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Questions and Answers of Marketing Strategy Planning

Compute the specific form of ˆ‡u for (4.36).(4.36) u(c1, c2) = VCic2 and h(c1, c2) =Pici + P2C2
Verify that the utility function is of the HARA-type.
Convince yourself that (4.30) is consistent with our economic intuition.
Verify that this solution fulfills (4.28) with α = 1/γ and b = β/γ.
What is the fourth order expansion term if ∈ ∼ N(0, σ2)?
Verify that for u(w) = w equality has to hold in (4.12).(4.12) u(E[ W]) > E[u(W)]
Convince yourself that the characteristic function of the compensated compound Poisson-process is given by 4xc(1, u) = exp(At L (eka iux)f(v)dy) %3D -0-
Use P({τ > 0}) = 1 to prove this statement.
Prove that Cov[dWQ1, dWQ2] = ρdt still holds.
Can you see why the first equality is correct?
Use the fact that φ(z∗) = φ∗(−z) to convince yourself that this statement is true.
Confirm that a stable tree implies S = sn+l) (п) (n+1) ++2
Verify that the hedge-ratio in the above toy market is Δ = 1/2.
Confirm the above mentioned bounds for ST < K1 and ST > K2.
Provide a formal argument for this statement.
Verify the last statement.
Verify the last statement.
Compute the eigenvector associated with λ = 2/3.
Verify that X has expectation value μ and variance σ2.
Use (9.15) and (9.17) to establish this argument formally.(9.15)(9.17) D = (1 + g)Dj-1 + €; 1+g S, =*(1 + g)*D, = I+8 D, k=1
Verify that this process satisfies the martingale condition (9.22).(9.22) В, %3 о' ЕТ В+к1F1 = 6 E[ Brk\Fi]
Show that for X = a + bY the correlation of X and Y is ρ = 1.
Why are all returns on the CML perfectly negative correlated with the SDF?
Verify the first and second equality in (9.105).(9.105) r=- log E[M,|Fi-1] =p+ y(g + (1 – ø)(5 - $i-1)) – ro?(1 + A(s81-1))² =p+ yg - (1 – $), =p+ yg - 2
Make the substitution FA(x) = u to prove (10.29).(10.29) (u)>
Show that the single bet of Samuelson’s fair coin flip gamble is not favorable, but the double bet has already positive prospective value.
Use this relation to verify the last equality in (10.43).(10.43) Eļu(X)] = log(2"-)=log 2 = log 2 . = log 2 2" 2" n= 2" n=1 n=1
What is the time T value of a time t deposit of one unit of currency for t < T?
What is the payoff function of a forward contract?
Sketch an arbitrage argument for the price of a butterfly spread to be bounded by 0 ≤ Πt ≤ e−r(T−t)ΔK for t ≤ T.
Apply the appropriate arbitrage arguments to confirm (11.26).(11.26) (eTK - So)* < Po(K, T)
What is the σ-algebra FT?
Show that the definition of q implies 1- q' = T(1 – q) 1+rAt %3D
How should the put price behave under dividend payments?
Verify that the distributions of ΔWt and √ΔtZt coincide.
Can you see why the О(ΔS3)-term is unaffected?
Confirm that the payoff transform of a put is regular for v < 0.
Show that eYt is indeed a Q-martingale by using the law of iterated expectations.
Confirm that this transformation is algebraically correct.
Prove this statement by differentiating both sides of (16.72).(16.72) w'(t) 1 B(t) = w(t)
Can you see why the constant factor of the weights has changed?
Check that (17.6) satisfies (17.5).(17.6)(17.5) -At G(1) = e¯t G(s + t) = G(s)G(1)
Prove this statement by computing the characteristic function, associated with the probability mass function (17.8).(17.8)
Confirm that E[ξt|Ft–1] = 0
Use (17.23) to show that P(Nt= n|Ft) is indeed a probability.(17.23) f(Ax,F1-1) = >f(Ax;|N, = n, F1-1) · P(N, =n\F1-1) 1 (Ax, – µ, – n®)²' 1 Σ exp 2 n! V27(h, + n8²) h, + nổ? II
Verify that the jump part of dxt is indeed YtdNt.
What is the barrier probability for the minimum not to cross xl?
Which process has characteristic exponent Wи) 3D іща — }u?о?) Po?|
Use Euler’s identity to verify this statement.
Use Taylor-expansion to confirm this statement.
Confirm this result with the help of (18.23).(18.23) B(t, T) = C ενaT-0 + Pe4NT-0) n=1
Confirm the result for the coefficients an,1.
Show that for a zero-coupon bond, convexity is the squared time to maturity.
Verify that this is a finite difference approximation of (18.42).(18.42) log P(1, T) A(t, T) =
Prove this statement with the help of (18.42).(18.42) At, T) =- log P(t, T) ƏT
Is the contract in Figure 18.12 a payer or receiver swap from the perspective of A?Figure 18.12 fixed B floating
Compute d+/− for the bond volatility σP(t) = σ · (T− t).
State the receiver swap parity relation.
Convince yourself that rS(T0) is the original par rate.
Show that FÌƒt in (19.53) is also a Q-martingale.(19.53) F, = P (1 - Lu, T. T+ 1/4) L(t, T, T + 1/4)
Confirm this result by using f(t, T) = Y (t, T) +∂y/∂T(T – t).
Use Itô-isometry to compute Var[r(s)|Ft],.
Prove equation (20.34).(20.34) V2KO P(r(0)
Which modifications have to be made, if the principal is not normalized?
Use B(T, T) = 0 to confirm this result.
Use (20.76) to verify this result.(20.76) Rt, T) = r(1) – 0;(T-1n²
Show that Xt is not a martingale.
Verify that σf(t, T) = σe−κ(T−t) generates a Markovian model.
Prove the equality by writing pn(t)/pn(t) as a telescoping product.
Use linearity of the DC-operator to prove (21.35).(21.35) |dW$) = \dW°) – R(DC log(S/U))dt
Show that Cov[dL1, dL3] = σ1L1σ3L3ρ12ρ23dt.
Prove the second equality in (21.88).(21.88) Σ Vim (To, To)|(1 + AtL(To) (m), V(1, To) = PA(1)-
Show that the corrected solution log L̂n(T0) contains indeed a drift function, frozen at the midpoint between t and T0.
Verify that integrating the complex function of Example A.5 along a simple closed curve C3, enclosing all three singularities, yields Az)dz= 2ni(1 – sin 1)
Check that (A.45) is correct.(A.45)
Show that the integral equals 1/6 – i/2, when calculated along the straight line γ(t) = t + it.
How is cos z represented in terms of complex exponentials?
Which Cauchy–Riemann-equation was violated for f′(z) = z∗?
Show that ii is a real number. Use that i has an exponential representation.
Use the Euler-identity to confirm (A.17).(A.17) 1= e° = e2ak
Show that r2 = |z|2 = z∗z holds.
Verify that the real and imaginary part of a complex number is obtained by Re[z] = }(z+ z*) and Im[z] = }(z – z*) %3D %3D
Convince yourself that the last equality indeed holds.
Confirm this linear expansion.
Confirm this equation.
Confirm Î”Cby differentiating (13.78) with respect to St.(13.78) C,(K, T) = S,P(d1) – e«T-1) KD(d2)
Verify the first equality in (13.120).(13.120) ӘР Ә(К- S) =-1 = as Is=S as
Verify the second solution for c2.
Confirm that is the correct generalized formula for a European binary call option. C(K, T) = e¯«T-1Þ(d_)
Confirm that (13.83) is correct.(13.83) av + (r – q)S- as రా్వాతా - rV=0
Verify the last formula from put-call parity.
Verify that Φ(x) = 1 − Φ(−x) holds.
Convince yourself that for x = log S the payoff function is equivalently formulated as C(K, T) = 0(x – log K)
Verify (13.68) for n = 2.(13.68) d' „Ax) → (-iuyAu) (х) (и)
Confirm that x(t) = eἱt, with ἱ = √–1, is also a solution.
Convince yourself that is also a representation of the Î´-function as Îµ †’ 0. pe(x) = Tere şe-{()* V2πε2
Calculate the price for a European put from the result in Example 12.1 and put-call parity. Confirm that the American put option is indeed more valuable.
Verify the last equation by using geometric series’ results.
Show that E[|Xt+1– Xt|Ft] =2t holds.
Check that the variance of X is indeed one.

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