Wittke, Manuel: Essays about Option Valuation under Stochastic Interest Rates. - Bonn, 2011. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.

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@phdthesis{handle:20.500.11811/4865,

urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-24859,

author = {{Manuel Wittke}},

title = {Essays about Option Valuation under Stochastic Interest Rates},

school = {Rheinische Friedrich-Wilhelms-Universität Bonn},

year = 2011,

month = apr,

note = {This thesis consists of three essays on the valuation of options under stochastic interest rates.

In the first essay we examine multivariate models where the stochastic process of a log-normally distributed underlying depends on the evolution of correlated interest rate processes. There the correlation structure can change by constant factor volatilities, which influences not only the values of financial instruments but also their hedge strategies. In this model class we propose a unified framework for the pricing and hedging of chooser options under stochastic interest rates. Chooser options are exotic derivatives who give the owner the right to enter at their exercise date a call or a put option with the same underlying. The chosen multivariate framework allows to derive closed form solutions of the arbitrage price for different specifications of chooser options such as different strike prices or time to maturities.

The second essay deals with the so called convexity correction of swap rates. A convexity correction needs to be computed in the case of constant maturity swaps (CMS) for example. The expectation is taken under a different measure than the assets martingale measure. Then, the expectation is not the forward value but the forward value plus a convexity correction. One approach in the literature suggests, that the convexity correction is the price of a static portfolio of plain-vanilla swaptions. This portfolio approach has the advantage that the volatility cube can be incorporated by using a stochastic or local volatility models, but it is the solution of an integral over an infinite number of strike prices. We propose an algorithm to approximate the replication portfolio with a finite number and therefore a discrete set of strike prices. The accuracy of the method is examined using numerical examples and different valuation models as well as different sets of strike prices.

The modeling of multi-asset options within an interest rate model is the topic of the third essay. There, we consider the joint dynamic of a basket of n-assets with the application to CMS spread options in mind. Therefore we use a Swap Market Model (SMM) with deterministic volatility and the SABR model with stochastic volatility. Using the Markovian Projection methodology we approximate multivariate SMM/SABR dynamic with a univariate SMM/SABR dynamic to price caps and floors in closed form. This enables us to consider not only the asset correlation but, in the case of the SABR model, as well the skew, the cross-skew and the decorrelation in our approximation. If for example, spread options are considered the latter is not possible in alternative approximations. We illustrate the method by considering the example where the underlyings are two constant maturity swap rates. There we examine the influence of the swaption volatility cube on CMS spread options and compare our approximation formulae to results obtained by Monte Carlo simulation and a copula approach.},

url = {http://hdl.handle.net/20.500.11811/4865}

}

urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-24859,

author = {{Manuel Wittke}},

title = {Essays about Option Valuation under Stochastic Interest Rates},

school = {Rheinische Friedrich-Wilhelms-Universität Bonn},

year = 2011,

month = apr,

note = {This thesis consists of three essays on the valuation of options under stochastic interest rates.

In the first essay we examine multivariate models where the stochastic process of a log-normally distributed underlying depends on the evolution of correlated interest rate processes. There the correlation structure can change by constant factor volatilities, which influences not only the values of financial instruments but also their hedge strategies. In this model class we propose a unified framework for the pricing and hedging of chooser options under stochastic interest rates. Chooser options are exotic derivatives who give the owner the right to enter at their exercise date a call or a put option with the same underlying. The chosen multivariate framework allows to derive closed form solutions of the arbitrage price for different specifications of chooser options such as different strike prices or time to maturities.

The second essay deals with the so called convexity correction of swap rates. A convexity correction needs to be computed in the case of constant maturity swaps (CMS) for example. The expectation is taken under a different measure than the assets martingale measure. Then, the expectation is not the forward value but the forward value plus a convexity correction. One approach in the literature suggests, that the convexity correction is the price of a static portfolio of plain-vanilla swaptions. This portfolio approach has the advantage that the volatility cube can be incorporated by using a stochastic or local volatility models, but it is the solution of an integral over an infinite number of strike prices. We propose an algorithm to approximate the replication portfolio with a finite number and therefore a discrete set of strike prices. The accuracy of the method is examined using numerical examples and different valuation models as well as different sets of strike prices.

The modeling of multi-asset options within an interest rate model is the topic of the third essay. There, we consider the joint dynamic of a basket of n-assets with the application to CMS spread options in mind. Therefore we use a Swap Market Model (SMM) with deterministic volatility and the SABR model with stochastic volatility. Using the Markovian Projection methodology we approximate multivariate SMM/SABR dynamic with a univariate SMM/SABR dynamic to price caps and floors in closed form. This enables us to consider not only the asset correlation but, in the case of the SABR model, as well the skew, the cross-skew and the decorrelation in our approximation. If for example, spread options are considered the latter is not possible in alternative approximations. We illustrate the method by considering the example where the underlyings are two constant maturity swap rates. There we examine the influence of the swaption volatility cube on CMS spread options and compare our approximation formulae to results obtained by Monte Carlo simulation and a copula approach.},

url = {http://hdl.handle.net/20.500.11811/4865}

}