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## Black-Scholes model with time-varying coefficients

In this section, we explain how our method can be applied in practice through the simple example of the Black-Scholes model with time-varying coefficients. So, we assume the underlying risky asset follows the dynamics dYt = Yt(btdt + ‡tdWt), where bt and ‡t are continuous deterministic functions. We also assume bt and ‡t do not vanish. Using the theory of linear-quadratic optimal control, we give an explicit solution for the problem of designing optimal rebalancing times in this specific setting.

### Asymptotically optimal sequence

Here, we construct a sequence of continuous trading strategies (bún, cún) which can be considered

essentially asymptotically optimal. This is not necessarily asymptotically optimal in the precise sense of Definition 2.3.1 because each (bún, cún) is not necessarily satisfying the technical conditions in Section 2.2.2. It means that the strategy associated with (bún, cún) is not well-defined in general. However in the proof of Theorem 2.3.2 in Section 4.5, we show that there exists a sequence (bn, cn) oe Sa which is an approximation to (bún, cún) in a suitable sense and in fact asymptotically optimal. Therefore, from practical point of view, one may use (bún, cún) defined below as an optimal strategy.

#### Implementation and numerical experiments

Here we describe the implementation of our new results in the specific case that the payoff is convex and the benchmark is given by a constant enlargement of volatility. Let f be convex and q = p– be the solution of (2.1.1). Consider the case ” © 0. By (2.2.5) and (2.3.2), we have a = –⁄ˆ2 s p–. Hence “(s, t) = 1 – a(s, t). The situation can be divided into three cases: 0 < – Æ 1, 1 < – < 2 and – Ø 2. The case where 0 < – Æ 1 corresponds to Fig. 2.4d. The optimal strategy X is essentially given by uniformly on [0, T] in probability as Ÿ æ 0. This means that the hedging error under (2.3.17) is of a smaller order than Ÿ while, as explained above, Leland’s strategy has a nondegenerate limit of order Ÿ. The strategy (2.3.17) can be implemented as, say, Xt = Yj for t oe (j”, (j + 1)”].

**Tracking with combined regular and impulse control**

Instead of giving directly a general result, which would lead to a cumbersome presentation, we focus in this section on the tracking problem with combined regular and impulse control. This allows us to illustrate our key ideas. Other situations, such as singular control, are discussed in Section 3.3. In the case of combined regular and impulse control, a tracking strategy (u, ·, ›) is given by a progressively measurable process u = (ut)tØ0 with values in Rd and (·, ›) = {(·j, ›j), j oe Nú}, with (·j) an increasing sequence of stopping times and (›j) a sequence of F·j -measurable random variables with values in Rd. The process (ut) represents the speed of the agent. The stopping time ·j represents the timing of j-th jump towards the target and ›j the size of the jump. The tracking error obtained by following the strategy (u, ·, ›) is given by Xt = ≠X¶t + ⁄ t 0 usds + ÿ 0<·jÆt ›j .

**Table of contents :**

**INTRODUCTION **

**Part I : Option pricing and hedging **

1.1 Discrete hedging with directional views

1.2 Option replication with modified volatility

**Part II : Asymptotic optimal tracking **

2.1 Formulation of the tracking problem

2.2 Asymptotic framework

2.3 Main results

2.4 Relation with other asymptotic studies

**Part III : Portfolio selection with capital gains taxes **

3.1 Preliminary : the model of [BST10]

3.2 Expansion around tax-deflated model

3.3 Capital gains taxes with recursive utility and regime-switching

3.4 Joint impact of capital gains taxes and transaction costs

**I OPTION PRICING AND HEDGING **

**1 Discrete hedging with directional views **

1.1 Introduction

1.2 Assumptions and admissible strategies

1.2.1 Assumptions on the dynamics and admissibility conditions

1.2.2 Comments on the admissibility conditions

1.2.3 Examples of admissible discretization rules

1.3 Asymptotic optimality: a preliminary approach

1.4 Asymptotic expectation-error optimization

1.5 Black-Scholes model with time-varying coefficients

1.5.1 Explicit formulas

1.5.2 Numerical study

Appendix 1.A Proofs

1.A.1 Proof of Proposition 1.2.1

1.A.2 Proof of Proposition 1.2.2

1.A.3 Proof of Proposition 1.2.3

1.A.4 Proof of Theorem 1.4.1

Appendix 1.B Linear-quadratic optimal control

**2 Option pricing with modified volatility and proportional costs **

2.1 Introduction

2.2 Framework

2.2.1 Benchmark strategy

2.2.2 Continuous control strategies

2.2.3 Hedging error

2.3 Main results

2.3.1 Limit theorem of hedging error

2.3.2 Minimum conditional variance

2.3.3 Asymptotically optimal sequence

2.3.4 Implementation and numerical experiments

2.4 Heuristic derivation

2.4.1 Local probability model

2.4.2 Relation with Leland’s strategy

2.4.3 Relation with Whalley and Wilmott’s strategy

Appendix 2.A Proof of Theorem 2.3.1

Appendix 2.B Proof of Theorem 2.3.2

**II ASYMPTOTIC OPTIMAL TRAKCING **

**3 Asymptotic lower bounds **

3.1 Introduction

3.2 Tracking with combined regular and impulse control

3.2.1 Asymptotic framework

3.2.2 Lower bound

3.3 Extensions of Theorem 3.2.1 to other types of control

3.3.1 Combined regular and singular control

3.3.2 Impulse control

3.3.3 Singular control

3.3.4 Regular control

3.4 Interpretation of lower bounds and examples

3.4.1 Martingale problem associated to controlled Brownian motion

3.4.2 Time-average control of Brownian motion

3.4.3 Explicit examples in dimension one

3.5 Proof of Theorem 3.2.1

3.5.1 Reduction to local time-average control problem

3.5.2 Proof of Theorem 3.2.1

3.5.3 Proof of Lemma 3.5.3

3.6 Proof of Theorem 3.3.1

3.7 Proof of Propositions 3.4.1-3.4.5

3.7.1 Verification theorem in Rd

3.7.2 Verification of Proposition 3.4.4

Appendix 3.A Kummer confluent hypergeometric function 1F1

Appendix 3.B Proof of Lemma 3.7.2

Appendix 3.C Tightness function

Appendix 3.D Convergence in probability, stable convergence

**4 Feedback strategies and applications **

4.1 Introduction

4.2 Combined regular and impulse control

4.2.1 Feedback strategies

4.2.2 Asymptotic performance

4.3 Extensions to other types of control

4.3.1 Combined regular and singular control

4.3.2 When only one control is present

4.4 Applications

4.4.1 Explicit optimal strategies in dimension one

4.4.2 Discretization of hedging strategies

4.4.3 Impacts of small market frictions

4.5 Proofs

4.5.1 Proof of Theorem 4.2.1

4.5.2 Proof of Theorem 4.3.1

4.5.3 Proof of Theorem 4.3.4

Appendix 4.A Convergence of integral functionals

Appendix 4.B Separability of (A,B)

4.B.1 Diffusion with Rebirth

4.B.2 Diffusion with Reflection

**III PORTFOLIO SELECTION WITH CAPTIAL GAINS TAXES **

**5 Recursive utility and regime switching**

5.1 Introduction

5.2 Model formulation

5.3 Preliminary analysis

5.3.1 First properties of the value function

5.3.2 Certainty equivalent wealth loss (CEWL)

5.3.3 The value of deferring capital gains realization

5.3.4 HJB equation

5.3.5 Trading and no-trading regions

5.4 Asymptotic analysis

5.4.1 Small parameter

5.4.2 Asymptotic expansion

5.4.3 Economic analysis: single-regime case

5.4.4 Economic analysis: two-regime case

5.5 Numerical results

5.5.1 Single-regime case

5.5.2 Two-regime Case

5.6 Why is the order of deferral value O(Á8/3)?

5.6.1 Expansion around the tax-deflated model

5.6.2 Strategies based on barriers and reduced model

5.6.3 Solution of the optimal control problem (5.6.17)-(5.6.18)

Appendix 5.A Formal derivation of Asymptotic Expansion 5.4.1

5.A.1 Expansion at solution of tax-defalted model

5.A.2 An approximation to the optimal consumption strategy

**6 Joint impact of capital gains taxes and transaction costs **

6.1 Main result

6.1.1 HJB equation

6.1.2 Benchmark model

6.1.3 Asymptotic expansion

6.2 Probabilistic interpretation

6.3 Homothetic case

6.4 Asymptotic development

6.4.1 Fast variables

6.4.2 Corrector equations

**Bibliography**