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## Options on realized variance and density expansion (Chapter 4)

We consider the problem of pricing options on realized variance when volatility is modelled as a diusion process with general drift and constant diusion coecient. Dene for every 2 (0; 1], the joint process (Y t ;Z t )t2[0;T ] of the integrated variance and the instantaneous volatility dY t = g(Z t ) dt dZ t = 2b(Z t ) dt + cdWt .

### Perturbation theory and interest rate derivatives pricing in the Levy Libor model (Chapter 5)

In this chapter, we study the pricing of interest rate derivatives in the Levy Libor market model (LLMM) developed in (Eberlein and Ozkan, 2005) by writing the LLMM as a perturbation of the standard log-normal LMM. Let 0 T0 < ::: < Tn be a tenor structure and denote by L = (L1; : : : ;Ln)> the column vector of the forward Libor rates Lj t := LTj t . We assume that the dynamics of L is given by the SDE dLt = Lt(b(t; Lt)dt + (t)dXt) ; where Xt is a compensated d-dimensional Levy process with non-zero diusive part under the terminal measure QTn, (t) a deterministic n d volatility matrix and b(t; Lt) is the drift vector such that Lj t is an QTj -martingale. Under this model, the price of a European derivative with payo g(LTk) satises.

#### Trajectorial large deviations for ane stochastic volatility model

In this section, we prove a trajectorial LDP for (Xt) when the time horizon is large. Dene, for 2 (0; 1] and 0 t T, the scaling X t = Xt=. We proceed by proving rst a LDP for X t in nite dimension, that we extend, in a second step to the whole trajectory of (X t )0tT .

**Finite-dimensional LDP**

Let = f0 < t1 < ::: < tn = tg, by convention t0 = 0, and dene ; () = log IE h e Pn k=1 kX tk i ; for 2 Rn. We start by formulating our main technical assumption. Assumption 4. One of the following conditions is veried.

1. The interval support of F is J = [u; u+] and w(u) = w(u+).

2. For every u 2 R, ~ w() = 1, i.e, the generalized Riccati equations have only one (stable) equilibrium. The following Lemma gives an intuition on Assumption 4.

**The Wishart stochastic volatility model**

Let (St)t0 be a n-dimensional vector stochastic process with dynamics dSt = Diag(St) r1 dt + a>X1=2 t dZt ; Si0 > 0; i = 1; : : : ; n; (3.2.1) where 1 = (1; :::; 1)>, Diag(St)ij = 1fi=jgSi t , Zt is n-dimensional standard Brownian motion and the stochastic volatility matrix X is a Wishart process with dynamics dXt = (In + bXt + Xtb) dt+X1=2 t dWt+(dWt)>X1=2 t ; X0 = x: (3.2.2) with > n 1, a 2 Mn invertible, b; x 2 S+; n and W is a n n matrix standard Brownian motion independent of Z. Note again that Xt 2 S+ n (Bru, 1991, Prop. 4). Let us also assume that a is such that a>a 2 S+; n . Remark 3.2.1. The model (S;X) dened in (3.2.1) and (3.2.2) is a (quite large) subclass of the one dened in (3.1.1) and (3.1.2). Indeed, dening ~Xt := a>Xt a, we have a>X1=2 t dZt = ~X 1=2 t d ~ Zt, where ~ Zt is another n-dimensional standard Brownian motion.

**Table of contents :**

**1 Introduction **

1.1 Option pricing

1.1.1 The Black and Scholes model

1.1.2 Beyond the BS model

1.1.3 Non-equity derivatives

1.1.4 Pricing options with asymptotic methods

1.2 Summary of the thesis

1.3 The main results of the thesis

1.3.1 Ane stochastic volatility models, large deviations and optimal sampling (Chapters 2 and 3)

1.3.2 Options on realized variance and density expansion (Chapter 4)

1.3.3 Perturbation theory and interest rate derivatives pricing in the Levy Libor model (Chapter 5)

**2 Pathwise large deviations for ane stochastic volatility models **

2.1 Introduction

2.2 Model description

2.3 Large deviations theory

2.4 Trajectorial large deviations for ane stochastic volatility model

2.4.1 Finite-dimensional LDP

2.4.2 Innite-dimensional LDP

2.5 Variance reduction

2.6 Numerical examples

2.6.1 European and Asian put options in the Heston model .

2.6.2 European put on the Heston model with negative exponential jumps

**3 Large deviations for Wishart stochastic volatility model **

3.1 Introduction

3.2 The Wishart stochastic volatility model

3.3 Long-time large deviations for the Wishart volatility model

3.3.1 Reminder of large deviations theory

3.3.2 Long-time behaviour of the Laplace transform of the log-price

3.3.3 Long-time large deviation principle for the log-price process

3.4 Asymptotic implied volatility of basket options

3.4.1 Asymptotic price for the Wishart model

3.4.2 Implied volatility asymptotics

3.5 Variance reduction

3.5.1 The general variance reduction problem

3.5.2 Asymptotic variance reduction

3.6 Numerical results

3.6.1 Long-time implied volatility

3.6.2 Variance reduction

**4 An asymptotic approach for the pricing of options on realized variance **

4.1 Introduction

4.2 Expansion of marginal densities

4.3 Denition and properties of the integrated variance process .

4.3.1 The integrated variance process

4.3.2 Hamiltonian equations and optimal control

4.3.3 Derivatives of the energy

4.4 Asymptotic expansion of the density

4.5 Pricing options on realized variance

4.5.1 Asymptotics for the price of options on realized variance

4.5.2 Asymptotics for the Black and Scholes implied volatility of options on realized variance

**5 Approximate option pricing in the Levy Libor model **

5.1 Introduction

5.2 Presentation of the model

5.2.1 The driving process

5.2.2 The model

5.3 Option pricing via PIDEs

5.3.1 General payo

5.3.2 Caplet

5.3.3 Swaptions

5.4 Approximate pricing

5.4.1 Approximate pricing for general payos under the terminal measure

5.4.2 Approximate pricing of caplets

5.4.3 Approximate pricing of swaptions

5.5 Numerical examples