Questions and Answers of Mathematical Models Of Financial Derivative

A sequential barrier option has two-sided barriers. Unlike the usual doublebarrier options, the order of breaching of the barrier is specified. The second barrier is activated only after the first
Considering the antithetic variates method [see (6.3.7a,b)], explain why Note that the amount of computational work to generate c̄AV [see (6.3.8)] is about twice the work to generate ĉ. By
Discuss how to implement the secant method in the root-finding procedure of solving the optimal exercise price S∗ti from the following algebraic equationin the Grant–Vora–Weeks algorithm (Fu et
It has been generally believed that the extension of the Tilley algorithm to multiasset American options is not straightforward. Discuss the modifications to the bundling and sorting procedure
For the 30/360 day count convention of the time period [D1,D2), with D1 included but D2 excluded, the year fraction is givenwhere di,mi and yi represent the day, month and year of date Di,i = 1, 2.
Let Ft(T0,Ti) denote the forward price at time t for buying at time T0 a unit-par zero-coupon bond with maturity Ti. Show that the forward swap rate [see (7.1.4)] can be expressed aswhere αi is the
Define the instantaneous forward rate to be lim S→R+ Lt[R, S]. Show that lim L[R, S] = S→R+ Ə - In B(t, R). ƏR
Suppose the price of a bond is dependent on the price of a commodity, denoted by Qt . Let the stochastic process followed by Qt be governed byBy hedging bonds of different maturities, show that the
Consider two securities, both of them are dependent on the interest rate. Suppose security A has an expected return of 4% per annum and a volatility of 10% per annum, while security B has a
Suppose the dynamics of the short rate r(t) is governed by the governing differential equation for the price of a zero coupon bond B(r,t) is given by [see (7.2.8)] the governing differential
From the bond price representation formula (7.2.10), use Ito’s differentiation to showwhere R(t,T ) is the yield to maturity. Also, try to relate the market price of interest rate risk λ(r,t) to
Suppose the forward rate as a function of time t evolves as where μ(t,T ) is a deterministic function of t and T . Show that the forward rate is normally distributed, where Explain why F(t,T
Suppose the duration D of a coupon-bearing coupon bond B at the current time t is defined by where ci,i = 1, 2, ··· ,n, is the ith coupon on the bond paid at time ti,F is the face value. Here, R
Recall that show that the forward rate is given bywhere the stochastic discount factor is defined by B(r, t) = exp ) = exp(- [* F(t, u) du) = E'o[exp(- [² r(u) du)].
Consider the Hull–White model where the short rate process follows where Zt is a Brownian process under the risk neutral measure Q. Using the relationshow thatAccordingly, the mean and variance of
Consider the yield to maturity R(t, T) corresponding to the Cox–Ingersoll– Ross model. Show that [see (7.2.32a,b)] Explain why an increase in the current short rate increases yields for all
Consider the yield curve associated with the Vasicek model [see (7.2.26)]. where F(t, t + τ) is the forward rate and EtQ is the expectation under Q conditional on the filtration Ft. Show that the
Consider the linear stochastic differential equation Show that the mean E[r(t)] is governed by the following deterministic linear differential equation:while the variance var(r(t)) is governed
An extended version of the Vasicek model takes the form (Hull and White, 1990) Let λ(t) denote the time dependent market price of risk. Show that the bond price equation is given by Suppose we
Suppose the forward rate F̂(t, τ) & is defined in terms of running time t and time to maturity τ (instead of maturity date T), whereUnder the one-factor HJM framework, we write σF̂ &(t, τ) =
Consider the following discrete version of the Vasicek model when the short rate r(t) follows the discrete mean reversion binary random walk Let V (t) denote the value of an interest rate contingent
Consider the pricing of a futures contract on a discount bond, where the short rate rt is assumed to follow the Vasicek process defined by (7.2.20). On the expiration date TF of the futures, a bond
Show that the steady state density function of the short rate at time T in the Cox–Ingersoll–Ross model is given by (Cox, Ingersoll and Ross, 1985) where ω = 2α/ρ2 and ν = 2αγ/ρ2. Show
Consider the extended CIR model where the short rate, rt ≥ 0, follows the processfor some smooth deterministic functions α(t),β(t) and σ(t) > 0, and Zt is a Brownian process under the risk
Consider the continuous time equivalent of the Ho–Lee model as a degenerate case of the Hull–White model, where the diffusion process for the short rate rt under the risk neutral measure Q is
Consider the Hull–White model where the short rate is defined by Suppose we define a new variable x(t) where Also, show that the bond price B(t,T ) can be expressed as (Kijima and Nagayama, 1994)
Hull and White (1994) proposed the following two-factor short rate model whose dynamics under the risk neutral measure are governed by where u has an initial value of zero and follows the
Consider a swap with reset dates T0,T1, ··· ,Tn−1 and payment dates T1,T2, ··· ,Tn. A trigger swap is a contract where the holder has to enter into a swap with fixed swap rate K over the
Assume that the T-maturity discounted bond price process B(t,T) follows the one-factor Gaussian HJM under the risk neutral measure Q:A caption is a call option on a cap, whose terminal payoff at time
Consider a European call option with strike price X maturing at T on a futures whose underlying asset is a T’-maturity discount bond. Derive the value of this option under the Gaussian HJM term
Consider the following general formulation of the quadratic term structure model (Jamshidian, 1996), where the short rate is defined by where x(t) = (x1(t)··· xm(t))T is an m-component vector,
For the two-factor CIR model proposed by Longstaff and Schwartz (1992), the short rate r(t) is defined by where α and β are positive constants, and α = β. Under the risk neutral measure, the
Suppose the dynamics of the short rate r(t) are governed by where the short rate mean reverts to a drift rate θ(t), which itself reverts to a fixed mean rate θ̅, dZr dZθ = ρdt, and all other
Find the lower and upper bounds on the difference of the values of the American put and call options on a commodity with cost of carry b.
Consider an American call option whose underlying asset price follows a Geometric Brownian process. Show that C(2S, T) - C(S, T) ≤ (2-1) S, λ ≥ 1.
Explain why an American call (put) futures option is worth more (less) than the corresponding American call (put) option on the same underlying asset when the cost of carry of the underlying asset is
When q ≥ r, explain why an American call on a continuous dividend paying asset, which is optimally held to expiration, will have zero value at expiration (Kim, 1990).
Let P(S,τ ; X,r,q) denote the price function of an American put option. Show that P(X,τ ; S,q,r) also satisfies the Black–Scholes equation: together with the auxiliary conditions:Note that the
From the put-call symmetry relation for the prices of American call and put options derived in Problem 5.7, show that Give financial interpretation of the results.Problem 5.7Let P(S,τ ; X,r,q)
Consider the pair of American call and put options with the same time to expiry τ and on the same underlying asset. Assume the volatility of the asset price to be at most time dependent. Let SC and
Suppose the continuous dividend paid by an asset is at the constant rate d but not proportional to the asset price S. Show that the American call option on the above asset would not be exercised
Let H denote the barrier of a perpetual American down-and-out call option. The governing equation for the price of the perpetual American barrier option C∞(S;r,q) is given bywhere S∗∞ is the
We can also derive the binomial formula using the riskless hedging principle (see Sect. 3.1.1). Suppose we have a call that is one period from expiry and we would like to create a perfectly hedged
Deduce the price formula for a European put option with terminal payoff: max(X − S, 0) for the n-period binomial model.
Consider a portfolio containing Δt units of the risky asset and Mt dollars of the riskless asset in the form of a money market account. The portfolio is dynamically adjusted so as to replicate an
The following statement is quoted from Black (1989): “. . . the expected return on a warrant (call option) should depend on the risk of the warrant in the same way that a common stock’s expected
Consider a self-financing portfolio that contains αt units of the underlying risky asset whose price process is St and βt dollars of the money market account with riskless interest rate r. Suppose
Suppose the cost of carry of a commodity is b. Show that the governing differential equation for the price of the option on the commodity under the Black– Scholes formulation is given by where V
Suppose the price process of an asset follows the diffusion process Show that the corresponding governing equation for the price of a derivative security V contingent on the above asset takes the
Let the dynamics of the stochastic state variable St be governed by the Ito processFor a twice differentiable function f (St), the differential of f (St) is given byPerform parts integration of the
To derive the backward Fokker–Planck equation, we consider where u is some intermediate time satisfying t0 From the forward Fokker–Planck equation derived in Problem 3.8, we obtain.Problem
Let Q be the martingale measure with the money market account as the numeraire and Q∗ denote the equivalent martingale measure where the asset price St is used as the numeraire. Suppose St follows
From the Black–Scholes price function c(S,τ) for a European vanilla call, show that the limiting values of the call price at vanishing volatility and infinite volatility are the lower and upper
Show that when a European option is currently out-of-the-money, then higher volatility of the asset price or longer time to expiry makes it more likely for the option to expire in-the-money. What
Show that when the European call price is a convex function of the asset price, the elasticity of the call price is always greater than or equal to one. Give the financial argument to explain why the
Suppose the greeks of the value of a derivative security are defined by (a) Find the relation between Θ and Γ for a delta-neutral portfolio where Δ = 0. (b) Show that the theta may become
Let Pα(τ) denote the European put price normalized by the asset price, that is,We would like to explore the behavior of the temporal rate of change of the European put price. The derivative of
Show that the value of a European call option satisfies The call price function is a linear homogeneous function of S and X, that is, c(S, T; X) = S дс Әс ;(S, T; X) + X - as ax (S, T; X).
Consider a European capped call option whose terminal payoff function is given by where X is the strike price and M is the cap. Show that the value of the European capped call is given by where
Consider the value of a European call option written by an issuer whose only asset is α (and zero otherwise. Show that the value of this European call option is given by (Johnson and Stulz, 1987)
Deduce the corresponding put-call parity relation when the parameters in the European option models are time dependent, namely, volatility of the asset price is σ(t), dividend yield is q(t) and
Explain why the option price should be continuous across a dividend date though the asset price experiences a jump. Using no arbitrage principle, deduce the following jump condition: where V denotes
Consider futures on an underlying asset that pays N discrete dividends between t and T and let Di denote the amount of the ith dividend paid on the ex-dividend date ti. Show that the futures price is
Suppose the dividends and interest incomes are taxed at the rate R but capital gains taxes are zero. Find the price formulas of the European put and call on an asset which pays a continuous dividend
A forward start option is an option that comes into existence at some future time T1 and expires at T2 (T2 > T1). The strike price is set equal the asset price at T1 such that the option is
Consider a chooser option that entitles the holder to choose, on the choice date Tc periods from now, whether the option is a European call with exercise price X1 and time to expiration T1 − Tc or
Show that the first term in the last integral in (3.4.27) can be expressed as where x = ln ST2 and y = ln ST1 . Through comparison with the second term in (3.4.27), show that the above integral
Explain why the sum of prices of the call-on-a-call and call-on-a-put is equal to the price of the call with expiration T2. Show that the price of a European call-on-a-put is given by 3 Option
Find the price formulas for the following European compound options:(a) Put-on-a-call option when the underlying asset pays a continuous dividend yield q;(b) Call-on-a-put option when the underlying
In the Merton model of risky debt, suppose we define which gives the volatility of the value of the risky debt. Also, we denote the credit spread by s(τ ; d), where s(τ ; d) = Y(τ) − r. Show
A firm is an entity that consists of its assets and let At denote the market value of the firm’s assets. Assume that the total asset value follows a stochastic process modeled by where μ and σ2
Consider the exchange option that gives the holder the right but not the obligation to exchange risky asset S2 for another risky asset S1. Let the price dynamics of S1 and S2 under the risk neutral
Suppose the terminal payoff of an exchange rate option is FT1{FT >X}. Let Vd (F, t) denote the value of the option in domestic currency, show that V₁(F, t) = e¯¹ƒ(T−¹) FEQƒ [1{F7>X}\Ft =
Let FS\U denote the Singaporean currency price of one unit of U.S. currency and FH\S denote the Hong Kong currency price of one unit of Singaporean currency. We may interpret FS\U as the price
By writing Pn(τ) = e−λτ [see (3.5.20)], show thatFurthermore, by observing that show that V (S,τ) satisfies the governing equation (3.5.19). Also, show that V (S,τ) and VBS(S, τ) satisfy the
Show that the total transaction costs in Leland’s model (Leland, 1985) increases (decreases) with the strike price X when X ∗ (X > X∗), where X* = Ser+2)(T-1) S (1²-164₁ o Use the
Suppose the transaction costs are proportional to the number of units of asset traded rather than the dollar value of the asset traded as in Leland’s original model. Find the corresponding
Suppose ln J is normally distributed with standard deviation σJ. Show that the price of a European vanilla option under the jump-diffusion model can be expressed as (Merton, 1976) where V (S, T)
Consider the expression for dΠ given in (3.5.17). Show that the variance of dΠ is given by Suppose we try to hedge the diffusion and jump risks as much as possible by minimizing var(dΠ). Show
Suppose V (σ) is the option price function with dependence on volatility σ. Show that where σ1 is given by (3.5.25). Hence, deduce that V” > 0 if σ1 > σimp and V” imp, where σimp is
Assume that the time dependent volatility function σ(t) is deterministic. Suppose we write σimp (t, T) as the implied volatility obtained from the time-t price of a European option with maturity T
We would like to compute d(ST −X)+, where St follows the Geometric Brownian processThe function (ST − X)+ has a discontinuity at ST = X. Rossi (2002) proposed to approximate (ST − X)+ by the
By applying the following transformation on the dependent variable in the Black–Scholes equation while the auxiliary conditions are transformed to become Consider the following diffusion equation
Under the risk neutral measure Q, the stochastic process of the logarithm of the asset price xt = ln St and its instantaneous volatility σt are assumed to be governed by where dZx dZσ = ρdt. All
Consider the European zero-rebate up-and-out put option with an exponential barrier: B(τ) = Be−γτ , where B(τ) > X for all τ . Show that the price of this barrier put option is given
Suppose the asset price follows the Geometric Brownian process with drift rate r and volatility σ under the risk neutral measure Q. Find the density function of the asset price ST at expiration time
Consider a contingent claim whose value at maturity T is given by min(ST0 ,ST ), where T0 is some intermediate time before maturity, T0 T and ST0 are the asset price at T and T0, respectively.
Consider a one-year forward contract whose underlying asset is a coupon paying bond with maturity date beyond the forward’s expiration date. Assume the bond pays coupon semi-annually at the coupon
Consider an interest rate swap of notional principal $1 million and remaining life of nine months, the terms of the swap specify that six-month LIBOR is exchanged for the fixed rate of 10% per annum
Suppose two financial institutions X and Y are faced with the following borrowing rates Suppose X wants to borrow British sterling at a fixed rate and Y wants to borrow U.S. dollars at a floating
Consider an airlines company that has to purchase oil regularly (say, every three months) for its operations. To avoid the fluctuation of oil prices on the spot market, the company may wish to enter
A financial institution X has entered into a five-year currency swap with another institution Y. The swap specifies that X receives fixed interest rate at 4% per annum in euros and pays fixed
Show that a dominant trading strategy exists if and only if there exists a trading strategy satisfying V0 1(ω) ≥ 0 for all ω ∈ Ω. Consider the dominant trading strategy H = (ho h₁ ... hm)
This problem examines the role of a financial intermediary in arranging two separate interest rate swaps with two companies that would like to transform a floating rate loan into a fixed rate loan
Consider a portfolio with one risky security and a risk free security. Suppose the price of the risky asset at time 0 is 4 and the possible values of the t = 1 price are 1.1, 2.2 and 3.3 (three
Show that if the law of one price does not hold, then every payoff in the asset span can be bought at any price.
Construct a securities model such that it satisfies the law of one price but admits a dominant trading strategy.Construct a securities model where the initial price vector lies in the row space of
Given the discounted terminal payoff matrixand the current price vector Ŝ∗ (0) = (134). (a) By presenting a counter example, show that the law of one price does not hold for this one-period
Define the pricing functional F(x) on the asset span S by F(x) = {y : y = S∗(0)h for some h such that x = S∗(1)h, where x ∈ S}. Show that if the law of one price holds, then F is a linear
Given the discounted terminal payoff matrixand the current price vector Ŝ∗(0) = (123), find the state price of the Arrow security with discounted payoff ek ,k = 1, 2, 3. Does the securities

Showing 100 - 200 of 295