U.U.D.M. Project Report 2009:5 Optimal Liquidation of a Mispriced Option Spread Bing Lu Examensarbete i matematik, 30 hp Handledare och examinator: Erik Ekström Mars 2009 Department of Mathematics Uppsala University Optimal Liquidation of a Mispriced Option Spread Bing Lu Master's Thesis at the Department of Mathematics Uppsala University Supervisor: Erik EkstromÄ Abstract Assuming an overestimated implied volatility, the optimal liquidation strategy of a digital option is investigated in this thesis, which extends the paper Optimal Liquidation of an Option Spread by utilizing more realistic models for the underlying asset. Five models for the price of the underlying asset are set up by GBM and the Bachelier model with added liquidation cost, interest rate and positive drift. Additionally, the optimal exercising strategy of a bull call spread with the Bachelier model for the underlying security is also studied. For the six di®erent cases above, the values for the options are formulated as optimal stopping problems which are transformed into free boundary problems. For the free boundary problems with in¯nite time horizon, the values of the options are solved analytically and the optimal strike prices are chosen. We also analyze the dependence of the size of the continuation region on some paramters. For the free boundary problems with ¯nite time horizon that can only be solved numerically, the free boundaries are drawn by ¯nite di®erent method and the free boundary equations are derived. Contents 1 Introduction 4 1.1 Overview . 4 1.2 Volatility . 6 1.3 Vertical Spread . 7 1.4 Digital Option . 8 2 Review of Optimal Stopping and Free Boundary Problems 9 2.1 Optimal Stopping Problem for Continuous Time . 9 2.2 Transformation From Optimal Stopping Problem to Free Bound- ary Problem . 12 3 Adding Liquidation Cost 14 3.1 The Optimal Liquidation Problem . 14 3.2 The Optimal Choice of Strike Price . 18 3.3 The Dependence of Size of Continuation Region on Parameters 19 4 Adding Interest Rate 22 4.1 The Optimal Liquidation Problem . 22 4.2 The Optimal Choice of Strike Price . 26 5 Adding Liquidation Cost and Interest Rate 27 5.1 The Optimal Liquidation Problem . 27 5.2 The Optimal Choice of Strike Price . 29 5.3 Comparison of the Free Boundaries . 30 6 Adding Drift 33 6.1 Finite Di®erence Method . 33 6.2 The Optimal Liquidation Problem . 35 6.3 Free Boundary Equation . 36 1 7 Geometric Brownian Motion 39 7.1 The Optimal Liquidation Problem . 39 7.2 Free Boundary Equation . 41 8 The Optimal Liquidation of A Bull Call Spread 44 9 Conclusion 47 Appendices 49 A Matlab Source Code 50 2 List of Figures 1.1 The payo® function of a digital option with overestimated volatility before maturity . 7 1.2 The payo® function of a digital option and a bull spread at maturity . 8 2.1 The superharmonic function V with the obstacle function G . 13 3.1 The dependence of B on ¯ .................... 20 3.2 The dependence of B on " .................... 21 5.1 The free boundaries in the four cases . 31 6.1 The free boundary with the Bachelier model with added drift 37 7.1 The free boundary with GBM for the underlying asset . 41 8.1 The payo® function of a bull call spread . 45 8.2 The free boundary of a bull call spread with the Bachelier model for the underlying asset . 46 3 Chapter 1 Introduction 1.1 Overview Regarded as the most crucial element in option pricing, volatility is well- known to be di±cult 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 pro¯t by taking advantage of a misvalued future volatility is to invest in a vertical spread, which is trading strategy involving a simultaneous purchase and sale of same type of two options with the same maturity but di®erent 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]. De¯ne a digital option with strike price K paying g(XT ) at terminal time T , where ½ 1 if x ¸ K g(x) = 0 if x < K The digital option closely resembles the bull call spread, which is long in n numbers of call options with strike K and short in n numbers of call with strike K + 1=n as n is large. Therefore, for simplicity, the digital option which can approximately hedge all kinds of vertical spreads is utilized as the contract function. When the underlying price stays above the strike K, with the assump- tion of an overestimated implied volatility, the market price of the digital option is too low (refer to Figure 1.1). Thus the investor would like to hold the option until the terminal time T . What if the underlying price 4 drop o® dramatically below strike K? The con¯dence of the holders for the price soaring back to K will disappear. Therefore, we predict there is an enduring limit under which the investors would give up keeping the posi- tion and liquidate it before maturity. Also, the enduring limit, denoted as free boundary, optimal stopping boundary or optimal exercising boundary is time dependent. One of the critical missions of this thesis is to study the free boundary of the option. As we have already set up a digital option as the contract function, the other essential element is the model of the price of the underlying asset. In paper [2], the Bachelier model is used as the price of the underlying security and the interest rate is assumed to be zero. As an extension of [2], this thesis will utilize more complex models for the underlying security and add liquidation cost for the earlier exercising. There are ¯ve cases we are interested in. 1. The Bachelier model with liquidation cost. 2. The Bachelier model with interest rate. 3. The Bachelier model with liquidation cost and interest rate. 4. The Bachelier model with added positive drift. 5. Geometric Brownian Motion. Except the ¯ve models above, we will take a look at the case that a bull call spread is provided as contract and the Bachelier model as price of the underlying asset. The ¯rst three cases can be studied analytically and solved explicitly, and the last three are mainly studied numerically. The procedures of dealing with the ¯rst three models are quite similar. First, formulating the value of the option as an optimal stopping problem in ¯nite time horizon; Second, converting the optimal stopping problem in ¯nite time horizon into an op- timal stopping problem in in¯nite time horizon by time and space change. Third, transforming the optimal stopping problem in in¯nite horizon into a free boundary problem with in¯nite horizon and solving it explicitly. Since the transformation from optimal stopping problem to free boundary prob- lem plays an extremely important role in the study, the theories in that ¯eld are reviewed in the next chapter. Additionally, the relations between the size of the continuation region and some parameters are investigated. To maximize the expected pro¯t, the optimal strike price is chosen. We also compare and analyze the free boundaries in the di®erent cases. For the last three models, the values of the options are formulated as the optimal stopping problems in ¯nite time horizon, then the optimal stop- ping problems are reduced into the free boundary problems with ¯nite time horizon. It seems impossible to solve them analytically, so we study them numerically using the ¯nite di®erence method. The free boundaries are illus- 5 trated in Figure 5.1 with comparison to the free boundary in paper [2]. The free boundary equations are derived for the cases of GBM and the Bachelier model with added drift. First, let's start with volatility. 1.2 Volatility In ¯nance, volatility is generally de¯ned as the standard deviation of the continuously compounded returns of a ¯nancial security or market index. A bigger volatility indicates the price of the instrument can increase or decrease dramatically in a short period. On the other hand, a smaller volatility implies the security's price can not change dramatically but varies smoothly during a period of time. Volatility does not imply directions. Historical volatility is calculated using the historical returns. Implied volatility is the volatility set by the market price of the derivative based on an option pricing model. Generally, options with the same underlying asset but di®erent maturity and strike prices will yield di®erent implied volatility. Given the known parameters such as stock price, current time, expiration time, interest rate and strike price, implied volatility can be derived from a model such as the Black-Scholes model. People usually believe that the bullish market is less risky than the bearish market. Hence, the implied volatility is relatively smaller in the bullish market and relatively larger in the bearish market. Due to incomplete information and imperfect market, the implied volatil- ity can hardly agree exactly with the future realized volatility. The di®erence between the two can be taken advantage to make pro¯t by investing in some ¯nancial derivatives. There are some options and variance swaps regarding volatility as the trading asset.