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## Option pricing and hedging using asymmetric risk measure: Asymptotic optimality

In finance, the pricing and hedging of contingent claims are major concerns. In theoretical point of view, those problems is well established (see [KS98], for instance). In practice, traders hedge their contracts by trading only at discrete times, say t0 “ 0 ă t1 ă . . . ă tN “ T, yielding a residual risk. Here, they intend to hedge the claim HT at time T using d hedging securities whose price are denoted by X “ pXp1q, . . . ,Xpdqq. For that, we define the local risk En associated with the trading times tn and tn`1 given by.where V denotes the valuation process; ϑ “ pϑp1q, . . . , ϑpdqq denotes the hedging process with ϑpiq being the number of shares invested in the i-th hedging instrument. We aim to find the valuation/hedging rules pV, ϑq minimizing this residual risk using a risk function ℓ. From the existing results (for instance, [Sch99], for the quadratic local risk minimization), we choose a function ℓ penalizing profits (En ă 0) and losses (En ą 0) asymmetrically. In this context, we study the integrated local risk RN under the form.

### Asymptotic asymmetric risk measure: Application to physical asset valuation

In Chapter 2, we provide valuation and hedging policies for future incomes due to the production of power plants using an asymmetric risk valuation. Since the deregulation of energy markets, several spot and future markets were created to exchange electricity. After that, power plant owners start to face the problem of evaluating their plant production, which depends on the electricity spot price in the future.

In practice, power plant owners perform a discrete hedging at trading times, t0 “ 0 ă t1 ă . . . ă tN “ T using a hedging security X in order to reduce the risk associated to a random income HT at a future time T. This discrete hedging produces a residual risk.ϑ denotes the number of shares invested in the hedging security;

• X is a forward contract Fp¨, Tq with delivery time T. Then, they search to find the valuation/hedging rule taking into account the local balance En

penalizing En ă 0) through an asymmetric risk function ℓ: where V stands for the valuation process under the replication constraint VT “ HT . Notice that plant owners:

• want to obtain a residual risk E with a small standard-deviation;

• prefer profitable scenarios where En ą 0.

In reality, power plant generates some fixed costs whether it is producing electricity or not. Those costs are called here fixed costs:

• they include the investment and depreciation costs;

• they exclude the fuel costs (coal, gas, uranium).

To record in their balance book, power producers are constrained to find today an equivalent price to the future income generated by their power plant. In fact, they do not face any financial risk due to this income. On the other hand, because they can not increase their income by influencing the electricity spot price S, they prefer to obtain a certain value due to the selling of their future production instead to receive the random positive amount HT “ gpST q.

Economically speaking, when the electricity spot price S is high, we require the power plant to face a higher demand. In indeed, a high price is produced by an increasing demand. Therefore, we increase the production by starting more often the power plant. That is the reason why we assume a dependence of the costs c on the electricity spot price S.

#### A non-intrusive stratified resampler for multi-factor models: Application in energy market

Stochastic dynamic programming equations are related to the resolution of non-linear problems (stochastic controls or nonlinear PDEs) arising in almost all areas of science, from water reservoir management to finance. In Chapter 4, we aim to solve dynamic programming equations (DPE) related to a financial valuation in energy market. Here, we are concerned, for example, in the pricing of Bermudan or Swing options, where the underlying assets are forward contracts. Then, we plan to develop a non-intrusive algorithm to solve those DPE using only the observable data without a full model calibration.

**Table of contents :**

Introduction générale

General Introduction

**I Asymmetric risk measure for pricing and hedging of options **

**1 Option pricing and hedging using asymmetric risk measure: Asymptotic optimality **

1.1 Introduction

1.2 Notations and assumptions

1.3 Asymptotic risk: Existence of the limit RN,γ

1.3.1 Asymptotic optimality through the PDE nonlinearity

1.3.2 Optimal PDE nonlinearity: Study in dimension 1

1.4 Asymptotic risk: Proof of the main result

1.4.1 Rescaling and conditioning

1.4.2 Stochastic expansion and approximation of sensitivities

1.4.3 Aggregation and passage to the limit

1.4.4 Proof of the stochastic expansion in Subsection 1.4.2.

1.4.5 Proof of the almost sure convergence in Subsection 1.4.3

1.5 Numerical results

1.5.1 Optimal PDE valuation of European options

1.5.2 Asymptotic risk: Dependence on the PDE nonlinearity

1.5.3 Optimal PDE valuation/hedging: Comparison with the discrete-time solution

1.6 Appendix

1.6.1 Technical results about Section 1.4

1.6.2 Proof of the convergence of the sensibilities

**2 Asymptotic asymmetric risk measure: Application to physical asset valuation **

2.1 Introduction

2.2 Setting

2.2.1 An asymmetric risk valuation with cost management

2.2.2 Main assumptions

2.2.3 Main results

2.2.4 Asymptotic risk with cost management

2.2.5 Optimal PDE non-linearity

2.3 Existence of the asymptotic risk.

2.4 Convex-concave staggered cost models

2.4.1 Increasing costs

2.4.2 Decreasing costs

2.5 Numerical experiments

2.5.1 Optimal valuation/hedging rule for increasing costs

2.5.2 Optimal valuation/hedging rule for decreasing costs

2.6 Conclusions and extensions

**II Numerical methods in stochastic control **

**3 Polynomial conditional McKean-Vlasov control problems: Some probabilistic numerical methods **

3.1 Introduction

3.2 Notations and Assumptions

3.2.1 Main assumptions

3.2.2 Markovian embedding

3.3 Probabilistic numerical methods

3.3.1 Value and Performance iteration

3.3.2 Approximation of conditional expectations

Regression Monte Carlo

Quantization

3.3.3 Training points design

Exploitation only strategy

Explore first, exploit later

3.3.4 Optimal control searching

Low cardinality control set

High cardinality/continuous control space

3.3.5 Upper and lower bounds

3.3.6 Pseudo-codes

Pseudo-code for a Regress-Later-based algorithm

Pseudo-code for a Control Randomization-based algorithm

Pseudo-code for a Quantization-based algorithm

3.4 Applications

3.4.1 Portfolio Optimization under drift uncertainty

3.4.2 Interbank systemic risk with partial observation

3.5 Numerical results

3.5.1 Portfolio Optimization

3.5.2 Interbank systemic risk

3.6 Discussion

**4 A non-intrusive stratified resampler for multi-factor models: Application in energy market **

4.1 Introduction

4.2 Setting

4.2.1 Dynamic programming equation

4.2.2 Modeling of the observable and hidden processes

4.2.3 Non-Intrusive resampling

4.3 The NISR scheme

4.3.1 Extraction

4.3.2 Stratification

4.3.3 NISR-regression Monte Carlo algorithm

4.4 Applications to Energy Market

4.4.1 Bermudan options

4.4.2 Swing options

**Bibliography**