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LIBOR models with stochastic volatility are extensions of the LFM where the instantaneous volatility of relevant rates evolves according to a diffusion process driven by a Brownian motion that is possibly instantaneously correlated with those governing the rates’ evolution. Formally, the general forward rate is assumed to evolve under its canonical measure according to where and are deterministic functions, ∈ {1/2, 1}, and are adapted processes, and and are possibly correlated Brownian motions.

In order to accommodate market option prices while preserving analytical tractability, uncertain-volatility models (UVMs) have been recently proposed in the financial literature as an easy-to-implement alternative to SVMs. UVMs are based on the assumption that the asset’s volatility is stochastic in the simplest possible way, modelled by a random variable rather than a diffusion process. Precisely, the dynamics of a general forward rate under the associated forward measure is assumed to be given by where is a discrete random variable independent of the Brownian motion , which can take values , , ... , , with probabilities , , ... , , respectively. A UVM, therefore, can be viewed as a Black (1976) model where the volatility is not constant and one assumes several possible scenarios for its value, which is to be drawn immediately after time zero.

In this chapter, we present a sample of financial products we believe to be representative of a large portion of the interest-rate market. We will use different models (mostly the LFM and the G2++ model) for different problems, and try to clarify the advantages of each model. All the discounted payoffs will be calculated at time = 0.

In this chapter, we explain how one can model both a first (domestic) and a second (foreign) interest-rate curve, each by a two-factor additive Gaussian short-rate model, in order to Monte Carlo price a quanto constant-maturity swap and similar contracts, which we will present in the following sections.

European governments have been issuing inflation-indexed bonds since the beginning of the 80’s, but it is only in the very last years that these bonds, and inflation-indexed derivatives in general, have become more and more popular.

Given a set of dates , ... , , an Inflation-Indexed Swap (IIS) is a swap where, on each payment date, Party A pays Party B the inflation rate over a predefined period, while Party B pays Party A a fixed rate. The inflation rate is calculated as the percentage return of the CPI index over the time interval it applies to. Two are the main IIS traded in the market: the zero coupon (ZC) swap and the year-on-year (YY) swap.

An Inflation-Indexed Caplet (IIC) is a call option on the inflation rate implied by the CPI index. Analogously, an Inflation-Indexed Floorlet (IIF) is a put option on the same inflation rate. In formulas, at time , the IICF payoff is where is the IICF strike, is the contract year fraction for the interval [, ], is the contract nominal value, and = 1 for a caplet and = −1 for a floorlet.

In this section, we consider an example of calibration to Euro market data as of October 7, 2004. Precisely, we test the performance of the JY model and the two market models as far as the calibration to inflation-indexed swaps is concerned, with some model parameters being previously fitted to at-the-money (nominal) cap volatilities. The zero-coupon and year-on-year swap rates we consider are plotted in Figure 16.1.

We have seen in the previous chapters that the pricing of II derivatives is typically addressed by resorting to a foreign-currency analogy. In fact, one can convert nominal values into real ones simply by dividing by the current value of the reference CPI. The most relevant application of the foreign-currency analogy is due to Jarrow and Yildrim (2003), who modelled the dynamics of the CPI along with nominal and real rates, both assumed to follow a one-factor Gaussian process in the HJM framework.

In the recent years, there has been an increasing interest for hybrid structures whose payoff is based on assets belonging to different markets. Among them, derivatives with an inflation component are getting more and more popular. In this chapter, we tackle the pricing issue of a specific hybrid payoff when no smile effects are taken into account. The valuation of more general structures is to be dealt with on a case by case basis and is likely to involve numerical routines as Monte Carlo.