Fixed income markets are vast markets, involving a large number of actors including financial institutions, state actors, asset managers and corporations. An import part of these markets are contracts written on the xIBOR rates. This report is concerned with the trying to provide prices for options written on these rates, in particular for swaptions that are not at-the-money (atm) utilizing prices in the cap market. Different methods have been suggested in the literature for solving this problem. In particular we study the method suggested by Hagan et al where one calibrates a SABR model to the caplet surface with the same expiry as the swaption. One then assumes that the swaption contract with the same expiry follows the same SABR dynamics as the caplet, but with a recalibrated initial volatility to fit the atm point. We also study the approach suggested by Rebonato and Jäckel. They derive a model for swaption prices based on the individual volatilities of the forward rates that the underlying interest rate swap consists of, as well as the correlation between the forward rates.  Both of these approaches are studied empirically for the STIBOR market. The data set span between 2016 and 2021 and consists of the yield curve, flat cap volatilities and swaption volatilities. We use the 1Y1Y and 5Y5Y swaption surfaces, where the prices are not only quoted atm, to verify our model. We conclude that despite the SABR model being able to fit the caplet prices well, the method suggested by Hagan does not capture the swaption smile. The Rebonato and Jäckel approach also falls short of capturing the smile and produces similar results as the Hagan et al method. This is suggested to be due to the Hagan method capturing the caplet smile well, and the constant correlation assumption made in this thesis.

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With the discontinuation of Interbank Offered Rates(IBOR), traders found some al-ternative reference rates to replace IBOR. Backward-looking rates are widely accepted new benchmark interest rates. In this thesis, we introduce and subsequently proceed to explore backward-looking rate model and continue doing some re-valuation of interest rate instruments under the backward-looking rates framework.

<p>In this paper the Heath, Jarrow and Morton (HJM) framework is applied in the programming language Java for the estimation of the future spot rate. The subcase of an exponential model for the diffusion coefficient (volatility) is used for the pricing of interest rate derivatives (caps).</p>

In this paper the Heath, Jarrow and Morton (HJM) framework is applied in the programming language Java for the estimation of the future spot rate. The subcase of an exponential model for the diffusion coefficient (volatility) is used for the pricing of interest rate derivatives (caps).

This thesis consists of three papers in the area of interest rate derivatives modelling. The pricing and hedging of (exotic) interest rate derivatives is one of the most demanding and complex problems in option pricing theory and is of great practical importance in the market. Models used in production at various banks can broadly be divided in three groups: 1- or 2-factor instantaneous short/forward rate models (such as Hull &amp; White (1990) or Cheyette (1996)), LIBOR/swap market models (introduced by Brace, Gatarek &amp; Musiela (1997), Miltersen, Sandmann &amp; Sondermannn (1997) and Jamshidian (1997)) and the one or two-dimensional Markov-functional models of Hunt, Kennedy &amp; Pelsser (2000)). In brief and general terms the main characters of the above mentioned three modelling frameworks can be summarised as follows. Short/forward rate models are by nature computationally efficient (implementations may be done using PDE or lattice methods) but less flexible in terms of fitting of implied volatility smiles and correlations between various rates. Calibration is hence typically performed in a ‘local’ (product by product based) sense. LIBOR market models on the other hand may be calibrated in a ‘global’ sense (i.e. fitting close to everything implying that one calibration may in principle be used for all products) but are of high dimension and an accurate implementation has to be done using the Monte Carlo method. Finally, Markov-functional models can be viewed as designed to combine the computational efficiency of short/forward rate models with flexible calibration properties. The defining property of a Markov-functional model is that each rate and discount factor at all times can be written as functionals of some (preferably computationally simple) Markovian driving process. While this is a property of most commonly used interest rate models Hunt et al. (2000) introduced a technique to numerically determine a set of functional forms consistent with market prices of vanilla options across strikes and expiries. The term a ‘Markov-functional model’ is typically referring to this type of model as opposed to the more general meaning, a terminology that is adopted also in this thesis. Although Markov-functional models are indeed a popular choice in practice there are a few outstanding points on the practitioners’ wish list. From a conceptual point of view there is still work to be done in order to fully understand the implications of various modelling choices and how to efficiently calibrate and use the model. Part of the reason for this is that while the properties of the short/forward rate and the LIBOR market models may be understood from their defining SDEs this is less clear for a Markov-functional model. To aid the understanding of the Markov-functional model Bennett &amp; Kennedy (2005) compares one-dimensional LIBOR and swap Markov-functional models with the one-factor separable LIBOR and swap market models and concludes that the models are similar distributionally across a wide range of viable market conditions. Although this provides good intuition there is still more work to be done in order to fully understand the implications of various modelling choices, in particular in a two or higher dimensional setting. The first two papers in this thesis treat extensions of the standard Markov-functional model to be able to use a higher dimensional driving process. This allows a more general understanding of the Markov-functional modelling framework and enables comparisons with multi-factor LIBOR market models. From a practical point of view it provides more powerful modelling of correlations among rates and hence a better examination and control of some types of exotic products. Another desire among practitioners is to develop an efficient way of using a process of stochastic volatility type as a driver in a Markov-functional model. A stochastic volatility Markov-functional model has the virtue of both being able to fit current market prices across strikes and to provide better control over the future evolution of rates and volatilities, something which is important both for pricing of certain products and for risk management. Although there are some technical challenges to be solved in order to develop an efficient stochastic volatility Markov-functional model there are also many (more practical) considerations to take into account when choosing which type of driver to use. To shed light on this the third paper in the thesis performs a data driven study in order to motivate and develop a suitable two-dimensional stochastic volatility process for the level of interest rates. While the main part of the paper is general and not directly linked to any complete interest rate model for exotic derivatives, particular care is taken to examine and equip the process with properties that will aid use as a driver for a stochastic volatility Markov-functional model. / <p>Diss. Stockholm :  Stockholm School of Economics, 2011. Introduction together with 3 papers</p>

No / We show that the unified HJM-based approach of constructing Gaussian dynamic term structure models developed by Li, Ye, and Yu (2016) nests most existing GDTSMs as special cases. We also discuss issues of interest rate derivatives pricing under this approach and using integration to construct Markov representations of HJM models.

Models of the term structure of interest rates play a central role in the modern theory of pricing bonds and other interest rate claims. Term structure models based on the principle of no-arbitrage, especially those of the Heath-Jarrow-Morton (1992) class, have become very popular recently, both with academics and practitioners. Surprisingly however, although the implied volatility function plays a crucial role in these no-arbitrage term structure models, there is little systematic evidence to guide optimal model specification within this broad class. We study the implied volatility in the Heath-Jarrow-Morton framework using Eurodollar futures options data. We estimate a daily time series of forward rates within the HJM framework such that, by construction, the predicted futures prices from our model exactly match the observed futures prices. Next, we estimate a daily time series of volatility parameters such that the sum of squared errors between futures options prices predicted by the model and observed futures options prices is minimized. We use the six different volatility specifications suggested by Amin and Morton (1994) within the HJM class of models to price interest rate claims. Since the volatilities are the only unobservables, we use these models to infer the volatilities from the market prices of Eurodollar futures options over the 1987-1998 periods. The minimized sum of squared errors in the option prices is used as the measure of accuracy of each specific model. Each model differs from the others in its ability to match the market option prices and the time required for the computation. We compare the performances of the six volatility specifications in the accuracy-versus-computation time tradeoff. We document the systematic biases between the model and market prices as a function of option type, maturity, and moneyness. We also examine alternative numerical implementations of HJM models using the six volatility specifications. In particular, we analyze the impact on accuracy and computation time of using different numbers of time-steps. We also examine the effect of using time-steps of varying lengths within the same estimation procedure, and of ordering the time-steps in different ways. / Ph. D.

The market for interest rate derivatives has in recent decades grown considerably and the need for proper valuation models has increased. Interest rate derivatives are instruments that in some way are contingent on interest rates such as bonds and swaps and most financial transactions are in some way exposed to interest rate risk. Interest rate derivatives are commonly used to hedge this risk. This study focuses on the Black Derman &amp; Toy model and its capability of pricing interest rate derivatives. The purpose was to simulate the model numerically using daily Euro-Bunds and options data to identify if the model can generate accurate prices. A second purpose was to simplify the theory of building a short rate binomial tree, since existing theory explains this step in a complex way. The study concludes that the BDT model have difficulties valuing the extrinsic value of options with longer maturities, especially out-of-the money options.

The globilisation phenomena is causing an increasing interaction between different markets and sectors. This has led to the evolution of derivative instruments from ”single asset” instruments to complex derivatives that have underlying assets from different markets, sectors and sub-sectors. These are the so-called hybrid products that have multi-assets as underlying instruments. This article focuses on interest rate hybrid products. In this article an analysis of the application of stochastic interest rate models and stochastic volatility models in pricing and hedging interest rate hybrid products will be explored.