+X2dW .
3.6. Consider a general SDE dX = μ(X)dt+σ(X)dW . Assuming a steady state PDF exists for the ensemble of realizations, show that the steady state PDF may be written as p(x) =A D(x)expμ(x)D(x)dx, where D(x) = 12σ(x)2 and A is a normalization constant. 4- 3- 2- 0 1 2 3 4 5 6 7 8 9 10 time t
2.5. Use Ito’s formula to find an SDE satisfied by the Ito process X = cos(t2+W) .Write the SDE in terms of only t, X, and the differentials dt and dW.
3.8. Generalize the proof of the Fokker–Planck equation (3.2) to include variable volatilityσ(x). This generalization is quite difficult because of the need to keep track of all the details, and also you will need to showξ = x∓σ√h+(σσ−μ)h+· · · .
1.11. Use a one-period binomial model to estimate that the hedge ratio H = 2 and the fair value of the call option is $5.57 when the initial asset price is $35, the exercise price of the call option is also $35, the rate of interest for bonds is 10%, and the period is one year. Assume that the
3.9. Reanalyze Example 3.7, but suppose instead that the death rate μn= βn2 due to increased competition for limited resources by larger populations of individuals. Use its Fokker–Planck equation to then argue that the corresponding SDE is approximately dN = (αN−βN2)dt+αN+βN2dW.
3.10. Consider the PDE ∂u∂t + ∂u∂x+ 12∂2u∂x2 =0 . One solution is u(t,x) = x2−2xt+t2−t.Write down the corresponding SDE, and its solution X(t), and then verify the Feynman–Kac formula for solutions with initial condition X(s) = y .
2.18. Consider in turn each of the four options whose expiration values are plotted in Figure 2.6. Use the qualitative arguments of Section 2.3 to sketch by hand the option value, as a function of S, as predicted by the Black–Scholes equation at a significantly earlier time.
2.17. Using a self-financing portfolio and assuming the interest rate on bonds is the constant r, rederive the Black–Scholes equation (2.6) from (2.9).
2.9. Use the simple version (2.3) of Ito’s formula to generate, by substituting a variety of functions f(t,W), a variety of SDEs whose analytic solutions you know.
2.10. Use Ito’s formula (2.4) for the following:• Since X = t+W(t) satisfies dX = dt+dW, confirm that Y = sinhX satisfies dY = 12 Y +1+Y2 dt+1+Y2dW .• For the above Ito process Y, deduce dZ for Ito process Z = Y3 .Recall that d dx sinhx = coshx , d dx coshx = sinhx, and cosh2x = 1+sinh2x .
2.7. Let I0(t) = 1 , I1(t) = W(t) , I2(t) = W(t)2−t , I3(t) = W(t)3−3tW(t), and I4(t) = W(t)4−6tW(t)2+3t2 . Use Ito’s formula to show dIn = nIn−1dW .Describe the analogue with classic calculus. Use guesswork, checked with Ito’s formula, to determine corresponding I5 and I6.27
2.11. Consider the Black–Scholes equation (2.6) with zero volatility:∂C∂t+rS∂C∂S= rC.Show by algebraic differentiation that C = f(Se−rt)ert satisfies this PDE for any differentiable function f.Now suppose you are considering some option with specified value at the expiry time T, say
2.12. In November 2007 the companyWesfarmers purchased the Coles–Myer group. As part of the settlement, Coles–Myer shares were transformed into Wesfarmers Partially Protected Shares (WPPS). With a Floor Price specified to be $36, a Cap Price of $45, and the Lapse Date meaning the date four
2.13. When one purchases, as an asset, shares in a company, one also expects some regular dividends as part of the financial benefit of holding the shares. Generalize the arguments of Section 2.3 to derive a modified Black–Scholes equation when the asset in the portfolio also returns dividends at
2.14. This exercise is challenging. Extend the previous modification to the case where the dividend has both a deterministic and stochastic component; over a small time interval the dividend’s contribution to increasing the value of the portfolio would beδΔt+ρΔW.
2.15. Use the trinomial model to value the following option. Comment on the match with the qualitative graphical solution.You, a bookmaker, have a client, Bob, who wants to bet $100 that Telstra shares will rise to be above $5 in one year’s time. The share’s current value is $4. Suppose the
2.16. Modify the trinomial MATLAB/SCILAB code for Example 2.10 to value a knock out option (Example 2.9) which is as for Example 2.10 but additionally pays a rebate of $1 if the asset price ever falls to $30.
2.6. Use Ito’s formula to show that the following are the solutions to the SDEs you solved numerically in Exercise 1.7:1. X = (1+t)2(X0+t+W(t)) satisfies dX =2X 1+t+(1+t)2 dt+(1+t)2dW;2. X = (1+W(t))2 satisfies dX = dt+2√XdW with X0 = 1 ;3. X = log(1+W(t)) satisfies dX=−12 e−2Xdt+e−XdW
3.11. Corresponding to the Black–Scholes equation is the SDE (3.8). Use many realizations of its solution to numerically estimate the options of Exercises 1.11, 1.12, and 1.19. Roughly what errors do you expect in your estimates?
3.12. Solve analytically 2du dx +xd2u dx2 = 0 on 1 < x < 5 such that u(1) = 4 and u(5) =0 . Use the corresponding SDE to estimate u(2), u(3) and u(4) and their errors.Compare. Say we use m= 100 realizations with a time step of h = 0.001 : discuss how the errors improve with increasing m and/or
1.20. Clearly explain the crucial features of a Wiener process that empower us to model noisy, fluctuating dynamics. Explain how and why a Wiener process is transformed to model general noisy, fluctuating signals.
1.10. Reconsider Example 1.10 of the valuation of a forward contract on Telecom shares.What bond rate would cause the current value of the forward contract to be precisely$1?
1.19. Alice buys a put option from Bob (see Definition 1.13), strike price X, for some fair price P which you are to determine by arguments similar to those employed for call options. Investigate one step of a binomial model for determining the value of the put option, then apply your result to a
1.18. Alice buys a put option from Bob (see Definition 1.13), strike price X = $39 , for some fair price P which you are to determine by arguments similar to those employed for call options. Develop one step of a binomial model estimating the value of the put option.1. First, show that at the
1.17. Alice buys a put option from Bob (see Definition 1.13), strike price X = $57 , for some fair price P which you are to determine by arguments similar to those employed for call options. Develop one step of a binomial model to estimate the value of the put option.Definition 1.13. A European put
1.16. Modify the MATLAB/SCILAB code of Algorithm 1.6 to use an n-step binomial lattice for estimating the value of the call option (where n is a supplied parameter).Show that it is not until n ≥ 512 (or thereabouts) that the estimated initial value of the call option converges to $3.40 to the
1.15. Suppose that volatility corresponds to +25% and −20% per year, and that the bond interest rate is 12% per year. What would be the corresponding factors u,d, and R for each monthly step of a twelve-step monthly binomial lattice model?
1.14. An asset is initially priced at $50. The exercise price of a call option on this asset is also $50. Assume that the asset price moves up by 25% or down by 20% per year. The call option is offered for sale at $8. Use a one-period binomial model to estimate the interest rate for bonds that
1.13. What interest rate for bonds would be needed for the call option of Example 1.9 to be correctly valued at $4 ?
3.7. Derive from first principles the Fokker–Planck equation∂p∂t=∂2∂x2(12 x2p)for the PDF p(t,x) of solutions X(t) of the SDE dX = XdW(this SDE is the case of the financial SDE dX = αXdt+βXdW for no stock drift,α = 0 , and unit stock volatility, β = 1).• First, assume small time
2.1. Use the probability distribution function p(z) = exp(−z2/2)/√2π for N(0,1) random variables Z and integration by parts to deduce that E[Zp] = (p−1)(p−3) . . .1 for even integers p. Hence deduce forWiener processesW(t) that E[W(t)p] = (p−1)(p−3) . . .1· tp/2 (for even integers p)
2.2. Show that almost surelyn−1 j=0 (ΔWj)4→0 as the time step Δtj=h→0 for a fixed final time T = nh. HINT: Use thatE[|Z|4] = 3 for random variates Z ∼ N(0,1) .
3.13. Verify, perhaps using computer algebra, that the PDFs for theWiener and Ornstein–Uhlenbeck processes satisfy the Kolmogorov backward equation (3.11).
4.1. Prove the linearity (4.3) of the Ito integral for step functions.
4.2. Prove the union (4.4) of the Ito integral for step functions.
4.3. By considering the differential d(W3 − 3tW), deduce the Ito integral I(ω) = T 0W(t,ω)2−tdW(t,ω), where W(t,ω) is a Wiener process. Hence verify the martingale property and Ito isometry for this Ito integral.
4.5. Prove that (a+b)2 ≤ 2a2+2b2 for any real numbers a and b by considering (a+b)2+(a−b)2. Similarly prove (a+b+c)2 ≤ 3a2+3b2+3c2 for any real a,b, and c.
4.6. Reconsider E[W(t,ω)k] for a Wiener process W(t,ω). Consider d(W(t,ω)k) by Ito’s formula and the martingale property of Ito integrals to deduce E[W(T,ω)k] = 12 k(k−1)T 0E[W(t,ω)k−2]dt for k ≥ 2 .Hence determine E[W(t,ω)2], E[W(t,ω)4], and E[W(t,ω)6] . Compare your solution with
4.7. For a nonrandom function g(t), that is, g has no dependence upon the realizationsωof a Wiener processW(t,ω), use Ito’s formula to show g(t)dW(t,ω) = g(t)W(t,ω)−W(t,ω)dg dt dt.
4.8. Let In(t,ω) = tn/2Hn(W(t,ω)/√t), where Hn is the nth Hermite polynomial,(see (Kreyszig 1999, pp. 246–247) or (Abramowitz and Stegun 1965, Chap. 22)), and the first few Hermite polynomials are H0(x) = 1, H1(x) = x, H2(x) = x2−1, and H3(x) = x3−3x. What are I0(t,ω),. . . ,I3(t,ω)?
2.4. Use Ito’s formula to deduce the differential dX for the stochastic process X(t) =2+t+exp(W(t)) .
2.3. Argue that almost surelyn−1 j=0 |ΔWj|p→ 0 as h→ 0 for p ≥ 3. HINT: Use that E[|Z|p] is finite for Z ∼ N(0,1) .
1.12. Use a one-period binomial model to estimate the hedge ratio and the fair value of the call option when the initial asset price is $50, the exercise price of the call option is also $50, the rate of interest for bonds is 10%, and the period is one year. Assume that the asset price moves up by
4.4. Argue from basic principles that T0 W(t,ω)dW(t,ω) = 12W(T,ω)2−Tby forming a partition over the time interval [0,T], approximating the integral as In(ω) =n−1 j=0 WjΔWj , and showing that these almost surely tend to I(ω) =12W(T,ω)2−Tby considering I−In.1. Argue that I−In=
Two boys, Charlie and Doug, take turns rolling two dice with Charlie going first. If Charlie rolls a 6 before Doug rolls a 7 he wins. What is the probability that Charlie wins?
5% of men and 0.25% of women are color blind. Assuming that there are an equal number of men and women, what is the probability that a color-blind person is a man?
To improve the reliability of the channel described in the last example, we repeat each digit in the message three times. What is the probability that 111 was sent given that (a) we received 101? (b)We received 000?
Suppose X1, . . . , Xn are independent exponential(λ). Show that min{X1, . . . , Xn} = exponential (nλ)
Suppose X and Y have joint density f (x, y) = 6y when x > 0, y > 0, and x + y < 1.(a) Find the marginal densities of X and Y and(b) The conditional density of X given Y = y.
At Llenroc College, 63% of freshmen who are premed switch to a liberal arts major, while 18% of liberal arts majors switch to being premed. If the incoming freshman class is 60% premed and 40% liberal arts majors, what fraction graduate as premed?
A person is flipping a coin repeatedly. Let Xn be the outcome of the two previous coin flips at time n; for example, the state might be HT to indicate that the last flip was T and the one before that was H. (a) Compute the transition probability for the chain.(b) Find p2.
A taxicab driver moves between the airport A and two hotels Band C according to the following rules. If he is at the airport, he will go to one of the two hotels next with equal probability. If at a hotel then he returns to the airport with probability 3/4 and goes to the other hotel with
Consider a gambler’s ruin chain with N = 4. That is, if 1 ≤ i ≤ 3, p(i, i + 1) = 0.4, and p(i, i − 1) = 0.6, but the endpoints are absorbing states: p(0, 0) = 1 and p(4, 4) = 1. Compute p3 (1, 4) and p3 (1, 0).
An outdoor restaurant in a resort town closes when it rains. From past records it was found that from May to September, when it rains one day the probability that it rains the next is 0.4; when it does not rain one day it rains the next with probability 0.1. (a) Write the transition matrix. (b) If
Market research suggests that in a 5-year period 8% of people with cable television will get rid of it and 26% of those without it will sign up for it. Compare the predictions of the Markov chain model with the following data on the fraction of people with cable TV: 56.4% in 1990, 63.4% in 1995,
A sociology professor postulates that in each decade 8% of women in the workforce leave it and 20% of the women not in it begin to work. Compare the predictions of his model with the following data on the percentage of women working: 43.3% in 1970, 51.5% in 1980, 57.5% in 1990, and 59.8% in 2000.
The following transition probability describes the migration patterns of birds between three habitats:If there are 1,000 birds in each habitat at the beginning of the first year, how many do we expect to be in each habitat at the end of the year? At the end of the second year? 1 1 0.75 2 0.07 23 3

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Elementary Probability For Applications 1st Edition Rick Durrett - Solutions

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