Regarded as the most crucial element in **option pricing**, volatility is well-known to be diﬃcult to estimate accurately. In the real world, by the reason of incomplete information and imperfect market, the implied volatility of an asset can rarely coincide with its true volatility. It is usually assumed that an implied volatility is overestimated. One important way to make proﬁt by taking advantage of a misvalued future volatility is to invest in a vertical spread, which is trading strategy involving a simultaneous purchase an sale of same type of two options with the same maturity but diﬀerent strike prices. A vertical spread can be constructed by both put and call option due to the put-call parity. This thesis is based on the paper [2] studying the optimal liquidation strategy of a digital option with the Bachelier model for the underlying asset. It is also an extension of paper [2].

*Contents*

1 Introduction

1.1 Overview

1.2 Volatility

1.3 Vertical Spread

1.4 Digital Option

2 Review of Optimal Stopping and Free Boundary Problems

2.1 Optimal Stopping Problem for Continuous Time

2.2 Transformation From Optimal Stopping Problem to Free Boundary Problem

3 Adding **Liquidation Cost**

3.1 The Optimal Liquidation Problem

3.2 The Optimal Choice of Strike Price

3.3 The Dependence of Size of Continuation Region on Parameters

4 Adding Interest Rate

4.1 The Optimal Liquidation Problem

4.2 The Optimal Choice of Strike Price

5 Adding Liquidation Cost and Interest Rate

5.1 The Optimal Liquidation Problem

5.2 The Optimal Choice of Strike Price

5.3 Comparison of the Free Boundaries

6 Adding Drift

6.1 Finite Difference Method

6.2 The Optimal Liquidation Problem

6.3 Free Boundary Equation

7 Geometric Brownian Motion

7.1 The Optimal Liquidation Problem

7.2 Free Boundary Equation

8 The Optimal Liquidation of A **Bull Call Spread**

9 Conclusion

Appendices

A Matlab Source Code

Author: Lu, Bing

Source: Uppsala University Library

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