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Essays on Pricing Derivatives by taking into account Volatility and Interest Rates Risks

donderdag, 13 september, 2012 - 16:00
Salle Solvay, Bâtiment NO
Campus Plaine, Université Libre de Bruxelles
Gregory Rayée

In Chapter 1, we present a new approach to evaluate barrier type options based on a method
known as the Vanna-Volga method. This new approach allows for a fast and easy calibration
which is directly done on the barrier options market. It allows to price these options with a tool
in accordance with the barrier options market. We also compare our results with those coming
from the Dupire and Heston models. Furthermore, we study the sensitivity of the Vanna-Volga
method with respect to the market data. We give a new theoretical justification for the Vanna-
Volga method. More precisely, we show that the Vanna-Volga option's price can be seen as a
first-order Taylor expansion of the Black-Scholes option price around the at-the-money

In Chapter 2, we study a model able to capture the market implied volatility effects and which
also takes into account the market variability of the interest rates. This relaxes the assumption of
constant interest rates present in the Black-Scholes model and solves a second main problem
encountered in the latter, which can have large consequences in the valuation and hedging
strategies especially for long maturity products. More precisely, we work in the foreign exchange
(FX) market, with a local volatility model for the dynamics of the foreign exchange spot rates in
which the domestic and foreign interest rates are also assumed stochastic. We derive the
expression of the local volatility and various results particularly useful for the calibration of the
model. Finally, we derive useful results for the calibration of hybrid volatility models where the
volatility of the FX spot rate is a mix of a local volatility and a stochastic volatility and we
develop a calibration method for this model.

In Chapter 3, we apply the local volatility model with stochastic interest rates developed in the
previous chapter to the pricing of life insurance derivatives. Since the maturity of such options is
the retirement age, they can be considered as long maturity products. For the calibration of the
local volatility, we use a method developed in Chapter 2. Since we study exotic products, we
also compare the prices obtained in different models, namely the local volatility, stochastic
volatility and finally the constant volatility model all combined with stochastic interest rates.
Finally, in Chapter 4 we work with Lévy type models for the underlying dynamics. The idea
underlying the Lévy model is the use of a more general stochastic process than the standard
Brownian motion which allows to be in agreement with the observed market probability
distribution at maturity. In a financial crisis period, this model is especially popular since it has
the particularity to allow for jumps in the dynamics. In this chapter, we are interested specifically
in the evaluation of discretely monitored arithmetic Asian type options whose payoff is based on
the discrete arithmetic mean of the underlying during the life of the option. As for many exotic
options, it is not possible to derive an analytical pricing formula even in the simple case of the
Black-Scholes model. In this case the only way to price such options is by using numerical
methods. In Chapter 4, we develop a method based on Monte-Carlo simulations and we use two
types of control variates to improve the convergence. We also develop a method based on a
conditioning approach to obtain a lower bound for the Asian option price. The efficiency of this
last method outperforms the efficiency of the other methods and the results are relatively close to
the Monte-Carlo value of the corresponding Asian.

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