- Assume deterministic interest rates. Following our discussion in Section 12.1.4, derive the quanto effect of having a stochastic funding spread assuming that the asset follows a standard lognormal
- Implement via Monte Carlo a long-dated FX model with stochastic interest rates and stochastic vol. The parameterisation in Question 2 of Chapter 10 is reasonable for interest rates and spot FX
- Repeat the comparison in Question 3 of Chapter 10, except this time leave correlations constant (i.e. 0% between spot FX and interest rates). Instead, increase both interest rates vols to 1%. Next
- Implement via Monte Carlo the CEV parameterisation of spot FX with Gaussian rates in Section 10.2.1. For spot FX , use 85, for CEV vol use 0.15 x 851−β Try it with β = 0.2, 0.5, 0.9. What values
- Use the same parameters as in Question 2 except where specified otherwise. Compute the 10-year forward starting volatilities for expiries 1y, 2y,..., 10y. Now, increase the domestic-spot correlation
- Consider the same parameterisation as in Question 1 but to be more specific, say the HJM processes correspond to a Hull-White model with constant vol of 0.7% and mean reversion speed of 2%. Take
- Consider a lognormal Libor Market Model. For simplicity, assume that Libor rates apply over 1y periods. Take current Libor rates to be flat at 3%. Using the discussion on the volatility triangle in
- For a Heston model, take mean reversion of 15%. Calibrate η, ρ, θ (t) to the prices of strangles and risk reversals over 5 years (using the characteristic function method for pricing European
- Consider the Taylor expansion of the integrand of the cumulative normal function for x ≈ 0, i.e. Thereby, find an approximation for the price of a lognormal option close-to-the-money. Attempt
- Compute the forward delta of a call option based on the normal formula. (The price is given in Section 1.2.4.) Contrast this with the forward delta for an at-the-money (i.e. forward = strike) call
- Consider a very steep skew. Would SABR always be able to fit this?
- Consider the Craig-Sneyd discretisation for three space variables. By performing the steps sequentially, show that the scheme is consistent (i.e. we do indeed recover a discretised form of the
- Find a self-similarity solution, the Fokker-Planck equation can be transformed into a standard diffusion, so solving the above diffusion equation with the given initial and boundary conditions, is
- Build a simple Monte Carlo framework based on stochastic vol. Convince yourself that the fair value strike of a vol swap is not very different from the at-the-money vol.
- Consider the Hull-White model with SDE drt = κ,(t)(θ(t) – rt)dt + σ(t)dWt. Following the same logic as we did in the text, determine θ(t). Can you see why in practice calculating θ(t) directly
- Risk reversals and strangles are described in Section 2.3.2 as packages of calls and puts with strikes on either side of the forward. Using the approximation to the SABR formula per Section 6.2.3,
- Take the actual SABR formula per Section 6.2.1 with σ0 = 0.5%, β = 0, ρ = −25%, ν = 30%, and F = 4%. By considering an expiry of 20 years and strikes at 10 basis points increments from 1 % to
- Show how the Black-Scholes PDE with final condition V(S,T) — VT(S) can be transformed into the initial condition. The exercise is more to convince you that the Black-Scholes PDE is a standard
- Consider the diffusion equation that the scheme is stable. Show also that the implicit scheme is unconditionally stable. ди Ət = 82 u а ах² Let u ||
- Suppose that each Libor rate is driven by a separate Wiener process dLi (t) = σi (Li(t),t) dWit (under the measure with numeraire being the discount bond maturing at the pay date of the Libor
- Consider a two-factor short rate model described by dxt = where ƒt,T is the HJM instantaneous forward rate. -Kr(t) xidt + oz(t)dWF, dyt -Ky(t)yidt + oy(t)dW!', rt = y(t) + x + yt and dWƒdW? =
- Consider again the Hull-White model. Using ƒtT = ETt [rT],obtain the volatility of the SDE of the discount bond D(t,T) = e–∫Tt ƒt,sds .
- Consider the construction of a trinomial tree for the general short rate model of form dyt = κ(θ(t) − yt)dt + σdWt, rt = g (yt). The approach typically involves constructing the tree for dxt =
- Consider again the Hull-White model with constant mean reversion κ = 1 %. Suppose also interest rates are flat at 4% so that discount bond prices are given by D(0,T) = e_0.04T First find the SDE for
- Consider the correlation matrix Ω. We can perform an eigenvector decomposition so that Ω = SAST, where A is a diagonal matrix of eigenvalues and S is the eigenvector matrix whose determinant is 1.
- Consider a two-factor lognormal Libor Market Model. Ignore the drift, so that we can write dLit = . . . dt + σiLit (ai1dWt1 + ai2dWt2), where a21i + a21i = 1 and dWt1 dWt2 = 0. Now, let us suppose
- The Longstaff-Schwartz algorithm involves a choice of basis functions. Suppose we have n + 1 dependent variables {yi}ni=0 to fit corresponding to the explanatory variables {xi}ni=0 Show that a
- Consider the following parameterisation based on Gaussian interest rates and lognormal spot FX : dSt = (rt – rƒt) Stdt + quanto Libor rate, i.e. where the Libor rate LT (set at time T, with
- Assume lognormal dynamics for the underlying dSt = rStdt +σStdWt and constant interest rates r. Compute the price of a cliquet with payoff max (ST2/ST1 − 1,0) at time T2 > T1.
- Consider the Hull-White model drt = κ(θ(t) − rt)dt + σdWt Find corr (ƒt,T,ƒu,T) where ƒt,T = ETt [rT] and t < u < T.
- Consider the equation for the par swap rate under OIS discounting in Section 12.1.3. If the OIS-Libor spread is zero, show that this reduces to the classical equation D(0,T0) – D (0, TN) = Suppose
- Due to the huge supply of yen, the funding bias for USD/JPY is such that it is cheaper to borrow yen versus dollars (i.e.sJPY BRL ≠ 0) Consider now a cross currency swap with one leg paying
- Consider a flat Libor discount curve given by D(0, T) =e −0.03T. Suppose that the OIS discount curve is D (0,T) = e −0.028T. Compute the swap rate (annual fixed coupons, semi-annual floating
- Notwithstanding the discussion in Section 12.1.4, since we forecast based on Libor, the martingale equation only holds for risky discounting at the credit-worthiness of Libor counterparts. As such,
- The Ho-Lee model assumes that the short rate has stochastic differential equation drt = θ (t)dt + crdWt. Note that the money market account is defined via BT = e∫0T rudu and the discount bond
- Consider a contract with payoff max at time T. From your answer to Question 2, what is the value of such a payoff?Question 2,Suppose that two assets SA and SB have stochastic differential equations
- Consider the simple convexity adjustment (Question 3 of Chapter 3) and apply it to the 10-year and 2-year CMS rates setting in 30 years’ time. (Suppose for this example that both swap rates are
- For a derivative V(XT) on an underlying XT as seen at time T, its value is given by V(Xt) = e –r (T– t) E[V(XT)] = e –r (T– t) ∫∞–∞ V(x)p(x)dx, where p(•) is the probability
- Derive Dupire’s formula in Section 5.1.2 in terms of put prices rather than call prices.Section 5.1.2, More interesting is the question of how to recover (St, t) to fit all given market prices. Let
- Derive the density for the Gumbel and Clayton copulae.
- For simplicity, assume the Ho-Lee dynamics drt = θ(t)dt+σdWt. The forward rate is given via ƒ(t,T) = ETt [rT]. (All this will become clear when we discuss short rate models in Chapter 8.)
- For a cash-settled swaption, the cash-annuity is defined by This is j ust some function of RT. Do a second order Taylor-expansion of Ac(RT) about Ro. Notice that under the T-forward measure, we get
- Assume interest rates are lognormal. Compute the convexity adjustment for the fair value LIA rate. What would the convexity adjustment be if rates are normal? (In this question, assume semi-annual
- Consider the Black-Scholes model. What is the gamma of a digital option with payoff 1ST>K at time T? Why would you not want to delta-hedge such an option?
- Derive the formula for the Black-Scholes vega for a call option. What about a digital call option with payoff 1ST>K at time T?
- Suppose we have a curve built from swaps with annual coupons and spanning maturities Suppose the swap rates are monotonically increasing (i.e. Si j for Ti j). Define annually compounded zero
- Compute the Black-Scholes delta and gamma for a call-option with payoff max(ST — K,0) at time T.
- Suppose you are given quotes on put options on interest rates of different strikes for a given maturity (i.e. payoffs are max( K – RT,0). For the 1% strike, the price is 0.002 and for the 2%
- Find P(min0Wt ≤ m.
- Find cov(Wt, Wt) for t < T.
- Suppose that two assets SA and SB have stochastic differential equations dStA,B = rStA,B dt + σ A,BStA,B dWtA,B where dWtA dWtB = ρdt. Derive the formula for the option that pays max at time
- Consider the SDE dXt = μXtdt + σXtdWt for the FX rate. By considering the domestic and foreign money market accounts with SDEs dBt = rtBtdt and dBƒt = rƒt Bƒt dt and using Girsanov’s
- Show that the formula for the price of a put option with payoff of PT = max(K – ST, 0) at time T is Pt = e-r(T-t) KN (−d₁ + o√Tt) - StN(-d₁), where d₁= = log ()+(r+²) (T-t) o√T-t
- Consider a derivative V(St,rt,t) dependent on the (nondividend-paying) stock price St and short rate rt with SDEs given by dSt = μ (t)Stdt + σStdWt and drt = κ(θ(t) – rt)dt +

Copyright © 2024 SolutionInn All Rights Reserved.