## Conditions For The Rate Process To Be Markovian

In his 1992 paper, Carverhill 1992 posits a process for discount bond prices of the form p f r Kt dr v t, T dzit 20.5 where all the symbols have the now familiar meaning, and Carverhill's rather idiosyncratic choice of symbols such as x for volatilities has been changed for consistency with this work. It will be remembered from Chapter 17 that specification 20.5 corresponds to a log-normal discount bond normal rates assumption, and that, in this particular case, the absolute volatility at time...

## Exact Fitting Of The Model To The Term Structure Of Volatilities

Let us denote the continuously compounded yield of a T-maturity discount bond by nP r,t,T A' t, T rB t, T R t,T -- --T-t 3'17 where A' t, T denotes the logarithm of A t,T and the dependence of the discount bond price on the short rate has been emphasised in the notation. Ito's lemma can therefore be applied to R t, T seen as a function of r to obtain whence B t, T can be easily obtained as Notice that absolute, rather than percentage, volatilities have been used in the expressions above. At...

## Info

xMXxiX 2MX2 Xi kMXk Xi What still remains to be specified is how to choose the weights J,. Notice that it is not strictly necessary to determine these weights, which could be all set equal to one. If the 'simpler' option is indeed very similar to the option we want to evaluate, this choice would produce almost as good a result as the more accurate least-square procedure described in the following. In general, however, the factors are designed to account for all sources of imperfections in the...

## List of symbols and abbreviations

Financial and statistical quantities Bit, T Rolled-up money-market account where 1 is reinvested at the prevailing instantaneous short rate from time t to time T. Bnd t,T time-r value of a coupon-bearing bond of maturity T. Correl a, b Correlation between a and b. Covar a, b Covariance between a and b. ,,g x T 3' i Expectation taken at time t on the basis of the information filtration 'Sit and in the probability measure Q of the time-r value of quantity x. f t, s Instantaneous forward rate seen...

## Bestfit Calibration Of The Onedimensional Hw Model To Market Data

The most 'natural' and intuitive procedure to fit the HW Extended Vasicek model to cap prices is to choose the couple of values for the reversion speed a and the absolute volatility a that 'best fit' a desired series of caps. The cap fitting procedure is, however, not quite as straightforward as one might surmise, both from the technical and from the conceptual point of view. Cap prices and percentage volatilities, in fact, are quoted in the market on the basis of the Black model, which assumes...

## Exotic interestrate instruments description and valuation issues

In this chapter important classes of exotic interest-rate instruments are presented with a view to illustrating their financial rationale to describing them as simply as possible in terms directly amenable to the treatment to be found in the subsequent chapters and to highlighting the valuation issues that must be taken into account when applying in practice any of the models described in Part Four. No attempt will be made to indicate 'the right way' to price any of these complex instruments,...

## The CIR and Vasicek models

11.1 general features of desirable interest-rate processes In the previous chapters some fundamental properties of interest-rate models have been analysed. In particular, the conditions obtained in Chapter 6 see also Appendices A and B specify how the price of various securities can stochastically vary over time as a function of the underlying factor so as to preclude the possibility of arbitrage. Nothing specific, however, has been said about the dynamics of the underlying driving variables....

## Preface to the First Edition

In every review of interest-rate derivative instruments it appears to be rigueur to present a table, or the ubiquitous bar chart, displaying the exponential-like growth of this type of product in terms of underlying notional, outstanding deals, or some other measure of volume. The correctness of these statistics and the importance of these measures are undeniable even more significant, however, has been the qualitative change in the type of options which have recently begun to be actively...

## I1111

Figure 4.4 The effects of a small number of factors on the pricing of a European swaption in order to obtain the correct 'average' decorrelation necessary to recover the correct market price curve labelled 'TrueCorrelation' , the first forward rates end up being too strongly correlated, and the last forward rates too strongly de-correlated with the first curve labelled 'ModelCorrelation' swaption this state of affairs, however, has the consequence that correct pricing with constant...

## Pricing European Swaptions With Instantaneous Volatilities

Let us consider the case of a discrete trading horizon made up of two dates only, and constrain, for simplicity of expositions, all quantities, such as volatilities or correlations, to be piece-wise constant, i.e. to be constant at possibly different levels over each of the two possible 'time steps'. Let us then consider the case of two forward rates f and f2 expiring at times t and t2, respectively, of average volatility of 20 more precisely, of unconditional variance 20 2 t and 20 2 t2 , and...

## Oneway floaters

One-way-floaters often referred to as ratchet caps and floors are instruments whereby Party A pays LIBOR flat to Party B, and receives from the latter LIBOR 4- X capped and or floored by two different stochastic strikes. More precisely, if r is the tenor in years, at the first reset time, t , the equal and opposite cash flows originating from the LIBOR rates and paid r years thereafter on both legs will be set. As a net result, as far as the first reset is concerned, party A will effectively...

## Instantaneous And Average Volatilities

If one compares see Figures 4.1 and 4.2 market implied volatilities for caplets and European swaptions, it is well known that the latter trade at levels which are significantly lower than the average of the underlying forward rates.1 Since, as shown in Chapter 1 Equation 1.40 , a swap rate can be written as a linear combination of these underlying forward rates, this empirical observation has in general been accounted for by invoking a less-than-perfect correlation amongst different portions of...

## Trigger swaps

A typical example of a trigger swap is an agreement whereby party A agrees to pay a fixed rate F' to party B against receiving LIBOR flows until a particular reference rate typically 6- or 12-month LIBOR resets on one of several pre-specified dates above or below a barrier level. If the barrier is reached or breached the swap terminates. The name stems from the fact that the resetting of the reference rate triggers the termination of the swap. In general, the full Taxonomy of trigger swaps...

## Li Bo R1narrears Swaps

A LIBOR-in-arrears swap is a collection of LIB OR-in- arrears FRAs. Each of these FRAs will pay on reset date the difference between the strike, K, and the realisation at time freset of a particular deposit rate R, giving rise to a payoff at reset time equal to PayoffarTCars reset Preset, freset r - K X. 2.1 Therefore a r-tenor LIBOR-in-arrears FRA and a r-tenor plain-vanilla FRA with the same reset date differ only because of the specification of the timing of their payoffs, which occur r...

## Part One The Need For Yield Curve Option Pricing Models

1 Definition and valuation of the underlying instruments 3 1.2 Definition of spot rates, forward rates, swap rates and par coupon rates 5 1.3 The valuation of plain-vanilla swaps and FRAs 8 1.4 Obtaining the discount function from a set of spanning forward or swap rates 14 1.5 The valuation of caps, floors and European swaptions 15 1.6 Determination of the discount function the case of bonds linear models 21 1.7 Determination of the discount function the case of bonds non-linear models 25 1.8...