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introduces quantitative finance

Questions and Answers of Introduces Quantitative Finance

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.
Consider an option which expires at time T. The current value of the option is V(S, t). It is possible to synthesize the option using vanilla European calls, all with expiry at time T. We assume that
C(S, t) and P(S, t) are the values of European call and put options, with exercise price E and expiry at time T. Show that a portfolio of long the call and short the put satisfies the Black-Scholes
Prove the following bounds on European call options C(S, t), on an underlying share price S, with no dividends:(a) CA ≥ CB, where CA and CB are calls with the same exercise price E and expiry dates
Prove the following bounds on European put options P(S, t), with expiry at time T, on an underlying share price S, with no dividends:(a) P ≤ Ee−r(T−t),(b) P ≥ Ee−r(T−t) − S,(c) 0 ≤ P2
Prove the following bounds on European call options C(S, t), with expiry at time T, on an underlying share price S, with no dividends:(a) C ≤ S,(b) C ≥ max(S − Ee−r(T−t), 0),(c) 0 ≤ C1
What is the most general solution of the Black–Scholes equation with each of the following forms?(a) V(S, t) = A(S),(b) V(S, t) = B(S)C(t).
Check that the following are solutions of the Black–Scholes equation:(a) V(S, t) = S,(b) V(S, t) = ert.Why are these solutions of particular note?
Two shares follow geometric Brownian motions, i.e.dS1 = μ1S1 dt + σ1S1 dX1,dS2 = μ2S2 dt + σ2S2 dX2.The share price changes are correlated with correlation coefficient ρ. Find the stochastic
The change in a share price satisfiesdS = A(S, t) dX + B(S, t) dt,for some functions A,B, what is the stochastic differential equation satisfied by f(S, t)?Can A,B be chosen so that a function g(S)
If dS = μS dt + σS dX, use Itô's lemma to find the stochastic differential equation satisfied by f(S) = log(S).
If S follows a lognormal random walk, use Itô's lemma to find the differential equations satisfied by(a) f(S) = AS + B,(b) g(S) = Sn,(c) h(S, t) = Snemt,where A,B, m and n are constants.
Find u(W, t) and v(W, t) wheredW(t) = u dt + v dX(t)and(a) W(t) = X2(t),(b) W(t) = 1 + t + eX(t),(c) W(t) = f(t)X(t),where f is a bounded, continuous function.
Consider a function f(t) which is continuous and bounded on [0, t]. Prove integration by parts, i.e. | f(t)dX(t) = f(t)X(t) – -| X(t)df(t).
By considering X2(t), show that X(t)dX(t) = }X (t) – Įt.
Compare interest rate data with your share price data. Are there any major differences? Is the asset price modeldS = μS dt + σS dXalso suitable for modeling interest rates?
Using daily share price data, find and plot returns for the asset. What are the mean and standard deviation for the sample you have chosen?
What is the distribution of the price increase for the share movement described in Question 1?Question 1A share has an expected return of 12% per annum (with continuous compounding) and a volatility
A share has an expected return of 12% per annum (with continuous compounding) and a volatility of 20% per annum. Changes in the share price satisfy dS = μS dt + σS dX. Simulate the movement of the
A share price is currently $75. At the end of three months, it will be either $59 or $92. What is the risk-neutral probability that the share price increases? The risk-free interest rate is 4% p.a.
A share price is currently $180. At the end of one year, it will be either $203 or $152. The risk-free interest rate is 3% p.a. with continuous compounding. Consider an American put on this
A share price is currently $15. At the end of three months, it will be either $13 or $17. Ignoring interest rates, calculate the value of a three-month European option with payoff max(S2 − 159, 0),
A share price is currently $63. At the end of each three month period, it will change by going up $3 or going down $3. Calculate the value of a six month American put option with exercise price $61.
A share price is currently $45. At the end of each of the next two months, it will change by going up $2 or going down $2. Calculate the value of a two month European call option with exercise price
A share price is currently $92. At the end of one year, it will be either $86 or $98. Calculate the value of a one-year European call option with exercise price $90 using a single-step binomial tree.
A share price is currently $80. At the end of three months, it will be either $84 or $76. Ignoring interest rates, calculate the value of a three-month European call option with exercise price $79.
Starting from the approximations for u and v, check that in the limit δt →0 we recover the Black–Scholes equation.
Solve the three equations for u, v and p using the alternative condition p = 1/2 instead of the condition that the tree returns to where it started, i.e. uv = 1.
Using the notation V(E) to mean the value of a European call option with strike E, what can you say about ∂V/∂E and ∂2V/∂E2 for options having the same expiration? Consider call and butterfly
A three-month, 80 strike, European call option is worth $11.91. The 90 call is $4.52 and the 100 call is $1.03. How much is the butterfly spread?
A share currently trades at $60. A European call with exercise price $58 and expiry in three months trades at $3. The three month default-free discount rate is 5%. A put is offered on the market,
What is the difference between a payoff diagram and a profit diagram? Illustrate with a portfolio of short one share, long two calls with exercise price E.
Find the value of the following portfolios of options at expiry, as a function of the share price:(a) Long one share, long one put with exercise price E,(b) Long one call and one put, both with
A particular forward contract costs nothing to enter into at time t and obliges the holder to buy the asset for an amount F at expiry T. The asset pays a dividend DS at time td, where 0 ≤ D ≤ 1
A spot exchange rate is currently 2.350. The one-month forward is 2.362. What is the one-month interest rate assuming there is no arbitrage?
You put $1000 in the bank at a continuously compounded rate of 5% for one year. At the end of this first year rates rise to 6%. You keep your money in the bank for another eighteen months. How much
The dollar sterling exchange rate (colloquially known as ‘cable’) is 1.83, £1 = $1.83. The sterling euro exchange rate is 1.41, £1 = € 1.41. The dollar euro exchange rate is 0.77, $1 =
A company whose stock price is currently S pays out a dividend DS, where 0 ≤ D ≤ 1. What is the price of the stock just after the dividend date?
A company makes a three-for-one stock split. What effect does this have on the share price?

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