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Markov Functional Model
Fast Greeks for Markov-Functional Models Using Adjoint Pde Methods
This paper demonstrates how the adjoint PDE method can be used to compute Greeks in Markov-functional models. This is an accurate and efficient way to compute Greeks, where most of the model sensitivities can be computed in approximately the same time as a single sensitivity using finite difference. We demonstrate the speed and accuracy of the method using a Markov-functional interest rate model, also demonstrating how the model Greeks can be converted into market Greeks.
, N.Denson, M.J.Joshi (2010)
Markov Functional Modeling of Equity, Commodity and other Assets
In this short note we show how to setup a one dimensional single asset model, e.g. equity model, which calibrates to a full (two dimensional) implied volatility surface. We show that the efficient calibration procedure used in LIBOR Markov functional models may be applied here too. In a addition to the calibration to a full volatility surface the model allows the calibration of the joint asset-interest rate movement (i.e. local interest rates) and forward volatility. The latter allows the calibration of compound or Bermudan options. The Markov functional modeling approach consists of a Markovian driver process x and a mapping functional representing the asset states S(t) as a function of x(t). It was originally developed in the context of interest rate models, see [Hunt Kennedy Pelsser 2000]. Our approach however is similar to the setup of the hybrid Markov functional model in spot measure, as considered in [Fries Rott 2004]. For equity models it is common to use a deterministic Num?raire, e.g. the bank account with deterministic interest rates. In our approach we will choose the asset itself as Num?raire. This is a subtle, but crucial difference to other approaches considering Markov functional modeling. Choosing the asset itself as Num?raire will allow for a very efficient numerically calibration procedure. As a consequence interest rates have to be allowed to be stochastic, namely as a functional of x too. The Black-Scholes model with deterministic interest rates is a special case of such a Markov functional model. The most general form of this modeling approach will allow for a simultaneous calibration to a full two dimensional volatility smile, a prescribed joint movement of interest rates and a given forward volatility structure.
, C.P.Fries (2009)
An N-Dimensional Markov-Functional Interest Rate Model
This paper develops an n-dimensional Markov-functional interest rate model, i.e. a model driven by an n-dimensional state process and constructed using Markov-functional techniques. It is shown that this model is very similar to an n-factor LIBOR market model hence allowing intuition from the LIBOR market model to be transferred to the Markov-functional model. This generalizes the results of Bennett & Kennedy from one-dimensional to n-dimensional driving state processes. The model is suitable for pricing certain type of exotic interest rate derivative products whose payoffs depend on the LIBORs at their setting dates. Specifically we investigate the pricing of TARNs and find that the n-dimensional Markov-functional model is faster and can be calibrated more easily to a target correlation structure than an n-factor LIBOR market model.
, L.Kaisajuntti, J.Kennedy (2008)
A Hybrid Markov-Functional Model with Simultaneous Calibration to Interest Rate and FX Smile
In this paper we present a Markov functional hybrid interest rate/fx model which allows the calibration of a given market volatility surface in both dimension simultaneously. We extend the approach introduced in [FriesRott] by introducing a functional for the FX which allows a fast, yet accurate calibration to a given market fx volatility surface. This calibration procedure comes as an additional step to the known calibration of the LIBOR functional.
, C.P.Fries, F.Eckstaedt (2006)
A Comparison of Markov-Functional and Market Models: The One-Dimensional Case
The LIBOR Markov-functional model is an efficient arbitrage-free pricing model suitable for callable interest rate derivatives. We demonstrate that the one-dimensional LIBOR Markov-functional model and the separable onefactor LIBOR market model are very similar. Consequently, the intuition behind the familiar SDE formulation of the LIBOR market model may be applied to the LIBOR Markov-functional model. The application of a drift approximation to a separable one-factor LIBOR market model results in an approximating model driven by a one-dimensional Markov process, permitting efficient implementation. For a given parameterisation of the driving process, we find the distributional structure of this model and the corresponding Markov-functional model are numerically virtually indistinguishable for short maturity tenor structures over a wide variety of market conditions, and both are very similar to the market model. A theoretical uniqueness result shows that any accurate approximation to a separable market model that reduces to a function of the driving process is effectively an approximation to the analogous Markov-functional model. Therefore, our conclusions are not restricted to our particular choice of driving process. Minor differences are observed for longer maturities, nevertheless the models remain qualitatively similar. These differences do not have a large impact on Bermudan swaption prices. Under stress-testing, the LIBOR Markov-functional and separable LIBOR market models continue to exhibit similar behaviour and Bermudan prices under these models remain comparable. However, the drift approximation model now appears to admit arbitrage that is practically significant. In this situation, we argue the Markov-functional model is a more appropriate choice for pricing.
, M.N.Bennett, J.E.Kennedy (2005)
A Comparison of Single Factor Markov-functional and Multi Factor Market Models
We compare single factor Markov-functional and multi factor market models for hedging performance of Bermudan swaptions. We show that hedging performance of both models is comparable, thereby supporting the claim that Bermudan swaptions can be adequately risk-managed with single factor models. Moreover, we show that the impact of smile can be much larger than the impact of correlation. We use the constant exercise method for calculating risk sensitivities of callable products in market models, which is a modification of the least-squares Monte Carlo method. The hedge results show the constant exercise method enables proper functioning of market models as risk-management tools.
, R.Pietersz, A.J.Pelsser (2004)
Markov-Functional Interest Rate Models
We introduce a general class of interest rate models in which the value of pure discount bonds can be expressed as a functional of some (low-dimensional) Markov process. At the abstract level this class includes all current models of practical importance. By specifying these models inMarkov-functional form, we obtain a specification which is efficient to implement. An additional advantage of Markov-functional models is the fact that the specification of the model can be such that the forward rate distribution implied by market option prices can be fitted exactly, which makes these models particularly suited for derivatives pricing. We give examples of Markov-functional models that are fitted to market prices of caps/floors and swaptions.
, P.J.Hunt, J.Kennedy, A.Pelsser (1999)
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