This thesis considers the price and characteristics of an exotic option called the Volatility-Cap-Target-Level(VCTL) option. The payoff function is a simple European option style but the underlying value is a dynamic portfolio which is comprised of two components: A risky asset and a non-risky asset. The non-risky asset is a bond and the risky asset can be a fund or an index related to any asset category such as equities, commodities, real estate, etc. The main purpose of using a dynamic portfolio is to keep the realized volatility of the portfolio under control and preferably below a certain maximum level, denoted as the Volatility-Cap-Target-Level (VCTL). This is attained by a variable allocation between the risky asset and the non-risky asset during the maturity of the VCTL-option. The allocation is reviewed and if necessary adjusted every 15th day. Adjustment depends entirely upon the realized historical volatility of the risky asset. Moreover, it is assumed that the risky asset is governed by a certain group of stochastic differential equations called affine jump-diffusion models. All models will be calibrated using out-of-the money European call options based on the Deutsche-Aktien-Index(DAX). The numerical implementation of the portfolio diffusions and the use of Monte Carlo methods will result in different VCTL-option prices. Thus, to price a nonstandard product and to comply with good risk management, it is advocated that the financial institution use several research models such as the SVSJ- and the Seppmodel in addition to the Black-Scholes model. Keywords: Exotic option, basket option, risk management, greeks, affine jumpdiffusions, the Black-Scholes model, the Heston model, Bates model with lognormal jumps, the Bates model with log-asymmetric double exponential jumps, the Stochastic-Volatility-Simultaneous-Jumps(SVSJ)-model, the Sepp-model.

The purpose of this thesis is to compare option pricing models. We have investigated the constant volatility models Black-Scholes-Merton (BSM) and Merton’s Jump Diffusion (MJD) as well as the stochastic volatility models Heston and Bates. The data used were option prices from Microsoft, Advanced Micro Devices Inc, Walt Disney Company, and the S&P 500 index. The data was then divided into training and testing sets, where the training data was used for parameter calibration for each model, and the testing data was used for testing the model prices against prices observed on the market. Calibration of the parameters for each model were carried out using the nonlinear least-squares method. By using the calibrated parameters the price was calculated using the method of Carr and Madan. Generally it was found that the stochastic volatility models, Heston and Bates, replicated the market option prices better than both the constant volatility models, MJD and BSM for most data sets. The mean average relative percentage error for Heston and Bates was found to be 2.26% and 2.17%, respectively. Merton and BSM had a mean average relative percentage error of 6.90% and 5.45%, respectively. We therefore suggest that a stochastic volatility model is to be preferred over a constant volatility model for pricing options. / Syftet med denna tes är att jämföra prissättningsmodeller för optioner. Vi har undersökt de konstanta volatilitetsmodellerna Black-Scholes-Merton (BSM) och Merton’s Jump Diffusion (MJD) samt de stokastiska volatilitetsmodellerna Heston och Bates. Datat vi använt är optionspriser från Microsoft, Advanced Micro Devices Inc, Walt Disney Company och S&P 500 indexet. Datat delades upp i en träningsmängd och en test- mängd. Träningsdatat användes för parameterkalibrering med hänsyn till varje modell. Testdatat användes för att jämföra modellpriser med priser som observerats på mark- naden. Parameterkalibreringen för varje modell utfördes genom att använda den icke- linjära minsta-kvadratmetoden. Med hjälp av de kalibrerade parametrarna kunde priset räknas ut genom att använda Carr och Madan-metoden. Vi kunde se att de stokastiska volatilitetsmodellerna, Heston och Bates, replikerade marknadens optionspriser bättre än båda de konstanta volatilitetsmodellerna, MJD och BSM för de flesta dataseten. Medelvärdet av det relativa medelvärdesfelet i procent för Heston och Bates beräknades till 2.26% respektive 2.17%. För Merton och BSM beräknades medelvärdet av det relativa medelvärdesfelet i procent till 6.90% respektive 5.45%. Vi anser därför att en stokastisk volatilitetsmodell är att föredra framför en konstant volatilitetsmodell för att prissätta optioner.