- Consider the European put option with expiration date Ti−1 and strike price X written on a Ti-maturity discount bond, 0 < X < 1. Recall that the ith caplet is equivalent to the put bond option,
- Consider a floor on composition, where the composition is defined as Here, Li is reset at time Ti and αi+1 is the accrual factor over the time interval (Ti,Ti+1]. The payment of this floor at
- Show that the process ξtT,t ≥ 0, defined in (8.1.7), is a positive Q-martingale and the expectation of ξtT under Q is one. Here, Q is the risk neutral measure.
- The expression for a(t,T ) derived from in (7.2.38b) involves ∅(t). It may be desirable to express a(t,T ) solely in terms of the initial bond prices B(0,T) for all maturities. Show that (Hull and
- Show that the bond price for the Cox–Ingersoll–Ross model [see (7.2.32a,b)] is a decreasing convex function of the short rate and a decreasing function of time to maturity. Further, show that the
- Explain why the estimator θ i1···ijlow,j defined by (6.3.18)–(6.3.19) is biased low.Upward bias is eliminated since the continuation value and the early exercise decision are determined from
- Judge whether the simulation estimator on the option price given by the Grant– Vora–Weeks algorithm is biased high or low or unbiased.
- Consider the Bermudan option pricing problem, where the Bermudan option has d exercise opportunities at times t1 2 d = T , with t1 ≥ 0. Here, the issue date and maturity date of the Bermudan option
- Use the Fourier method to deduce the von Neumann stability condition for (i) the Jarrow–Rudd binomial scheme (see Problem 6.1), (ii) the Kamrad– Ritchken trinomial scheme, and (iii) the explicit
- Consider the modified binomial formula employed for the numerical valuation of an American put on a nondividend paying asset [see (6.1.14)], deduce the optimal exercise price at time close to expiry
- Instead of the tree-symmetry condition: u = 1/d [see (6.1.1c)], Jarrow and Rudd (1983) chose the third condition to be p = 1/2. By solving together with (6.1.1a,b), show that
- Consider the shout call option discussed in Sect. 5.4.2 (Dai, Kwok and Wu, 2004). Explain why the value of the shout call is bounded above by the fixed strike lookback call option with the same
- Let Cdo(S, τ ; X,H,r,q) and Puo(S, τ ; X,H,r,q) denote the price function of an American down-and-out barrier call and an American up-and-out barrier put, respectively, both with constant barrier
- Let S∗C(∞) denote limτ→∞ S∗C (τ), where S∗C (τ) is the solution to the integral equation defined in (5.2.19). By taking the limit τ → ∞ of the above integral equation, solve for
- n the two-dividend American call option model, we assume discrete dividends of amount D1 and D2 are paid out by the underlying asset at times t1 and t2, respectively. Let S̃t denote the asset price
- Suppose discrete dividends of amount D1,D2, ··· ,Dn are paid at the respective ex-dividend dates t1,t2, ··· ,tn and let tn+1 denote the date of expiration T. Show that the risky component is
- Give a mathematical proof to the following inequality which arises from the Black approximation formula for the one-dividend American call (see Sect. 5.1.5). Here, td and T are the ex-dividend date
- Consider the one-dividend American call option model. Explain why the exercise price S∗d , which is obtained by solving (5.1.24), decreases when the dividend amount D increases. Also, show that
- Show that for any random variable X, we have and apply the result to show the result in (4.3.29).
- Apply the exchange option price formula (see Problem 3.34) to price the floating strike Asian call option based on the knowledge of the price formula of the fixed strike Asian call option. The
- The terminal payoff of the lookback spread option is given by Show that the price of the European lookback spread option can be expressed as (Wong and Kwok, 2003)(i) currently at- or in-the-money,
- Prove the following put-call parity relation between the prices of the fixed strike lookback call and floating strike lookback put: Give a financial interpretation why cfix is insensitive to M when
- Suppose we use a straddle (combination of a call and a put with the same strike m) in the rollover strategy for hedging the floating strike lookback call and write Find an integral representation of
- As an alternative approach to derive the value of a European floating strike lookback call, we consider where St = S,mt T0 = m and τ = T − t. We may decompose the above expectation calculation
- Let fmin(y) and fmax(y) denote the density function of yT and YT , respectively. Show that
- Using the method of path counting, Sidenius (1998) showed that g+(x, T ) defined in (4.1.44b) has the following analytic solutionFind the closed form price formula of the European upper-barrier
- Suppose the holder of the chooser option can make the choice of either a call or a put at any time between now and a later cutoff date Tc. Is it optimal for the holder to make the choice at some time
- Show that the payoff function of a chooser option on the date of choice Tc can be alternatively decomposed into the following form:[see (3.4.22)]. Find the alternative representation of the price
- Consider the price functions of European call and put options on an underlying asset which pays a dividend yield at the rate q, show that their deltas and thetas are given by where d̂1 and d̂1 are
- Show that τ is a stopping time if and only if {τ ≤ t} ∈ Ft . {t ≤t} = {t = 0} U {t=1} U... U {t=t} and {t=t} = {t ≤t}n{t ≤t - 1}.
- Consider EQ[Ã(tn;t)2] defined in (4.3.68). Show that when t 0 we have (Levy, 1992) where B₁ = B3 = Eq[Ã (In; t)²] = S² (n + 1)² 1- e(2r+o²)(n+1) At (1 — er4¹)[1 – e(2r+0²)41]' - er 4t
- We defineShow that the price functions of various European barrier put options are given bywhere pE(S, τ ; X) is the price function of the European put option, B is the barrier and X is the strike
- Consider a European down-and-out call option where the terminal payoff depends on the payoff state variable S1 and knock-out occurs when the barrier state variable S2 breaches the downstream barrier
- Consider a typical term in g(x,t) where ξ can be either 2n(u − ℓ) or 2ℓ + 2n(u − ℓ). Show that the above term can be rewritten asHence, show that Use the above result to derive the price
- Consider a European down-and-out partial barrier call option where the barrier provision is activated only between the option’s starting date (time 0) and t1. Here, t1 is some time earlier than the
- Let the exit time density q+(t; x0,t0) have dependence on the initial state X(t0) = x0. We write τ = t − t0 so that q+(t; x0,t0) = q+(x0,τ). Show that the partial differential equation
- Let P(x,t; x0,t0) denote the transition density function of the restricted Brownian process Wμt = μt + σZt with two absorbing barriers at x = 0 and x = ℓ. Using the method of separation of
- Consider a discretely monitored down-and-out call option with strike price X and barrier level Bi at discrete time ti,i = 1, 2, ··· ,n. Show that the price of this European barrier call option is
- Using the following form of the distribution function of mTt [see (4.2.4a)]how that P(m ≤ mTt ) becomes zero when S = m. ( -10 + + μ²) - (2) ² ³ x (1 5 +
- Suppose the terminal payoff function of the partial lookback call and put options are max(ST − λmTT0 , 0), λ > 1 and max (λMTT0 − ST , 0) , 0
- The holder of a European in-the-money call option may suffer loss in profits if the asset price drops substantially just before expiration. The “limited period” fixed strike lookback feature may
- Use (4.2.21) to derive the following partial differential equation for the floating strike lookback put option Solve the above Neumann boundary value problem and check the result with the put price
- Let p(S, t; δt) denote the value of a floating strike lookback put option with discrete monitoring of the realized maximum value of the asset price, where δt is the regular interval between
- The dynamic fund protection feature in an equity-linked fund product guarantees a predetermined protection level K to an investor who owns the underlying fund. Let St denote the value of the
- Explain why callEuropean ≥ Asian callarithmetic ≥ Asian callgeometric.An average price is less volatile than the series of prices from which it is computed.
- We define the geometric average of the price path of asset price Si, i = 1, 2, during the time interval [t,t + T ] byConsider an Asian option involving two assets whose terminal payoff is given by
- Deduce the following put-call parity relation between the prices of European fixed strike Asian call and put options under continuously monitored geometric averaging c(S, G, t) - P(S, G,
- Suppose continuous arithmetic averaging of the asset price is taken from t = 0 to T, T is the expiration time. The terminal payoff function of the floating strike call and put options are,
- Show that the put-call parity relations between the prices of floating strike and fixed strike Asian options at the start of the averaging period are given by By combining the above put-call parity
- Consider a self-financing trading strategy of a portfolio with a dividend paying asset and a money market account over the time horizon [0,T ]. Under the risk neutral measure Q, let the dynamics of
- Consider the European continuously monitored arithmetic average Asian option with terminal payoff: max(AT − X1ST − X2, 0), whereAt the current time t > 0, the average value At over the time
- Under the risk neutral measure Q, let St be governed byshow that (Milevsky and Posner, 1998) Defining d St | St = (r- q) dt + o d Z₁. A(t, T) = 1 T-t T s Su du,
- Suppose we define the flexible geometric average GF (n) of asset prices at n evenly spaced time instants byand Si is the asset price at time ti. Here, ωi is the weighting factor associated with
- Let Zt denote the standard Brownian process. Show that the covariance matrix of the bivariate Gaussian random variable (Zt ∫01Zu du) is given by t(1 - {/) E[(Z₁. [' Zu du)' (Zi. f' Zu du)] =
- Let S(ti) denote the asset price at time ti,i = 1, 2, ··· ,N, where 0 = t0 N = T. Define the discretely monitored arithmetic average and geometric average byLet cA(0; X) and cG(0; X) denote the
- Find the value of an American vanilla put option when (i) riskless interest rate r = 0, (ii) volatility σ = 0, (iii) strike price X = 0, (iv) asset price S = 0.
- We would like to show by heuristic arguments that the American price function P(S,τ) satisfies the smooth pasting condition at the optimal exercise price S∗(τ). Consider the behaviors of the
- Consider an American call option with a continuously changing strike price X(τ) where dX(τ)/dτDefine the following new set of variables:Show that the governing equation for the price of the above
- Consider an American call option on an asset that pays discrete dividends at anticipated dates t1 2 1,D2, ··· ,Dn, and T = tn+1 be the time of expiration. Show that it is never optimal to exercise
- Consider the one-dividend American put option model where the discrete dividend at time td is paid at the known rate λ, that is, the dividend payment is λStd . Show that the slope of the optimal
- Show that the delta of the price of an American put option on an asset which pays a continuous dividend yield at the rate q is given by Examine the sign of the delta of the early exercise premium
- Bunch and Johnson (2000) gave the following three different definitions of the optimal exercise price of an American put. 1. It is the value of the asset price at which one is indifferent between
- Consider an American put option on an asset which pays no dividend. Show that the early exercise premium e(S, τ; X) is bounded by e¯r§ rX rX * e-¹² N(−ã£) d² ≤ e(S, t; X) ≤ rx
- By considering the corresponding integral representation of the early exercise premium of an American commodity option with cost of carry b, show that (a) When b ≥ r, r is the riskless interest
- Consider an American up-and-out put option with barrier level B(τ) = B0e−ατ and strike price X. Assuming that the underlying asset pays a continuous dividend yield q, find the integral
- Consider a down-and-in American call Cdi (S, τ ; X,B), where the down-and-in trigger clause entitles the holder to receive an American call option with strike price X when the asset price S falls
- The exercise payoff of an American capped call with the cap L is given by max(min(S, L) − X, 0), L > X. Let S∗cap(τ) and S∗(τ) denote the early exercise boundary of the American capped call
- Consider an American call option with the callable feature, where the issuer has the right to recall throughout the whole life of the option. Upon recall by the issuer, the holder of the American
- Unlike usual option contracts, the holder of an installment option pays the option premium throughout the life of the option. The installment option is terminated if the holder chooses to discontinue
- Suppose an American put option is only allowed to be exercised at N time instants between now and expiration. Let the current time be zero and denote the exercisable instants by the time vector t =
- The approximate equation for f in the quadratic approximation method becomes undefined when K(τ) = 1 − e−rτ = 0, which corresponds to r = 0. Following a similar derivation procedure as in the
- Show that the approximate value of the American commodity put option based on the quadratic approximation method is given byS*.">Explain why the formula holds for all values of b.Show that P(S, T)
- Show that where p∗(τ) is defined in (5.4.2). To prove the results in (5.4.6a,b,c), it suffices to consider the sign behavior ofConsider the following two cases (Dai, Kwok and Wu, 2004). (a) For
- Let Wn∞ (S; X) = limτ→∞ erτUn(S, τ ; X), where Un(S, τ; X) is the value of the n-reset put option [see (5.4.20)]. For r n∞ (S) is given by (Dai, Kwok and Wu, 2003)The auxiliary
- For the reset-strike put option, assuming r ≤ q, show that the early reset premium is given by (Dai, Kwok and Wu, 2004) How do we modify the formula when r > q? where e(s, t)= Seqt T): di.t-u T N
- The reload provision in an employee stock option entitles its holder to receive X/S∗ units of newly “reloaded” at-the-money options from the employer upon exercise of the stock option. Here, X
- Consider a landowner holding a piece of land who has the right to build a developed structure on the land or abandon the land. Let S be the value of the developed structure and H be the constant rate
- Consider an American installment option in which the buyer pays a smaller upfront premium, while a constant stream of installments at a certain rate per unit time are paid subsequently throughout the
- Suppose the underlying asset is paying a continuous dividend yield at the rate q, the two governing equations for u,d and p are modified as Show that the parameter values in the binomial model are
- Show that Note that By considering the Taylor expansion of n(p′ ln u/d + ln d) and np′ (1 − p′)(ln u/d)2 in powers of Δt, show thatwhere nΔt = τ . lim (n, k, p') = N(d₁) n→∞ where
- Consider the nodes in the binomial tree employed for the numerical valuation of an American put option on a nondividend paying asset. The (n,j)th node corresponds to the node which is n time steps
- Consider the pricing of the callable American put option by binomial calculations, let us write Show that binomial scheme (6.1.15) can be modified to become Give the financial interpretation of the
- Show that the total number of multiplications and additions in performing n steps of numerical calculations using the trinomial and binomial schemes are given by Scheme trinomial binomial Number of
- Suppose we let p2 = 0 and write p1 = −p3 = p in the trinomial scheme. By matching the mean and variance of ζ(t) and ζa(t) accordinglyshow that the parameters v and p obtained by solving the above
- Boyle (1988) proposed the following three-jump process for the approximation of the asset price process over one period: where S is the current asset price. The middle jump ratio m is chosen to be
- Suppose we let y = ln S, the Kamrad–Ritchken trinomial scheme can be expressed as Show that the Taylor expansion of the above trinomial scheme is given by Given the probability values stated in
- Show that the width of the domain of dependence of the trinomial scheme (see Fig. 6.5) increases as √n, where n is the number of time steps to expiry.Fig. 6.5 (X,, 0) (x, nΔt) (x, 0) ηΔι (x+, 0)
- Consider the five-point multinomial scheme defined in (6.1.22) and the corresponding four-point scheme (obtained by setting λ = 1), show that the total number of multiplications and additions in
- Consider a three-state option model where the logarithmic return processes of the underlying assets are given by where vi = λσi √Δt, i = 1, 2, 3. Following the Kamrad–Ritchken approach, find
- Consider the window Parisian feature. Associated with each time point, a moving window is defined with m) consecutive monitoring instants before and including that time point. The option is knocked
- Consider the European put option with the automatic strike reset feature, where the strike price is reset to the prevailing asset price on a prespecified reset date if the option is out-of-the-money
- Construct the FSG scheme for pricing the continuously monitored European style floating strike lookback call option. In particular, describe how to define the terminal payoff values. How can we
- Suppose we would like to approximate df/dx at x0 up to O(Δx2) using function values at x0,x0 − Δx and x − 2Δx, that is, where α−2,α−1 and α0 are unknown coefficients to be determined.
- Consider the following difference operators, show that they approximate the corresponding differential operator up to second-order accuracy (i) d² f dx2 хо = 2f(xo) - 5f(xo - Ax) + 4f(x - 24x) -
- Show that the leading local truncation error terms of the following Crank– Nicolson scheme Perform the Taylor expansion at (jΔx, (n + 1/2 )Δτ). v"+1 - V₁² _ 0² (V²+1 -
- Consider the following form of the Black–Scholes equation: where V (S,τ) is the option price and S is the asset price. The two-level sixpoint implicit compact scheme takes the form: where Show
- Let p(S,M,t) denote the price function of the European floating strike lookback put option. Define x = ln M/S and V (x,t) = p(S,M,t)/S. The pricing formulation of V (x,t) is given byThe final and
- To obtain a consistent binomial scheme for the floating strike lookback put option, we derive the binomial discretization at j = 0 using the finite volume approach. First, we integrate the governing
- Suppose we use the FTCS scheme to solve the Black–Scholes equation so that Show that the sufficient conditions for nonappearance of spurious oscillations in the numerical scheme are given by
- The penalty method is characterized by the replacement of the linear complementarity formulation of the American option model by appending a nonlinear penalty term in the Black–Scholes equation.

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