EARLY EXERCISE OPTIONS WITH DISCONTINUOUS PAYOFF

Transcription

1 EARLY EXERCISE OPTIONS WITH DISCONTINUOUS PAYOFF A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 217 Min Gao Candidate School of Mathematics

2 Contents Abstract 1 Declaration 11 Copyright Statement 12 Acknowledgements 13 1 Introduction 14 2 The American Binary Options Introduction About the optimal stopping problem Cash-or-nothing options Asset-or-nothing options Summary The British Asset-or-Nothing Put Option Introduction Basic motivation for the British asset-or-nothing put option The British asset-or-nothing put option: Definition and basic properties The arbitrage-free price and the rational exercise boundary Financial analysis of the British asset-or-nothing put option The British Binary Put Options Introduction Basic motivation for the British binary option

3 4.3 Definition and basic properties The arbitrage-free price and the rational exercise boundary Financial analysis of the British cash-or-nothing put option The British Binary Call Options Introduction Basic motivation for the British binary call options The British cash-or-nothing call option The arbitrage-free price and the rational exercise boundary Financial analysis of the British cash-or-nothing call option The British asset-or-nothing call option The arbitrage-free price and the rational exercise boundary Comparison of the British cash-or-nothing and asset-or-nothing call options The Barrier Binary Options Introduction Preliminaries The American knock-in binary option The American knock-out binary options The American knock-out asset-or-nothing options Financial analysis of the American barrier binary option Bibliography 112 Word count xxxxx 3

4 List of Tables 3.1 Returns observed upon exercising the British asset-or-nothing put options (with µ c =.13 and µ c =.2) above the strike price K. The returns are calculated by R(t, x) = G µc (t, x)/v (, 11). The parameter set is same as in Figure 3.3 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is Returns observed upon exercising the British asset-or-nothing put options (with µ c =.2) at and below the strike price K. The returns are calculated by R(t, x) = G µc (t, x)/v (, 11) and R A (t, x) = x/v A (, 11) respectively. The parameter set is same as in Figure 3.3 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is Returns observed upon exercising the British asset-or-nothing put options (with µ c =.2) above the strike price K compared with returns received upon selling the American asset-or-nothing put option in the same contingency. The returns are calculated by R B (t, x) = G µc (t, x)/v (, 11) and R A (t, x) = V A (t, x)/v A (, 11). The parameter set is same as in Figure 3.3 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is Returns observed upon exercising the British cash-or-nothing put option at and above the strike price K. The returns are calculated by R(t, x) = G µc (t, x)/v (, K). The parameter set is same as in Figure 4.2 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is K

5 4.2 Returns observed upon exercising the British cash-or-nothing put option (with µ c =.13) at and below the strike price K compared with the American cash-or-nothing put option in the same contingency. The returns are calculated by R(t, x) = G µc (t, x)/v (, 11) and R A (t, x) = G A (t, x)/v A (, 11) respectively. The parameter set is same as in Figure 4.2 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is Returns observed upon exercising the British cash-or-nothing put option (with µ c =.13) at and above K compared with selling the American cash-or-nothing put option in the same contingency. The returns are calculated by R(t, x) = G µc (t, x)/v (, 11) and R A (t, x) = V A (t, x)/v A (, 11) respectively. The parameter set is same as in Figure 4.2 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is Returns observed upon exercising the British cash-or-nothing call option at and below the strike price K.The returns are calculated by R(t, x) = G µc (t, x)/v (, K). The parameter set is same as in Figure 5.2 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is K Returns observed upon exercising the British cash-or-nothing call option (with µ c =.5) above the strike price K compared with returns received upon selling the European cash-or-nothing call option in the same contingency. The returns are calculated by R(t, x) = G µc (t, x)/v (, K) and R E (t, x) = V E (t, x)/v E (, K) respectively. The parameter set is same as in Figure 5.2 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is K Returns observed upon exercising the British cash-or-nothing call option (with µ c =.5) at and below the strike price K compared with returns received upon selling the European cash-or-nothing call option in the same contingency. The returns are calculated by R(t, x) = G µc (t, x)/v (, K) and R E (t, x) = V E (t, x)/v E (, K) respectively. The parameter set is same as in Figure 5.2 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is K

6 5.4 Returns observed upon exercising the British binary call options (with µ c =.5) above the strike price K. The returns are calculated by R(t, x) = G µc (t, x)/v (, K). The parameter set is same as in Figure 5.5 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is K Returns observed upon exercising the British binary call options (with µ c =.5) at and below the strike price K. The returns are calculated by R(t, x) = G µc (t, x)/v (, K). The parameter set is same as in Figure 5.5 i.e. K = 1, T = 1, r =.1, σ =.4 and the initial stock price is K. 92 6

7 List of Figures 2.1 The payoffs and values of the European cash- and asset-or-nothing calls and puts with parameters K = 1, r =.5, =.2, T = 1 and t = The payoffs of Gap options with different parameters The payoff of a supershare option A Comparison of the values of the European and American binary options for t given and fixed with the parameters K = 1, r =.5, σ =.2 and T = A computer drawing showing how the rational exercise boundary of the British asset-or-nothing put option changes as one varies the contract drift with µ c > r A computer drawing showing how the rational exercise boundary of the British asset-or-nothing put option changes as one varies the contract drift with µ c < A computer drawing comparing the rational exercise boundary of the British asset-or-nothing put option with K = 1, T = 1, r =.1, σ =.4 when the contract drift µ c =.13 and µ c = A computer comparison of the value of the British/American asset-ornothing put options for t given and fixed A computer drawing showing how the rational exercise boundary of the British cash-or-nothing put option changes as one varies the contract drift A computer drawing showing the rational exercise boundary of the British cash-or-nothing put option with K = 1, T = 1, r =.1, σ =.4 when µ c =.13 and µ c =

8 5.1 A computer drawing showing how the rational exercise boundary of the British cash-or-nothing call option changes as one varies the contract drift A computer drawing showing the rational exercise boundary of the British cash-or-nothing call option with K = 1, T = 1, r =.1, σ =.4 when µ c =.3 and µ c = A computer drawing showing the rational exercise boundary of the British cash-or-nothing call option with K = 1, T = 1, r =.1, µ c =.5 when σ =.2 and σ = A computer drawing showing how the rational exercise boundary of the British asset-or-nothing call option changes as one varies the contract drift A computer drawing comparing the rational exercise boundary of the British cash-or-nothing call option and asset-or-nothing call option A computer comparison of the values of the European barrier cash-ornothing call(cnc) and asset-or-nothing call(anc) options for t given and fixed A computer comparison of the values of the European barrier cash-ornothing put (CNP) and asset-or-nothing put (ANP) options for t given and fixed A computer comparison for the values of the European and the American down-in cash-or-nothing call options with parameters r =.1, σ =.4, K = 1, T = 1, Barrier = 6 and t = A computer comparison for the values of the European and the American up-in cash-or-nothing put options with parameters r =.1, σ =.4, K = 1, T = 1, Barrier = 15 and t = A computer comparison for the values of the European and the American up-out cash-or-nothing put options with parameters r =.5, σ =.1, K = 1, T = 1, L = 15 and t = A computer comparison for the values of the European and the American up-out asset-or-nothing put options with parameters r =.5, σ =.1, K = 1, T = 1, L = 15 and t =

9 6.7 A computer comparison for the values of the European and the American down-out cash-or-nothing call options with parameters r =.5, σ =.1, K = 1, T = 1, L = 6 and t = A computer comparison for the values of the European and the American down-out asset-or-nothing call options with parameters r =.5, σ =.1, K = 1, T = 1, L = 6 and t =

10 The University of Manchester Min Gao Candidate Doctor of Philosophy Early Exercise Options With Discontinuous Payoff December 5, 217 The main contribution of this thesis is to examine binary options within the British payoff mechanism introduced by Peskir and Samee [45]. This includes British cashor-nothing put, British asset-or-nothing put, British binary call and American barrier binary options. We assume the geometric Brownian motion model and reduce the optimal stopping problems to free-boundary problems under the Markovian nature of the underlying process. With the help of the local time-space formula on curves [41], we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a non-linear integral equation. We begin by investigating the binary options of American-type which are also called one-touch binary options. Then we move on to examine the British binary options. Chapter 2 reviews the existing work on all different types of the binary options and sets the background for the British binary options. We price and analyse the Americantype (one-touch) binary options using the risk-neutral probability method. In Chapters 3 4 and 5, we present the British binary options where the holder enjoys the early exercise feature of American binary options whereupon his payoff is the best prediction of the European binary options payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, if the option holder finds that the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British binary option is the protection feature as not only can the option holder exercise at unfavourable stock price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. Chapters 3 and 4 focus on the British binary put options and Chapter 5 on call options. We also analyse the financial meaning of the British binary options and show that with the contract drift properly selected the British binary options become very attractive alternatives to the classic European/American options. Chapter 6 extends the binary options into barrier binary options and discusses the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we examine the method in [7] in the application to the knock-in binary options. For the American knock-out binary options, the smoothfit property does not hold when we apply the local time-space formula on curves. We transfer the expectation of the local time term into a computational form under the basic properties of Brownian motion. Using standard arguments based on Markov processes, we analyse the properties of the value function. The results of Chapter 3 are contained in the publication [14] and the results of Chapter 4 are presented in preprint [13]. Condensed versions of Chapters 5 and 6 will be submitted for publication. 1

11 Declaration No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 11

12 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the Copyright ) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the Intellectual Property ) and any reproductions of copyright works in the thesis, for example graphs and tables ( Reproductions ), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and commercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see in any relevant Thesis restriction declarations deposited in the University Library, The University Library s regulations (see and in The University s Policy on Presentation of Theses. 12

13 Acknowledgements There are many people I would like to show my sincere thanks. Firstly, to my parents for their encouragement and support through my life. To my friends who gave me a hug in the tough days and for all the happy times we spent together. To fellow PhD students Dr Yerkin Kitapbayev, Dr Shi Qiu, Dr Jay Unadkat, Dr Daniel Liversey, Dr Yi Pan, Tianren Bu and Daniel Wilson for the insightful discussions on the optimal stopping problem. To the probability group for organising the weekly seminar to broaden my research region. Last but not the least, to my supervisor Prof. Goran Peskir, it is appreciated for your accurate guidance on my tough day in mathematical problems and the endless support to create new financial instruments in my research. 13

14 Chapter 1 Introduction Optimal stopping theory is one of the most developed and exciting parts in modern probability. The theory originates from Wald s sequential analysis in 1947 (see [55]). A general optimal stopping problem in discrete time was formulated by Snell in [54]. Snell also characterised the value function as the smallest supermartingale that dominates the gain process (so-called Snell s envelope). Dynkin [11] introduced a general optimal stopping problem for Markov process and formulated the solution by superharmonic characterisation that states that the value function of optimal stopping problem is the smallest superharmonic function dominating the gain function. Dynkin studied the case of discrete time and indicated that the case of continuous time can get an analogous result. Chow et al. continued the research in [6, Chapter 4-5] using both methods from [54] and [11]. Myneni [4] priced the American put option using the results of Karoui and Karatzas [33] that established the Reisz decomposition of the value function related to an optimal stopping time problem. The Reisz decomposition divides the value function into two parts, a martingale and a potential that corresponds to the early exercise premium (EEP) component of the option value. McKean [37] discovered that an optimal stopping problem can be reduced to a free-boundary problem from mathematical analysis. Grigelionis and Shiryaev [2] started a systematic research following the work of [37] to reduce the optimal stopping problems to free-boundary problems. Most of the theoretical developments above solved particular situations were thoroughly shown and studied by Peskir and Shiryaev in [47]. Peskir also used a local time-space formula on curves [41] to show that the free boundary is the unique solution of the nonlinear integral equation. The local 14

15 CHAPTER 1. INTRODUCTION 15 time-space formula on curves [41] and surfaces [44] were also applied in [43] and [48] to price the American, Russian, and Asian options. In this thesis, we consider the optimal stopping problem of the form V (t, x) = sup E t,x [e rτ G(t + τ, X t+τ )], (1.1) τ T t where x R and the supremum is taken over all stopping times τ of X. The Markov process X = (X t ) t solves the following stochastic differential equation dx t = µ(t, X t )dt + σ(t, X t )db t, (1.2) where µ and σ > are continuous and B is a standard Brownian motion. value function V stands for the arbitrage-free price of the option and G is the gain function. The What we need to do first is to exhibit the stopping time at which the supremum is attained and then to find the value for any point (t, x) in the twodimensional state space as explicitly as possible. At maturity time T the terminal condition V (T, x) = G(T, x) always holds for all x R. A stopping set D is introduced as D = {(t, x) [, T ) (, ) : V (t, x) = G(t, x)}. (1.3) Similarly, a continuation set C is defined as C = {(t, x) [, T ) (, ) : V (t, x) > G(t, x)}, (1.4) and the stopping time τ D = inf{t : X t D} is optimal in the problem (1.1) (see [47]). The key tool to get the value function is the application of a local-time formula derived by Peskir in [41]. Since Ito s formula applies to all C 2 functions, the local time-space formula is an extension of Ito s formula (see [17]) suitable for functions not C 2 in the entire space. Let X = (X t ) t is a continuous semimartingale and b : R + R is a continuous function of bounded variation. Suppose a continuous function F : R + R R is C 1,2 on C and C 1,2 on

16 CHAPTER 1. INTRODUCTION 16 D. Then the change-of-variable formula [41] holds F (t, X t ) =F (, X ) + + t t t 1 2 (F t(s, X s +) + F t (s, X s ))ds 1 2 (F x(s, X s +) + F x (s, X s ))dx s (1.5) where l b s(x) is given by t l b 1 s(x) = P lim ξ 2ξ F xx (s, X s ) I(X s b(s))d X, X s (F x (s, X s +) F x (s, X s )) I(X s = b(s))dl b s(x) s I(b(r) ξ < X r < b(r) + ξ)d X, X r (1.6) and dl b s(x) refers to integration with respect to the continuous increasing function s l b s(x). In contrast to smooth payoff patterns of standard options, binary options have discontinuous payoffs and switch completely one way or the other depending on whether the underlying asset satisfies some condition. Binary options are mostly suitable for investment and are commonly used in foreign exchanges, equities, and commodities of financial markets. It can also be viewed as a rebate of barrier options. We discuss the binary options in Chapters 2, 3, 4 and 5, and extend them to barrier binary options in Chapter 6. The use of barrier options, binary options and other path-dependent options has increased dramatically in recent years especially by large financial institutions for the purpose of hedging, investment, and risk management. The pricing of European barrier and binary options in closed-form formulae has been addressed in a range of literature (see [24], [38], [51] and reference therein). Rubinstein and Reiner [52] valued a wide variety of binary options and more complex barrier binary options. Ingersoll [27] presented a simple and unified approach to value a variety of options using three types of digital contracts. It is particularly interesting since it shows that these three types of binary options can be combined to price more complex contracts. Hui [25] discussed up-and-down out binary option and American binary knock-out option in [25] and derived analytical solutions under the Black-Scholes option-pricing environment. Pooley et al. discussed three numerical methods in [49] to price options with the discontinuous payoff. Haug [23] has presented analytic valuation formulae for American up-and-in put and down-and-in call options in terms of standard American

17 CHAPTER 1. INTRODUCTION 17 options. It is extended by Dai and Kwok [7] to more types of American knock-in options in terms of integral representations. Jun and Ku [3] derived a closed-form valuation formula for a digital barrier option with exponential random time and provided analytic valuation formulae of American partial barrier options in [31]. Hui [25] used the Black-Scholes environment and derived the analytical solution for knock-out binary option values. Gao, Huang and Subrahmanyam [12] proposed an early exercise premium presentation for the American knock-out calls and puts in terms of the optimal free boundary. Over the last few years, a new type of option known as the British option was introduced by Peskir and Samee [45]. This kind of option entitles the holder the early exercise right like American option whereupon his payoff is the best prediction of the European option payoff under the hypothesis that the true drift equals a contract drift. The value of the contract drift is chosen by the option holder to represent the level of tolerance and protection. Based on the observed price movements, the option holder finds that if the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British option is the protection feature as not only can the option holder exercise to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. This option type is shown to be a very attractive alternative to classic European or American option when the contract drift is properly selected. This thesis prices and analyses the exotic American-type options with discontinuous payoff with the application of the optimal stopping theory. All chapters are self-contained and have detailed introductions. In the following paragraphs, we will highlight the main contributions of each chapter. In Chapter 2 we review the existing work on all different types of the binary options and set a background for the British binary options. We price and analyse the American-type (one-touch) binary options from the risk-neutral probability method. Chapter 3 is based on [14] and addresses the British asset-or-nothing put option since we believe that this is the most interesting case from a mathematical point of view. It mainly includes two parts: analytical solution and financial analysis. Optimal

18 CHAPTER 1. INTRODUCTION 18 stopping problems have been solved for different cases since with different value of the contract drift µ c it leads to different exercising boundaries and continuation sets. Using a local time-space calculus on curves [41] we derive a closed form expression for the arbitrage-free price in terms of the British asset-or-nothing put option and show that the rational exercise boundary can be characterised as the unique solution to a nonlinear integral equation. The second part of Chapter 3 provides a financial analysis of the returns of the British asset-or-nothing put option in comparison with its American and European versions. In Chapters 4 and 5 we continue to study the British mechanism in the context of a cash-or-nothing put option and the binary call options. From the local time-space formula, we derive the early exercise premium representation of the option value and show that the free boundary can be characterised as the unique solution for the system of two integral equations. We also list other kinds of binary options and analyse the financial meaning of the British binary options using the results above. In Chapter 6 we extend the binary options into barrier binary options and discuss the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Following the method in [7] we show that the price function of a knock-in American binary option can be expressed in terms of the price functions of simple barrier options and American options. For the knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. By the properties of Brownian motion and convergence theorems, we show how to calculate the expectation of the local time. In the financial analysis, we briefly compare the values of the American and European barrier binary options. Chapter 3 has been accepted for publication in International Journal of Theoretical and Applied Finance and Chapter 4 appears as a preprint [13] which is under consideration of publication. The condensed versions of Chapters 5 and 6 respectively will be submitted for consideration of publication.

19 Chapter 2 The American Binary Options 2.1 Introduction In the past decades, there is an increasing number and variety of derivative contracts in the financial market. American binary options, also known as one-touch digital options, are a kind of simple derivatives in financial markets. They can also be viewed as barrier options with a rebate but no other payoff. There is a wide variety of financial assets that use binary contracts as simple building blocks because their payoffs are either on or off. The term binary and digital are generic and can be applied to contracts whose payoff events are complex. Generally, binary options have the following types: (i) cash-or-nothing calls and puts; (ii) asset-or-nothing calls and puts; (iii) gap options; (iv) supershares. (i) For a cash-or-nothing call option, it pays off nothing if the asset price ends up below the strike price at time T and pays a fixed amount $1 if it ends up above the strike price. A cash-or-nothing put is defined analogously to the call. It pays off $1 if the asset price is below the strike price and nothing if it is above the strike price. In some other work, it pays a fixed amount but for simplicity we make the payoff 1 or. The payoff functions are (see [26]) P CN call (X T ) = if X T < K 1 if X T K (2.1) 19

20 CHAPTER 2. THE AMERICAN BINARY OPTIONS 2 Payoff Payoff 1 1 V E CNC V E CNP PCN call PCN put K Stock Price K Stock Price Cash-or-Nothing Call(CNC) Cash-or-Nothing Put(CNP) Payoff Payoff K V E ANC PAN call K V E ANP PAN put K Stock Price K Stock Price Asset-or-Nothing Call(ANC) Asset-or-Nothing Put(ANP) Figure 2.1: The payoffs and values of the European cash- and asset-or-nothing calls and puts with parameters K = 1, r =.5, =.2, T = 1 and t =. P CN put (X T ) = if X T K 1 if X T < K (2.2) where X T is the price of the underlying asset at time T and K is the strike price. According to the geometric Brownian motion (GBM) model, in a risk-neutral world, the value at time t of a cash-or-nothing call is e rt Φ(d 1 ) and of a put is e rt Φ( d 1 ), where d 1 = ln X K σ2 + (r )(T t) 2 σ T t (2.3) and Φ is the cumulative distribution function of the standard normal distribution. Panels CNC and CNP in Figure 2.1 show the payoffs and values of the European cash-or-nothing call and put as functions of the underlying asset with t given and fixed. (ii) For an asset-or-nothing call option, it pays nothing if the asset price ends up below the strike price at time T and pays the asset price if it ends up above the strike price. An asset-or-nothing put pays the asset price if the asset price is below the strike price

21 CHAPTER 2. THE AMERICAN BINARY OPTIONS 21 Payoff Payoff K t K s Stock Price K t-k s Payoff K s K t Stock Price Gap call option with K s <K t K t-k s Payoff Gap call option with K s >K t K s-k t K s K t Stock Price K t K s Stock Price Gap put option with K s >K t K s-k t Gap put option with K s <K t Figure 2.2: The payoffs of Gap options with different parameters. and nothing if it is above the strike price (see [26]). The payoff functions are if X T < K Pcall AN (X T ) = if X T K P AN put (X T ) = X T (2.4) if X T K. (2.5) if X T < K X T In the GBM model, the value of an asset-or-nothing call is X t Φ(d 2 ) and of a put is X t Φ( d 2 ), where d 2 = d 1 + σ T t. (2.6) Panels ANC and ANP in Figure 2.1 show the payoffs and values of the European asset-or-nothing call and put as functions of the underlying asset with t given and fixed. (iii) For the gap options, it has a strike price K s and a trigger price K t. There is a gap between the price at which the option can be exercised and the price at which it would produce a payoff to the holder. The strike price K s determines the size of the option s payoff, while the trigger price K t determines whether the payoff would be made or not. The strike price K s can be larger or smaller than the trigger price K t.

22 CHAPTER 2. THE AMERICAN BINARY OPTIONS 22 Payoff U/L 1 L U Stock Price Figure 2.3: The payoff of a supershare option. If they are equal, the gap option reduces to an ordinary call or put option. It has the following payoffs P gap call (X if X T < K t T ) = (2.7) X T K s if X T K t if X P gap T > K t put (X T ) = (2.8) K s X T if X T K t The value of a gap option can be attained by subtracting the value of a cash-or-nothing option from the value of an asset-or-nothing option. The values for the gap call c and put p in the GBM model can be expressed as (see [52] and reference therein) c gap (t, X t ) = X t Φ(d 2 ) e r(t t) K s Φ(d 1 ) (2.9) p gap (t, X t ) = e r(t t) K s Φ( d 1 ) X t Φ( d 2 ), (2.1) where d 1 and d 2 are defined in (2.3) and (2.6) with the strike price K replaced by K t. The payoffs with different parameter settings are shown in Figure 2.2. Strictly speaking, a gap option cannot be called an option since its payoff and premium can be negative. Therefore, it is not our main topic in this thesis. (iv) In a supershare option, there is a lower boundary L and upper boundary U (see Figure 2.3). If the underlying is between these boundaries at expiry date the payoff is the fraction of the asset price over the lower boundary. On the other hand, if the underlying is outside these boundaries the payoff is zero (see [22]). Mathematically,

23 CHAPTER 2. THE AMERICAN BINARY OPTIONS 23 the payoff is expressed as if X T < L P super (X T ) = X T /L if L X T U (2.11) if U < X T. The value of a supershare option in European style equals an asset-or-nothing call with strike L deducting an asset-or-nothing call with strike U, i.e. V (t, X t ) = X Xt t L (Φ(ln L + (r + σ2 )(T t) 2 σ T t Xt ln U ) Φ( + (r + σ2 )(T t) 2 σ )). (2.12) T t The cash- and asset-or-nothing options are in American style and they should be exercised as soon as the strike price is reached (also known as One-touch options). There is no gain for holding the option for longer. We lose profit from the discounted payoff and get higher risk if the asset price moves in the other direction. It follows that the strike price is the optimal stopping boundary. The American digital option was mentioned in [29, Chapter 8], which studied an option getting the payoff at expiry if at any point during the life of the option a given barrier has been breached. Rubinstein and Reiner (see [52]) studied a wide variety of binary options and more complex barrier binary options which we will give more details in Chapter 6. Ingersoll (see [27]) studied three types of digitals: European cash-or-nothing, asset-or-nothing and American cash-or-nothing and combined these three kinds of digitals to determine more complex contracts. Hakansson (see [22]) proposed the notation of supershares to investors in Pooley et al. discussed three numerical methods in [49] to price options with the discontinuous payoff. The main purpose of this chapter is to review the existing research about binary options (mainly focus on cash- and asset-or-nothing options) and the optimal stopping problem and to value the American binary options from risk-neutral expectation method and to set the background for the next chapter. 2.2 About the optimal stopping problem The following facts are based on the book of optimal stopping and free boundary problems (see [47]).

24 CHAPTER 2. THE AMERICAN BINARY OPTIONS The stock price X = (X t+s ) s follows a geometric Brownian motion: with X t started at zero. by: dx t+s = rx t+s ds + σx t+s db s (2.13) = x under P t,x and B = (B s ) s denotes the standard Brownian motion The process X is strong Markov (diffusion) with the infinitesimal generator given 2. For the optimal stopping problem V (t, x) = it follows that V admits the representation: L X = rx x + σ2 2 x2 2 x. (2.14) 2 sup E t,x (e rτ G(t + τ, X t+τ )), (2.15) τ T t V (t, x) = E t,x (e rτ D G(t + τ D, X t+τd )), (2.16) where τ D is the first entry time of X into D given by τ D = inf{t : X t D}. (2.17) The function V solves the following free-boundary problem: where C is the continuation set. V t + L X V = rv in C, (2.18) V D = G D, (2.19) V x C = G x C (smooth fit) (2.2) 3. Let X = (X t ) t is a continuous semimartingale and b : R + R is a continuous function of bounded variation, amd set: C = {(t, x) [, T ) (, ) : x < b(t)} (2.21) D = {(t, x) [, T ) (, ) : x > b(t)}. (2.22)

25 CHAPTER 2. THE AMERICAN BINARY OPTIONS 25 Suppose a continuous function F : R + R R is C 1,2 on C and C 1,2 on D. Then the change-of-variable formula holds [41]: F (t, X t ) =F (, X ) + + t t t 1 2 (F t(s, X s +) + F t (s, X s ))ds 1 2 (F x(s, X s +) + F x (s, X s ))dx s (2.23) where l b s(x) is given by t l b 1 s(x) = P lim ξ 2ξ F xx (s, X s ) I(X s b(s))d X, X s (F x (s, X s +) F x (s, X s )) I(X s = b(s))dl b s(x), s I(b(r) ξ < X r < b(r) + ξ)d X, X r (2.24) and dl b s(x) refers to integration with respect to the continuous increasing function s l b s(x). Furthermore, if (2.13) is the case, then (2.23) becomes F (t, X t ) =F (, X ) t t t (F t + L X F )(s, X s ) I(X s b(s))ds (F x (s, X s )σ(x s ) I(X s b(s))db s (F x (s, X s +) F x (s, X s )) I(X s = b(s))dl b s(x). (2.25) Moreover, the condition that F : R + R R is C 1,2 on C and C 1,2 on D is equal to the following 1. F t + L X F is locally bounded on C D; 2. x F (s, x) is convex or concave on [b(s) δ, b(s)] and convex or concave on [b(s), b(s) + δ] for each s [, t] with some δ > ; 3. s F x (s, b(s)±) is continuous on [, t] with values in R. Then the change-of-variable formula (2.25) still holds. 2.3 Cash-or-nothing options We focus on the cash-or-nothing call option for concreteness. A cash-or-nothing put option can be handled similarly. We use the discounted risk-neutral expectation to

26 CHAPTER 2. THE AMERICAN BINARY OPTIONS 26 price the cash-or-nothing call option. To compute this expectation we need to know the distribution of the maximum and the first hitting time for a Brownian motion with drift. Therefore, we will first develop a better understanding of Brownian motion, stopping times and Girsanov s theorem. We start from setting background in the optimal stopping theory for better understanding for the next chapter and then price the American cash-or-nothing call option from the risk neutral probability measure. 1. The arbitrage-free price of American call option is given by V CNC (t, x) = where X = (X t+s ) s solves sup E t,x (e rτ I(X t+τ K)), (2.26) τ T t dx t+s = rx t+s ds + σx t+s db s (2.27) with X t = x > under P t,x. Recall that B = (B s ) s is a standard Brownian motion process, T > is the maturity of the options, r > is the interest rate, and σ > is the volatility coefficient. The unique solution of (2.27) is given by X t+s = x exp(σb s + (r σ 2 /2)s) (2.28) for t and x are given and fixed. The process X is strong Markov (diffusion) with the infinitesimal generator given by L X = rx x + σ2 2 x2 2 x. (2.29) 2 2. We aim to get the arbitrage-free price V CNC from (2.26) and the optimal stopping time τ where V CNC is attained in (2.26). From earlier work (see [37]), we define the continuation set: the stopping set: and the optimal stopping time: C = {(t, x) [, T ) (, ) : V CNC (t, x) > P CN call (x)}, (2.3) D = {(t, x) [, T ] (, ) : V CNC (t, x) = P CN call (x)}, (2.31) τ D = inf{ s T t : X t+s D}. (2.32)

27 CHAPTER 2. THE AMERICAN BINARY OPTIONS 27 From (2.1), (2.26), (2.31) and (2.32), it suggests that the stopping time τ K = inf{ s T t : X t+s K} (2.33) is optimal in (2.26) and the supremum is attained at this stopping time. If the initial stock price X t = x > K then we should exercise immediately to avoid any future discount of the payoff. Hence we only consider the situation x < K. 3. We will make use the following probability density function of first hitting time (see [5, Page 622] ) P x (τ K dt) = ln K x σ 2πt (K 3 x )v exp( (ln K x )2 2σ 2 t for τ K = inf{t : X t = K} with X = x < K and v = r σ v2 σ 2 t )dt (2.34) 2 Lemma 1. For drifted Brownian motion B v s = B s + vs where B s is a standard Brownian motion, the cumulative distribution function (CDF) of the maximum process M t = max s t (B v s ) is P(M t m) = Φ( m vt ) e 2vm m vt Φ( ). (2.35) t t Proof. There are many ways to get the result. Here we use Girsanov s theorem and a change of measurement. We refer to [32, Section 2.8] for more technical details. From the reflection principle, for the standard Brownian motion B t we have P(M t m, B t w) = P(B t 2m w) = 1 2πt 2m w e y2 2t dy (2.36) where M t function = max s t (B s ). By differentiating (2.36), it is easy to get the joint density f(m, w) = = 2 P w m 2(2m w) t e (2m w)2 2t. 2πt (2.37) By Girsanov s theorem we define a new measure by dq = e vbt v2 2 t dp = e vbv t + v2 2 t dp. (2.38)

28 CHAPTER 2. THE AMERICAN BINARY OPTIONS 28 Under Q, the drifted Brownian motion Bt v = B t + vt is a standard Brownian motion. Q(M t h) = Following (2.38) and (2.39), we get h h f(m, w)dmdw (2.39) P(M t h) = h h = 1 2πt h = 1 2πt [ h e vw v2 2 t f(m, w)dmdw e vw v2 2 t [e w2 2t e (2h w)2 2t ]dw h e (w vt)2 2t dw e 2vh = Φ( h vt ) e 2vh h vσt Φ( ) t t e (w vt 2h)2 2t dw] (2.4) 4. Our main result of this section is stated now. Theorem 2.1. The stopping time (2.33) is optimal for the optimal stopping problem (2.26). The arbitrage-free price of the American cash-or-nothing call option can be expressed as follows: V CNC (t, x) = x σ2 + (r + )(T t) 2 Φ(log(x/K) K σ ) T t + ( K σ2 log(x/k) (r + )(T t) 2 x )2r/σ2 Φ( σ ) T t (2.41) for all (t, x) [, T ] (, K). Proof. 1. Let us first show that (2.33) is optimal in (2.26). Since (2.26) is a discounted price, the larger τ is, the less value we will get. As for the payoff, it is either $1 or nothing. Therefore, the optimal stopping time is the first time that the stock price hits K, which is represented mathematically as (2.33). To prove this, let τ define any stopping time. We need to show that E t,x (e rτ K I(X t+τk K)) E t,x (e rτ I(X t+τ K)). (2.42)

29 CHAPTER 2. THE AMERICAN BINARY OPTIONS 29 Actually, E t,x (e rτ K I(X t+τk K)) =E t,x (e rτ K (I(X t+τk K, t + τ K < T )+ I(X t+τk K, t + τ K T ))) (2.43) =E t,x (e rτ K (I(t + τ K < T ) + I( ))) =E t,x (e rτ K I(t + τ K < T )). On the other hand, E t,x (e rτ I(X t+τ K)) =E t,x (e rτ I(X t+τ K, τ < τ K ) + e rτ I(X t+τ K, τ τ K )) =E t,x (e rτ I(X t+τ K, τ τ K )) =E t,x (e rτ I(X t+τ K, τ τ K, t + τ K < T )+ (2.44) e rτ I(X t+τ K, τ τ K, t + τ K T )) =E t,x (e rτ I(X t+τ K, τ τ K, t + τ K < T )) E t,x (e rτ K I(t + τ K < T )). Hence we conclude that τ K is optimal in (2.26). 2. Following step 1, then (2.26) becomes E t,x (e rτ K I( max s T t X t+s K)). (2.45) We define M(T t) = max s T t (X t+s ). The result becomes E t,x (e rτ K I(M(T t) K)). (2.46) In fact, (2.46) is equal to E t,x (e rτ K I(τ K T t)). (2.47) Thus, the value function changes into a problem of the first hitting time for geometric Brownian motion. The distribution of the first hitting time is given by (2.34). Then the value function (2.26) turns into V CNC (t, x) = ln K x σ 2π (K x )v T t where v = r 1. By changing σ 2 2 v = r + 1, we get σ 2 2 V CNC (t, x) = x K (s 3/2 exp( (v 2 σ 2 /2 + r)s (ln K x )2 2σ 2 s T t ))ds, (2.48) P x (τ K dt), (2.49)

30 CHAPTER 2. THE AMERICAN BINARY OPTIONS 3 where τ K = inf{ s T t : X t+s K} and X t+s = x exp(σb s + v σ 2 s). It equals because x K P x(τ K T t), (2.5) P x (τ K T t) = P x ( max s T t X s K) = P x ( max s T t (B s + v σs)) 1 σ ln K x )). (2.51) From Lemma 1, the cumulative distribution function of the M t = max s t (B v s ) equals P(M t m) = Φ( m v σt ) e 2v σm Φ( m v σt ). (2.52) t t Following (2.5), (2.51) and (2.52), we get V CNC (t, x) = x σ2 + (r + )(T t) 2 Φ(log(x/K) K σ ) T t + ( K σ2 log(x/k) (r + )(T t) 2 x )2r/σ2 Φ( σ ). T t (2.53) On the other hand, for the cash-or-nothing put (CNP) option, it is quite similar. The arbitrage-free price is and the stopping time is V CNP (t, x) = sup E t,x (e rτ I(X t+τ < K)) (2.54) τ T t τ K = inf{ s T t : X t+s < K}. (2.55) What we need to do is to substitute (2.52) with the CDF of the minimum of the geometric Brownian motion. Lemma 2. The CDF of the minimum process N t = min s t (B v s ) is P(N t n) = Φ( n vt ) + e 2vn Φ( n + vt ), (2.56) t t where B v s = B s + vs and B s is a standard Brownian motion.

31 CHAPTER 2. THE AMERICAN BINARY OPTIONS 31 Proof. Fortunately, the fact that the negative of a Brownian motion with drift is also a Brownian motion with drift, i.e. N t = min s t (B s + vs) = min s t ( (B s + vs)) = max s t ( B s vs), (2.57) where B s is a standard Brownian motion. We denote B v t = B t vt, then P(N v t n) = P(M v t m) = Φ( n vt ) + e 2vm Φ( n + vt ). t t (2.58) Theorem 2.2. The stopping time (2.55) is optimal for the optimal stopping problem (2.54). The arbitrage-free price of the American cash-or-nothing put option can be expressed as follows: for all (t, x) [, T ] (K, ). V CNP (t, x) = x σ2 (r + )(T t) 2 Φ(log(K/x) K σ ) T t + ( K σ2 log(x/k) + (r + )(T t) 2 x )2r/σ2 Φ( σ ) T t (2.59) Proof. Following the proof of cash-or-nothing call option that τ K is the optimal stopping time, the value (2.54) equals It equals x K P(τ K T t). (2.6) P x (τ K T t) = P x ( min s T t X s K) = P x ( min s T t (B s + vs)) 1 σ ln K x )). (2.61) The CDF of the random variable N t = min s t (B v s ) is given by Lemma 2. The rest of the proof is handled in a similar way as Theorem Asset-or-nothing options First, we analyse the put option in detail. If the strike price K is larger than the initial point X t = x, we would exercise the option at once since the process e rs X t+s is a

32 CHAPTER 2. THE AMERICAN BINARY OPTIONS 32 martingale. It means that if we wait until the asset price hits K, the discounted payoff equals the initial price x. So we do not need to wait. In the following content, we only consider the condition K < x. The arbitrage-free price for the asset-or-nothing call option is defined by V ANC (t, x) = sup E t,x (e rτ X t+τ I(X t+τ K)). (2.62) τ T t The arbitrage-free price for the asset-or-nothing put option is V ANP (t, x) = Our main result of this section is stated as follows. sup E t,x (e rτ X t+τ I(X t+τ K)). (2.63) τ T t Theorem 2.3. The arbitrage-free price of the American asset-or-nothing put and call options are equal to for all (t, x) [, T ] (K, ) and V ANP (t, x) = KV CNP (t, x) (2.64) V ANC (t, x) = KV CNC (t, x) (2.65) for all (t, x) [, T ] (, K), where V CNP (t, x) and V CNC (t, x) are defined respectively in (2.62) and (2.63). Proof. These expressions follow immediately from the fact that e rs X t+s is a martingale. Therefore, it is always optimal to exercise when the underlying asset reaches the strike price and then the option holder gets the payoff K. 2.5 Summary This Chapter mainly discusses the American cash- and asset-or-nothing options. We evaluate them from the risk-neutral probability measure. Figure 2.4 shows a comparison of the European and American binary option values. Generally, American options are more expensive since the American feature gives the holder the additional flexibility of early exercise. For the cash- and asset-or-nothing call options, the values for both European and American version converge when the underlying price is at a very low/high level. For the binary puts, the difference is still significant when the stock price is large. This phenomenon can be reduced with different parameter settings.

33 CHAPTER 2. THE AMERICAN BINARY OPTIONS 33 European CNP American CNP European CNC American CNC Value Value 1 1 K cash-or-nothing put option Stock Price European ANP American ANP K cash-or-nothing call option Stock Price European ANC American ANC Value Value K K K asset-or-nothing put option Stock Price K asset-or-nothing call option Stock Price Figure 2.4: A Comparison of the values of the European and American binary options for t given and fixed with the parameters K = 1, r =.5, σ =.2 and T = 1.

34 Chapter 3 The British Asset-or-Nothing Put Option 3.1 Introduction The purpose of this chapter is to introduce and examine the British payoff mechanism of [45] and [46] in the context of the asset-or-nothing put option that is one type of the binary options. There are various types of binary options that may be considered to this end: (i) calls and puts; (ii) cash-or-nothing and asset-or-nothing. This leads to an extensive programme of research that we open in this chapter by focusing on the asset-or-nothing put option as we believe that this is the most interesting case from a mathematical point of view. Following the economic rationale of the British put and call options, we introduce a new asset-or-nothing put option that endogenously provides its holder with a protection mechanism against unfavourable stock price movements. This mechanism is intrinsically built into the option contract using the concept of optimal prediction [1]. We refer to such contracts as British for the reasons outlined in [45] and [46] where the British put and call options were introduced. Similar to the British put and call options, the British asset-or-nothing put option not only provides a unique protection against unfavourable stock price movements (see more details in section 3.5) but also enables the option holder to obtain higher returns when the stock price movements are favourable in both liquid and illiquid markets. This reaffirms the fact noted in [2], [3], [18], [19] and [35] for the value of British barrier, British Asian, British Russian 34

35 CHAPTER 3. THE BRITISH ASSET-OR-NOTHING PUT OPTION 35 and British lookback options that the British feature of optimal prediction can provide both protection against unfavourable price movements as well as securing high returns when movements are favourable. These combined features are especially appealing as the problems of liquidity and return are addressed completely endogenously. The chapter mainly consists of two parts: analytical solution and financial analysis. According to the financial theory [53], the arbitrage-free price of the option is a solution to an optimal stopping problem with the gain function as the payoff of the option. Following the notion compound option by [16] where an analytical solution for the compound option was derived, we refer to a more recent paper [21] for an informative review. We then use a local time-space calculus on curves [41] to derive a closed form expression for the arbitrage-free price in terms of the optimal stopping boundary and show that the free boundary itself is the unique solution to a nonlinear integral equation. We perform the analysis of returns of the British asset-or-nothing put option with the American asset-or-nothing put option. After observing the returns upon exercising, we conclude the remarkable protection feature of the British asset-or-nothing put option. This chapter is organised as follows. In Section 3.2 we present a basic motivation for the British asset-or-nothing put option. In Section 3.3 we firstly formally define the British asset-or-nothing put option and present some of its basic properties. We continue in Section 3.4 to derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary can be characterised as the unique solution to a nonlinear integral equation. In Section 3.5 we provide a financial analysis using the results above, making a comparison with American asset-or-nothing put option. 3.2 Basic motivation for the British asset-or-nothing put option The basic economic motivation for the British asset-or-nothing put option is parallel to that of British put and call options [45, 46]. In this section, we briefly review the key elements of the motivation on the asset-or-nothing put option. 1. Consider the financial market consisting a risky stock X and a riskless bond B