# Interest Rates Models : Analysis of Stochastic Modeling and Derivative Pricing

Interest Rates Models : Analysis of Stochastic Modeling and Derivative Pricing

Interest Rates Models : Analysis of Stochastic Modeling and Derivative Pricing : Markov-functional models correspond to a particular class of pricing models, on which zero-coupon bond prices are functionals of some low-dimensionally driven Markov process in the “boundary curve of time,” the largest time horizon chosen to price certain related products.

That functional form is intentionally constructed to facilitate the calibration process of the model to relevant interest-rate options’ market prices.

Its key advantage is the easy construction of the correct distributions of certain relevant market rates, while at the same time retaining low dimensionality. While not the most popular model, it is still used by many banks worldwide because of its flexibility and relatively low computational cost.

To model interest rates for derivative pricing, we use the risk-neutral pricing framework, in which the key assumption is the possibility of setting up a hedging portfolio.

The most popular model for short-rates is the Heath-Jarrow-Morton (HJM) model: a general framework from which several other short-rate models, such as the Hull & White model, are derived.

Nowadays, short-rate models are not predominant in pricing interest-rate derivatives for several reasons.

One important issue is that short-rates are unobservable in the markets. It’s known that no short-rate model can lead to Black’s pricing formula for caplets.

Moreover, we require models to be calibrated properly to a large number of market products, primarily to caps and swaptions, the two largest interest-rate derivatives markets.

To that end, practitioners prefer to model instead the joint dynamics of several forward interest swap rates, both of which are directly observable in the markets.

A popular forward rate model used in banking is the Libor Market Model, also known as the Lognormal Forward-Libor Market Model or sometimes as BGM, which is able to model the correlation structure among the different LIBOR forward rates.

For USD, these are USD LIBOR 1M, 3M, 6M, 1Y, 2Y, etc. The Libor Market Model is used to price interest rate products whose pay-off may be written as an expression involving the different forward rates, e.g. caps and swaps, and needs to be calibrated properly to satisfy risk-neutrality and lack of arbitrage.

The model parameters can thus be calibrated against historical data, current market prices of caps and caplets, or other ad-hoc methods.

Unfortunately, this model is not compatible with the other half of the world of the interest-rate derivative products, the swaptions.

This is due to the incompatibility between the choice of numéraries and corresponding measures, among other reasons. For such products, we need construct a different, independent method based on modeling the joint dynamics of the various forward swap rates.

Japan’s Chronic Deflation

This is known as the Lognormal Forward Swap Model, or LSM.

While correlation structures may not matter for caps, they are critical in pricing a large number of swaptions. There are also other models on the rise for pricing exotic products like Bermudan Swaptions, such as the Linear Gaussian Model, or LGM.

While AI and ML are not used to directly predict IR rates in pricing and hedging, they could very well be used for many different purposes.

In particular, deep networks can be used for calibration by learning the model parameters-pricing map, which is generally an immense computational task when done numerically.

Some have also used unsupervised methods, such as mixture models and clustering, to estimate the probability densities of model parameters (Kanevski and Timonin 2010).

Written by Gihyen Eom & Alexander Fleiss

Edited by Calvin Ma, Michael Ding & Tianyi Li

Sources

Kanevski, M., & Timonin, V. (2010). Machine Learning Analysis and Modeling of Interest Rate Curves. European Symposium on Artificial Neural Networks.

Interest Rates Models : Analysis of Stochastic Modeling and Derivative Pricing