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In this first chapter we present the main definitions that will be used throughout the book. We will introduce the basic concepts in a rigorous way while providing at the same time intuition and motivation for their introduction. However, before starting with the definitions, a remark is in order.

The fundamental economic assumption in the seminal paper by Black and Scholes (1973) is the absence of arbitrage opportunities in the considered financial market. Roughly speaking, absence of arbitrage is equivalent to the impossibility to invest zero today and receive tomorrow a nonnegative amount that is positive with positive probability. In other words, two portfolios having the same payoff at a given future date must have the same price today. By constructing a suitable portfolio having the same instantaneous return as that of a riskless investment, Black and Scholes could then conclude that the portfolio instantaneous return was indeed equal to the instantaneous risk-free rate, which immediately led to their celebrated partial differential equation and, through its solution, to their option-pricing formula.

The theory of interest-rate modeling was originally based on the assumption of specific one-dimensional dynamics for the instantaneous spot rate process . Modeling directly such dynamics is very convenient since all fundamental quantities (rates and bonds) are readily defined, by no-arbitrage arguments, as the expectation of a functional of the process . Indeed, the existence of a risk-neutral measure implies that the arbitrage-free price at time of a contingent claim with payoff at time is given by with denoting the time -conditional expectation under that measure. In particular, the zero-coupon-bond price at time for the maturity is characterized by a unit amount of currency available at time , so that = 1 and we obtain

In the present chapter we introduce two major two-factor short-rate models. Before starting with the actual models, we would like to motivate two-factor models by pointing out the weaknesses of the one-factor models of the previous chapter. This is the purpose of this introductory section.

Modeling the interest-rate evolution through the instantaneous short rate has some advantages, mostly the large liberty one has in choosing the related dynamics. For example, for one-factor short-rate models one is free to choose the drift and instantaneous volatility coefficient in the related diffusion dynamics as one deems fit, with no general restrictions. We have seen several examples of possible choices in Chapter 3. However, short-rate models have also some clear drawbacks. For example, an exact calibration to the initial curve of discount factors and a clear understanding of the covariance structure of forward rates are both difficult to achieve, especially for models that are not analytically tractable.

In this chapter we consider one of the most popular and promising families of interest-rate models: The market models.

In this chapter we present some numerical examples concerning the goodness of fit of the LFM to both the caps and swaptions markets, based on market data. We study several cases based on different instantaneous-volatility parameterizations. We will also point out a particular parameterization allowing for a closed-form-formulas calibration to swaption volatilities and establishing a one to one correspondence between swaption volatilities and LFM covariance parameters.

In this chapter we test the analytical approximations leading to closedform formulas for both swaption volatilities and terminal correlations under the Libor market model (LFM), by resorting to Monte Carlo simulation of the LFM dynamics. We aim at establishing whether the approximations based on drift freezing and approximating lognormal distributions are accurate. We adopt two different contexts.

We have seen in previous chapters that Black’s formula for caplets is the standard in the cap market. This formula is consistent with the LFM, in that it comes as the expected value of the discounted caplet payoff under the related forward measure when the forward-rate dynamics is given by the LFM.

In the equity or foreign-exchange markets, under the assumption that a whole surface of option prices (in strike and maturity) is available for the underlying asset, Dupire (1994, 1997) has derived a candidate LVM that is compatible with the given implied-volatility surface. Balland and Hughston (2000) and Brigo and Mercurio (2003), see Section 10.13, have addressed a similar issue in the interest-rate case, where a single caplet maturity is available for each market forward rate.