No abstract

Models which postulate lognormal dynamics for interest rates which are compounded according to market conventions, such as forward LIBOR or forward swap rates, can be constructed initially in a discrete tenor framework. Interpolating interest rates between maturities in the discrete tenor structure is equivalent to extending the model to continuous tenor. The present paper sets forth an alternative way of performing this extension; one which preserves the Markovian properties of the discrete tenor models and guarantees the positivity of all interpolated rates.Practitioners have long priced caps, floors and other interest rate derivative contracts by Black/ Scholes-like 1 formulae, a practice usually attributed to the work of Black (1976). Initially, this approach was very much focused on the pricing of individual contracts, without regard to the arbitrage relationships between different fixed income instruments. The seminal paper of Ho and Lee (1986) instigated further research into models of the entire yield curve and fitting initial data, where the yield curve was given either terms of zero coupon bond prices 2 or continuously compounded short or forward rates. The latter approach was taken by Heath, Jarrow and Morton (1992) (HJM), who developed a general framework for models of the term structure of interest rates, deriving drift conditions for instantaneous forward rates which must be satisfied in any arbitrage-free model. Thus the fundamental objects of these models were mainly mathematical constructs, not market observables. The first work to move away from this paradigm was Sandmann and Sondermann (1989, 1994), who construct a binomial term structure model with the short rate compounded according to market conventions (e.g. three-month LIBOR) as the fundamental model variable. Taking the modelling of market observable interest rates to continuous time and bringing theoretical developments back into line with widespread market practice, Miltersen, Sandmann and Sondermann (1997) 3 (MSS) embedded the pricing of caps and floors by Black/Scholes-like formulae in a consistent HJM framework. The critical assumption for this result is that relative volatility 4 of observable market forward rates such as LIBOR is deterministic. 5 Brace, Gatarek and Musiela (1997) (BGM) resolved key questions in the construction of such a model, in particular concerning existence and measure relationships. Given the deterministic volatility assumption, they explicitly identified forward LIBOR as lognormal martingales under the forward measure to the end of the respective accrual periods. Their label Market Models

Abstract. If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner.