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mathematical models of financial derivative

Questions and Answers of Mathematical Models Of Financial Derivative

Consider the securities model with determine the value of k such that the law of one price holds. Taking this particular value of k in Ŝ∗ (0), does the securities model admit dominant trading
Show that if there exists a dominant trading strategy, then there exists an arbitrage opportunity. How to construct a securities model such that there exists an arbitrage opportunity but dominant
Show that h is an arbitrage if and only if the discounted gain G∗ satisfies (i) G∗ ≥ 0 and (ii) E[G∗] > 0. Here, E is the expectation under the actual probability measure P,P(ω) > 0.
Suppose the set of risk neutral measures for a given securities model is nonempty. Show that if the securities model is complete, then the set of risk neutral measures must be singleton. Consider the
Suppose a betting game has three possible outcomes. If a gambler bets on outcome i, then he receives a net gain of di dollars for one dollar betted, i = 1, 2, 3. The payoff matrix thus takes the form
Consider the following securities modeldo risk neutral measures exist? If not, explain why. If yes, find the set of all risk neutral measures. S* (1; 2) = 3 4 25 2 4 S* (0) = (24),
Let P be the true probability measure, where P(ω) denotes the actual probability that the state ω occurs. Define the state price density by the random variable L(ω) = Q(ω)/P(ω), where Q is a
Suppose the set of risk neutral measures for a given securities model is nonempty. Show that if the securities model is complete, then the set of risk neutral measures must be singleton. Under
Let Πu and Πd denote the state prices corresponding to the states of asset value going up and going down, respectively. The state prices can also be interpreted as state contingent discount rates.
Suppose u > d > R in the discrete binomial model. Show that an investor can lock in a riskless profit by borrowing cash as much as possible to purchase the asset, and selling the asset after one
Consider the sample space Ω = {−3, −2, −1, 1, 2, 3} and the algebra F = {∅,{−3, −2},{−1, 1},{2, 3},{−3, −2, −1, 1},{−3, −2, 2, 3},{−1, 1, 2, 3}, Ω}. For each of the
Let X, X1, ··· , Xn be random variables defined on the filtered probability space (Ω, F, P). Prove the following properties on conditional expectations: (a) E[XIB] = E[IBE[X|F]] for all B ∈
Let X = {Xt ; t = 0, 1, ··· ,T} be a stochastic process adapted to the filtration F = {Ft ; t = 0, 1, ··· , T}. Does the property: E[Xt+1 − Xt|Ft] = 0,t = 0, 1, ··· ,T − 1 imply that X
Consider the binomial experiment with the probability of success p, 0 < p < 1. We let Nk denote the number of successes after k independent trials. Define the discrete process Yk by Nk − kp, the
Consider the two-period securities model shown in Fig. 2.5. Suppose the riskless interest rate r violates the restriction r Fig. 2.5. S(0;92) = 4 S(1; 0,0) = 3 1 S(1; 0,0) = 5 S(2; ₁) = 4 - S(2;
Let X be a normal random variable with mean mX and variance σ2X. Show that the higher central moments of the normal random variable are given byFor the log-normal random variable Z = exp(αX), α is
Suppose τ1 and τ2 are stopping times, show that max(τ1, τ2) and min(τ1, τ2) are also stopping times.
Suppose Z(t) is the standard Brownian process, show that the following processes defined by are also Brownian processes.To show that Xi(t) is a Brownian process, i = 1, 2, 3, it suffices to show
Suppose {X(t),t ≥ 0} is the standard Brownian process, its corresponding reflected Brownian process is defined by Show that Y(t) is also Markovian and its mean and variance are, respectively,
Consider the Brownian process with drift defined by where Z(t) is the standard Brownian process, find E[X(t)|X(t0)], var(X(t)| X(t0)) and cov(X(t1), X(t2)). X(t)= ut +oZ(t), X(0) = 0,
Let the stochastic process X(t),t ≥ 0, be defined by where Z(t) is the standard Brownian process. Show that = [₁ eu(1-4), S 10 X(t) = ea(t-u) dz(u),
Let Z(t) denote the standard Brownian process. Show that (a) dZ(t)2+n = 0, for any positive integer n,(b)for any positive integer n, (c) E[Z4(t)] = 3t2, (d) E[eαZ(t)] = eα2t/2. S to Z(t)" dz(t)
Assume that the price of an asset follows the Geometric Brownian process with an expected rate of return of 10% per annum and a volatility of 20% per annum. Suppose the asset price at present is
Let Z(t),t ≥ 0, be the standard Brownian process, f (t) and g(t) be differentiable functions over [a,b]. Show that Interchange the order of expectation and integration, and observe b E[ f*
Let Z(t),t ≥ 0, be the standard Brownian process. Show that has zero mean and variance σ2(T − t)3/3. T [" t [Z(u) - Z (t)] du
Suppose the stochastic state variables S1 and S2 follow the Geometric Brownian processes whereLet ρ12 denote the correlation coefficient between the Brownian processes dZ1 and dZ2. Let f = S1S2,
Show that N2(a, b; ρ) + N2(a, −b; −ρ) = N(a). Also, show that N₂ (a, b; p): = [" Lance n(x) N b - px 1- p dx.
Define the discrete random variable X by where the sample space Ω = {ω1,ω2,ω3},P(ω1) = P(ω2) = P(ω3) = 1/3. Find a new probability measure P̃ such that the mean becomes EP̃ [X] = 3.5
Suppose the function F(x, t) satisfies with terminal condition: F(X(T),T) = h(X(T)). Show that F(x, t) = e−r(T −t)Et[h(X(T))|X(t) = x], t a F - at a F + m(x, t). Әx + o²(x, t) 0²
Consider a forward contract on an underlying commodity, find the portfolio consisting of the underlying commodity and a bond (bond’s maturity coincides with forward’s maturity) that replicates
Let uμ(x, t) denote the solution to the partial differential equationwith u(x, 0+) = δ(x). From (2.3.12), it is seen that By applying the change of variable: x = y + μt, show that the above
Let P and Q be two probability measures on the same measurable space (Ω, F) and let f = dQ/dP denote the Radon–Nikodym derivative of Q with respect to P. Show thatwhere G is a sub-sigma-algebra of
Given that the process St is a Geometric Brownian process, it follows thatwhere Zt is a P-Brownian process. Find another measure P̃ by specifying the Radon–Nikodym derivative dP̃/dP̃ such that
Suppose the strike price is growing at the riskless interest rate, show that the price of an American put option is the same as that of the corresponding European counterpart. Show that the early
Show that the lower and upper bounds on the difference between the prices of the American call and put options on a foreign currency are given by where Bf (τ) and B(τ) are bond prices in the
Show that the put prices (European and American) are convex functions of the asset price, that is, where S1 and S2 denote the asset prices and X denotes the strike price.Let S1 = h1X and S2 = h2X,
Show that a portfolio of holding various single-asset options with the same date of expiration is worth at least as much as a single option on the portfolio of the same number of units of each of the
Suppose the strike prices X1 and X2 satisfy X2 > X1, show that for European calls on a nondividend paying asset, the difference in the call values satisfieswhere B(τ) is the value of a pure
A box spread is a combination of a bullish call spread with strike prices X1 and X2 and a bearish put spread with the same strike price. All four options are on the same underlying asset and have the
A strangle is a trading strategy where an investor buys a call and a put with the same expiration date but different strike prices. The strike price of the call may be higher or lower than that of
How can we construct the portfolio of a butterfly spread that involves put options with different strike prices but the same date of expiration and on the same underlying asset? Draw the
A strip is a portfolio created by buying one call and writing two puts with the same strike price and expiration date. A strap is similar to a strip except it involves long holding of two calls and
Consider a forward contract whose underlying asset has a holding cost of cj paid at time tj, j = 1, 2, ···, M − 1, where time tM is taken to be the maturity date of the forward. For notational
Consider a European call option on a foreign currency. Show that Give a financial interpretation of the result. Deduce the conditions under which the value of a shorter-lived European foreign
Deduce from the put-call parity relation that the price of a European put on a nondividend paying asset is bounded above by Then deduce that the value of a perpetual European put option is zero.
Portfolio A: One European call option plus X dollars of money market account. Portfolio B: One American put option, one unit of the underlying asset and borrowing of loan amount D. The loan is in
Assume that the dynamics of the short rate process under the risk neutral measure is governed by with dZ1(t)dZ2(t) = ρdt. Show that the time-t price of a unit par discount bond is given byLet
Empirical evidence reveals that the long rate and the spread (short rate minus long rate) are almost uncorrelated. Suppose we choose the stochastic state variables in the two-factor interest rate
Consider the multifactor extension of the CIR model, where the short rate r(t) is defined by Here, Xi(t), i = 1, ··· ,n, are uncorrelated processes of the one-factor CIR type as governed byunder
Consider the three-factor stochastic volatility model [see (7.3.17)], by assuming constant market prices of risk λr,λr and λv, show that the bond price function B(t, T) satisfies the partial
Suppose Li(t)satisfies the LLM model as in Problem 8.30. Here, we would like to find the stochastic differential equation of Li(t) under the spot LIBOR measure QM̃ whose numeraire is the discrete
Let the exchange rate process X(t), T -maturity domestic and foreign bond processes Bd (t, T ) and Bf (t, T ) be the same as those defined in Problem 8.9. Find the time-t value of the LIBOR spread
Consider the two-factor Gaussian model, which is a combination of the Ho–Lee and Vasicek models. Let the volatility structure in the HJM framework be given by Show that the bond price B(t, T) is
Using the analytic representation of F(t, T) in (7.4.7), show that the discount bond price can be expressed as B(t, T) = B(0, T) B(0, t) T + [ n xp(-2[/" ['a',(u, s) duds į (u, s) dz, (u) ds]). T
Consider the multifactor extension of the Inui–Kijima model. Let σiF (t, T ), i = 1, 2, ··· ,n, satisfyfor some deterministic function ki(T ) and initial conditionDefine where Also, show that
Under the one-factor Inui–Kijima model [see (7.4.23)], we would like to solve for B(t, T) in terms of r(t), ∅(t), F(0,t) and other parameter functions. Defineshow that T = $₁² e - 5₁² K(s)
Let σB(t, T) denote the volatility structure of the return of a discount bond. The Gaussian term structure models are characterized by (i) deterministic σB(t, T) and (ii) a Markov short rate
Let L(t,T ) denote the time-t LIBOR process Lt(T, T + δ] over the period (T, T + δ], and σiL(t, T ) be its ith component of volatility function [see (7.4.26)]. From the relationand the
For the one-factor Inui–Kijima model, suppose the short rate volatility depends on its level and show that the forward rates are positive with probability one. k(t) F (0, t) + a -F(0, t) > 0, Ət
The dynamics of the instantaneous forward rate F(t,T ) under the risk neutral measure Q is governed by where (Z1(t)···Zm(t))T is an m-dimensional Brownian process under Q. Let (W1(t)···
The price function of a European call option under stochastic interest rates can also be solved using the partial differential equation approach. Let the asset value process St and the short rate
Show that the risk neutral measure Q and the T-forward measure QT are identical if and only if the short rate rt is deterministic. Also, show that the instantaneous forward rate is given bywhere B(t,
Consider a contingent claim whose payoff XT′ is known at time T′ but it is payable at a later time T , T >T′. Show that the time-t value of the contingent claim is given bywhere B(T′,T) is
Let R(t,t+δ) denote the yield to maturity over the period (t, t+δ] of a discount bond maturing at t + δ, and f (0,t,t + δ) be the forward rate observed at time zero over the period (t, t +δ].
Under the risk neutral measure Q, the dynamics of the price process of an asset S(t) and the discount bond price process are governed by where Z1(t) and Z2(t) are uncorrelated standard Q-Brownian
The forward rate over the future period (T1,T2] as observed at the earlier time t can be computed from bond prices by the formulaOn the other hand, the futures rate is given by the expectation of the
We would like to examine the credit yield spreads of the floating rate debt and fixed rate debt under the Merton risky bond model with stochastic interest rate (Ikeda, 1995). Let At and rt denote the
Let X(t) denote the exchange rate process in units of the domestic currency per one unit of the foreign currency, and let rd (t) and rf (t) denote the domestic and foreign riskless interest rate,
Under the risk neutral measure, St follows the Geometric Brownian process whereLet B(t,T) be the time-t value of the T-maturity discount bond. Under the forward measure QT where B(t,T) is used as the
Let the dynamics of the short rate r(t) be governed by the extended Vasicek model Show that the value of the European call option with strike price X maturity at T on a T’-maturity discount bond
Under the risk neutral measure Q, assume that the bond price B(t,T) is related to the short rate rt byDefine the following probability measures QT and QT∗ , whereConsider a European call option on
Let Y(t; τ) denote the yield at time t for a discount bond with a fixed time to maturity τ . The average of the constant maturity yield Y(t; τ) over a prespecified time period (0,T ] is given byWe
Suppose the dynamics of Li(t) under the forward measure QTk is governed by (8.3.23), show that the distribution of the LIBOR Li(T ) under QTk admits the following lognormal approximation (Daniluk and
Consider a European call bond option maturity on T0 whose underlying bond pays Ai ≥ 0 at time Ti, 1 ≤ i ≤ n, where 0 0 1 n. Assume that the zero-coupon bond price B(t,T ) follows the one-factor
We would like to price the floor on the composition defined in Problem 8.22 using the LIBOR Market model. Now, we assume that the LIBOR Li(t) follows the arithmetic Brownian process:Problem
Suppose the forward rate F(t,T ) under the risk neutral measure is governed by Suppose the forward rate F(t,T ) under the risk neutral measure is governed by and consider the coupon bond with n
Suppose the forward rate volatility under the one-factor HJM model takes the form show that the Jamshidian decomposition technique (Jamshidian, 1989) cannot be used to price an option on a coupon
Suppose the short rate rt is governed by the Vasicek modelwhere Zt is a Brownian process under the risk neutral measure Q. Show that the stochastic differential equation of the swap rate Kt[T0,Tn]
Consider the forward payer swap settled in advance, that is, each reset date is also a settlement date. The LIBOR Li(t) reset at Ti is used to determine the cash flow at Ti. Suppose the payments made
Suppose we define the modified forward LIBOR Lmi (t) and futures LIBOR Lf i (t) byrespectively. Here, QTi and Q are the Ti-forward measure and risk neutral measure, respectively. Assuming that the
Suppose the forward LIBOR L(t,T) satisfies the following stochastic differential equation under the risk neutral Q-measurewhere σi(t, T), i = 1, 2, ··· ,n are deterministic volatility functions
Consider the Lognormal LIBOR Market (LLM) model for the LIBOR Li(t), i = 0, 1, ··· ,n − 1, defined on the tenor structure {T0,T1, ··· ,Tn} where 0 0 1 n. Let vi(t) denote the scalar
Use the following relations,to show the result in (8.3.23). and dz(t) = dz' (t) + OB (t, T;) dt = dZk (t) + OB (t, Tk) dt di+1Li(t) 1+&i+1L;(t) OB (t, Ti) - OB(t, Ti+1)= of (t. T),
The no arbitrage bond price process is assumed to follow the m-dimensional Gaussian HJM model. Let t < T0
Suppose the cap rate of a cap and the floor rate of a floor are both set equal to L. Let C(t; T0, ··· ,Tn) and F(t; T0, ··· ,Tn) denote the time-t value of the cap and floor, respectively.
Under the risk neutral measure, suppose the dynamics of the domestic interest rate rd and foreign interest rate rf follow the mean-reversion processes:Let ρdf and ρfX denote the correlation
Suppose we write the price function of the swaption as [see (8.4.9)], the resulting expression reveals a hedging strategy of the swaption using discount bonds with varying maturities. The
Consider the time-t value of the LIBOR-in-arrears payment [see (8.3.14a,b)], show that where Cj (t; Tj ,Tj+1,X) is the time-t value of the caplet with strike X, resetting at Tj and paying max(Lj
The holder of a reverse floater is entitled to receive the LIBOR Li(Ti) while pay max(K − Li(Ti), K’) at time Ti+1,i = 1, 2, ··· ,n − 1, with respect to a unit principal, K > 0 and K’ > 0.
Consider the family of forward swap rates Kt[Ti,Tn],i = 0, 1, ··· ,n−1, with the common terminal payment date Tn. We would like to express the dynamics of Kt[Ti,Tn] under the terminal forward
Under the one-factor HJM framework, show that where EtQ denotes the expectation under the risk neutral measure Q conditional on information Ft. Explain why the forward rate F(t, T) is a biased
Consider a Constant Maturing Swap (CMS) caplet whose payoff at payment date Tp takes the formwhere the par swap rate over the tensor {T0, ··· ,Tn} is set at T0,Tp > T0 and scap is some pre-set
A differential swap may involve three currency worlds: interest payments are calculated based on the floating LIBOR of the first two currencies but the actual payments are denominated in a third
Following the n-factor HJM framework, show that the dynamics of r(t) under the risk neutral measure [see (7.4.8b)] can be expressed as where F(t,T) is defined in (7.4.7). Can you provide a financial
Following the n-factor HJM framework, let the forward rate F(t, T) be governed by the following dynamics Show that the covariance of the increments of F(t,T1) and F(t,T2) is given byDeduce that

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