Arbitrage-Free Interpolation in Models of Market Observable Interest Rates

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

An Equity-Interest Rate Hybrid Model with Stochastic Volatility and the Interest Rate Smile

“…We start with the interpolation technique introduced in [Sch02]. In this approach a linear interpolation which produces a piecewise deterministic short rate for t ∈ (T m(t)−1 , T m(t) ) is used.…”

Incorporating an Interest Rate Smile in an Equity Local Volatility Model

“…The LMM is however defined on a discrete tenor structure, so it does not fit immediately to the continuous interest rate model. A way of generalizing the Libor market model from the discrete to a continuous tenor was proposed in [29], where via an arbitrage-free interpolation between the Libors the zero-coupon bond for continuous time was defined. In this section we generalize this idea and use the HJM arbitragefree condition to extract the short-rate from the interpolated zero-coupon bonds.…”

“…Based on the arbitrage-free interpolation proposed in [29], here we extract a multi-factor shortrate process from the SV-LMM and include it into the LVM framework for the equity component. We extract the short-rate via a continuous tenor structure from the LMM.…”

“…In Section 3 we recall some necessary results from the HJM framework. Also in this section we use the arbitrage-free interpolation from [29] of the zero-coupon bonds (ZCB) under the LMM and derive the corresponding short-rate processes. Then, we perform a number of numerical experiments, that show the impact of the interest rate smile.…”

“…In [29], it was shown that for a given discrete tenor model, and an arbitrage-free interpolation of the short dated ZCBs, the continuous tenor model is completely specified. We extend this interpolation in which a control of the amount of external volatility for the ZCB can be included.…”