A probabilistic approach to the valuation of general floating rate notes with an application to interest rate swaps

Résumé du document

The aim of this article is to get an exact formula to price Floating Rate Notes. The authors will go further in order to price Interest Rate Swaps, in a stochastic term structure environment. Main assumptions of the article are the following, the market is complete, and the probability (explained thereafter) is unique. This article uses martingales, which are an easier way to calculate than solving Partial Derivatives Equations.

Extraits

[...] II/ A probabilistic approach to the pricing of floating-rate notes Given a stochastic intertemporal economy the probability space (Ω represents the uncertainty, and the filtration . We assume the existence of the risk-neutral probability measure Q equivalent (i.e P and Q have the same insignificant sets). Thanks to this “risk-adjusted probability” Value of the FRN = Value of the Zero-coupon bond, maturity at time t=0 A. The model (continuous time framework) Forward Rate T+θ) : ρ T+θ) = Instantaneous forward-rate*: ρ = lim = - *it is the determined forward rate beginning in T and ending an infinite time after T (used in Heath-Jarrow-Merton model) Money market instrument such that: with : instantaneously risk-free short-term interest rate Absence of Arbitrage Opportunities S.D.E: (under with σ(t,T) : volatility of (deterministic or stochastic) such that σ(t,t)=0 Wt: Q-standard Wiener process associated to such that dWt=ε√dt ( N(0;dt) Therefore, the yield-to-maturity θ has the following expression: With dWs a Q-Geometrical Brownian Motion B. [...]

[...] El-Karoui and Geman choose a current-yield base approach of the term structure, which reduces pricing and IRS to determining the value of a floater since the price of the fixed-rate bond can be seen on the current term structure. Floating VS Floating leg: It can be priced as the exchange of two FRNs plus the swap value (=basis risk). Thanks to the calculations done and results obtained before, we can build a swap volatility term structure corresponding to the index J. [...]

[...] Empirical testing Two FRNs: TRE2000 : tied to an index as an annualized 7yrs, with h=1 TME2001 : tied to the same base-rate, with coupon unknown still payment Deriving the theoretical values of the two FRNs, we denote them respectively V1 and V2. Y1 and Y2 are the prices practitioners can read in the current term structure of interest rates. We notice through the plots that V2-Y2>V1-Y1 because of the lag of TRE2000. ( Y is a lower bound of the market price. Best representation of the term structure volatility: We notice that TME2001 Implied volatility is greater than TRE2000 one, this situation can be explained by the lag of TME2001. [...]

[...] A probabilistic approach to the valuation of general floating rate notes with an application to interest rate swaps The aim of this article is to find out an exact formula to price Floating Rate Notes. The authors will go further in order to price Interest Rate Swaps (fixed leg VS floating leg), in a stochastic term structure environment. Main assumptions of the article are the following: - The market is complete (quoted primary assets are linked to derivatives ( possibility to duplicate the derivative) - The probability Q (explained thereafter) is unique This article uses martingales, which are an easier way to calculate than solving Partial Derivatives Equations (Ramaswamy&Sundaresan, 1986) Description and classical results Issued in the 70s, FRNs enable the issuer and the bondholder to share the risk. [...]

[...] We obtain a linear relationship between implied volatility and the difference which is independent of time ( It emphasizes the assumption that interest rates volatility is the main factor to explain the excess of market prices over the sum of discounted forward rates. The spread between long-term and short-term interest rates cannot be considered as an explanatory factor for V. Summary and conclusions Thanks to the “change of probability” method, El-Karoui and Geman provide us an easier approach than Partial Derivative Equations (Sundaresan) to price all the types of FRNs. [...]