Listado de contenidos y flujo de la propuesta.

The following examples of exotic swap market products are adapted from Musiela and Rutkowski (1997) and Hull (2003):

• Float-for-floating swap: Two floating interest rates are swapped, e.g. the three-month LIBOR rate and the yield on a given government bond.

• Amortizing swap: The notional principal is reduced from period to period following a pre-specified scheme, e.g. so that the notional principle at any time reflects the outstanding debt on a loan with periodic instalments (as for an annuity or a serial bond).

• Step-up swap: The notional principal increases over time in a pre-determined way.

• Accrual swap: The scheduled payments of one party are only to be paid as long as the floating rate lies in some intervalI. Assume for concreteness that it is the fixed rate payments that have this feature. At the swap payment dateTi the effective fixed rate payment is then

HδKN1/N2, where N1is the number of days in the period betweenTi−1 andTi, where the floating rateltt+δ was in the intervalI, andN2is the total number of days in the period. The

interval _I may even differ from period to period either in a deterministic way or depending on the evolution of the floating interest rate so far.

2.10 Exercises 41

• Constant maturity swap: At the payment dates a fixed rate is exchanged for the (equi- librium) swap rate on a swap of a given, constant maturity, i.e. the floating rate is itself a swap rate.

• Extendable swap: One party has the right to extend the life of the swap under certain conditions.

• Forward swaption: A forward swaption gives the right to enter into a forward swap, i.e. the swaption expires at timet∗ _{before the starting date of the swapT}

0. The payoff is Hδ n X i=1 maxL˜δ,t_T0∗₋K,0BTi t∗ = n X i=1 BTi t∗ ! HδmaxL˜δ,t_T0∗ ₋K,0.

• Swap rate spread option: The payoff is determined by the difference between (equilibrium) swap rates for two different maturities. Recall that ˜lδT0(m) denotes the swap rate for a swap

with payment dates T1, . . . , Tm, whereTi =T0+iδ. An (m, n)-period European swap rate

spread call option with an exercise rateK yields a payoff at timeT0 of

max˜lTδ0(m)−˜l

δ

T0(n)−K,0

.

The corresponding put has a payoff of

maxK₋h˜lδT0(m)−˜l

δ T0(n)

i ,0.

• Yield curve swap: In a one-period yield curve swap one party receives at a given dateT a swap rate ˜lδ

T(m) and pays a rateK+ ˜lδT(n), both computed on the basis of a given notional principal H. A multi-period yield curve swap has, say, L payment dates T1, . . . , TL. At timeTl one party receives an interest rate of ˜lδTl(m) and pays an interest rate ofK+ ˜l

δ Tl(n).

In addition, several instruments combine elements of interest rate swaps and currency swaps. For example, in adifferential swap a domestic floating rate is swapped for a foreign floating rate.

2.10

Exercises

EXERCISE 2.1_{Show that the no-arbitrage price of a European call on a zero-coupon bond will satisfy}

max0, BtS−KBTt

≤CK,T,St ≤BtS(1−K)

provided that all interest rates are non-negative. Here,Tis the maturity date of the option,Kis the exercise

price, andS is the maturity date of the underlying zero-coupon bond. Compare with the corresponding

bounds for a European call on a stock, cf. Hull (2003, Ch. 8). Derive similar bounds for a European call on a coupon bond.

EXERCISE 2.2Give a proof of the put-call parity for options on coupon bonds in Theorem 2.4.

EXERCISE 2.3_{Let ˜l}δ_T

0(k) be the equilibrium swap rate for a swap with payment datesT1, T2, . . . , Tk,

where Ti=T0+iδ as usual. Suppose that ˜lδT0(1), . . . ,˜l

δ

T0(n) are known. Find a recursive procedure for

deriving the associated discount factorsBT1

T0, B

T2

T0, . . . , B

Tn

T0.

EXERCISE 2.4 Show the parity (2.34). Show that a payer swaption and a receiver swaption (with

2.10 Exercises 42

rate ˜Lδ,T0

t .

EXERCISE 2.5 Consider a swap with starting date T0 and a fixed rate K. For t ≤ T0, show that Vfl

t /Vtfix= ˜Lδ,Tt 0/K, where ˜L

δ,T0

Chapter 3

Stochastic processes and stochastic

calculus

In the previous chapter we saw that many interest rate dependent securities cannot be priced uniquely just by appealing to no-arbitrage arguments. To derive prices and hedging strategies we have to model the uncertainty about the term structure of interest rates at relevant future dates. In order to analyze the relation between interest rates and other macroeconomic variables such as aggregate consumption or production, we also have to take the uncertainty about the future values of these variables into account. For example, the uncertainty about future consumption will affect individuals’ supply and demand for bonds and, hence, affect the interest rates set today. In modern finance, stochastic processes are used to model the evolution of uncertain variables over time. Therefore, a basic knowledge of stochastic processes and how to do computations involving stochastic processes is needed in order to understand, evaluate, and develop models of the term structure of interest rates. This chapter is devoted to a relatively brief introduction to stochastic processes and the mathematical tools needed to do calculations with stochastic processes, the so-called stochastic calculus. We will omit many technical details that are not important for a reasonable level of understanding and focus on processes and results that will become important in later chapters. For more details and proofs, the reader is referred to the textbooks of Øksendal (1998) and Karatzas and Shreve (1988).

3.1

Probability spaces

The basic object for studies of uncertain events is a probability space, which is a triple (Ω,_F,P). Here, Ω is the state space, which is the set of possible states or outcomes of the uncertain object. For example, if one studies the outcome of a throw of a dice, the state space is Ω = _{1,2,3,4,5,6_}. An event is a set of possible outcomes, i.e. a subset of the state space. In the example with the dice, some events are_{1,2,3_},_{4,5_},_{1,3,5_},_{6_}, and_{1,2,3,4,5,6_}. The second component of a probability space, F, is the set of events to which a probability can be assigned, i.e. the set of “probabilizable” events. Hence,_F is a set of subsets of the state space! It is required that

(i) the entire state space can be assigned a probability, i.e. Ω_{∈ F};

(ii) if some eventF _⊆Ω can be assigned a probability, so can its complementFc

≡Ω_\F, i.e. 43