- Calculate Halton sequences for bases 2, 3 and 4. Compare the results (you may wish to plot one base against another). What do you notice? How should this affect the bases you choose for a
- Repeat the above using Halton sequences.Previous questionValue a European call option using a Monte Carlo simulation. Use the simulation to estimate the value of an integral, as opposed to simulating
- Value a European call option using a Monte Carlo simulation. Use the simulation to estimate the value of an integral, as opposed to simulating the random walk for the asset.
- Why is it difficult to use Monte Carlo simulations to value American options?
- Modify the above spreadsheet to value various exotic options such as barriers, Asians and lookbacks.
- Modify the spreadsheet in the above to ouput the option’s delta.
- Simulate the risk-neutral random walk for an asset using a spreadsheet package, or otherwise. Use these data to calculate the value of a European call option.
- Alter your compound option program to value a chooser option which allows you to buy a call or a put at expiry.
- Write a program to value compound options of the following form:(a) Call on a call;(b) Call on a put;(c) Put on a call;(d) Put on a put.
- Write a program to value a down-and-out call option, with barrier below the strike price.
- Adjust your program to value call options with the forward price as underlying.
- Write a program to value European call and put options by solving the Black–Scholes equation with suitable final and boundary conditions. Include a constant, continuous dividend yield on the
- Sudden, large movements in a stock are usually accompanied by an increase in implied volatilities. Incorporate a vega term into the one-asset CrashMetrics model. What is the role of actual volatility
- Extend the example CrashMetrics spreadsheet to incorporate many underlyings, all related via an index. How would interest rate products be incorporated?
- We have seen how to apply the CreditMetrics methodology to a single risky bond, to apply the ideas to a portfolio of risky bonds is significantly harder since it requires the knowledge of any
- Construct the intermediate steps in the derivation of the equation for the value of a risky bond (when default is governed by a Poisson process): av av + at + (u – Aw) - (r+p)V = 0. ar ar?
- Repeat the analyses of the above two problems but with the extra assumption that on default their is a recovery rate of 10%, i.e. you will receive $10 at maturity if there has been default.
- The same government and company as in the above question now issue five-year bonds with $100 principal and prices for government bond and corporate bond of $82.15 and $78.89 respectively. What does
- A risk-free government, zero-coupon bond, with a principal of $100, maturing in two years has a value of $91.75. A risky corporate zero coupon bond with the same principal and maturity is worth
- What criticisms of Value at Risk as described here can you think of? Consider distributions other than Normal, discontinuous paths and non linear instruments.
- Assuming a Normal distribution, what percentage of returns are outside the negative two standard deviations from the mean? What is the mean of returns falling in this tail? (This is called the
- What are the economic significances of α and β in the Capital Asset Pricing Model and how are they measured or estimated in practice?
- Where should we be on the efficient frontier in question 1 if we wish to minimize the chance of a return less than 0.05?
- Find the efficient frontier for the assets in the table above, when asset D is replaced by a risk-free asset, E, which has a mean of 0.10 over our time horizon of two years. Asset E is uncorrelated
- Work out the efficient frontier for the following set of assets:The correlation coefficients between the four assets are given by Asset A 0.08 0.12 B 0.10 0.12 0.10 0.15 0.14 0.20 9.
- Build a spreadsheet that simulates the above, in which you keep betting the same fraction of your accumulated wealth.
- A die is weighted so that the probability of getting a 6 is 0.3, of getting a 1 is 0.1, and of getting 2, 3, 4 or 5 is 0.15 each. You bet on 6, receiving odds of 5 to 1 (fair odds for an unbiased
- Using forward rate data, perform a principal component analysis. What are the three main components in the forward price movements and what are their weights?
- Perform a simulation, using the method of section 19.7, to value an option on a zero-coupon bond. You will need to decide upon a suitable form for ν(t, T ), the forward rate volatility. Does your
- Derive the equation for the evolution of the forward rate:fromand (3o²4, 7) – Hit. T) at -Tot, a (t, T) dX, dF(t; T) =
- Find the approximate value of a cashflow for a floorlet on the one-month LIBOR, when we use the Vasicek model.
- When an index amortizating rate swap has a lockout period for the first year, we must solvewith jump conditionwhereg(r, i) =1 if ti < 1,and with final conditionV(r, P, T ) = (r − rf )P.In this
- How would a collar be valued practically? What is the explicit solution for a single payment?
- Derive a relationship between a floorlet and a call option on a zero-coupon bond.
- Write down the problem we must solve in order to value a puttable bond.
- Use market data for zero-coupon bond prices to fit a Vasicek model for the interest rate.
- Use market data for the price of zero-coupon bonds to fit a Ho & Lee model. Refit the model with data for a week later in time. Compare the two curves for η ∗(t).The second curve for η∗
- Differentiate Equation (17.2) twice to solve for the value of η ∗(t). What is the value of a zero-coupon bond with a fitted Vasicek model for the interest rate?Equation (17.2) 1 3 :- n"(s)B(s;
- Substitute the fitted function for A(t; T ), using the Ho & Lee model, back into the solution of the bond pricing equation for a zero-coupon bond,Z(r, t; T ) = eA(t;T )−r(T−t).What do you
- What form does the bond pricing equation take when the interest rate satisfies the Vasicek modeldr = (η − γ r)dt + β1/2dX?Solve the resulting equations for A and B in this case, to findand 1 z(B
- What final condition (payoff) should be applied to the bond pricing equation for a swap, cap, floor, zero-coupon bond, coupon bond and a bond option?
- Simulate random walks for the interest rate to compare the different named models suggested in this chapter.
- SubstituteZ(r, t; T ) = eA(t;T )−rB(t;T ),into the bond pricing equationWhat are the explicit dependencies of the functions in the resulting equation? av + (u – Aw) av - rV = 0. ar %3D at ar2
- A swap allows the side receiving floating to close out the position before maturity. How does the ‘fair’ value for the fixed rate side of the swap compare to that for a swap with no call/put
- An index amortizing rate swap has a principal which decreases at a rate dependent on the interest rate at settlement dates. Over a payment date, the principal changes from P to g(r)P, where g(r) is a
- Consider a swap with the following specification:The floating payment is at the six-month rate, and is set six months before the payment (swaplet) date. The swap expires in five years, and payments
- Figure 14.24 is a term sheet for a step-up note paying a fixed rate that changes during the life of the contract. Plot the price/yield curve for this product today, ignoring the call feature. What
- Solve the equationwith final data V(T) = 1. This is the value of a coupon bond when there is a known interest rate, r(t). What must we do if interest rates are not known in advance? dV +K(t) = r(t)V,
- What assumption do we make when we duration hedge? Is this a reasonable assumption to make?
- Zero-coupon bonds are available with a principal of $1 and the following maturities:(a) 1 year (market price $0.93),(b) 2 years (market price $0.82),(c) 3 years (market price $0.74).Calculate the
- A coupon bond pays out 2% every year on a principal of $100. The bond matures in six years and has a market value of $92. Calculate the yield to maturity, duration and convexity for the bond.
- A zero-coupon bond has a principal of $100 and matures in four years. The market price for the bond is $72. Calculate the yield to maturity, duration and convexity for the bond.
- Construct a spreadsheet to examine how $1 grows when it is invested at a continuously compounded rate of 7%. Redo the calculation for a discretely compounded rate of 7%, paid once per annum. Which
- A coupon bond pays out 3% every year, with a principal of $1 and a maturity of five years. Decompose the coupon bond into a set of zero coupon bonds.
- Why might we prefer to treat a European up-and-out call option as a portfolio of a vanilla European call option and a European up-and-in call option?
- Prove put-call parity for simple barrier options:CD/O + CD/I − PD/O − PD/I = S − Ee−r(T−t),where CD/O is a European down-and-out call, CD/I is a European down and-in call, PD/O is a
- Price the following double knockout option: the option has barriers at levels Su and Sd, above and below the initial asset price, respectively. The option has payoff $1, unless the asset touches
- Formulate the following barrier option pricing problems as partial differential equations with suitable boundary and final conditions:(a) The option has barriers at levels Su and Sd, above and below
- Formulate the following problem for the accrual barrier option as a Black–Scholes partial differential equation with appropriate final and boundary conditions:The option has barriers at levels Su
- Check the value for the down-and-in call option using the explicit solutions for the down-and-out call and the vanilla call option.
- Why do we need the condition Sd < E to be able to value a down-and-out call by adding together known solutions of the Black–Scholes equation (as in question 1)? How would we value the option in
- Check that the solution for the down-and-out call option,VD/O, satisfies Black–Scholes, whereand C(S, t) is the value of a vanilla call option with the same maturity and payoff as the barrier
- What is the explicit formula for the price of a quanto which has a put payoff on the Nikkei Dow index with strike at E and which is paid in yen. S$ is the yen-dollar exchange rate and SN is the level
- Set up the following problems mathematically (i.e. what equations do they satisfy and with what boundary and final conditions?) The assets are correlated.(a) An option that pays the positive
- Check that if we use the pricing formula for European non-path dependent options on dividend-paying assets, but for a single asset (i.e. in one dimension), we recover the solution found in Chapter 8:
- Using tick data for at least two assets, measure the correlations between the assets using the entirety of the data. Split the data in two halves and perform the same calculations on each of the
- N shares follow geometric Brownian motions, i.e.dSi = μiSi dt + σiSi dXi,for 1 ≤ i ≤ N. The share price changes are correlated with correlation coefficients ρij. Find the stochastic
- Find the value of the power European call option. This is an option with exercise price E, expiry at time T, when it has a payoff:Λ(S) = max(S2 − E, 0).Note that if the underlying asset price is
- Prove put-call parity for European compound options:CC + PP − CP − PC = S − E2e−r(T2−t),where CC is a call on a call, CP is a call on a put, PC is a put on a call and PP is a put on a
- How would we value the chooser option in the above question if EC was non-zero?
- A chooser option has the following properties:At time TC < T, the option gives the holder the right to buy a European call or put option with exercise price E and expiry at time T, for an amount
- Take the How to Hedge spreadsheet on the CD and rewrite using VB, C++, or other code. Now modify the code to do the following.(a) Allow for arbitrary fixed period between rehedges. Observe how the
- Collect real option data from the Wall Street Journal, the Financial Times or elsewhere, calculate implied volatilities and plot them against expiration, against strike, and, in a three-dimensional
- Using real, daily data, for several stocks, plot a time series of volatility using several models.(a) Divide the data into yearly intervals and estimate volatility during each year.(b) Use a
- The fundamental solution, uδ , is the solution of the diffusion equation on−∞ < x < ∞ and τ > 0 with u(x, 0) = δ(x). Use this solution to solve the more general problem:with u(x, 0)
- Use put-call parity to find the relationships between the deltas, gammas, vegas, thetas and rhos of European call and put options.
- Find the partial differential equation satisfied by ρ, the sensitivity of the option value to the interest rate.
- Show that the vega of an option, v, satisfies the differential equationwhere Γ = ∂2V/∂S2 . What is the final condition? av av +rs- - v +os?r = 0, at as? se
- Show that for a delta-neutral portfolio of options on a non-dividend paying stock, Π, O +3o?s?r = rI.
- Consider a delta-neutral portfolio of derivatives, Π. For a small change in the price of the underlying asset, δS, over a short time interval, δt, show that the change in the portfolio value,
- A forward start call option is specified as follows: at time T1, the holder is given a European call option with exercise price S(T1) and expiry at time T1 + T2. What is the value of the option for 0
- The range forward contract is specified as follows: at expiry, the holder must buy the asset for E1 if S < E1, for S if E1 ≤ S ≤ E2 and for E2 if S > E2. Find the relationship between E1
- Consider an asset with zero volatility. We can explicitly calculate the future value of the asset and hence that of a call option with the asset as the underlying. The value of the call option will
- Using the explicit solutions for the European call and put options, check that put-call parity holds.
- Consider a European call, currently at the money. Why is delta hedging self-financing in the following situations?(a) The share price rises until expiry,(b) The share price falls until expiry.
- Find the implied volatility of the following European call. The call has four months until expiry and an exercise price of $100. The call is worth $6.51 and the underlying trades at $101.5, discount
- Consider the pay-later call option. This has payoff Λ(S) = max(S − E, 0) at time T. The holder of the option does not pay a premium when the contract is set up, but must pay Q to the writer at
- Find the explicit solution for the value of a European supershare option, with expiry at time T and payoffΛ(S) = H(S − E1) −H(S − E2),where E1 < E2.
- Find the explicit solution for the value of a European option with payoff Λ(S) and expiry at time T, where S A(S) = S if S> E 0 if S< E.
- If f(x, τ ) ≥ 0 in the initial value problemwithu(x, 0) = 0, and u → 0 as |x| → ∞,then u(x, τ ) ≥ 0. Hence show that if C1 and C2 are European calls with volatilities σ1 and σ2
- Show that ifwithu(x, 0) = u0(x) > 0,then u(x, τ ) > 0 for all τ .Use this result to show that an option with positive payoff will always have a positive value. a?u on - 0 0, du at
- Using a change of time variable, reduceto the diffusion equation when c(τ ) > 0.Consider the Black–Scholes equation, when σ and r can be functions of time, but k = 2r/σ2 is still a constant.
- Reduce the following parabolic equation to the diffusion equation.where a and b are constants. a2u du +a-+b, ax2 Tax du
- Solve the following initial value problem for u(x, τ ) on a semi-infinite interval, using a Green’s function:withu(x, 0) = u0(x) for x > 0, u(0, τ ) = 0 for τ > 0.Define v(x, τ) asv(x, τ
- Check that uδ satisfies the diffusion equation, where 1 us
- Suppose that u(x, τ ) satisfies the following initial value problem:withu(−π, τ ) = u(π, τ ) = 0, u(x, 0) = u0(x).Solve for u using a Fourier sine series in x, with coefficients depending on
- The solution to the initial value problem for the diffusion equation is unique (given certain constraints on the behavior, it must be sufficiently smooth and decay sufficiently fast at infinity).
- Consider an option with value V(S, t), which has payoff at time T. Reduce the Black– Scholes equation, with final and boundary conditions, to the diffusion equation, using the following
- Compare the equation for futures to Black–Scholes with a constant, continuous dividend yield. How might we price options on futures if we know the value of an option with the same payoff with the
- Find the random walk followed by a European option, V(S, t). Use Black Scholes to simplify the equation for dV.