Option Pricing Under Jump-Diffusion Processes: Calibration to the Bitcoin Options Market

Home , Jump diffusion

INSTITUTO SUPERIOR DE CIÊNCIAS DO TRABALHO E DA EMPRESA FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA

DEPARTAMENTO DE FINANÇAS DEPARTAMENTO DE MATEMÁTICA

Option pricing under jump-diffusion processes: Calibration to the bitcoin options market

Fernando Correia da Silva

Mestrado em Matemática Financeira

Dissertação orientada por: Professor Doutor João Pedro Nunes

2018

Acknowledgements

The writing of this thesis was a challenging but rewarding project. Firstly, I would like to thank my su- pervisor, Professor Doutor João Pedro Nunes, for his guidance throughout the work of this thesis. His feedback and suggestions were very constructive and allowed me to better consolidate the contents of the jump-diffusion processes. I am also grateful to the course teachers for the teachings that introduced me to the world of ﬁnancial mathematics. Finally, I want to thank my family, especially my dear son Miguel, who helped me with his computer skills.

In memory of my parents. Resumo

Depois do trabalho ímpar de Black e Scholes (1973), a maior parte da literatura sobre avaliação de opções financeiras veio pressupor que o ativo subjacente segue um processo de difusão, a saber, um movimento Browniano geométrico. No entanto, as evidências empíricas revelam a existência de vari- ações extremas nos preços. Para incorporar eventos raros ou extremos, não capturados por modelos de difusão pura, vários modelos de difusão com saltos foram introduzidos na literatura financeira. Kou (2002) criou um modelo de difusão com saltos que explica dois fenómenos empíricos - a leptocurtose assimétrica e o “sorriso de volatilidade”. Além disso, o modelo de Kou conduz a soluções analíticas para muitos dos problemas de avaliação de opções. Assim, o objetivo desta dissertação é estudar o modelo de Kou e a sua aplicação à avaliação de opções. Na parte de análise empírica, o objetivo é investigar se o modelo de Kou faz um bom fit aos dados de opções financeiras sobre bitcoin.

Keywords: Avaliação de opções, modelos de difusão com saltos, modelo de Kou, bitcoin

Códigos de classiﬁcação JEL: C21, G12 Abstract

Since the seminal work of Black and Scholes (1973), most of standard literature in option pricing assumes that the underlying asset follows diffusion type process, namely a geometric Brownian motion. However, empirical evidence reveals extreme price variations. So, to incorporate rare or extreme events, not captured by diffusion models, several jump diffusion models have been introduced to the ﬁnance literature. Kou (2002) has created a double exponential jump diffusion model that explains two empirical phenomena - the asymmetric leptokurtic feature and the “volatility smile”. In addition to that, Kou’ s model leads to analytical solutions to many option pricing problems. Thus, the aim of this dissertation is to study Kou’s option pricing model. For the empirical analysis, the goal is to investigate the goodness of the Kou’ s model calibration, using the recent bitcoin options market.

Keywords: Option pricing, Jump-diffusion models, Kou’ s model, bitcoin

JEL Codes: C21, G12 Contents

List of Figures v

List of Tables vii

1. Introduction 1

2. Option pricing models 2 2.1. The diffusion models and their shortcomings ...... 2 2.2. Advantages of the jump-diffusion models ...... 4

3. Lévy processes 6 3.1. Overview ...... 6 3.2. Mathematical tools ...... 7 3.3. Double exponential jump-diffusion model ...... 13

4. Pricing bitcoin options under Kou’s model 21 4.1. Bitcoin options ...... 21 4.2. Calibration of Kou’ s model ...... 26 4.3. Pricing bitcoin options under Kou’s model ...... 27 4.4. Results ...... 38

5. Conclusions 43

A. Derivations 52

B. Data and results 77

C. MATLAB code 101

iv List of Figures

3.1. Sample path of a Poisson process ...... 8 3.2. Sample path of a compound Poisson process ...... 10

4.1. Bitcoin closing prices - 18/07/2010 to 30/06/2018 ...... 22 4.2. Bitcoin prices - histogram and the descriptive statistics ...... 22 4.3. Bitcoin daily log-returns - 18/07/2010 to 30/06/2018...... 23 4.4. Bitcoin log-returns - histogram and descriptive statistics ...... 23 4.5. Bitcoin option prices on 24/08/2018 ...... 25 4.6. Kou vs Black-Scholes - calls on 16/12/2017 with 13 days to maturity ...... 27 4.7. Kou vs Black-Scholes - puts on 16/12/2017 with 13 days to maturity ...... 28 4.8. Kou vs Black-Scholes - calls on 16/12/2017 with 377 days to maturity ...... 28 4.9. Kou vs Black-Scholes - puts on 16/12/2017 with 377 days to maturity ...... 29 4.10. Kou vs Black-Scholes - calls on 05/02/2018 with 18 days to maturity ...... 29 4.11. Kou vs Black-Scholes - calls on 05/02/2018 with 326 days to maturity ...... 30 4.12. Kou vs Black-Scholes - puts on 05/02/2018 with 18 days to maturity ...... 30 4.13. Kou vs Black-Scholes - puts on 05/02/2018 with 326 days to maturity ...... 30 4.14. Kou vs Black-Scholes - calls on 28/06/2018 with 1 days to maturity ...... 31 4.15. Kou vs Black-Scholes - calls on 28/06/2018 with 29 days to maturity ...... 31 4.16. Kou vs Black-Scholes - calls on 28/06/2018 with 64 days to maturity ...... 32 4.17. Kou vs Black-Scholes - calls on 28/06/2018 with 92 days to maturity ...... 32 4.18. Kou vs Black-Scholes - calls on 28/06/2018 with 183 days to maturity ...... 32 4.19. Kou vs Black-Scholes - calls on 28/06/2018 with 547 days to maturity ...... 33 4.20. Kou vs Black-Scholes - puts on 28/06/2018 with 1 days to maturity ...... 33 4.21. Kou vs Black-Scholes - puts on 28/06/2018 with 29 days to maturity ...... 34 4.22. Kou vs Black-Scholes - puts on 28/06/2018 with 64 days to maturity ...... 34 4.23. Kou vs Black-Scholes - puts on 28/06/2018 with 92 days to maturity ...... 34 4.24. Kou vs Black-Scholes - puts on 28/06/2018 with 183 days to maturity ...... 35

v 4.25. Kou x Black-Scholes - puts on 28/06/2018 with 547 days to maturity ...... 35 4.26. Surface volatility - call options on 16/12/2017 ...... 36 4.27. Surface volatility - call options on 05/02/2018 ...... 37 4.28. Surface volatility - call options on 28/06/2018 ...... 37

A.1. Possible path from (2,2) to (1,0) ...... 64

B.1. Bitcoin options market prices on 16/12/2018 ...... 81 B.2. Bitcoin options market prices on 05/02/2018 ...... 85 B.3. Bitcoin options market prices on 28/06/2018 ...... 93 List of Tables

4.1. APE values by date of data collection - all options ...... 39 4.2. APE values by date of data collection - calls only ...... 39 4.3. APE values by date of data collection - puts only ...... 39 4.4. APE values by time to maturity - all options on 16/12/2017 ...... 39 4.5. APE values by time to maturity - all options on 05/02/2018 ...... 40 4.6. APE values by time to maturity - all options on 28/06/2018 ...... 40 4.7. APE values by time to maturity - call options on 16/12/2017 ...... 40 4.8. APE values by time to maturity - put options on 16/12/2017 ...... 41 4.9. APE values by time to maturity - call options on 05/02/2018 ...... 41 4.10. APE values by time to maturity - put options on 05/02/2018 ...... 41 4.11. APE values by time to maturity - call options on 28/06/2018 ...... 41 4.12. APE values by time to maturity - put options on 28/06/2018 ...... 42

B.1. Model calibration parameters ...... 77 B.2. Output of the Kou model - 16/12/2017 ...... 95 B.3. Output of the Kou model - 05/02/2018 ...... 97 B.4. Output of the Kou model - 28/06/2018 ...... 100

vii 1. Introduction

Despite the Black and Scholes (1973) model, [23], has been successful applied in the world of mathematical finance, many empirical studies show several deviations in the market from this model. Real prices display properties that contradict the assumptions of this model. The so-called stylized facts reveal that prices do not strictly follow a geometric Brownian motion, as the Black-Scholes model assumes. In fact, using the Black-Scholes model in option pricing produces inconsistent results with the options market price. Those shortcomings of the classical model led researchers to consider a variety of asset pricing models with non-Gaussian increments. There are many extensions of the Black-Scholes model to explain these phenomena. One of the most important is based on Lévy’ s exponential processes, which have emerged as an improvement for financial modeling, since they take into account the stylized characteristics of the markets. Indeed, jumps occur in financial time series data. This problem was picked up by many authors, [31], [53] and [60]. A special case of models based on Lévy processes is the double exponential jump-diffusion model, proposed by Kou, [56], [57], [60]. The study of Kou’ s model is the core of this thesis, that is structured as follows. The literature review is comprised in Chapter 2. Here we present the shortcomings and advantages of jump-diffusion models. Chapter 3 be- gins with an overview and the mathematical tools of Lévy jump-diffusion processes and then the Kou (2002) model is explored in detail. In Chapter 4 we present the numerical results. For the empirical analysis, the goal is to investigate the goodness of the Kou’ s model fit, using the recent bitcoin options market. We compare the derived model prices to observed market quotes. Conclusions for our research are included in Chapter 5. Some derivations and the data, as well as the Matlab code for the European option pricing under Kou’ s model, are included in the Appendices A, B and C.

1 2. Option pricing models

Since the seminal work of Black and Scholes, [23], most of the standard literature in option pricing assumes that the underlying assets follow a diffusion type process, namely a geometric Brownian motion. A Brownian motion is a random process with independent and stationary increments that follows a Gaussian distribution. In the world of stochastic processes used to model price ﬂuctuations, Brownian motion is “undoubtedly the brightest star”, [31, Chapter 1]. However, the modelling of risky asset by stochastic processes with continuous paths, based on Brownian motions, suffers from several defects. As documented in a signiﬁcant number of papers written by academics and practitioners, both normality and continuity assumptions are contradicted by the data in several pieces of evidence, [45].

2.1. The diffusion models and their shortcomings

Some features of financial data have received much attention from both practitioners and people who come from a more theoretical background. Cont (2001), in [29], presents a set of statistical facts which emerge from the empirical study of asset returns and which are common across a wide range of instruments, markets and time periods. Such properties are classified as “stylized facts”. According to [29] and [82], a few relevant phenomena of the real-world financial markets are:

• Discontinuous trajectories;

• Skewness and leptokurtosis of the distribution of returns;

• Volatility smile in option pricing;

• Volatility clustering.

The existence of these phenomena, observed due to the analysis of market data, means that asset prices do not strictly follow a Geometric Brownian motion, [61]. For example, if we use the normal distribution to model the ﬁnancial returns, probably we will underestimate the number and magnitude of crashes. In the Brownian motion model, the trajectories are continuous. That is, in a short interval of time, the

2 stock price can only change by a small amount, [71]. However, the path continuity assumption does not seem reasonable in view of the possibility of sudden price variations (jumps) resulting from market crashes, gaps or opening jumps. Jumps may clearly be identified in equity data and they caused by various economic, political and social factors, [66]. There are many reasons for the stock market to display discontinuities. A simple one could be discrepancies between the closing and opening prices, especially if some important news, positive or negative, have become public during the market’ s clo- sure. In fact, the inability to trade continuously implies jumps in prices, [45]. The classical models simply ignore the leptokurtic feature of asset returns, while the empirical evidence of this feature is commonly known, [55]. The modeling of risky asset prices through a Brownian motion relies on the use of the Gaussian distribution which tends to underestimate the probabilities of extreme events, [31] and [76]. Evidence shows that the skewness and kurtosis of stock returns differ from the normal distribution. In most cases, real-world returns are leptokurtic and display slightly negative skewness, [17] and [12]. The leptokurtic property, also known as “fat tails” property, implies that the tails of the distribution of historical returns are thicker than those predicted by the Gaussian distribution. This means that extreme market events occur more often in the real world than is predicted within the Black-Scholes framework. This has important implications for option pricing [21]. However, this feature is more accentuated when the holding period becomes shorter and becomes particularly clear on high frequency data, [45]. The implied volatility is a “wrong number which plugged into the wrong formula gives the right answer”, [81]. According to the Black-Scholes model, one should expect options that expire on the same date to have the same implied volatility regardless of the strikes. But option prices exhibit the famous volatility smile as well as prices higher than predicted by the Black-Scholes formula for short-dated options. Research has focused at implied Black Scholes volatility since implied volatility has become a key concept for option pricing. In trading, option prices are often quoted by their implied volatility, [21]. Another stylized fact of financial time series is the presence of volatility clustering. This is another feature that is not considered by the Black-Scholes model. Two empirical circumstances — the squared returns or absolute values of returns are correlated and the returns themselves seem to have approxi- mately no correlation — yield the phenomenon called the volatility clustering effect. This effect cannot be incorporated into any financial model that relies on the assumption that stock returns have independent increments, [29] and [58]. It may be worth mention here that, as early as 1965, Eugene Fama had tested the random-walk model of stock price behavior. He concluded that the empirical distributions of price changes should have longer tails than does the normal distribution and the independence assumption of the random-walk model seems to be an adequate description of reality, that is, “the past history of the series cannot be used to increase the investor’ s expected profits”, [44, pp. 87]. Furthermore, Mandelbrot (1963), [67], had also noted that “large changes tend to be followed by large changes, of

3 either sign, and small changes tend to be followed by small changes”, in short, the volatility clustering. The importance of the correct speciﬁcation of asset returns probability laws is very well understood, due to the implications on derivative pricing. The asset returns behavior has been studied by many authors, and many models have been suggested. Some of them have captured a reasonable part of this behavior, such the leptokurtic feature, implied volatility smile and volatility clustering effect. For a list of some of them see [55]. Unfortunately, most currently existing models fail to reproduce all these statistical features at once, [29]. This conclusion was the main reason for developing more complex models. Each of them has some advantages, but there does not exist any model that could meet all the aspects related to the price movements. Depending on a problem, the data used and available tools, we have to choose the most appropriate model.

2.2. Advantages of the jump-diffusion models

The well-known weaknesses of diffusion-based models of option prices have led to a variety of models involving randomly occurring discontinuous jumps (in addition to a “diffusive component” driven by a Brownian motion). These models are known as “jump diffusions” and form a subset of the set of models driven by a Lévy process. In general, a Lévy process is a process with stationary independent increments which is continuous in probability, [22] and [83]. As we have already seen, in a model with continuous paths, like a diffusion model, the probability that the stock moves by a large amount over a short period of time is very small. Therefore, in such models, the prices of short term out- of-the-money options should be much lower than what one observes in real markets. On the other hand, if stock prices are allowed to jump, even when the time to maturity is very short, there is a non-negligible probability that, after a sudden change in the stock price, the option will move in the money. The jump-diffusion models are developed to overcome this kind of drawbacks, but probably the strongest argument for using discontinuous models is simply the presence of jumps in observed prices [89]. Models with jumps allow for a more realistic representation of price dynamics and greater flexibility in modelling and have been the focus of much recent work [33]. In that sense, a generic class of processes, called Lévy processes, have shown to be an adequate context for the modelling of the asset returns, which allows us to obtain a good fit with real data without the need to introduce extreme parameter values [43], [26] and [74]. On the other hand, the mathematical tools behind these processes are very well established and known, [91], because the two basic building blocks of every jump-diffusion model are the Brownian motion (the diffusion part) and the Poisson process (the jump part). So, the jump-diffusion models are “an essential and easy-to-learn tool for option pricing”, [89]. Examples of such models are the Merton jump-diffusion model with Gaussian jumps and the Kou model with double exponential jumps. The first of these models extended the Black-Scholes model to a model that attempts to capture the negative skewness and excess kurtosis of the log stock price density

4 by a simple addition of a compound Poisson jump process. In the Merton jump diffusion process there are two sources of randomness. Changes in the asset price consist of a normal component (continuous diffusion) that is modeled by a Brownian motion with drift and an abnormal component (discontinuous) that is modeled by a compound Poisson process. Asset price jumps are assumed to be independent and identically distributed. The Poisson process causes the asset price to jump randomly and the jump sizes are normally distributed [69]. Kou (2002) introduced a similar model, in which the distribution of jump sizes is an asymmetric double exponential. The study of the Kou (2002) model is the core of this thesis and is developed in the following chapters. Both these models possess certain features that they share with observed market prices and which are not present in the popular Black–Scholes (1973) model. The advantage of the Kou´s model compared to the Merton model is that, due to the memoryless property of exponential random variables, “large explicit formulas” for many important types of options, path-dependent options included, may be obtained [31] and [60]. Empirical studies have indicated that the double exponential jump-diffusion model ﬁts the asset price process better than the normal jump-diffusion model does [79], [16] and [66]. Therefore, we have chosen the Kou model, [60], for this course ﬁnal work.

5 3. Lévy processes

In this Chapter, we ﬁrst introduce the necessary knowledge to understand the basic assumptions of Lévy processes, with respective matematical tools, and then present a detailed derivation of a jump diffusion model, the Kou (2002) model, which is the main theme of this thesis.

3.1. Overview

Historically, Lévy processes have played a central role in the study of stochastic processes because they model a wide variety of physical, biological, engineering and economic scenarios [62]. In mathematical finance, Lévy processes have becoming extremely fashionable because they can describe the observed reality of financial markets in a more accurate way than models based on Brownian motion, [74]. A general Lévy process can not only generate continuous paths via a Brownian motion and rare and large events via a compound Poisson process, but it can also generate frequent jumps of different sizes, [92]. Lévy processes consist of a combination of a linear drift, a Brownian motion and an independent jump process, [64]. Therefore, the simplest Lévy process is the linear drift, a deterministic process. Brownian motion is the only non-deterministic Lévy process with continuous sample paths. The Poisson and Compound Poisson processes are pure jump processes, [74]. There are two types of Lévy processes: jump-diffusion and infinite activity Lévy processes, [58]. In the first category, the total change in price is a composition of two components: the normal evolution of prices, which is modeled by a diffusion process, and unusual significant discontinuous changes in prices modeled by a jump process. Here, the jumps represent rare events (for example, crashes and large drawdowns) and in any given finite interval there are only finite many jumps, [31] and [62]. One could also consider pure jump processes of finite activity without diffusion component, but they do not lead to a realistic description of price dynamics, [31] and [75]. The second category consists of processes with infinite number of jumps in every interval. In these processes it is not necessary to introduce a Brownian component because the dynamics of jumps is already rich enough to generate nontrivial local behavior, [25] and [58]. This gives a more realistic description of the price process at various time scales, [45] and [65]. Anyway, this second category is not a subject for this thesis. Two important examples of models based on a Lévy process are Merton (1976), [71], who solved for

6 option prices with log-normally distributed jumps, and Kou (2002), [60], who proposed the same, but where the jump size is double-exponentially distributed. To obtain an option pricing formula, the authors relied upon particular properties of those distributions. Merton’s solution relies upon a product of lognormal variates being lognormally distributed. Kou’ s derivation stresses the importance of the memoryless property of the exponential distribution, [64]. Other models based in Lévy processes are:

• Generalized Hyperbolic model — see [39] and [40];

• Normal Inverse Gaussian, as a special case of the Generalized Hyperbolic — see [13] and [38];

• The Carr-Madan-Geman-Yor Lévy process — see [25];

• The Meixner process — see [84] and [85];

• Pareto-Beta Jump Diffusion model — see [78] and [80].

These models help explain some, but not all, of the deviations from the benchmark model (the Geo- metric Brownian motion). Any model for stock returns with independent increments, such as Lévy processes, cannot incorporate the volatility clustering effect. As the jump-diffusion model is a special case of a Lévy process, it cannot deal with the volatility clustering effect straightway. In this case, one possibility is to combine Lévy jump-diffusion processes with other processes to incorporate this stylized fact — see for example [14], [27] and [36].

3.2. Mathematical tools

Mathematically, a stochastic process {Lt}t≥0 is called a Lévy process if it has almost surely right- continuous paths with left limits and if its increments are independent and time-homogeneous, [31], [45], [74], [88].

Deﬁnition 3.2.1. A stochastic process {Lt}t≥0 on a probability space (Ω, F, P) is a Lévy process if: • The process starts at zero:

L0 = 0;

• The process has independent increments:

Lt − Ls is independent of {Lu}u≤s, for 0 ≤ s < t;

• The process has stationary increments:

Lt − Ls is equal in distribution to Lt−s, for 0 ≤ s < t;

• The process is continuous in probability:

∀ > 0, limh→0 P [|Lt+h − Lt| ≥ ] = 0.

7 Deﬁnition 3.2.2. A stochastic process {Nt}t≥0 is a Poisson process if it can be deﬁned as: X Nt = 1t≥Tn (3.1) n≥1

Pn where Tn = i=1 τi and {τi}i≥1 are a sequence of independent exponential random variables with parameter λ.

The Poisson process is a counting process, that is, it quantiﬁes the number of events (i.e., the jumps) that occurred randomly up to time t. It has jumps of size one only and its paths are constant between two jumps, [31], [76]. Figure 3.1 shows a path of a Poisson process.

Figure 3.1.: Sample path of a Poisson process

Proposition 3.2.1. Let {Nt}t≥0 be a Poisson process. For any t > 0, Nt follows a Poisson distribution with parameter λt: (λt)n ∀n ∈ , [N = n] = exp (−λt) (3.2) N P t n! Proof. See [31, Chapter 2.5].

Proposition 3.2.2. The distribution of the increments of a Poisson process has the Poisson distribution with parameter λ (t − s):

(λ (t − s))n [N − N = n] = exp (−λ (t − s)) , n ∈ (3.3) P t s n! N

Proof. See [87, Chapter 11].

The expected number of jumps given by a Poisson process is proportional to the length of the interval considered and the intensity λ is the proportionality constant, [48].

8 Proposition 3.2.3. Let {Nt}t≥0 be a Poisson process with parameter λ. The number of expected jumps over an interval [0, t] is given by:

E [Nt] = λt (3.4)

The variance of {Nt}t≥0 have the same expression of the expected value:

V ar [Nt] = λt (3.5)

Proof. See Appendix A.

The Poisson distribution allows us to answer questions like the number of events that occur in an interval of time. On the other hand, if we want to know the elapsed time between two events, then the exponential distribution gives the answer.

Deﬁnition 3.2.3. A random variable ξ has exponential distribution with parameter λ, and we denote ξ ∼ Exp(λ), if its density function can be written as:

−λx fξ (x) = λe , x ≥ 0 (3.6)

An important property of the exponential distribution is the memoryless property.

Proposition 3.2.4. If ξ is a random variable with exponential distribution, then ξ verify the following property, known as the memoryless property: If ξ ∼ Exp (λ), then P (ξ > x + h | ξ > x) = P (ξ > h)

Proof. See [31, Chapter 2.5].

This property says that, if we know that the jump has not occurred at time t, the probability of arriving h unit of time later that t is the same as in the beginning t = 0. In other words, knowing that the jump has not arrived, we do not have any information to know when it will occur. The exponential distribution is the only continuous distribution that has the memoryless property. If we want to model more jumps we need to introduce a sequence of independent random variables with exponential distribution each of one with parameter λ. The Poisson process itself is not suitable to model asset prices since the constraint that the jump size is always equal to one is too restrictive. Consequently, there is some interest in considering jump processes that can have random jumps size.

Deﬁnition 3.2.4. A stochastic process {Yt}t≥0 is a compound Poisson process with intensity λ > 0 if it can be deﬁned as: N Xt Yt = Ji (3.7) i=1

9 where jump sizes Ji are independent and identically distributed random variables with a given distribution F and {Nt}t≥0 is a Poisson process with intensity λ, independent of {Ji}i≥1.

A compound Poisson process is a process whose jumps size is not any more one like the simple Poisson process. A compound Poisson process is a piecewise constant process where the jump times follow a Poisson process and the jump sizes are independent and identically distributed random variables with a given distribution. Figure 3.2 shows a path of a compound Poisson process.

Figure 3.2.: Sample path of a compound Poisson process

The main key to understand the compound Poisson process is the conditional expectation concept.

Deﬁnition 3.2.5. The conditional expectation of a stochastic process {Yt}t≥0 is given by:

∞ X E [Yt] = E [Yt|Nt = n] P [Nt = n] (3.8) n=0 where P [Nt = n] gives the probability that exactly n events (jumps) have occurred by time t, which is deﬁned in equation (3.2), and E [Yt|Nt = n] is the conditional expectation for Yt given that there has been n events (jumps) until time t.

Proposition 3.2.5. Let {Yt}t≥0 be a compound Poisson process with intensity λ, as deﬁned in 3.2.4. The expected value of this process can be computed as the product of the mean number of jumps times and the mean jump size:

E [Yt] = λtE [J] The variance of this process has the following expression:

2 V ar [Yt] = λtE J

10 Proof. See Appendix A.

The paths of a Lévy jump-diffusion process can be described by:

N ! Xt Lt = bt + σWt + Ji − tλk (3.9) i=1 where b ∈ R, σ ∈ R+, {Wt}t≥0 is a standard Browniam motion, {Nt}t≥0 is a Poissson process with parameter λ and {Jk}k≥1 is a independent and identically distributed sequence of random variables with probability distribution F (describes the distribution of the jumps which arrive according to the Poisson process). All sources of randomness are mutually independent and k = E [J] is ﬁnite.

In general, a Lévy process {Lt}t≥0 can be characterized by its characteristic function. Deﬁnition 3.2.6. The characteristic function of a random variable X is the function ϕ deﬁned by:

ϕ (u) = E [exp {iuX}] , u ∈ R (3.10)

Deﬁnition 3.2.7. The characteristic exponent function of a Lévy jump-diffusion process {Lt}t≥0 is a continuous function ψ such that:

E [exp {iuLt}] = exp {tψ (u)} , u ∈ R (3.11) where ψ is given by: u2σ2 Z ψ (u) = iub − + eiux − 1 ν (dx) (3.12) 2 R where ν is a Lévy measure.

Lévy’ s measure is responsible for the richness of the class of Lévy processes and carries useful information about the structure of the process [74, pp. 15]. Intuitively speaking, the Lévy measure represents the expected number of jumps of a certain height in a time interval of length 1, [74, pp. 14]. The Lévy measure of a Lévy jump-diffusion is ν (dx) = λF (dx), where F is the distribution function of the jump size. From that, we can deduce that the expected number of jumps is λ. A Lévy process has ﬁnite variation or not, depending on the Lévy measure (and on the presence or absence of a Brownian part) [74, pp. 15]. Since the characteristic function, ϕ, of a random variable determines its distribution, we have a “char- acterization” of the distribution of the random variables underlying the Lévy jump-diffusion process [31, Chapter 3]: 2 2 Z u σ iux ϕLt (u) = exp t iub − + e − 1 ν (dx) (3.13) 2 R

11 This is the so-called Lévy-Khintchine representation of a Lévy jump-diffusion process, which shows that a Lévy jump-diffusion process is “made of three parts” [31, Chapter 3.5], [74, Chapter 1.4] and [83]. The opposite way is the issue of the Lévy-Itô decomposition. The Lévy-Itô decomposition says that we can decompose any Lévy jump-diffusion process into three independent Lévy processes, [74, Chapter 1.6], [83, Chapter 4] and [62, Chapter 2]:

• a deterministic linear process whit parameter b;

• a Brownian motion with coefﬁcient σ;

• a compound Poisson process with arrival rate λ and jump magnitude F (dx);

Another important tool is the Itô’ s formula, [76], [77].

Theorem 3.2.1. Let (Xt)t≥0 be a diffusion process with jumps, deﬁned as the sum of a drift term, a Brownian stochastic integral and a compound Poisson process:

N Z t Z t Xt Xt = X0 + asds + bsdWs + 4Xi (3.14) 0 0 i=1

hR T 2 i where at and bt are continuous processes with E 0 bt dt < ∞. 1,2 Then, for any C function f : [0,T ] × R → R, the process Yt = f (t, Xt) can be represented as

Z t ∂f ∂f f (t, Xt) − f (0,X0) = (s, Xs) + as (s, Xs) ds 0 ∂s ∂x

Z t 2 Z t 1 2 ∂ f ∂f + bs 2 (s, Xs) ds + bs (s, Xs) dWs 2 0 ∂x 0 ∂x X + f XTi− + 4Xi − f XTi− (3.15)

{i≥1,Ti≤t} or in the differential notation

∂f ∂f 1 ∂2f dY = df (t, X ) = (t, X ) dt + a (t, X ) dt + b2 (t, X ) dt t t ∂t t t ∂x t 2 t ∂x2 t

∂f + b (t, X ) dW + f X + 4X − f X (3.16) t ∂x t t t− t t− Proof. See [62, Chapter 4], [74, Chapter 1.11] and [31, Chapter 8.3] .

Itô’ s formula for the general Lévy process can be found in [31].

12 There are many models based on Lévy jump-diffusion processes. However, different exponential Lévy models proposed in the ﬁnancial modelling literature simply correspond to different choices for the Lévy measure [30]. In models based on a Lévy process the asset price St is represented as Lt St = S0e where Lt is the Lévy process. In other words, saying that an asset price process St is modeled as an exponential of a Lévy process Lt simply means that its log-return follows a Lévy process, that is, ln St = L . For a jump-diffusion process, the log-return will be a Lévy process S0 t such that: N Xt Lt = µt + σWt + Ji and X0 ≡ 0 (3.17) i=1

PNt where {Wt}t≥0 is a standard Brownian motion, {Nt}t≥0 is a Poisson process with rate λ, i=1 Ji is a compound Poisson process with jump intensity λ, and µ and σ are the drift and volatility of the diffusion part, respectively. The jump sizes {J1,J2, ...} are independent and identically distributed random variables.

3.3. Double exponential jump-diffusion model

Kou (2002) proposed the double exponential jump-diffusion model to incorporate the leptokurtic feature and “volatility smile” in option pricing [60]. Kou’ s model is very simple but has rich theoretical implications. This model can explain those two empirical phenomena and, simultaneously, it leads to analytical solutions to many option pricing problems, like pricing European call and put options, interest rate derivatives and path-dependent options [61] and [55]. There are many models that can incorporate some of the three properties — leptokurtic feature, “volatility smile” and analytical tractability. The jump-diffusion model proposed by Kou can incorporate all three properties under a uniﬁed framework [56, pp. 2-3]. This double exponential jump-diffusion model is an exponential Lévy model with ﬁnite jump intensity, in which the price of the underlying asset is modeled by two parts. A continuous part, in which the logarithm of the asset price is assumed to follow a Brownian motion, and a jump part constituted by a compound Poisson process. The jump times are driven by a Poisson process and the jumps size have a two-sided exponential distribution. Two singular properties of the double exponential distribution — the leptokurtic feature and the memoryless property of the double exponential distribution — are crucial for the model [60], [55]. The leptokurtic feature of the jump size distribution is inherited by the return distribution and, with the double exponential jump-diffusion model, it is possible to get not only an higher peak but also heavier tails (in particular, the left tail) for the asset return distribution [60], [55]. The memoryless property of the exponential distribution explains why the closed form solutions for various option pricing problems

13 are feasible under this model, while it seems impossible for many other model, including the Merton normal jump diffusion model [60]. To model the asset price St under the risk-neutral measure Q, the following stochastic differential equation is used: Nt ! dSt X = (r − δ − λζ) dt + σdW + d (V − 1) (3.18) S t i t i=1 where η η ζ = E [V − 1] = p 1 + q 2 − 1 (3.19) η1 − 1 η2 + 1 and r is the interest rate, δ is the continuous dividend yield, σ is the volatility of the returns, {Wt}t≥0 is a standard Brownian motion, {Nt}t≥0 is a Poisson process with rate λ, {Vi}i≥1 is a sequence of independent and identically distributed nonnegative random variables such that X = ln (V ) has an asymmetric double exponential distribution with density

−η1x η2x fX (x) = pη1e 1{x≥0} + qη2e 1{x<0}, η1 > 1, η2 > 0 (3.20)

Here p represents the probability of upward jumps and q = 1−p represents the probability of downward jumps. The inverse of the parameters η1 and η2 represent the means of the right tail and left tail of the distribution. All three sources of randomness {Wt}t≥0, {Nt}t≥0 and {Xi}i≥1 are assumed to be independent. According to [57, pp. 4], each random variable X can be decomposed by:

( + d ξ with probability p X = (3.21) −ξ− with probability q

+ − where ξ ∼ Exp (η1) and ξ ∼ Exp (η2).

Proposition 3.3.1. Let ξ be a random variable with exponential distribution, X a random variable with double exponential distribution and V = eX . Then, we have:

• ξ+ = 1 and ξ− = 1 E η1 E η2 2 2 • V ar ξ+ = 1 and V ar ξ− = 1 η1 η2

2 p q 1 1 p q • [X] = − and V ar [X] = pq + + 2 + 2 E η1 η2 η1 η2 η1 η2

• [V ] = eX = p η1 + q η2 E E η1−1 η2+1

Proof. See Appendix A.

14 Proposition 3.3.2. The solution of the stochastic differential equation (3.18) is given by:

( N ) σ2 Xt S = S exp r − δ − − λζ t + σW + X (3.22) t 0 2 t i i=1

Proof. See Appendix A.

X Note that, since Vi = e i , the equation (3.22) can be written as:

N σ2 Yt S = S exp r − δ − − λζ t + σW V (3.23) t 0 2 t i i=1

In Kou model, the Lévy process under Q is equal to

N σ2 Xt L = r − δ − − λζ t + σW + X (3.24) t 2 t i i=1

Proposition 3.3.3. In the Kou model the characteristic exponent function under Q of the Lévy process is 2 2 2 σ σ u η1 η2 ψLt (u) = iu r − δ − − λζ − + λ p + q − 1 (3.25) 2 2 η1 − iu η2 + iu Proof. See Appendix A.

Proposition 3.3.4. The discounted price process is a martingale under the risk-neutral measure Q:

−rt E e St | F0 = S0

Proof. See [56, pp. 9] and Proposition A.0.1 in Appendix A.

To obtain closed-form pricing solutions for option pricing under the Kou (2002) model we need to compute the expectation of the discounted terminal payoff of the option under the risk-neutral measure Q. For European call options we have:

−rT −rT + c(S0,K,T ) = EQ e c(ST ,K,T ) = e EQ (ST − K) (3.26)

Taking the equation (3.23) with t=T and substituting the expression of ST into the equation (3.26), we get  + NT ! σ2 −rT r−δ− 2 −λζ T +σWT Y c(S0,K,T ) = e EQ  S0e Vi − K  (3.27) i=1

15 We know that the standard Brownian motion, {W } , has a normal distribution with mean 0 and t t≥0 √ variance t. If Z is a random variable with a standard normal distribution, then we have Wt = tZ in distribution, [56, pp. 9], which results directly from: Since 2 Wt ∼ N (0, t) =⇒ σWt ∼ N 0, σ t , and √ Z ∼ N (0, 1) =⇒ σ tZ ∼ N 0, σ2t , then √ d σWt = σ tZ.

Consequently, equation (3.27) can be rewritten as

 + √ NT ! σ2 −rT r−δ− 2 −λζ T +σ TZ Y c(S0,K,T ) = e EQ  S0e Vi − K  (3.28) i=1

Using the law of total probability,

 +  ∞ √ NT ! σ2 −rT X r−δ− 2 −λζ T +σ TZ Y c(S0,K,T ) = e EQ  S0e Vi − K | NT = n  P [NT = n] n=0 i=1 (3.29) Combining the equations (3.2) and (3.29), we get

+ ∞ " 2 √ n ! # n X r−δ− σ −λζ T +σ TZ Y (λT ) c(S ,K,T ) = e−rT S e 2 V − K e−λT 0 EQ 0 i n! n=0 i=1

∞ " √ +# σ2 Pn n X r−δ− T +σ TZ+ Xi (λT ) = e−rT S e−λζT e 2 i=1 − K e−λT (3.30) EQ 0 n! n=0

The randomness on the equation (3.30) comes from a normal random variable and n double exponential random variables. Following [57, pp. 24], the sum of n double exponential random variables can be decomposed in a mixed sum of exponential random variables — see Proposition A.0.2 in Ap- pendix A.

Combining equations (3.30) and (A.13), then the equation (3.30) comes

" 2 √ +# 0 r−δ− σ T +σ TZ (λT ) c(S ,K,T ) = e−rT S e−λζT e 2 − K e−λT 0 EQ 0 0!

16 ∞ n " √ # σ2 k + + n X X r−δ− T +σ TZ+P ξ (λT ) +e−rT P S e−λζT e 2 i=1 i − K e−λT n,kEQ 0 n! n=1 k=1 ∞ n " √ # σ2 k − + n X X r−δ− T +σ TZ−P ξ (λT ) + e−rT Q S e−λζT e 2 i=1 i − K e−λT (3.31) n,kEQ 0 n! n=1 k=1 where the probabilities Pn,k and Qn,k are deﬁned in [56, pp. 17]. We present an additional explanation in Proposition A.0.3 of Appendix A.

In the equation (3.31) we have now three expectations to compute. In the last two, the randomness comes from a normal random variable and from a sum of exponential random variables. To calculate these expected values we need to know the density of the sum of two random variables, one of them is a normal distributed random variable, Z, and the other is a sum of k exponential random variables, Pk + Pk − i=1 ξi or i=1 ξi . Proposition 3.3.5. The density of the sum of two random variables Z = X + Y is

Z +∞ fZ (z) = fZ (x, z − x) dx −∞

Proof. See Appendix A.

Proposition 3.3.6. Let ξi , i = 1, 2, ..., k, be independent and identically distributed exponential random variables with rate η > 0. The probability density function of their sum is

k−1 −ηt (ηt) fPk ξ (t) = ηe , t > 0, i=1 i (k − 1)! gamma distribution with parameters k and η.

Proof. See [76].

Proposition 3.3.7. The density function of the sum of k exponential random variables, with rate η, and the normal random variable is

2 k (ση) 1 t 2 −tη fZ+Pk ξ (t) = (ση) e √ e Hhk−1 − + ση i=1 i σ 2π σ

2 k (ση) 1 t 2 tη fZ−Pk ξ (t) = (ση) e √ e Hhk−1 + ση i=1 i σ 2π σ

where the functions Hhn (.) are deﬁned in [9] and [50].

Proof. See Appendix A.

17 Proposition 3.3.8. The expected values provided by [57, pp. 28]

+ 2 b+c(Z+Pk ξ ) b k (ση) 1 1 E ae i=1 i − K = ae (ση) e 2 √ Ik−1 h; c − η; − ; −ση Q σ 2π σ

2 k (ση) 1 1 −K(ση) e 2 √ Ik−1 h; −η; − ; −ση , k ≥ 1, c < η σ 2π σ and

+ 2 b+c(Z−Pk ξ ) b k (ση) 1 1 E ae i=1 i − K = ae (ση) e 2 √ Ik−1 h; c + η; ; −ση Q σ 2π σ

2 k (ση) 1 1 −K(ση) e 2 √ Ik−1 h; η; ; −ση , k ≥ 1, η > −c σ 2π σ

Proof. See Appendix A.

Now, the calculation of the expected values of the equation (3.31) comes straightforward. Using −λζT σ2 the results of Proposition 3.3.8, with a = S0e , b = r − δ − 2 T and c = 1, then the equation (3.31) becomes

−λT −(δ+λζ)T −rT c(S0,K,T ) = e S0e Φ(d+) − Ke Φ(d−)

 √ k ∞ n √ 2 σ2 σ T η1 (σ T η ) √ −rT X X −λζT r−δ− T 1 1  2 √ √ 2 √ +e Pn,k S0e e e Ik−1 h; 1 − η1; − ; −σ T η1 n=1 k=1 σ T 2π σ T

√ k √ 2 # n σ T η1 (σ T η ) √ 1 1 −λT (λT ) −K √ √ e 2 Ik−1 h; −η ; − √ ; −σ T η e σ T 2π 1 σ T 1 n!

 √ k ∞ n √ 2 σ2 σ T η2 (σ T η ) √ −rT X X −λζT r−δ− T 2 1  2 √ √ 2 √ +e Qn,k S0e e e Ik−1 h; 1 + η2; ; −σ T η2 n=1 k=1 σ T 2π σ T

√ k √ 2 # n σ T η2 (σ T η ) √ 2 1 −λT (λT ) − K √ √ e 2 Ik−1 h; η ; √ ; −σ T η e (3.32) σ T 2π 2 σ T 2 n!

By simplifying these terms, we ﬁnally came to the formula for call options pricing under the Kou model: −λT −(δ+λζ)T −rT c(S0,K,T ) = e S0e Φ(d+) − Ke Φ(d−)

18 √ 2 ∞ n σ2 (σ T η ) n √ k √ − δ+λζ+ T 1 1 X −λT (λT ) X 1 +S e 2 e 2 √ √ e P σ T η I h; 1 − η ; − √ ; −σ T η 0 n! n,k 1 k−1 1 1 σ T 2π n=1 k=1 σ T

√ 2 ∞ n (σ T η ) n √ k √ −rT 1 1 X −λT (λT ) X 1 −Ke e 2 √ √ e P σ T η I h; −η ; − √ ; −σ T η n! n,k 1 k−1 1 1 σ T 2π n=1 k=1 σ T

√ 2 ∞ n σ2 (σ T η ) n √ k √ − δ+λζ+ T 2 1 X −λT (λT ) X 1 +S e 2 e 2 √ √ e Q σ T η I h; 1 + η ; √ ; −σ T η 0 n! n,k 2 k−1 2 2 σ T 2π n=1 k=1 σ T

√ 2 ∞ n (σ T η ) n √ k √ −rT 2 1 X −λT (λT ) X 1 − Ke e 2 √ √ e Q σ T η I h; η ; √ ; −σ T η n! n,k 2 k−1 2 2 σ T 2π n=1 k=1 σ T (3.33) where:

• ζ is deﬁned in equation (3.19);

2 S0 σ ln K + r − δ ± 2 − λζ T • d± = √ ; σ T

2 • h = ln K + λζ − r + δ + σ T ; S0 2

• Pn,k and Qn,k are deﬁned in equations (A.14) and (A.15);

• Hhi (.) comes from equation A.21 or equation A.20;

• Ik−1 (x; α; β; θ) is deﬁned in equation (A.24) and evaluable by Proposition A.0.6;

• Φ represents the cumulative density function of the normal distribution.

19 Using the put-call parity:

−rT −rT p(S0,K,T ) − c(S0,K,T ) = Ke − S0 ⇐⇒ p(S0,K,T ) = Ke − S0 + c(S0,K,T ) (3.34) we can obtain the expression for the fair value of a European put option under the Kou model.

−(δ+λ+λζ)T −rT −λT p(S0,K,T ) = S0 e Φ(d+) − 1 + Ke 1 − e Φ(d−)

√ 2 ∞ n σ2 (σ T η ) n √ k √ − δ+λζ+ T 1 1 X −λT (λT ) X 1 +S e 2 e 2 √ √ e P σ T η I h; 1 − η ; − √ ; −σ T η 0 n! n,k 1 k−1 1 1 σ T 2π n=1 k=1 σ T

√ 2 ∞ n (σ T η ) n √ k √ −rT 1 1 X −λT (λT ) X 1 −Ke e 2 √ √ e P σ T η I h; −η ; − √ ; −σ T η n! n,k 1 k−1 1 1 σ T 2π n=1 k=1 σ T

√ 2 ∞ n σ2 (σ T η ) n √ k √ − δ+λζ+ T 2 1 X −λT (λT ) X 1 +S e 2 e 2 √ √ e Q σ T η I h; 1 + η ; √ ; −σ T η 0 n! n,k 2 k−1 2 2 σ T 2π n=1 k=1 σ T

√ 2 ∞ n (σ T η ) n √ k √ −rT 2 1 X −λT (λT ) X 1 − Ke e 2 √ √ e Q σ T η I h; η ; √ ; −σ T η n! n,k 2 k−1 2 2 σ T 2π n=1 k=1 σ T (3.35)

For the pricing formula of the American options see [60], [61], [59].

20 4. Pricing bitcoin options under Kou’s model

In this Chapter we present the practical part of our work. First, we give an overview of bitcoin options. Then, we calibrate the model to the market data. Finally, we present and analyze the results obtained with the Kou model using the Black-Scholes model as a benchmark.

4.1. Bitcoin options

To test the Kou model in option pricing, we use bitcoin as the underlying asset. Bitcoin, proposed by Nakamoto (2008), [72], is an electronic financial mechanism providing features that resemble an established currency system with its own money creation and transaction regime but relies on a decen- tralized organizational structure. The method for registration of bitcoin transactions was the inception of the Blockchain technology, a protocol where the relevant information is recorded in subsequent blocks on a ledger, that is shared by all the nodes of the network, [35]. The Blockchain represents all verified and valid transactions between users of the network and is the fundamental technology underlying cryptocurrencies, smart contracts and more in general smart services, [68]. Perhaps, in the future, the bitcoin options and other derivatives can be negotiated through smart contracts. We can find a short historical evolution of the bitcoin in [28]. However, several recent surveys show that users view bitcoin more as an asset than as a currency, [46], [93]. An interpretation for this is supported by the fact that bitcoin returns react on news events related to this digital currency. Their excessive volatility is more consistent with the behavior of a speculative investment than a currency, [20], [24]. Furthermore, the analysis of transaction data of bitcoin accounts shows that bitcoins are mainly used as a speculative investment and not as a medium of exchange [19]. Figure 4.1 shows the time series of bitcoin prices since 2010.

21 20,000

16,000

12,000

8,000

4,000

0 2010 2011 2012 2013 2014 2015 2016 2017 2018

Figure 4.1.: Bitcoin closing prices - 18/07/2010 to 30/06/2018

In the year 2017, bitcoin has become a subject of interest to economists, banks, governments and the general public. Bitcoin appeared as the latest technological and ﬁnancial phenomenon in almost all media. At the beginning of this year it was worth around 1,000 USD. Throughout 2017, bitcoin’s price accelerated exponentially, reaching 19,343.03 USD on 16/12/2017. Since then, bitcoin has suffered a period of sharp decline, lowering from 7,000 USD in early February 2018. Figure 4.2 shows the histogram and the descriptive statistics for the bitcoin prices.

2,000 Series: CLOSEPRICE Sample 7/18/2010 6/30/2018 1,600 Observations 2905

Mean 1265.605 Median 272.9500 1,200 Maximum 19343.04 Minimum 0.050000 Std. Dev. 2825.545 800 Skewness 3.206615 Kurtosis 13.68523

400 Jarque-Bera 18798.23 Probability 0.000000

0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Figure 4.2.: Bitcoin prices - histogram and the descriptive statistics

Remark the high volatility, the high skewness and the high kurtosis. The high volatility of the bitcoin prices mostly reﬂects the asset’ s rapid growth, not something inherent in the technology. It is mathematically impossible for bitcoin’s rapid growth to continue forever. Once it slows, there are good reasons to think volatility will decline with it [86]. And there are those who believe that bitcoin options is what we need to “tame this beast”. Paul Chou, chief executive and co-founder of LedgerX, on June

22 2015 to the Euromoney magazine, said that “bitcoin volatility has come down but having an options market can reduce that volatility further”. Figure 4.3 illustrates the process of the bitcoin log-returns since 2010.

-.2

-.4

-.6 2010 2011 2012 2013 2014 2015 2016 2017 2018

Figure 4.3.: Bitcoin daily log-returns - 18/07/2010 to 30/06/2018.

The historical volatility of the bitcoin log-returns is about 5.6% as shown in Figure 4.4. The negative skewness means the return has a heavier left tail than the right tail.

1,200 Series: RETURNS Sample 7/18/2010 6/30/2018 1,000 Observations 2904

800 Mean 0.003846 Median 0.001426 Maximum 0.424580 600 Minimum -0.491528 Std. Dev. 0.056684 Skewness -0.157733 400 Kurtosis 14.69406

200 Jarque-Bera 16558.91 Probability 0.000000

0 -0.500 -0.375 -0.250 -0.125 0.000 0.125 0.250 0.375

Figure 4.4.: Bitcoin log-returns - histogram and descriptive statistics

Figure 4.3 suggests the existence of jumps in bitcoin’ s returns process. But, how can we identify a jump in returns? Many empirical and theorical studies proved the existence of jumps and their impact on option pricing, particularly [71], [12], [36]. Jumps are empirically difﬁcult to identify, because only discrete data are available from continuous-time models in which the applications were studied. However, there are some jump detection tests [49], [15], [16], [63], [34], [37] and [54]. We chose to apply the Bipower nonparametric test, proposed in [16], to test the presence of jumps in the bitcoin

23 returns. For this purpose, we used the Matlab code available in the MathWorks webpage, [4]. The input sample data was the bitcoin log-returns from January 2017 and June 2018. The test result shows that the process of bitcoin’ s returns has jumps and gives us a jump size value equal to 0.3272. One can also ask whether a large movement in returns is generated by a jump in returns or by high volatility? Aït-Sahalia (2004), [10], replied to this question. According to him, it is possible to disentangle Brow- nian noise from jumps as long as the sampling frequency is large enough. The higher is the observation frequency, the higher is the probability that a jump can be recognized as such from the observation of a large log-return. And daily frequency is generally considered to be low enough to be largely unaffected by the market microstructure issues [11]. For him, big jumps are changes in asset returns that are rare and much larger than what can be explained by a diffusion process and small jumps can be revealed by the fact that returns of a particular size appear with higher frequency than would be expected from a diffusion process. Besides the high volatility of bitcoin prices, we showed that bitcoin log-returns form a process with jumps. So, now we have two possibilities. One is to use a stochastic volatility model and other is to apply a model with jumps. We decide to apply the last one because there are some evidences for mis- speciﬁcation in the volatility processes [12], [18], [73], [42]. Furthermore, jumps in returns and jumps in volatility play an important role in determining the dynamics of returns, especially in the periods of market stress, while diffusive stochastic volatility plays a secondary role [42]. As popularity in the cryptocurrency grows up the products to trade the underlying asset will widen. De- spite being relatively new, bitcoin option trading is available in a handful of countries, which include the USA. On 02/10/2017, a cryptocurrency trading platform operator, called LedgerX, won approval from the Commodity Futures Trading Commission to clear bitcoin options (European calls and puts), making it the ﬁrst USA federally regulated platform of its kind. Several offshore exchanges like Coin- Desk, from New York, USA, [1], and Deribit, from Netherlands, [2], offer options and futures on bitcoin. As we saw, bitcoin is one of the most volatile asset trading at this time, meaning it is very expensive to buy an option. Figure 4.5 is a price screen obtained from [2].

24 Figure 4.5.: Bitcoin option prices on 24/08/2018

The above screen for European calls and puts with 6 days to expire shows implied volatilities ranging from 75.0% to 211.4%, for calls, and from 68.5% to 181.8%, for puts, with strike prices from 4,000 USD to 10,000 USD. So, they are very expensive yet. On the other hand, we see an outrageously wide bid/ask spread. For example, the bid/ask spread for the 8,000 USD put is 61.77 USD. In addition, there is a lack of liquidity of this options. There are different versions of the option prices in the market, namely the bid and the ask prices. Neither the ﬁrst nor the second one does express the true value of option. Thus, we will use the arithmetic average of bid and ask as our option market data to calibrate the model. We used bitcoin call and put options market prices on 16/12/2017, 05/02/2018 and 28/06/2018, collected from [3], that are listed in Figures B.1, B.2 and B.3 of Appendix B. We chose these days because the ﬁrst and the third were the days where the price of the underlying reached its maximum and minimum values, respectively, since November, 2017 to July, 2018. The bitcoin mid prices were 19,343.03 USD, 6,914.26 USD and 5,848.26 USD, on those days. The bitcoin data was obtained from the historical daily closing prices available in [1]. The options are priced across different maturities. The treasury yield curve based on Treasury Bills rates is often used as a proxy of risk-free interest rates. In the Kou model, the interest rate is considered constant. So, as a risk-free interest rate r we take the Treasury Bills rate equal to 0.015 from [8]. The estimation of σ and the jump parameters is presented in the next section.

25 4.2. Calibration of Kou’ s model

Before obtaining any practical applicability, the model needs to be calibrated. An option pricing model is used as a device for capturing the features of option prices quoted on the market at a given instant. To achieve this, the parameters of the model are chosen to ﬁt the market prices of options, a procedure known as “calibration”, [31]. While the pricing problem is about computing option values given the model parameters, the calibration problem is about computing the model parameters given the option prices. Thus, the calibration problem is the inverse problem of the pricing problem.

The input data to use in the Kou model are the jump parameters λ, p, η1, η2 and the option parameters

S0, K, T, r, σ, δ, where λ is the intensity of jumps; p represents the probability of upward jumps; η1 is the magnitude of up-jumps; η2 is the magnitude of down-jumps; S0 is the time-0 stock price; K is the strike price; T is the option maturity; r is the interest rate; σ is the volatility; δ is the dividend yield. Given that bitcoin does not pay dividends, we will set δ equal to zero. There are many different methods for the parameters estimation and there are some studies to assess the performance of the double exponential jump-diffusion model for different risky assets by using several techniques of parameter estimation. We shall mention some of them here. Ramezani and Zeng (2004), [79], and Maekawa et al. (2008), [66], used the maximum likelihood estimation method to obtain parameter estimates for the double exponential jump-diffusion. The generalized method of moments has been investigated by Tuzov (2006), [90], in estimating parameters of the double exponential jump-diffusion model. Cont and Tankov (2009), [32], estimated the parameters of the double exponential jump-diffusion model by using the empirical characteristic exponent. In the study of Emenogu (2012), [41], the parameters of Kou’ s model were estimated by maximum likelihood estimation, by the empirical characteristic function method, by the cumulant matching method and by the generalized method of moments. Hu et al. (2006), [47], and Chen et al. (2017), [27], proposed a Markov chain Monte Carlo method estimation for the double exponential jump-diffusion model. Matsuda (2005), [70], calibrated the Merton (1976) jump-diffusion model using a regularization method with relative entropy like Cont and Tankov (2004), [31], did. Recently, Karimov (2017), [51], has conducted a numerical implementation and parameter estimation under the Kou model.

In the Kou model we have six unknow parameters θ = (µ, σ, λ, p, η1, η2), where µ is the drift component of returns and the other parameters were deﬁned above. However, the parameter µ is easily computed by the drift of the stochastic differential equation (3.18). So, we just have to calibrate the other ﬁve parameters. It should be noted that Kou, [60], does not explain how he has estimated the parameters in his model. In this thesis, our objective is to calibrate the model by minimizing the sum of squared weights for the selected options, subject to the following constraints: σ > 0, η1 > 1, η2 > 0 and 0 ≤ p ≤ 1. To do this, we used the fmincom Matlab function to solve the optimization problem and to obtain the estimated parameters. The output is presented in Appendix B.

26 4.3. Pricing bitcoin options under Kou’s model

The main purpose of financial modelling is to price financial derivatives. In most of the jump diffusion models it is not possible to get a closed-form solution for option prices. In this case, we would need to use numerical methods to find these option prices. However, in the Kou model this is not necessary, because there are closed-form solutions for all European-style options as we saw in the previous chapters. After estimating the parameters by the method described in the previous section, we introduce the values in equations (3.33) and (3.35) to obtain the model prices of the bitcoin options. Appendix C contains the Matlab code implementing the model. To obtain the Black-Sholes model prices we used the blsprice Matlab function available in [5]. Thus, in this section we present the comparison of the prices derived by the Kou model and by the Black-Sholes model with the market prices. Complete results of our calculations are presented in Appendix B and discussed in section 4.4. Here we display some graphs with the relationship between the market prices and both models prices. Then we evaluate the output of the model and build the surface volatility for a set of options. The model was applied to 44 options of 16/12/2014, 42 of 05/02/2018 and 102 of 06/28/2018. Figure 4.6 displays the fit of model prices to the market prices for call options with 13 days to expiry, for a range of strikes from 6,000 USD to 20,000 USD. The market prices go from 1,700 USD to 13,000 USD. For the only out-of-the-money call, the model price is lower than the market price, while in the deep in-the-money calls the model price is higher than the market price. For this set of options there are no significant differences betweeen the Kou and the Black-Scholes models.

Calls on 16/12/2017, 13 days to maturity Calls on 16/12/2017, 13 days to maturity 14000 14000 Market Market Kou Black-Scholes 12000 12000

10000 10000

8000 8000

6000 6000 Options price Options price

4000 4000

2000 2000

0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strikes ×104 Strikes ×104

Figure 4.6.: Kou vs Black-Scholes - calls on 16/12/2017 with 13 days to maturity

Figure 4.7 displays the ﬁt of model prices to the market prices for put options with the previous time to maturity and range strikes. The model prices are lower than the market prices and the ﬁt is better for the out-of-the-money puts than for the only in-the-money put.

27 Puts on 16/12/2017, 13 days to maturity Puts on 16/12/2017, 13 days to maturity 3000 3000 Market Market Kou Black-Scholes 2500 2500

2000 2000

1500 1500 Options price Options price 1000 1000

500 500

0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strikes ×104 Strikes ×104

Figure 4.7.: Kou vs Black-Scholes - puts on 16/12/2017 with 13 days to maturity

In Figure 4.8, the model prices are higher than market prices and there do not seem to be signiﬁca- tive differences between the two models.

×104 Calls on 16/12/2017, 377 days to maturity ×104 Calls on 16/12/2017, 377 days to maturity 1.6 1.6 Market Market Kou Black-Scholes 1.4 1.4

1.2 1.2

1 1 Options price Options price 0.8 0.8

0.6 0.6

0.4 0.4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.8.: Kou vs Black-Scholes - calls on 16/12/2017 with 377 days to maturity

Figure 4.9 shows a good ﬁt for the long-term put options.

28 ×104 Puts on 16/12/2017, 377 days to maturity ×104 Puts on 16/12/2017, 377 days to maturity 3.5 3.5 Market Market Kou Black-Scholes 3 3

2.5 2.5

2 2

1.5 1.5 Options price Options price

1 1

0.5 0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.9.: Kou vs Black-Scholes - puts on 16/12/2017 with 377 days to maturity

Figures 4.10 and 4.11 show the performance of the Kou model compared to the Black-Scholes model for the speciﬁed observation date and time to maturity.

Calls on 05/02/2018, 18 days to maturity Calls on 05/02/2018, 18 days to maturity 2500 2500 Market Market Kou Black-Scholes

2000 2000

1500 1500

1000 1000 Options price Options price

500 500

0 0 0.5 1 1.5 2 0.5 1 1.5 2 Strikes ×104 Strikes ×104

Figure 4.10.: Kou vs Black-Scholes - calls on 05/02/2018 with 18 days to maturity

29 Calls on 05/02/2018, 326 days to maturity Calls on 05/02/2018, 326 days to maturity 4000 4000 Market Market 3500 Kou 3500 Black-Scholes

3000 3000

2500 2500

2000 2000

Options price 1500 Options price 1500

1000 1000

500 500

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.11.: Kou vs Black-Scholes - calls on 05/02/2018 with 326 days to maturity

Puts on 05/02/2018, 18 days to maturity Puts on 05/02/2018, 18 days to maturity 14000 14000 Market Market Kou Black-Scholes 12000 12000

10000 10000

8000 8000

6000 6000 Options price Options price

4000 4000

2000 2000

0 0 0.5 1 1.5 2 0.5 1 1.5 2 Strikes ×104 Strikes ×104

Figure 4.12.: Kou vs Black-Scholes - puts on 05/02/2018 with 18 days to maturity

×104 Puts on 05/02/2018, 326 days to maturity ×104 Puts on 05/02/2018, 326 days to maturity 4.5 4.5 Market Market 4 Kou 4 Black-Scholes

3.5 3.5

3 3

2.5 2.5

2 2 Options price Options price 1.5 1.5

1 1

0.5 0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.13.: Kou vs Black-Scholes - puts on 05/02/2018 with 326 days to maturity

30 The adjustment to the call options on 28/06/2018 is shown in Figures 4.14, 4.15, 4.16, 4.17, 4.18 and 4.19.

Calls on 28/06/2018, 1 days to maturity Calls on 28/06/2018, 1 days to maturity 1200 1200 Market Market Kou Black-Scholes 1000 1000

800 800

600 600 Options price Options price 400 400

200 200

0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 Strikes ×104 Strikes ×104

Figure 4.14.: Kou vs Black-Scholes - calls on 28/06/2018 with 1 days to maturity

Calls on 28/06/2018, 29 days to maturity Calls on 28/06/2018, 29 days to maturity 1200 1200 Market Market Kou Kou 1000 1000

800 800

600 600 Options price Options price 400 400

200 200

0 0 5000 6000 7000 8000 9000 10000 11000 12000 13000 5000 6000 7000 8000 9000 10000 11000 12000 13000 Strikes Strikes

Figure 4.15.: Kou vs Black-Scholes - calls on 28/06/2018 with 29 days to maturity

31 Calls on 28/06/2018, 64 days to maturity Calls on 28/06/2018, 64 days to maturity 1500 1500 Market Market Kou Black-Scholes

1000 1000 Options price Options price 500 500

0 0 5000 5500 6000 6500 7000 7500 8000 8500 9000 5000 5500 6000 6500 7000 7500 8000 8500 9000 Strikes Strikes

Figure 4.16.: Kou vs Black-Scholes - calls on 28/06/2018 with 64 days to maturity

Calls on 28/06/2018, 92 days to maturity Calls on 28/06/2018, 92 days to maturity 1500 1500 Market Market Kou Black-Scholes

1000 1000 Options price Options price 500 500

0 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Strikes ×104 Strikes ×104

Figure 4.17.: Kou vs Black-Scholes - calls on 28/06/2018 with 92 days to maturity

Calls on 28/06/2018, 183 days to maturity Calls on 28/06/2018, 183 days to maturity 2000 2000 Market Market 1800 Kou 1800 Black-Scholes

1600 1600

1400 1400

1200 1200

1000 1000

800 800 Options price Options price

600 600

400 400

200 200

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.18.: Kou vs Black-Scholes - calls on 28/06/2018 with 183 days to maturity

32 Calls on 28/06/2018, 547 days to maturity Calls on 28/06/2018, 547 days to maturity 3000 3000 Market Market Kou Black-Scholes 2500 2500

2000 2000

1500 1500 Options price Options price 1000 1000

500 500

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.19.: Kou vs Black-Scholes - calls on 28/06/2018 with 547 days to maturity

Similarly, the adjustment to the put options on 28/06/2018 is shwon in Figures 4.20, 4.21, 4.22, 4.23, 4.24 and 4.25.

×104 Puts on 28/06/2018, 1 days to maturity ×104 Puts on 28/06/2018, 1 days to maturity 2 2 Market Market 1.8 Kou 1.8 Black-Scholes

1.6 1.6

1.4 1.4

1.2 1.2

1 1

0.8 0.8 Options price Options price

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 Strikes ×104 Strikes ×104

Figure 4.20.: Kou vs Black-Scholes - puts on 28/06/2018 with 1 days to maturity

33 Puts on 28/06/2018, 29 days to maturity Puts on 28/06/2018, 29 days to maturity 7000 7000 Market Market Kou Black-Scholes 6000 6000

5000 5000

4000 4000

3000 3000 Options price Options price

2000 2000

1000 1000

0 0 5000 6000 7000 8000 9000 10000 11000 12000 13000 5000 6000 7000 8000 9000 10000 11000 12000 13000 Strikes Strikes

Figure 4.21.: Kou vs Black-Scholes - puts on 28/06/2018 with 29 days to maturity

Puts on 28/06/2018, 64 days to maturity Puts on 28/06/2018, 64 days to maturity 3500 3500 Market Market Kou Black-Scholes 3000 3000

2500 2500

2000 2000

1500 1500 Options price Options price

1000 1000

500 500

0 0 5000 5500 6000 6500 7000 7500 8000 8500 9000 5000 5500 6000 6500 7000 7500 8000 8500 9000 Strikes Strikes

Figure 4.22.: Kou vs Black-Scholes - puts on 28/06/2018 with 64 days to maturity

Puts on 28/06/2018, 92 days to maturity Puts on 28/06/2018, 92 days to maturity 10000 10000 Market Market 9000 Kou 9000 Black-Scholes

8000 8000

7000 7000

6000 6000

5000 5000

4000 4000 Options price Options price

3000 3000

2000 2000

1000 1000

0 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Strikes ×104 Strikes ×104

Figure 4.23.: Kou vs Black-Scholes - puts on 28/06/2018 with 92 days to maturity

34 ×104 Puts on 28/06/2018, 183 days to maturity ×104 Puts on 28/06/2018, 183 days to maturity 4.5 4.5 Market Market 4 Kou 4 Black-Scholes

3.5 3.5

3 3

2.5 2.5

2 2 Options price Options price 1.5 1.5

1 1

0.5 0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.24.: Kou vs Black-Scholes - puts on 28/06/2018 with 183 days to maturity

×104 Puts on 28/06/2018, 547 days to maturity ×104 Puts on 28/06/2018, 547 days to maturity 4.5 4.5 Market Market 4 Kou 4 Black-Scholes

3.5 3.5

3 3

2.5 2.5

2 2 Options price Options price 1.5 1.5

1 1

0.5 0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 Strikes ×104

Figure 4.25.: Kou x Black-Scholes - puts on 28/06/2018 with 547 days to maturity

From these graphs there seems to be no signiﬁcant difference between the two models. Neverthe- less, in Figure 4.19 it is well visible the best performance of the Kou model for strikes of 15,000 USD and 25,000 USD.

Implied Volatility In the world of trading, volatility is most commonly known as the amount of risk or uncertainty a security has based on changes in the security’ s value. Securities with high volatility are deemed “risky” and securities with low volatility are deemed “safe”. For example, if a stock hardly moves, it will have low volatility. On the other hand, if a stock is moving all over the place, often by 10% every day, the volatility will be enormous because of the extreme and unpredictable price swings. The other common reference of “volatility” is in correlation to options pricing. In this deﬁnition, the meaning of volatility is the same, but it refers to how cheap or expensive options on an underlying asset are.

35 Implied volatility is the expected magnitude of a stock’s futures price changes, as implied by the stock’ s option prices. In general, implied volatility increases when the market is bearish, when investors believe that the asset’ s price will decline over time, and decreases when the market is bullish, when investors believe that the price will rise over time. This is due to the common belief that bearish markets are riskier than bullish markets. Implied volatility is a way of estimating the future ﬂuctuations of a security’ s worth based on certain predictive factors. Thus, as implied volatility rises, so does the price of an option. But implied volatility is all probability. It is only an estimate of future prices, rather than an indication of them. Implied volatility does not predict the direction in which the price change will go. Implied volatility can be determined by using an option pricing model. For example, if we use the Black-Scholes model, implied volatility is the only factor in the model that is not directly observable in the market. Thus, we can use market option prices to calculate implied volatility under such model. Finally, we will use the Matlab code, available in [6], in order to obtain the volatility surface for call options on each of the observation dates.

Implied Volatility Surface

1.35 )

T,M 1.3 ( σ 1.25

1.2

1.15 Implied Volatility 1.1

2.5 2 1 1.5 0.8 0.6 1 0.4 0.5 0.2 Time to Matutity T

S Moneyness M = K

Figure 4.26.: Surface volatility - call options on 16/12/2017

36 Implied Volatility Surface

1.8 ) 1.7 T,M (

σ 1.6

1.5

1.4

1.3 Implied Volatility 1.2

6 0.8 0.7 4 0.6 0.5 0.4 2 0.3 0.2 0.1 Time to Matutity T

S Moneyness M = K

Figure 4.27.: Surface volatility - call options on 05/02/2018

Implied Volatility Surface

3.5 )

T,M 3 ( σ 2.5

1.5

Implied Volatility 1

6 1.4 1.2 1 4 0.8 0.6 2 0.4 0.2 Time to Matutity T

S Moneyness M = K

Figure 4.28.: Surface volatility - call options on 28/06/2018

As we can see in Figures 4.26, 4.27 and 4.28, as the time to maturity decreases, the implied volatility increases. For short-term calls, the higher implied volatility is obtained when the strike price is in the neighborhood of the underlying asset price.

37 4.4. Results

In this section we present and discuss the complete results. The set of 22 call options observed on 12/16/2017 includes 86% in-the-money calls and 14% out-of-the-money calls — see Table B.2. This was a period of fast increase in the price of the bitcoin, reason why it is not surprising to have many in-the-money calls. The Kou model returns prices closer to the market price for all out-of-the-money puts. But, the Black-Scholes model outperforms the Kou model in 60% of the in-the-money calls. In the set of 21 call options observed on 05/02/2018 there are 24% in-the-money calls and 76% out-of- the-money calls — see Table B.3. The Kou model presents better results for the in-the-money calls and for the out-of-the-money calls and puts, while the prices of the Black-Scholes model are closer to the market prices for the in-the-money puts. The set of 51 call options, observed on 28/06/2018, includes 18% in-the-money calls and 82% out-of-the-money calls — see Table B.4. The Kou model presents better results for the puts options and the Black-Scholes model gives the best fit to the market prices for the call options. Given these overall results, we want to know which model is more efficient and if the Kou model brings significant improvements over the Black-Scholes model to pricing bitcoin options. As suggested by [52, pp. 435], we can check the quality of the fitting through the common error measures, such as Average Absolute Error, Average Percentage Error, Average Relative Percentage Error or Root Mean Square Error. We choose the Average Percentage Error (APE) measure, because it does not skew error rates approaching or at zero. The formula is:

N ˆ N 1 X Ci − Ci 1 X AP E = , with ω = C , N ω N i i=1 i=1 where N is equal to the number of quoted option prices, Ci is the market price and Cˆi is the model price. To evaluate how the Kou model manages the price of options on bitcoin, it was compared to the performance of the Black-Sholes model. Thus, the next step was to calculate the measure of ﬁt for each of our models. For this purpose, we used the Matlab code available in [7]. The results are displayed in following tables. As we can see in Table 4.1 errors do not exceed 5% which is a satisfactory result. These results also tell us that, in general, the Kou model outperforms the Black-Sholes model, since it has the lowest values for the measure used.

38 Date Kou model Black-Scholes model BTC options on 16/12/2017 3.86% 4.03% BTC options on 05/02/2018 2.77% 2.85% BTC options on 28/06/2018 4.23% 4.32%

Table 4.1.: APE values by date of data collection - all options

Tables 4.2 and 4.3 show the goodness of ﬁt for calls and puts separately, by observation date. The Kou model gives better results in four of those six subsets. It is also observed that for the options on 05/02/2018 and on 28/06/2018, both models provide better ﬁt in puts than in calls.

Date Kou model Black-Scholes model BTC call options on 16/12/2017 2.90% 2.87% BTC call options on 05/02/2018 9.53% 9.58% BTC call options on 28/06/2018 19.44% 19.85%

Table 4.2.: APE values by date of data collection - calls only

Date Kou model Black-Scholes model BTC put options on 16/12/2017 6.35% 7.01% BTC put options on 05/02/2018 1.82% 1.82% BTC put options on 28/06/2018 3.04% 3.11%

Table 4.3.: APE values by date of data collection - puts only

For each observation date, Tables 4.4, 4.5 and 4.6 display the results by time to maturity. For short and mid term options, the Kou model reveals a better ﬁt to the market prices. In long-term options there are no signiﬁcant differences. However, on 16/12/2017, the options with 104 and 195 days to maturity have all strikes lower than the underlying asset price. For these options, Kou model returns the best result against Black-Scholes model.

Time to maturity Kou model Black-Scholes model 13 days 5.74% 5.83% 104 days 6.36% 6.88% 195 days 3.68% 3.92% 286 days 3.05% 3.05% 377 days 2.19% 2.32%

Table 4.4.: APE values by time to maturity - all options on 16/12/2017

39 Time to maturity Kou model Black-Scholes model 18 days 3.60% 3.61% 53 days 2.81% 2.82% 144 days 3.04% 3.07% 235 days 3.42% 3.44% 326 days 2.40% 2.37%

Table 4.5.: APE values by time to maturity - all options on 05/02/2018

Time to maturity Kou model Black-Scholes model 1 days 4.80% 4.80% 29 days 10.34% 10.52% 64 days 12.57% 13.08% 92 days 5.96% 5.97% 183 days 2.40% 2.43% 274 days 1.95% 1.96% 365 days 4.75% 4.63% 456 days 6.26% 6.11% 547 days 2.73% 3.15%

Table 4.6.: APE values by time to maturity - all options on 28/06/2018

Table 4.7, 4.8, 4.9, 4.10, 4.11 and 4.12 show, for each of the observation dates, the quality of the ﬁt for calls and puts separately, by time to maturity. In all three days of observation, the Kou model performs better than Black-Scholes in short-term calls and puts. But it is in the short-term that both models present the worse performance, for calls in particular.

Time to maturity Kou model Black-Scholes model 13 days 3.29% 3.24% 104 days 2.69% 2.92% 195 days 1.10% 0.97% 286 days 2.82% 2.41% 377 days 3.83% 4.01%

Table 4.7.: APE values by time to maturity - call options on 16/12/2017

40 Time to maturity Kou model Black-Scholes model 13 days 40.05% 42.13% 104 days 29.65% 31.99% 195 days 15.95% 17.94% 286 days 4.03% 5.73% 377 days 0.81% 0.90%

Table 4.8.: APE values by time to maturity - put options on 16/12/2017

Time to maturity Kou model Black-Scholes model 18 days 14.28% 14.33% 53 days 14.83% 14.92% 144 days 6.71% 6.84% 235 days 3.29% 3.41% 326 days 12.57% 12.48%

Table 4.9.: APE values by time to maturity - call options on 05/02/2018

Time to maturity Kou model Black-Scholes model 18 days 2.26% 2.26% 53 days 1.39% 1.39% 144 days 2.40% 2.41% 235 days 3.48% 3.46% 326 days 1.27% 1.25%

Table 4.10.: APE values by time to maturity - put options on 05/02/2018

Time to maturity Kou model Black-Scholes model 1 days 42.51% 42.41% 29 days 17.77% 17.86% 64 days 6.48% 6.62% 92 days 6.73% 5.17% 183 days 15.07% 13.94% 274 days 10.57% 9.25% 365 days 23.23% 21.81% 456 days 34.37% 33.73% 547 days 23.14% 30.56%

Table 4.11.: APE values by time to maturity - call options on 28/06/2018

41 Time to maturity Kou model Black-Scholes model 1 days 3.81% 3.82% 29 days 9.44% 9.63% 64 days 15.03% 15.69% 92 days 5.85% 6.09% 183 days 1.86% 1.94% 274 days 1.13% 1.27% 365 days 2.40% 2.59% 456 days 2.70% 2.61% 547 days 1.51% 1.51%

Table 4.12.: APE values by time to maturity - put options on 28/06/2018

In short, the pattern is similar for the options on 16/12/2017 and 28/06/2018. The mid-term call options have the smallest errors, while big errors are in the long-term and short-term calls. For the put options, the model has the smallest errors in the long-term, while the short-term puts display the biggest errors. As time to maturity decreases, the errors for the put options increases. In the same way, we can see that the subset of calls on 05/02/2018 presents the same pattern, but the subset of puts has an undeﬁned pattern. With this measure of error, the performance between models becomes visible. In general, the Kou model ﬁts better than Black-Scholes model, especially for out-of-the-money bitcoin puts.

42 5. Conclusions

In this thesis, we have presented the Kou (2002) model for pricing options, [60], which is a jump- diffusion model where jump sizes have an asymmetric double exponential distribution. Being an arbitrage-free model based on a Lévy process it has the advantage to incorporate the stylized facts and leads to tractability solutions to option pricing thanks to the properties of the exponential distribution. As we have seen, although the pricing formula is too long, its derivation is a “masterpiece” and involves only simple mathematical tools. The first step was to calculate the expected value of a sum of a normal random variable and n double exponential random variables, whose distribution is unknown. However, the memoryless property allowed a decomposition of the sum of double exponential random variables. Indeed, the sum of independent and identically distributed double exponential random variables can be written as a mix of gamma random variables with the probability weights given by a combinatorial approach. The calculation of these probabilities is very interesting and ingenious. Then, the problem turned into knowing the distribution of a sum of a normal random variable and the sum of exponential random variables. The closed form of its density function can be obtained through special functions, which are easily evaluated via a linear recursion. A final step in deriving the pricing formula consists of algebric simplifications. Theoretically a jump diffusion model reveals a much better performance than a diffusion model and there are many empirical studies corroborating this. To conduct the empirical analysis we use options written on the bitcoin. The choice of bitcoin as the underlying asset was obvious because its behavior seems like a truth jump-diffusion process. Applying the Kou model to financial options with an underlying asset such as the bitcoin placed high expectations about results because the bitcoin has a high level of volatility. The outcomes partially meet expectations. Although the Kou model fits reasonably to the market prices of the bitcoin options, it does not present much better results than the Black- Scholes model. The existence of few available options may be the source of such modest results. The fact that there is low liquidity in this market can also be one of the reasons. Bitcoin options are a recent market and traded prices are still not the result of demand and supply. We also admit that the calibration method did not provide the best parameters for the model. We choose a nonlinear programming solver on cross sectional option prices that finds the minimum of a constrained nonlinear multivariable

43 function which depends too much on the initial guess. A final conclusion is that the goodness of fit test shows that bitcoin option prices under the Kou model are satisfactory, but there is no significant differences for the Black-Scholes model. Further investigation could go in two directions. One is combining stochastic volatility models with the Kou jump-diffusion model. This makes sense because the bitcoin has high volatility, as we saw, and the stochastic volatility models can capture the volatility clustering effects. Other way is to incorporate jumps in volatility, because jumps in volatility provide a factor that combines features from both jumps in returns and diffusive stochastic volatility, [42].

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51 A. Derivations

Proof. of Proposition 3.2.3

• Expected value ∞ ∞ n X X (λt) [N ] = n [N = n] = ne−λt E t P t n! n=0 n=0 ∞ (n−1) X (λt) = λte−λt = λte−λteλt = λt (n − 1)! n=1

• Variance 2 First, we need to compute E Nt :

∞ ∞ n X X (λt) N 2 = n2 [N = n] = n2e−λt E t P t n! n=0 n=0 ! ! λt (λt)2 (λt)3 (λt)2 (λt)3 = e−λt 0 + + 4 + 9 + ... = e−λtλt 1 + 2λt + 3 + 4 + ... 1! 2! 3! 2! 3! ( ! !) (λt)2 (λt)3 (λt)2 (λt)3 = e−λtλt 1 + λt + + + ... + λt + 2 + 3 + ... 2! 3! 2! 3! ( !) (λt)2 3(λt)3 = e−λtλt eλt + λt 1 + λt + 2 + + ... 2! 3! n o = e−λtλt eλt + λteλt = λt (1 + λt)

Now, we can calculate the variance

2 2 2 V ar [Nt] = E Nt − (E [Nt]) = λt (1 + λt) − (λt) = λt

52 Proof. of Proposition 3.2.5

• Expected value First, we compute E [Yt|Nt = n]:

" N # " n # " n # Xt X X E [Yt|Nt = n] = E Ji|Nt = n = E Ji|Nt = n = E Ji = nE [J1] (A.1) i=1 i=1 i=1

Now, ∞ X E [Yt] = E [E [Yt|Nt = n]] = P [Nt = n] E [Yt|Nt = n] n=1 ∞ ∞ X X = P [Nt = n] nE [J1] = E [J1] nP [Nt = n] n=1 n=1

= E [J1] E [Nt] = E [J1] λt = λtE [J1]

• Variance

First we compute V ar [Yt|Nt = n]:

" N # " n # " n # Xt X X V ar [Yt|Nt = n] = V ar Ji|Nt = n = V ar Ji|Nt = n = V ar Ji = nV ar [J1] i=1 i=1 i=1 (A.2) On the other hand, we know that

2 2 E Yt |Nt = n = V ar [Yt|Nt = n] + (E [Yt|Nt = n]) (A.3)

Combining equations (A.1), (A.2) and (A.3), then

2 2 2 2 E Yt |Nt = n = nV ar [J1] + (nE [J1]) = nV ar [J1] + n E [J1]

Now comes,

∞ 2 2 X 2 2 V ar [Yt] = E Yt − E [Yt] = P [Nt = n] E Yt |Nt = n − E [Yt] n=1

∞ X 2 2 2 = P [Nt = n] nV ar [J1] + n E [J1] − (λtE [J1]) n=1

53 ∞ ∞ X 2 X 2 2 = V ar [J1] nP [Nt = n] + E [J1] n P [Nt = n] − (λtE [J1]) n=1 n=1 2 2 2 = V ar [J1] E [Nt] + E [J1] E Nt − (λtE [J1])

2 2 2 = V ar [J1] λt + λtE [J1] = λt V ar [J1] + E [J1] = λtE J1

Proof. of Proposition 3.3.1

• Expected value of ξ+

+ −η1x Since ξ ∼ Exp (η1), then fξ+ (x) = η1e , x > 0, η1 > 0. Therefore, integrating by parts,

Z ∞ Z ∞ 1 + −η1x E ξ = xfξ+ (x) dx = xη1e dx = 0 0 η1

Similarly we can get the expected value of ξ−.

• Expected value of X

Z +∞ Z +∞ −η1x η2x E [X] = xfX (x) dx = x pη1e 1{x≥0} + qη2e 1{x<0} dx −∞ −∞

Z +∞ Z 0 −η1x η2x = xpη1e 1{x≥0}dx + xqη2e 1{x<0}dx 0 −∞ Z +∞ Z 0 −η1x η2x = pη1 xe dx + qη2 xe dx 0 −∞ Using integration by parts, we get

+∞ +∞ ! 1 −η1x Z 1 −η1x E [X] = pη1 −x e − − e dx η1 0 0 η1

0 0 ! 1 η2x Z 1 η2x +qη2 x e − e dx η2 −∞ −∞ η2 " #+∞!  " #0  1 2 1 2 −η1x η2x = pη1 0 + e + qη2 0 − e  η1 η2 0 −∞

54 2 2! 1 1 1 1 + − = pη1 + qη2 − = p − q = pE ξ − qE ξ (A.4) η1 η2 η1 η2

• Variance of X The purpose is to compute: 2 2 V ar [X] = E X − (E [X]) (A.5)

2 First, we need to ﬁnd E X :

Z +∞ Z +∞ 2 2 2 −η1x η2x E X = x fX (x) dx = x pη1e 1{x≥0} + qη2e 1{x<0} dx −∞ −∞

Z +∞ Z 0 2 −η1x 2 η2x = x pη1e dx + x qη2e dx 0 −∞ Z +∞ Z 0 2 −η1x 2 η2x = pη1 x e dx + qη2 x e dx 0 −∞ Using integration by parts, ! 1 +∞ Z +∞ 1 2 2 −η1x −η1x E X = pη1 −x e + 2 x e dx η1 0 0 η1

! 1 0 Z 0 1 2 η2x η2x +qη2 x e − 2 x e dx η2 −∞ −∞ η 2 Z +∞ Z 0 = 2p xe−η1xdx − 2q xeη2xdx 0 −∞ Using integration by parts again,

+∞ Z +∞ ! 2 1 −η x 1 −η x E X = 2p −x e 1 + e 1 dx η1 0 η1 0

! 1 0 1 Z 0 −2q x eη2x − eη2xdx η2 −∞ η2 −∞ 1 Z +∞ 1 Z 0 = 2p e−η1xdx + 2q eη2xdx η1 0 η2 −∞ +∞ 0 2 2 1 1 −η1x 1 1 η2x 1 1 = 2p − e + 2q e = 2p + 2q (A.6) η1 η1 0 η2 η2 −∞ η1 η2 Now, substituting the expressions of equations (A.4) and (A.6) into the right-hand side of equa-

55 tion (A.5), comes

2 2 2 2 2 1 1 p q V ar [X] = E X − (E [X]) = 2p + 2q − − η1 η2 η1 η2 " # 1 2 1 2 1 2 1 1 2 = 2p + 2q − p2 − 2pq + q2 η1 η2 η1 η1η2 η2

1 2 1 2 1 2 1 1 2 = 2p + 2q − p2 + 2pq − q2 η1 η2 η1 η1η2 η2 1 2 1 2 1 2 1 1 2 1 2 1 2 = p + q − p2 + 2pq − q2 + p + q η1 η2 η1 η1η2 η2 η1 η2 1 2 1 2 1 1 2 1 2 = p − p2 + q − q2 + 2pq + p + q η1 η2 η1η2 η1 η2 1 2 1 2 1 1 2 1 2 = p (1 − p) + q (1 − q) + 2pq + p + q η1 η2 η1η2 η1 η2 1 2 1 2 1 1 2 1 2 = pq + qp + 2pq + p + q η1 η2 η1η2 η1 η2 " # 1 2 1 2 1 1 2 1 2 = pq + + 2 + p + q η1 η2 η1η2 η1 η2

1 1 2 p q = pq + + 2 + 2 η1 η2 η1 η2

• Expected value of V Z +∞ X x E [V ] = E e = e fX (x) dx −∞ Z +∞ x −η1x η2x = e pη1e 1{x≥0} + qη2e 1{x<0} dx −∞ Z +∞ Z +∞ x −η1x x η2x = e pη1e 1{x≥0}dx + e qη2e 1{x<0}dx −∞ −∞ Z +∞ Z 0 x −η1x x η2x = pη1 e e dx + qη2 e e dx 0 −∞ Z +∞ Z 0 (1−η1)x (1+η2)x = pη1 e dx + qη2 e dx 0 −∞

56 1 +∞ 1 0 (1−η1)x (1+η2)x = pη1 e + qη2 e 1 − η1 0 1 + η2 −∞ 1 1 = pη1 0 − + qη2 − 0 , η1 > 1, η2 > 0 1 − η1 1 + η2

η1 η2 = p + q , η1 > 1, η2 > 0 η1 − 1 η2 + 1

(1−η )x Note that we need that η1 > 1. Otherwise, if η1 < 1, as x → + ∞, then e 1 → + ∞ and +∞ R (1−η1)x 0 e dx is not ﬁnite.

• Expected value of V − 1:

η η E [V − 1] = E [V ] − 1 = p 1 + q 2 − 1 η1 − 1 η2 + 1

Proof. of Proposition 3.3.2 The stochastic differential equation (3.18) under the risk neutral measure Q is equivalent to

N Z t Z t Xt St = S0 + (r − δ − λζ) Sudu + σSudWu + 4Si 0 0 i=1

Now we can apply the Itô formula in Theorem 3.2.1, with f(t, St) = ln (St), at = (r − δ − λζ) St and bt = σSt. Substituting these expressions into (3.14),

2 ∂f ∂f 1 2 ∂ f ∂f df(t, St) = dt+(r − δ − λζ) St dt+ (σSt) 2 dt+σSt dWt+[f (St + 4St) − f (St)] ∂t ∂St 2 ∂St ∂St 1 1 2 2 1 1 ⇔ df(St, t) = 0+(r − δ − λζ) St dt+ σ St − 2 dt+σSt dWt +ln (St + 4St)−ln (St) St 2 St St 1 2 St + 4St ⇔ df(St, t) = r − δ − σ − λζ dt + σdWt + ln 2 St 1 2 4St ⇔ df(St, t) = r − δ − σ − λζ dt + σdWt + ln + 1 2 St

1 ⇔ df(S , t) = r − δ − σ2 − λζ dt + σdW + ln (V ) t 2 t t

57 1 ⇔ d ln (S ) = r − δ − σ2 − λζ dt + σdW + ln (V ) t 2 t t Z t Z t Z t Z t 1 2 ⇔ d ln (Su)du = r − δ − σ − λζ du + σdWu + ln (Vu)du 0 0 2 0 0

Nt St 1 X ⇔ ln = r − δ − σ2 − λζ t + σW + ln (V ) S 2 t i 0 i=1

N 1 Yt ⇔ S = S exp r − δ − σ2 − λζ t + σW V t 0 2 t i i=1

R t PNu Note that 0 ln (Vu)dNu = i=1 ln (Vi) is the sum of n independent jump terms. Conditioned upon the event Nt = n, there must have been exactly n times, say τ i, i = 1, ..., n, between 0 and t such that dN(τ i) = 1.

Proof. of Proposition 3.3.3 By equation (3.13), we have

u2σ2 Z ψ (u) = iub − + eiux − 1 ν (dx) 2 R

σ2 where b = r − δ − 2 − λζ is the drift of equation (3.24). In Kou model, the Lévy measure ν is

−η1y η2y ν (y) = λ pη1e 1{y≥0} + qη2e 1{y<0} , η1 > 1, η2 > 0

Then,

u2σ2 Z +∞ Z 0 iux −η1x iux η2x ψ (u) = iub − + e − 1 λpη1e dx + e − 1 λqη2e dx 2 0 −∞

σ2u2 1 1 +∞ −(η1−iu)x −η1x = iub − + λpη1 − e + e 2 η1 − iu η1 0 1 1 0 (η2+iu)x η2x +λqη2 e + e η2 + iu η2 −∞ σ2u2 η η = iub − + λp 1 − λp + λq 2 − λq 2 η1 − iu η2 + iu σ2u2 η η = iub − + λ p 1 + q 2 − 1 2 η1 − iu η2 + iu

58 Thus, σ2 σ2u2 η η ψ (u) = iu r − δ − − λζ − + λ p 1 + q 2 − 1 2 2 η1 − iu η2 + iu

Proposition A.0.1. Assume we model the dynamic of an underlying asset as follows:

2 σ PNt α− 2 −λβ t+σWt+ i=1 Xi St = S0e (A.7)

PNt where {Wt}t≥0 is a Brownian motion and i=1 Xi is a compound Poisson process with {Xi}i≥1 independent and identically distributed random variables and {Nt}t≥0 is a Poisson process of rate λ. Also assume a constant risk-free rate r and no dividends. For this dynamic to be a valid risk-neutral dynamic, the parameters α and β should verify the following conditions:

• α = r

X1 • β = EQ e − 1

Proof. To prevent arbitrage opportunities, the discounted asset price process should be a martingale under the risk neutral measure Q — see Proposition 3.3.4:

−rt rt EQ e St | F0 = S0 ⇔ EQ [St | F 0] = S0e (A.8)

On the other hand, we can rewrite the equation A.7 as:

2 σ PNt αt − t+σWt −λβt+ Xi St = S0e e 2 e i=1 (A.9)

Taking the expected value of both terms of the equation A.9, under Q, and bearing in mind that the Wiener process is independent from the compound Poisson process, yields:

2 σ PNt αt − 2 t+σWt −λβt+ i=1 Xi EQ [St] = EQ S0e e e

2 σ h PNt i αt − 2 t+σWt −λβt+ i=1 Xi = EQ S0e EQ e EQ e

2 2 σ σ h PNt i αt − 2 t+ 2 t −λβt i=1 Xi = S0e e e EQ e

h PNt i αt −λβt i=1 Xi = S0e e EQ e (A.10)

59 The equation A.10 comes αt EQ [St] = S0e (A.11) if and only if h PNt i i=1 Xi λβt EQ e = e (A.12)

On the other side, comparing equations A.8 and A.11, comes too α = r. Moreover, ∞ h PNt i X h Pn i i=1 Xi i=1 Xi EQ e = EQ e | Nt = n Q [Nt = n] n=0 ∞ n ∞ n X Y X n (λt) = eXi | N = n [N = n] = eX1 e−λt EQ t Q t EQ n! n=0 i=1 n=0 n ∞ X1 −λt X λtEQ e −λt λt eX1 λ eX1 −1 t = e = e e EQ[ ] = e (EQ[ ] ) n! n=0

X1 λ(EQ[e ]−1)t λβt X1 X1 So, to get e = e , we must have β = EQ e − 1. Now, we can compute EQ e depending on the density of X1.

In the case of X1 having a double exponential distribution as in the Kou model, and taking into account Proposition 3.3.1, comes

η η X1 1 2 EQ e = p + q η1 − 1 η2 + 1 then, η η β = p 1 + q 2 − 1 η1 − 1 η2 + 1 which is the ζ of equation 3.19.

Proposition A.0.2. The sum of n double exponential random variables can be decomposed as:

n ( Pk + X d ξ , with probability Pn,k X = i=1 i k = 1, 2, ..., n (A.13) i Pk − i=1 − i=1 ξi , with probability Qn,k

+ − where ξi ∼ Exp(η1) and ξi ∼ Exp(η2). Pk + Pk − Note that i=1 ξi ∼ Gamma(k, η1) and i=1 ξi ∼ Gamma(k, η2).

Proof. See [57, pp. 24] However, the following we present the proof for n=2.

60 Let X1 and X2 be two random variables with double exponential distribution decomposed as

( + d ξ1 with probability p X1 = − −ξ1 with probability q and ( + d ξ2 with probability p X2 = − −ξ2 with probability q then, the sum of X1 and X2 can be decomposed by:  ξ+ with probability 2pq η2  1 η1+η2  + + 2 d  ξ1 + ξ2 with probability p X1 + X2 = −ξ− with probability 2pq η1  2 η1+η2  − − 2  −ξ1 −ξ2 with probability q

Let’ s see:

By the memoryless property of X1 — see Proposition 3.2.4, we have:

+ − + − + − + − P ξ1 −ξ2 > x | ξ1 > ξ2 = P ξ1 > ξ2 + x | ξ1 > ξ2

+ + − − − = P ξ1 > y + x | ξ1 > ξ2 , ξ2 > y P ξ2 > y + + = P ξ1 > y + x | ξ1 > y + = P ξ1 > x

Then, + − + − + P ξ1 −ξ2 > x | ξ1 > ξ2 = P ξ1 > x + − + − + wich means that ξ1 −ξ2 | ξ1 > ξ2 is equal in distribution to ξ1 .

By the same way, and using the memoryless property of X2, we have:

+ − + − + − + − P ξ1 −ξ2 > x | ξ1 < ξ2 = 1 − P ξ1 −ξ2 < x | ξ1 < ξ2

− + + − = 1 − P −ξ2 < −ξ1 + x | ξ1 < ξ2 − + − + + = 1 − P −ξ2 < y + x | ξ1 < ξ2 , −ξ1 < y P −ξ1 < y − − + + + = 1 − P −ξ2 < y + x | −ξ2 < −ξ1 , −ξ1 < y P −ξ1 < y − − = 1 − P −ξ2 < y + x | −ξ2 < y

61 − − = 1 − P ξ2 > −y − x | ξ2 > −y − − − = 1 − P ξ2 > −x = P ξ2 < −x = P −ξ2 > x

Then, + − + − − P ξ1 −ξ2 > x | ξ1 < ξ2 = P −ξ2 > x + − + − − wich means that ξ1 −ξ2 | ξ1 < ξ2 is equal in distribution to −ξ2 . In short, we can decompose the sum of double exponential random variables in a mixed sum of exponential random variables and:

+ − + − d + + − + − + ξ1 −ξ2 | ξ1 > ξ2 = ξ1 ⇐⇒ P ξ1 −ξ2 > x | ξ1 > ξ2 = P ξ1 > x , ∀x

+ − + − d − + − + − − ξ1 −ξ2 | ξ1 < ξ2 = −ξ2 ⇐⇒ P ξ1 −ξ2 > x | ξ1 < ξ2 = P −ξ2 > x , ∀x

Pk + Pk − Proposition A.0.3. The probabilities weight Pn,k and Qn,k assigned to i=1 ξi and to − i=1 ξi , Pn respectively, when we decompose i=1 Xi as in the equation (A.13) are deﬁned as

n−1 i−k n−i X n − k − 1n η η P = 1 2 piqn−i, 1 ≤ k ≤ n − 1 (A.14) n,k i − k i η + η η + η i=k 1 2 1 2 and

n−1 n−i i−k X n − k − 1n η η Q = 1 2 pn−iqi, 1 ≤ k ≤ n − 1 (A.15) n,k i − k i η + η η + η i=k 1 2 1 2

Proof. See [57, pp. 26]. However, the following is an explanation for n=4. In this case, we have four random variables with double exponential distribution:

( + d ξi , with probability p Xi = − i = 1, 2, 3, 4, −ξi , with probability q and the decomposition of its sum is:

4 ( Pk + X d ξ , with probability P4,k X = i=1 i k = 1, 2, 3, 4, i Pk − i=1 − i=1 ξi , with probability Q4,k

62 or disaggregating the terms:  ξ+, with probability P  1 4,1  + +  ξ1 + ξ2 , with probability P4,2   ξ+ + ξ+ + ξ+ + ξ+, with probability P  1 2 3 4 4,3  + + + + d  ξ1 + ξ2 + ξ3 + ξ4 , with probability P4,4 X1 + X2 + X3 + X4 = −ξ−, with probability Q  1 4,1  − −  −ξ1 − ξ2 , with probability Q4,2   −ξ− − ξ− − ξ−, with probability Q  1 2 3 4,3  − − − −  −ξ1 − ξ2 − ξ3 − ξ4 , with probability Q4,4

Consider now we have a random walk in a plane with the horizontal axis representing the number + − of ξi and the vertical axis representing the number of ξi . Suppose that the random walk only goes one step to the left or one step down and that stops once it reaches either the horizontal or the vertical axis. From any point P of the integer lattice points of this plane, the probability of random walk goes + − η2 one step down is P ξ > ξ = and the probability of random walk goes one step to the left η1 + η2 + − η1 is P ξ < ξ = , see Proposition A.0.4. η1 + η2

#ξ2

η1/ (η1 + η2) P 2

η2/ (η1 + η2) 1

O 1 2 3 4 #ξ1

To compute P4,k, with 1 ≤ k ≤ 4, (note that in the general case it would be: Pn,k, with 1 ≤ k ≤ n), is equivalent to consider the probability of the random walk ever reaching the point (k, 0) on the horizontal axis starting from the diagonal points (i, 4 − i), with 0 ≤ i ≤ 4.

To compute Q4,k, with 1 ≤ k ≤ 4, is equivalent to consider the probability of the random walk ever reaching the point (0, k) on the vertical axis starting from the diagonal points (4 − i, i), with 0 ≤ i ≤ 4.

63 Given the movement of our random walk, note that each point (k, 0) on the horizontal axis can only be reached from the diagonal points (i, 4 − i) with the constraint k ≤ i ≤ 3. The points (0, k) on the vertical axis can only be reached from points (4 − i, i) with the constraint k ≤ i ≤ 3.

If we take any point, P, on the set of diagonal points and another point, A, on the horizontal or vertical axis, we have (by the deﬁnition of conditional probability):

P [random walk reach A starting from the diagonal points ] X = P [going from P to A , starting from P] P [starting from P] P∈D X = P [going from P to A] P [starting from P] P∈D where D is the set of the diagonal points, which in our case is D = {(1, 3) , (2, 2) , (3, 1)}.

#ξ2

4 (1, 3) 3 (2, 2) 2 (3, 1) 1

O 1 2 3 4 #ξ1

Figure A.1.: Possible path from (2,2) to (1,0)

Figure A.1 shows an example of a possible path, starting from (2, 2) and reaching (1, 0), where the red dots are our target points and the blue dots are the starting points.

Let’ s start the calculation. + The probability weight P4,1 assigned to ξ1 is equivalent to the probability of the random walk reach to the point (1, 0) starting from the diagonal points (i, 4 − i), 1 ≤ i ≤ 3, which in our case are the points

64 (1, 3), (2, 2) and (3, 1), that involves the following sum:

P4,1 = P [(1, 3) → (1, 0)] P [(1, 3)] + P [(2, 2) → (1, 0)] P [(2, 2)] + P [(3, 1) → (1, 0)] P [(3, 1)] (A.16)

#ξ2 #ξ2 #ξ2 4 (1, 3) 4 4 3 3 (2, 2) 3 2 2 2 (3, 1) 1 1 1

O 1 2 3 4 #ξ1 O 1 2 3 4 #ξ1 O 1 2 3 4 #ξ1

The ﬁrst term on the right-hand side of equation (A.16) is equal to:

0 3 2 η1 η2 4 1 3 P [(1, 3) → (1, 0)] P [(1, 3)] = p q 0 η1 + η2 η1 + η2 1

The probability of random walk going from (1, 3) to (1, 0), which we have denote by P [(1, 3) → (1, 0)], 0 3 2 η1 η2 is given by 0 because the path (1, 3) → (1, 0) has 0 lefts and 3 downs. η1 + η2 η1 + η2 Recall that the probability of one step left is η1/ (η1 + η2) and the probability of one step down is

η2/ (η1 + η2). Any path that connecting (1, 3) → (1, 0) must reach (1, 1) first before it makes a final step to (1, 0) and all the paths from (1, 3) to (1, 1) must have exactly 0 lefts and 2 downs. Then, the 2 number of such paths is 0 . 4 1 3 The probability of starting from (1, 3), which we have denote by P [(1, 3)], is given by 1 p q because at the point (1, 3) we have 1 random variables ξ+ and 3 random variables ξ−. The probability to have one ξ+ is p and the probability to one have ξ− is q, then comes p1q3. Any path that starts at (1, 3) 1+3 must have 1 left and 3 downs to get (0, 0) and the total number of these paths is 1 . Thus, we have obtained the first term on the right side of equation (A.16).

The second term on the right-hand side of equation (A.16) is equal to:

1 2 2 η1 η2 4 2 2 P [(2, 2) → (1, 0)] P [(2, 2)] = p q 1 η1 + η2 η1 + η2 2

The probability of random walk going from (2, 2) to (1, 0), P [(2, 2) → (1, 0)] , is given by 1 2 2 η1 η2 1 because the path (2, 2) → (1, 0) has 1 lefts and 2 downs. Any path η1 + η2 η1 + η2

65 connecting (2, 2) → (1, 0) must reach (1, 1) ﬁrst before it makes a ﬁnal step to (1, 0) and all the paths 1+1 from (2, 2) to (1, 1) must have exactly 1 lefts and 1 downs. Then, the number of such paths is 1 . 4 2 2 The probability of starting from (2, 2), P [(2, 2)] , is given by 2 p q because at the point (2, 2) we have 2 random variables ξ+ and 2 random variables ξ−. The probability of having ξ+ is p and the probability of having ξ− is q, then p2q2 follows. Any path that starts at (2, 2) has 2 lefts and 2 downs 2+2 to get to (0, 0) and the total number of these paths is 2 . Thus, we have obtained the second term on the right side of equation (A.16).

The third term on the right-hand side of equation (A.16) is equal to:

2 1 2 η1 η2 4 3 1 P [(3, 1) → (1, 0)] P [(3, 1)] = p q 2 η1 + η2 η1 + η2 3

The probability of random walk going from (3, 1) to (1, 0), P [(3, 1) → (1, 0)] , is given by 2 1 2 η1 η2 2 because the path (3, 1) → (1, 0) has 2 lefts and 1 downs. Any path η1 + η2 η1 + η2 connecting (3, 1) → (1, 0) must reach (1, 1) ﬁrst before it makes a ﬁnal step to (1, 0) and all the paths 2+0 from (3, 1) to (1, 1) must have exactly 2 lefts and 0 downs. Then, the number of such paths is 2 . 4 3 1 The probability of starting from (3, 1) — P [(3, 1)] — is given by = 3 p q because at the point (3, 1) we have 3 random variables ξ+ and 1 random variable ξ−. The probability of having ξ+ is p and the probability of having ξ− is q, then p3q1 follows. Any path that starts at (3, 1) has 3 lefts and 1 downs 3+1 to get to (0, 0) and the total number of these paths is 3 . Thus, we have obtained the third term on the right side of equation (A.16).

Finally, putting all the terms together, we have the expression of equation (A.16):

0 3 2 η1 η2 4 1 3 P4,1 = p q 0 η1 + η2 η1 + η2 1

2 η 1 η 24 + 1 2 p2q2 1 η1 + η2 η1 + η2 2 2 η 2 η 14 + 1 2 p3q1 2 η1 + η2 η1 + η2 3 − − The probability weight Q4,2 assigned to −ξ1 −ξ2 is equivalent to the probability of the random walk reach to the point (0, 2) starting from the diagonal points (4 − i, i), 2 ≤ i ≤ 3, which in our case are the points (2, 2) and (1, 3), that involves the following sum:

Q4,2 = P [(2, 2) → (0, 2)] P [(2, 2)] + P [(1, 3) → (0, 2)] P [(1, 3)] (A.17)

66 #ξ2 #ξ2 4 (1, 3) 4 3 3 (2, 2) 2 2 1 1

O 1 2 3 4 #ξ1 O 1 2 3 4 #ξ1

The ﬁrst term on the right-hand side of equation (A.17) is equal to:

2 0 1 η1 η2 4 2 2 P [(2, 2) → (0, 2)] P [(2, 2)] = p q 0 η1 + η2 η1 + η2 2

The probability of random walk going from (2, 2) to (0, 2), which we have denote by P [(2, 2) → (0, 2)], 2 0 1 η1 η2 is given by to = 0 because the path (2, 2) → (0, 2) has 2 lefts and 0 η1 + η2 η1 + η2 downs. Any path that connecting (2, 2) → (0, 2) must reach (1, 1) first before it makes a final step to (0, 2) and all the paths with 2 lefts and 0 downs that first reach the vertical axis at (0, 2) have 1 lefts 1 and 0 downs. Then, the number of such paths is 0 . The probability of starting from (2, 2), which we 4 2 2 have denote by P [(2, 2)], is given by = 2 p q because at the point (2, 2) we have 2 random variables ξ+ and 2 random variables ξ−. The probability of having ξ+ is p and the probability of having ξ− is q, then p2q2 follows. Any path that starts at (2, 2) has 2 lefts and 2 downs to get to (0, 0) and the total 2+2 number of these paths is 2 . Thus, we have obtained the first term on the right-hand side of equation (A.17). The second term on the right-hand side of equation (A.17) is equal to:

1 1 1 η1 η2 4 1 3 P [(1, 3) → (0, 2)] P [(1, 3)] = p q 1 η1 + η2 η1 + η2 3

The probability of random walk going from (1, 3) to (0, 2), P [(1, 3) → (0, 2)] , is given by 1 1 1 η1 η2 1 because the path (1, 3) → (0, 2) has 1 lefts and 1 downs. All the paths η1 + η2 η1 + η2 with 1 lefts and 1 downs that ﬁrst reach the vertical axis at (0, 2) has 0 lefts and 1 downs. Then, the 1 number of such paths is 1 . 4 1 3 The probability of starting from (1, 3), P [(1, 3)] , is given by 3 p q because at the point (1, 3) we have 1 random variable ξ+ 3 random variables ξ−. The probability of having ξ+ is p and the probability of having ξ− is q, then p1q3 follows. Any path that starts at (1, 3) has 1 lefts and 3 downs to get (0, 0) 1+3 and the total number of these paths is 3 . Thus, we have obtained the second term on the right-hand side of equation (A.17).

67 Finally, putting all the terms together, we have the expression of equation (A.17):

2 0 1 1 1 η1 η2 4 2 2 1 η1 η2 4 1 3 Q4,2 = p q + p q 0 η1 + η2 η1 + η2 2 1 η1 + η2 η1 + η2 3

We leave the discovery of the other expressions of P4,k and Q4,k to the care of the reader.

+ − + − Proposition A.0.4. Let ξ and ξ are two random variables such that ξ ∼ Exp(η1) and ξ ∼

Exp(η2). Then, + − η2 P ξ > ξ = η1 + η2 and + − η1 P ξ < ξ = η1 + η2

Proof. A random variable X with a double exponential distribution can be written by ( ξ+, with probability p X =d ξ−, with probability q where p is the probability of upward jumps and q is the probability of downward jumps, such that p + q = 1. Using the conditional probability deﬁnition, we get

Z ∞ + − + − − − P ξ > ξ = P ξ > ξ | ξ = x P ξ ∈ dx 0 Z ∞ + −η2x = P ξ > x η2e dx 0 Z ∞ −η1x −η2x = e η2e dx 0 Z ∞ η −(η1+η2)x 2 = η2 e dx = 0 η1 + η2 and, similarly, we have + − η1 P ξ < ξ = η1 + η2

Proof. of Proposition 3.3.5

68 By the deﬁnition of a density joint function f(.), we have

Z +∞ Z +∞ fZ (z) = fX,Y (x, y) d(x, y) = fZ (x, z − x) dx −∞ −∞

Alternatively, by the deﬁnition of a cumulative joint function F(.), we have

FZ (z) = P [Z ≤ z] = P [X + Y ≤ z] = P [X ≤ z − y] = FX (z − y) or,

FZ (z) = P [Z ≤ z] = P [X + Y ≤ z] = P [Y ≤ z − x] = FY (z − x)

Therefore,

d d Z +∞ fZ (z) = FZ (z) = FX (z − y) = fX (z − y) = fX,Y (z − y, y) dy dz dz −∞ or, d d Z +∞ fZ (z) = FZ (z) = FY (z − x) = fY (z − x) = fX,Y (x, z − x) dx dz dz −∞

If X and Y are independent random variables, we have fX,Y (x, y) = fX (x) fY (y), and hence

Z +∞ fZ (z) = fX (x) fY (z − x) dx −∞ or, Z +∞ fZ (z) = fX (z − y) fY (y) dy −∞ In the case of Z = X − Y , with X and Y independent random variables, we have

Z +∞ fZ (z) = fX (x) fY (x − z) dx −∞ or, Z +∞ fZ (z) = fX (y − z) fY (y) dy −∞

Proof. of Proposition 3.3.7 Applying Proposition 3.3.5, and taking account that the variables are independent, comes

Z +∞ fZ+Pk ξ (t) = fZ (x) fPk ξ (t − x) dx (A.18) i=1 i −∞ i=1 i

69 We know that the probability density function of a standard normal random variable is

2 1 − x fZ (x) = √ e 2σ2 σ 2π

From Proposition 3.3.6 we also know that

k−1 −ηt (ηt) fPk ξ (t) = ηe , t > 0, i=1 i (k − 1)!

Therefore, equation (A.18) comes

Z t 2 k−1 1 − x (η (t − x)) 2σ2 −η(t−x) fZ+Pk ξ (t) = √ e ηe dx i=1 i −∞ σ 2π (k − 1)!

Z t 2 k−1 −tη k 1 − x ηx (t − x) = e η √ e 2σ2 e dx −∞ σ 2π (k − 1)!

2 2 2 Z t (x−σ η) k−1 −tη k (ση) 1 − (t − x) = e η e 2 √ e 2σ2 dx −∞ σ 2π (k − 1)!

x−σ2η 2 Using the change of variable y = σ , then x = σy + σ η =⇒ dx = σdy

t 2 −ση 2 k−1 k (ση) Z σ 1 y2 t − σy − σ η −tη 2 − 2 fPk ξ (t) = e η e √ e σdy i=1 i −∞ σ 2π (k − 1)!

t 2 k−1 Z −ση k−1 2 −tη k (ση) σ 1 σ t − y = e η e 2 √ −y + − ση e 2 dy (A.19) 2π (k − 1)! −∞ σ Taking the deﬁnition of Hh function provided, for instance, in [9] or [50]:

Z a 2 1 k−1 − y Hhk−1 (−a) = (a − y) e 2 dy (A.20) (k − 1)! −∞ and replacing it into the equation (A.19), comes

2 k−1 k (ση) σ t −tη 2 fZ+Pk ξ (t) = e η e √ Hhk−1 − + ση . i=1 i 2π σ

Identical reasoning leads to the other expression:

Z +∞ fZ−Pk ξ (t) = fZ (x) fPk ξ (x − t) dx, t > 0 i=1 i −∞ i=1 i

70 Z +∞ 2 k−1 1 − x −η(x−t) (η (x − t)) = √ e 2σ2 ηe dx t σ 2π (k − 1)!

Z +∞ 2 k−1 tη k 1 − x −ηx (x − t) = e η √ e 2σ2 e dx t σ 2π (k − 1)!

2 2 2 Z +∞ (x+σ η) k−1 tη k (ση) 1 − (x − t) = e η e 2 √ e 2σ2 dx t σ 2π (k − 1)!

x+σ2η 2 Using the change of variable y = σ , then x = σy − σ η =⇒ dx = σdy, and

2 +∞ 2 k−1 k (ση) Z 1 y2 σy − σ η − t tη 2 − 2 fZ−Pk ξ (t) = e η e √ e σdy i=1 i t σ 2π (k − 1)! σ +ση

2 k−1 Z +∞ k−1 2 tη k (ση) σ 1 t − y = e η e 2 √ y − ση − e 2 dy 2π (k − 1)! t σ σ +ση Using again the deﬁnition of Hh function,

Z +∞ 2 1 k−1 − y Hhk−1 (−a) = (a + y) e 2 dy (A.21) (k − 1)! −a comes 2 k−1 k (ση) σ t tη 2 fZ−Pk ξ (t) = e η e √ Hhk−1 + ση i=1 i 2π σ

Proof. of Proposition 3.3.8 We take the deﬁnition of expected value

+∞ Pk + Z + b+c(Z+ i=1 ξi) b+ct EQ ae − K = ae − K fZ+Pk ξ (t) dt −∞ i=1 i

K b+ct ln ( a )−b Since ae − K ≥ 0 =⇒ t ≥ c ≡ h, then

+∞ Pk + Z b+c(Z+ i=1 ξi) b+ct EQ ae − K = ae − K fZ+Pk ξ (t) dt h i=1 i

Z +∞ Z +∞ b+ct = ae fZ+Pk ξ (t) dt − K fZ+Pk ξ (t) dt h i=1 i h i=1 i Z +∞ " k # b ct X = ae e f Pk (t) dt − K Z + ξi ≥ h (A.22) Z+ i=1 ξi P h i=1

71 We now have a tail probability whose expression results from Proposition A.0.5. Additionally we have to use Proposition 3.3.7. Thus, equation (A.22) becomes

+ 2 Z +∞ Pk k (ση) 1 t b+c(Z+ i=1 ξi) b 2 ct −tη EQ ae − K = ae (ση) e √ e e Hhk−1 − + ση dt σ 2π h σ

2 k (ση) 1 1 −K(ση) e 2 √ Ik−1 h; −η; − ; −ση σ 2π σ

2 b k (ση) 1 1 = ae (ση) e 2 √ Ik−1 h; c − η; − ; −ση σ 2π σ

2 k (ση) 1 1 −K(ση) e 2 √ Ik−1 h; −η; − ; −ση σ 2π σ where the last equality follows from equation (A.24). The other expectation is computed as the same way:

+∞ Pk + Z + b+c(Z− i=1 ξi) b+ct EQ ae − K = ae − K fZ−Pk ξ (t) dt −∞ i=1 i

K b+ct ln ( a )−b Since ae − K ≥ 0 =⇒ t ≥ c ≡ h, then

+∞ Pk + Z b+c(Z− i=1 ξi) b+ct EQ ae − K = ae − K fZ−Pk ξ (t) dt h i=1 i

Z +∞ Z +∞ b+ct = ae fZ−Pk ξ (t) dt − K fZ−Pk ξ (t) dt h i=1 i h i=1 i Z +∞ " k # b ct X = ae e f Pk (t) dt − K Z − ξi ≥ h (A.23) Z− i=1 ξi P h i=1 Propositions 3.3.7 and A.0.5 lead equation (A.23) to become

+ 2 Z +∞ Pk k (ση) 1 t b+c(Z− i=1 ξi) b 2 ct tη EQ ae − K = ae (ση) e √ e e Hhk−1 + ση dt σ 2π h σ

2 k (ση) 1 1 −K(ση) e 2 √ Ik−1 h; η; ; −ση σ 2π σ

2 b k (ση) 1 1 = ae (ση) e 2 √ Ik−1 h; c + η; ; −ση σ 2π σ

2 k (ση) 1 1 −K(ση) e 2 √ Ik−1 h; η; ; −ση σ 2π σ

72 where the last equality comes from the equation (A.24).

Proposition A.0.5. The tail probabilities provided by [57, pp. 28]

" k # 2 X k (ση) 1 1 Z + ξ ≥ x = (ση) e 2 √ I x; −η; − ; −ση P i k−1 σ i=1 σ 2π and " k # 2 X k (ση) 1 1 Z − ξ ≥ x = (ση) e 2 √ I x; η; ; −ση P i k−1 σ i=1 σ 2π where Z +∞ αt Ik−1 (x; α; β; θ) = e Hhk−1 (βt − θ) dt, k ≥ 1 (A.24) x Proof. " k # X Z +∞ Z + ξi ≥ x = f Pk (t) dt P Z+ i=1 ξi i=1 x

Z +∞ 2 k (ση) 1 −tη t = (ση) e 2 √ e Hhk−1 − + ση dt x σ 2π σ

2 Z +∞ k (ση) 1 −tη t = (ση) e 2 √ e Hhk−1 − + ση dt σ 2π x σ

2 k (ση) 1 1 = (ση) e 2 √ Ik−1 x; −η; − ; −ση σ 2π σ because Z +∞ 1 −tη t Ik−1 x; −η; − ; −ση = e Hhk−1 − + ση dt σ x σ Similarly, " k # X Z +∞ Z − ξi ≥ x = f Pk (t) dt P Z− i=1 ξi i=1 x

Z +∞ 2 k (ση) 1 tη t = (ση) e 2 √ e Hhk−1 + ση dt x σ 2π σ

2 Z +∞ k (ση) 1 tη t = (ση) e 2 √ e Hhk−1 + ση dt σ 2π x σ

2 k (ση) 1 1 = (ση) e 2 √ Ik−1 x; η; ; −ση σ 2π σ because Z +∞ 1 tη t Ik−1 x; η; ; −ση = e Hhk−1 + ση dt σ x σ

73 Proposition A.0.6. Evaluation of the integral of the equation (A.24)

• If β > 0 and α 6= 0, then for all k ≥ 1, √ αλ k−1 k−1−i k 2 e X β β 2π αθ + α α I (x; α; β; θ) = − Hh (βx − θ)+ e β 2β2 Φ −βx + θ + k−1 α α i α β β i=0 (A.25)

• If β < 0 and α < 0, then for all k ≥ 1, √ αλ k−1 k−1−i k 2 e X β β 2π αθ + α α I (x; α; β; θ) = − Hh (βx − θ)− e β 2β2 Φ βx − θ − k−1 α α i α β β i=0 (A.26)

• If β > 0 and α = 0, then for all k ≥ 1,

1 I (x; α; β; θ) = Hh (βx − θ) k−1 β i

• If β ≤ 0 and α ≥ 0, then for all k ≥ 1,

Ik−1 (x; α; β; θ) = ∞

• If β = 0 and α < 0, then for all k ≥ 1,

αx Ik−1 (x; α; β; θ) = Hhi (−θ) e

Proof. See [57, pp. 27]

Proposition A.0.6 allows to develop the integrals Ik−1 of equation (3.33):

• α = 1 − η < 0 and β = − √1 < 0, then we use equation (A.26) 1 σ T

k−1 k−1−i 1 √ e(1−η1)h X 1 h √ I h; 1 − η ; − √ ; −σ T η = − √ Hh − √ + σ T η k−1 1 1 1 − η i 1 σ T 1 i=0 σ T (η1 − 1) σ T

k 1 √ √ 2 1 2 2 h √ √ (1−η1)σ T η1+ (1−η1) σ T − √ −σ T 2π e 2 Φ − √ + σ T η1 + (1 − η1) σ T σ T (η1 − 1) σ T

74 k−1 k−1−i e(1−η1)h X 1 h √ = − √ Hh − √ + σ T η 1 − η i 1 1 i=0 σ T (η1 − 1) σ T k √ √ √ 1 (1−η )σ2T η + 1 (1−η )2σ2T h − √ −σ T 2π e 1 1 2 1 Φ − √ + σ T σ T (η1 − 1) σ T

• α = −η < 0 and β = − √1 < 0, then we use equation (A.26) 1 σ T

k−1 k−1−i 1 √ e−η1h X 1 h √ I h; −η ; − √ ; −σ T η = √ Hh − √ + σ T η k−1 1 1 η i 1 σ T 1 i=0 σ T η1 σ T

k 1 √ √ 2 2 1 2 2 h √ √ −η1 σ T + η1 σ T − √ −σ T 2π e 2 Φ − √ + σ T η1 − η1σ T σ T η1 σ T

k−1 k−1−i e−η1h X 1 h √ = √ Hh − √ + σ T η η i 1 1 i=0 σ T η1 σ T k √ √ 1 −η 2σ2T + 1 η 2σ2T h − √ −σ T 2π e 1 2 1 Φ − √ σ T η1 σ T

• α = 1 + η > 0 and β = √1 > 0, then we use equation (A.25) 2 σ T

k−1 k−1−i 1 √ e(1+η2)h X 1 h √ I h; 1 + η ; √ ; −σ T η = − √ Hh √ + σ T η k−1 2 2 1 + η i 2 σ T 2 i=0 σ T (1 + η2) σ T

k 1 √ √ 2 1 2 2 h √ √ −(1+η2)σ T η2+ (1+η2) σ T + √ σ T 2π e 2 Φ − √ − σ T η2 + (1 + η2) σ T σ T (1 + η2) σ T

k−1 k−1−i e(1+η2)h X 1 h √ = − √ Hh √ + σ T η 1 + η i 2 2 i=0 σ T (1 + η2) σ T k √ √ √ 1 −(1+η )σ2T η + 1 (1+η )2σ2T h + √ σ T 2π e 2 2 2 2 Φ − √ + σ T σ T (1 + η2) σ T

• α = η > 0 and β = √1 > 0, then we use equation (A.25) 2 σ T

k−1 k−1−i 1 √ eη2h X 1 h √ I h; η ; √ ; −σ T η = − √ Hh √ + σ T η k−1 2 2 η i 2 σ T 2 i=0 σ T η2 σ T

75 k 1 √ √ 2 2 1 2 2 h √ √ −η2 σ T + η2 σ T + √ σ T 2π e 2 Φ − √ − σ T η2 − η2σ T σ T η2 σ T

k−1 k−1−i eη2h X 1 h √ = − √ Hh √ + σ T η η i 2 2 i=0 σ T η2 σ T k √ √ 1 −η 2σ2T + 1 η 2σ2T h + √ σ T 2π e 2 2 2 Φ − √ σ T η2 σ T

76 B. Data and results

In the Table B.1 we present the model calibration parameters.

Estimated parameters - Kou model Date σ λ p η1 η2 Options on 16/12/2017 1.2336 12.5467 0.2001 4.4535 8.1739 Options on 05/02/2018 1.1110 5.5289 0.2295 15.0473 3.3937 Options on 28/06/2018 0.8501 0.7998 0.4001 35.0112 6.9987

Table B.1.: Model calibration parameters

Figures B.1, B.2 and B.3 contain the options used in the Kou model application.

77 2017-12-16

As of 4pm ET on Saturday, December 16th, 2017.

Volume- Report Open Weighted Contract Contract Last Bid Last Ask Volume Date Interest Average Type Price

Day 2017- BTC 2017-12-16 $19,120.75 $19,186.00 27 10 $18,479.24 ahead 12-16 Swap swap

2017- BTC 2017-12-29 Options $30.00 $105.00 6 20 $80.00 12-16 Put $8000.00 contract

2017- BTC 2017-12-29 Options $310.00 $625.00 3 23 $565.00 12-16 Put $14000.00 contract

2017- BTC 2018-03-30 Options $1,090.00 $1,560.00 1 34 $1,160.00 12-16 Put $10000.00 contract

2017- BTC 2017-12-29 Options $12,500.00 $13,500.00 0 0 --- 12-16 Call $6000.00 contract

2017- BTC 2017-12-29 Options $10.00 $65.50 0 10 --- 12-16 Put $6000.00 contract

2017- BTC 2017-12-29 Options $11,500.00 $12,500.00 0 0 --- 12-16 Call $7000.00 contract

2017- BTC 2017-12-29 Options $20.00 $90.50 0 6 --- 12-16 Put $7000.00 contract

2017- BTC 2017-12-29 Options $10,500.00 $11,500.00 0 109 --- 12-16 Call $8000.00 contract

2017- BTC 2017-12-29 Options $8,750.00 $9,350.00 0 2 --- 12-16 Call $10000.00 contract

78 Volume- Report Open Weighted Contract Contract Last Bid Last Ask Volume Date Interest Average Type Price

2017- BTC 2017-12-29 Options $8.75 $225.00 0 3 --- 12-16 Put $10000.00 contract

2017- BTC 2017-12-29 Options $6,800.00 $7,450.00 0 15 --- 12-16 Call $12000.00 contract

2017- BTC 2017-12-29 Options $165.00 $415.00 0 4 --- 12-16 Put $12000.00 contract

2017- BTC 2017-12-29 Options $5,290.00 $5,940.00 0 4 --- 12-16 Call $14000.00 contract

2017- BTC 2017-12-29 Options $4,530.00 $5,180.00 0 0 --- 12-16 Call $15000.00 contract

2017- BTC 2017-12-29 Options $451.00 $892.00 0 4 --- 12-16 Put $15000.00 contract

2017- BTC 2017-12-29 Options $1,430.00 $1,970.00 0 0 --- 12-16 Call $20000.00 contract

2017- BTC 2017-12-29 Options $2,560.00 $3,200.00 0 1 --- 12-16 Put $20000.00 contract

2017- BTC 2018-03-30 Options $13,500.00 $14,500.00 0 0 --- 12-16 Call $5000.00 contract

2017- BTC 2018-03-30 Options $270.00 $510.00 0 0 --- 12-16 Put $5000.00 contract

2017- BTC 2018-03-30 Options $10,000.00 $10,500.00 0 0 --- 12-16 Call $10000.00 contract

2017- BTC 2018-03-30 Options $7,160.00 $7,640.00 0 0 --- 12-16 Call $15000.00 contract

2017- BTC 2018-03-30 Options $3,010.00 $3,540.00 0 0 --- 12-16 Put $15000.00 contract

2017- BTC 2018-06-29 Options $14,000.00 $15,000.00 0 0 --- 12-16 Call $5000.00 contract

2017- BTC 2018-06-29 Options $410.00 $595.00 0 0 --- 12-16 Put $5000.00 contract

79 Volume- Report Open Weighted Contract Contract Last Bid Last Ask Volume Date Interest Average Type Price

2017- BTC 2018-06-29 Options $10,500.00 $11,500.00 0 0 --- 12-16 Call $10000.00 contract

2017- BTC 2018-06-29 Options $1,820.00 $2,230.00 0 25 --- 12-16 Put $10000.00 contract

2017- BTC 2018-06-29 Options $8,340.00 $8,950.00 0 0 --- 12-16 Call $15000.00 contract

2017- BTC 2018-06-29 Options $4,350.00 $4,950.00 0 0 --- 12-16 Put $15000.00 contract

2017- BTC 2018-09-28 Options $14,000.00 $15,000.00 0 0 --- 12-16 Call $5000.00 contract

2017- BTC 2018-09-28 Options $535.00 $745.00 0 0 --- 12-16 Put $5000.00 contract

2017- BTC 2018-09-28 Options $11,000.00 $12,000.00 0 0 --- 12-16 Call $10000.00 contract

2017- BTC 2018-09-28 Options $2,250.00 $2,760.00 0 2 --- 12-16 Put $10000.00 contract

2017- BTC 2018-09-28 Options $9,230.00 $9,740.00 0 0 --- 12-16 Call $15000.00 contract

2017- BTC 2018-09-28 Options $4,970.00 $5,630.00 0 0 --- 12-16 Put $15000.00 contract

2017- BTC 2018-12-28 Options $14,500.00 $15,500.00 0 0 --- 12-16 Call $5000.00 contract

2017- BTC 2018-12-28 Options $645.00 $830.00 0 0 --- 12-16 Put $5000.00 contract

2017- BTC 2018-12-28 Options $11,500.00 $12,500.00 0 0 --- 12-16 Call $10000.00 contract

2017- BTC 2018-12-28 Options $2,860.00 $3,450.00 0 4 --- 12-16 Put $10000.00 contract

2017- BTC 2018-12-28 Options $9,860.00 $10,500.00 0 0 --- 12-16 Call $15000.00 contract

80 Volume- Report Open Weighted Contract Contract Last Bid Last Ask Volume Date Interest Average Type Price

2017- BTC 2018-12-28 Options $5,810.00 $6,500.00 0 0 --- 12-16 Put $15000.00 contract

2017- BTC 2018-12-28 Options $7,180.00 $7,890.00 0 40 --- 12-16 Call $25000.00 contract

2017- BTC 2018-12-28 Options $12,500.00 $14,000.00 0 0 --- 12-16 Put $25000.00 contract

2017- BTC 2018-12-28 Options $3,830.00 $4,570.00 0 0 --- 12-16 Call $50000.00 contract

2017- BTC 2018-12-28 Options $33,500.00 $35,500.00 0 0 --- 12-16 Put $50000.00 contract

81 Figure B.1.: Bitcoin options market prices on 16/12/2018 82 83 84 Figure B.2.: Bitcoin options market prices on 05/02/2018

85 86 87 88 89 90 91 92 93 Figure B.3.: Bitcoin options market prices on 28/06/2018 Time to Strike Mid price Kou model BS model ID Type maturity (USD) (USD) price (USD) price (USD) (days) 1 Call 13 $ 6000,00 $ 13000,00 $ 13347,34 $ 13346,23 2 Call 13 $ 7000,00 $ 12000,00 $ 12349,43 $ 12346,77 3 Call 13 $ 8000,00 $ 11000,00 $ 11353,00 $ 11347,37 4 Call 13 $ 10000,00 $ 9050,00 $ 9369,03 $ 9350,86 5 Call 13 $ 12000,00 $ 7125,00 $ 7413,81 $ 7377,72 6 Call 13 $ 14000,00 $ 5615,00 $ 5541,83 $ 5500,88 7 Call 13 $ 15000,00 $ 4855,00 $ 4669,66 $ 4636,39 8 Call 13 $ 20000,00 $ 1700,00 $ 1498,95 $ 1541,20 9 Put 13 $ 6000,00 $ 37,75 $ 1,10 $ 0,00 10 Put 13 $ 7000,00 $ 55,25 $ 2,66 $ 0,00 11 Put 13 $ 8000,00 $ 67,50 $ 5,70 $ 0,06 12 Put 13 $ 10000,00 $ 116,88 $ 20,66 $ 2,49 13 Put 13 $ 12000,00 $ 290,00 $ 64,37 $ 28,29 14 Put 13 $ 14000,00 $ 467,50 $ 191,32 $ 150,37 15 Put 13 $ 15000,00 $ 671,50 $ 318,62 $ 285,35 16 Put 13 $ 20000,00 $ 2880,00 $ 2145,23 $ 2187,49 17 Call 104 $ 5000,00 $ 14000,00 $ 14434,75 $ 14412,90 18 Call 104 $ 10000,00 $ 10250,00 $ 10192,01 $ 10137,37 19 Call 104 $ 15000,00 $ 7400,00 $ 7040,76 $ 7000,41 20 Put 104 $ 5000,00 $ 390,00 $ 70,39 $ 48,55 21 Put 104 $ 10000,00 $ 1325,00 $ 806,33 $ 751,69 22 Put 104 $ 15000,00 $ 3275,00 $ 2633,75 $ 2593,41 23 Call 195 $ 5000,00 $ 14500,00 $ 14667,34 $ 14632,11 24 Call 195 $ 10000,00 $ 11000,00 $ 11097,57 $ 11039,46 25 Call 195 $ 15000,00 $ 8645,00 $ 8535,97 $ 8486,51 26 Put 195 $ 5000,00 $ 502,50 $ 284,40 $ 249,17 27 Put 195 $ 10000,00 $ 2025,00 $ 1674,72 $ 1616,61 28 Put 195 $ 15000,00 $ 4650,00 $ 4073,22 $ 4023,76 29 Call 286 $ 5000,00 $ 14500,00 $ 14947,63 $ 14912,32 30 Call 286 $ 10000,00 $ 11500,00 $ 11866,78 $ 11810,90 31 Call 286 $ 15000,00 $ 9485,00 $ 9669,90 $ 9617,51 32 Put 286 $ 5000,00 $ 640,00 $ 546,18 $ 510,87 33 Put 286 $ 10000,00 $ 2505,00 $ 2406,90 $ 2351,02 34 Put 286 $ 15000,00 $ 5300,00 $ 5151,60 $ 5099,21 35 Call 377 $ 5000,00 $ 15000,00 $ 15151,86 $ 15201,07

94 Time to Strike Mid price Kou model BS model ID Type maturity (USD) (USD) price (USD) price (USD) (days) 36 Call 377 $ 10000,00 $ 12000,00 $ 12461,17 $ 12473,06 37 Call 377 $ 15000,00 $ 10180,00 $ 10540,46 $ 10540,85 38 Call 377 $ 25000,00 $ 7535,00 $ 7956,89 $ 7959,62 39 Call 377 $ 50000,00 $ 4200,00 $ 4675,85 $ 4702,49 40 Put 377 $ 5000,00 $ 737,50 $ 731,96 $ 781,17 41 Put 377 $ 10000,00 $ 3155,00 $ 2964,40 $ 2976,29 42 Put 377 $ 15000,00 $ 6155,00 $ 5966,83 $ 5967,21 43 Put 377 $ 25000,00 $ 13250,00 $ 13229,51 $ 13232,24 44 Put 377 $ 50000,00 $ 34500,00 $ 34564,13 $ 34590,77

Table B.2.: Output of the Kou model - 16/12/2017

Table B.2 compares the model prices with the corresponding bitcoin options market prices on 16/12/2017. The parameters of the Kou model are in Table B.1. The volatility σ used in the Black- Scholes model is 1.25 and was chosen as the optimal value of the range [0.01, 2.45] .

95 Time to Strike Mid price Kou model BS model ID Type maturity (USD) (USD) price (USD) price (USD) (days) 1 Call 18 $ 5000,00 $ 2194,50 $ 2028,39 $ 2027,91 2 Call 18 $ 7500,00 $ 690,38 $ 569,34 $ 568,52 3 Call 18 $ 10000,00 $ 161,13 $ 114,60 $ 114,22 4 Call 18 $ 15000,00 $ 8,88 $ 3,32 $ 3,30 5 Call 18 $ 20000,00 $ 113,25 $ 0,10 $ 0,10 6 Put 18 $ 5000,00 $ 182,75 $ 110,43 $ 109,95 7 Put 18 $ 7500,00 $ 1161,38 $ 1149,53 $ 1148,72 8 Put 18 $ 10000,00 $ 3051,38 $ 3192,95 $ 3192,56 9 Put 18 $ 15000,00 $ 7905,00 $ 8077,97 $ 8077,94 10 Put 18 $ 20000,00 $ 12900,00 $ 13071,05 $ 13 071,05 11 Call 53 $ 5000,00 $ 2570,00 $ 2364,26 $ 2363,17 12 Call 53 $ 10000,00 $ 719,00 $ 540,31 $ 539,05 13 Call 53 $ 15000,00 $ 174,75 $ 126,64 $ 126,07 14 Call 53 $ 25000,00 $ 106,75 $ 9,66 $ 9,58 15 Put 53 $ 5000,00 $ 596,38 $ 439,13 $ 438,03 16 Put 53 $ 10000,00 $ 3580,00 $ 3604,29 $ 3603,03 17 Put 53 $ 15000,00 $ 8227,13 $ 8179,74 $ 8179,18 18 Put 53 $ 25000,00 $ 17850,00 $ 18041,00 $ 18040,92 19 Call 144 $ 5000,00 $ 3180,00 $ 2999,00 $ 2997,11 20 Call 144 $ 10000,00 $ 1505,00 $ 1402,38 $ 1399,90 21 Call 144 $ 15000,00 $ 838,00 $ 734,86 $ 732,77 22 Call 144 $ 25000,00 $ 255,25 $ 254,19 $ 253,00 23 Put 144 $ 5000,00 $ 1231,88 $ 1055,23 $ 1053,35 24 Put 144 $ 10000,00 $ 4621,88 $ 4429,12 $ 4426,64 25 Put 144 $ 15000,00 $ 8665,00 $ 8732,09 $ 8730,00 26 Put 144 $ 25000,00 $ 18 548,63 $ 18 192,43 $ 18191,23 27 Call 235 $ 5000,00 $ 3600,00 $ 3453,14 $ 3450,81 28 Call 235 $ 10000,00 $ 2095,00 $ 2032,41 $ 2029,25 29 Call 235 $ 15000,00 $ 1346,25 $ 1324,09 $ 1321,01 30 Put 235 $ 5000,00 $ 1788,50 $ 1490,83 $ 1488,50 31 Put 235 $ 10000,00 $ 4939,00 $ 5022,04 $ 5018,88 32 Put 235 $ 15000,00 $ 9095,00 $ 9265,67 $ 9262,59 33 Call 326 $ 5 000,00 $ 3752,75 $ 3812,91 $ 3810,29 34 Call 326 $ 10000,00 $ 2535,00 $ 2534,97 $ 2531,37 35 Call 326 $ 15000,00 $ 1418,50 $ 1838,61 $ 1834,85

96 Time to Strike Mid price Kou model BS model ID Type maturity (USD) (USD) price (USD) price (USD) (days) 36 Call 326 $ 25000,00 $ 704,50 $ 1111,27 $ 1107,89 37 Call 326 $ 50000,00 $ 259,00 $ 462,00 $ 459,77 38 Put 326 $ 5000,00 $ 1983,75 $ 1832,11 $ 1829,49 39 Put 326 $ 10000,00 $ 5260,00 $ 5487,63 $ 5484,03 40 Put 326 $ 15000,00 $ 9661,50 $ 9724,73 $ 9720,98 41 Put 326 $ 25000,00 $ 18450,00 $ 18864,31 $ 18860,93 42 Put 326 $ 50000,00 $ 42750,00 $ 42882,34 $ 42880,12

Table B.3.: Output of the Kou model - 05/02/2018

Table B.3 compares the model prices with the corresponding bitcoin options market prices on 05/02/2018. The parameters of the Kou model are in Table B.1. The volatility σ used in the Black- Scholes model is 1.30 and was chosen as the optimal value of the range [0.01, 1.95] .

97 Time to Strike Market Kou model BS model ID Type maturity (USD) price (USD) price (USD) price (USD) (days) 1 Call 1 $ 5000,00 $ 1100,00 $ 848,77 $ 848,48 2 Call 1 $ 6000,00 $ 139,50 $ 46,56 $ 48,40 3 Call 1 $ 7000,00 $ 10,00 $ 0,00 $ 0,00 4 Call 1 $ 7500,00 $ 27,63 $ 0,00 $ 0,00 5 Call 1 $ 8000,00 $ 26,13 $ 0,00 $ 0,00 6 Call 1 $ 9000,00 $ 50,00 $ 0,00 $ 0,00 7 Call 1 $ 10000,00 $ 40,00 $ 0,00 $ 0,00 8 Call 1 $ 12500,00 $ 25,50 $ 0,00 $ 0,00 9 Call 1 $ 15000,00 $ 25,50 $ 0,00 $ 0,00 10 Call 1 $ 20000,00 $ 75,00 $ 0,00 $ 0,00 11 Call 1 $ 25000,00 $ 38,00 $ 0,00 $ 0,00 12 Put 1 $ 5000,00 $ 17,50 $ 0,30 $ 0,02 13 Put 1 $ 6000,00 $ 44,25 $ 198,05 $ 199,89 14 Put 1 $ 7000,00 $ 916,50 $ 1151,46 $ 1151,45 15 Put 1 $ 7500,00 $ 1388,50 $ 1651,43 $ 1651,43 16 Put 1 $ 8000,00 $ 1920,00 $ 2151,41 $ 2151,41 17 Put 1 $ 9000,00 $ 2900,00 $ 3151,37 $ 3151,37 18 Put 1 $ 10000,00 $ 3915,00 $ 4151,33 $ 4151,33 19 Put 1 $ 12500,00 $ 6415,00 $ 6651,23 $ 6651,23 20 Put 1 $ 15000,00 $ 8915,00 $ 9151,12 $ 9151,12 21 Put 1 $ 20000,00 $ 13950,00 $ 14150,92 $ 14150,92 22 Put 1 $ 25000,00 $ 18950,00 $ 19150,71 $ 19150,71 23 Call 29 $ 5000,00 $ 1168,00 $ 1056,75 $ 1061,37 24 Call 29 $ 6000,00 $ 521,00 $ 500,25 $ 508,32 25 Call 29 $ 7000,00 $ 175,50 $ 205,14 $ 212,63 26 Call 29 $ 7500,00 $ 105,88 $ 125,90 $ 132,16 27 Call 29 $ 8000,00 $ 46,00 $ 75,55 $ 80,44 28 Call 29 $ 9000,00 $ 85,50 $ 25,81 $ 28,38 29 Call 29 $ 10000,00 $ 82,50 $ 8,39 $ 9,56 30 Call 29 $ 12500,00 $ 53,00 $ 0,45 $ 0,57 31 Put 29 $ 5000,00 $ 87,63 $ 202,54 $ 207,15 32 Put 29 $ 6000,00 $ 453,00 $ 644,84 $ 652,92 33 Put 29 $ 7000,00 $ 1125,00 $ 1348,54 $ 1356,03 34 Put 29 $ 7500,00 $ 1528,00 $ 1768,71 $ 1774,97 35 Put 29 $ 8000,00 $ 1970,00 $ 2217,76 $ 2222,65

98 Time to Strike Market Kou model BS model ID Type maturity (USD) price (USD) price (USD) price (USD) (days) 36 Put 29 $ 9000,00 $ 2920,00 $ 3166,83 $ 3169,40 37 Put 29 $ 10000,00 $ 3910,00 $ 4148,22 $ 4149,39 38 Put 29 $ 12500,00 $ 6405,00 $ 6637,30 $ 6637,42 39 Call 64 $ 5000,00 $ 1401,50 $ 1278,41 $ 1286,45 40 Call 64 $ 6000,00 $ 795,50 $ 777,72 $ 788,80 41 Call 64 $ 7000,00 $ 435,50 $ 454,96 $ 466,48 42 Call 64 $ 8000,00 $ 228,00 $ 259,49 $ 269,61 43 Call 64 $ 9000,00 $ 148,88 $ 145,80 $ 153,78 44 Put 64 $ 5000,00 $ 273,00 $ 417,02 $ 425,06 45 Put 64 $ 6000,00 $ 716,50 $ 913,70 $ 924,78 46 Put 64 $ 7000,00 $ 1332,00 $ 1588,31 $ 1599,83 47 Put 64 $ 8000,00 $ 2130,00 $ 2390,22 $ 2400,33 48 Put 64 $ 9000,00 $ 3010,00 $ 3273,90 $ 3281,88 49 Call 92 $ 5000,00 $ 1492,50 $ 1421,02 $ 1430,97 50 Call 92 $ 6500,00 $ 820,00 $ 766,48 $ 779,99 51 Call 92 $ 7500,00 $ 510,00 $ 498,85 $ 512,26 52 Call 92 $ 10000,00 $ 177,25 $ 167,56 $ 176,81 53 Call 92 $ 12500,00 $ 76,50 $ 57,12 $ 62,15 54 Call 92 $ 15000,00 $ 66,50 $ 20,19 $ 22,69 55 Put 92 $ 5000,00 $ 419,50 $ 553,89 $ 563,84 56 Put 92 $ 6500,00 $ 1186,00 $ 1393,69 $ 1407,21 57 Put 92 $ 7500,00 $ 1882,00 $ 2122,28 $ 2135,70 58 Put 92 $ 10000,00 $ 4002,50 $ 4281,56 $ 4290,81 59 Put 92 $ 12500,00 $ 6430,00 $ 6661,69 $ 6666,72 60 Put 92 $ 15000,00 $ 8875,00 $ 9115,33 $ 9117,82 61 Call 183 $ 5000,00 $ 1864,00 $ 1780,91 $ 1795,20 62 Call 183 $ 7500,00 $ 981,00 $ 908,59 $ 927,31 63 Call 183 $ 10000,00 $ 432,00 $ 476,86 $ 493,88 64 Call 183 $ 12500,00 $ 283,50 $ 259,65 $ 273,12 65 Call 183 $ 15000,00 $ 172,00 $ 146,59 $ 156,63 66 Call 183 $ 20000,00 $ 224,00 $ 51,47 $ 56,70 67 Call 183 $ 25000,00 $ 119,50 $ 20,12 $ 22,82 68 Call 183 $ 50000,00 $ 110,00 $ 0,51 $ 0,66 69 Put 183 $ 5000,00 $ 691,00 $ 895,19 $ 909,48 70 Put 183 $ 7500,00 $ 2267,50 $ 2504,14 $ 2522,85

99 Time to Strike Market Kou model BS model ID Type maturity (USD) price (USD) price (USD) price (USD) (days) 71 Put 183 $ 10000,00 $ 4305,00 $ 4553,68 $ 4570,70 72 Put 183 $ 12500,00 $ 6555,00 $ 6817,73 $ 6831,20 73 Put 183 $ 15000,00 $ 8965,00 $ 9185,95 $ 9195,99 74 Put 183 $ 20000,00 $ 13850,00 $ 14053,36 $ 14058,60 75 Put 183 $ 25000,00 $ 18750,00 $ 18984,55 $ 18987,25 76 Put 183 $ 50000,00 $ 43550,00 $ 43777,64 $ 43777,78 77 Call 274 $ 5000,00 $ 2174,00 $ 2057,18 $ 2074,48 78 Call 274 $ 15000,00 $ 346,75 $ 332,58 $ 349,44 79 Call 274 $ 25000,00 $ 245,50 $ 84,06 $ 91,83 80 Put 274 $ 5000,00 $ 1022,00 $ 1152,94 $ 1170,24 81 Put 274 $ 15000,00 $ 9145,00 $ 9316,36 $ 9333,23 82 Put 274 $ 25000,00 $ 18930,00 $ 18955,87 $ 18963,64 83 Call 365 $ 5000,00 $ 2545,00 $ 2287,02 $ 2306,63 84 Call 365 $ 15000,00 $ 987,50 $ 536,39 $ 558,78 85 Call 365 $ 25000,00 $ 387,50 $ 186,14 $ 199,61 86 Put 365 $ 5000,00 $ 1885,00 $ 1364,32 $ 1383,93 87 Put 365 $ 15000,00 $ 9315,00 $ 9464,81 $ 9487,20 88 Put 365 $ 25000,00 $ 18930,00 $ 18965,68 $ 18979,15 89 Call 456 $ 5000,00 $ 1996,00 $ 2487,53 $ 2507,41 90 Call 456 $ 15000,00 $ 1344,00 $ 742,59 $ 768,58 91 Call 456 $ 25000,00 $ 561,50 $ 313,52 $ 332,20 92 Put 456 $ 5000,00 $ 2130,00 $ 1546,45 $ 1566,32 93 Put 456 $ 15000,00 $ 9800,00 $ 9615,85 $ 9641,84 94 Put 456 $ 25000,00 $ 18935,00 $ 19001,13 $ 19019,81 95 Call 547 $ 5000,00 $ 2088,00 $ 2741,71 $ 2685,23 96 Call 547 $ 15000,00 $ 1260,50 $ 1176,08 $ 972,60 97 Call 547 $ 25000,00 $ 754,50 $ 611,92 $ 479,16 98 Call 547 $ 50000,00 $ 335,00 $ 188,74 $ 139,14 99 Put 547 $ 5000,00 $ 2367,50 $ 1782,30 $ 1725,83 100 Put 547 $ 15000,00 $ 9955,00 $ 9994,39 $ 9790,91 101 Put 547 $ 25000,00 $ 18940,00 $ 19207,95 $ 19075,19 102 Put 547 $ 50000,00 $ 43000,00 $ 43229,05 $ 43179,45

Table B.4.: Output of the Kou model - 28/06/2018

Table B.4 compares the model prices with the corresponding bitcoin options market prices on 28/06/2018. The parameters of the Kou model are in Table B.1. The volatility σ used in the Black- Scholes model is 0.87 and was chosen as the optimal value of the range [0.01, 1.55] .

100 C. MATLAB code

% The Kou Model — European Call price function out = Kou_European_Call(S0, K, T, r, delta, sigma, lambda, p, eta1, eta2) q = 1 - p; zeta = (p * eta1) / (eta1 - 1) + (q * eta2) / (eta2 + 1) - 1; b_plus = ( log(S0/K) + ( r - delta + sigma^2/2 - lambda * zeta) * T ) / ( sigma * sqrt(T) ); b_minus = ( log(S0/K) + ( r - delta - sigma^2/2 - lambda * zeta) * T ) / ( sigma * sqrt(T) ); h = log(K/S0) + (lambda * zeta - r + delta + sigma^2/2) * T; d1 = exp( (sigma * eta1)^2 * T * 0.5 ) / ( sigma * sqrt(2 * pi * T) ); d2 = exp( (sigma * eta2)^2 * T * 0.5 ) / ( sigma * sqrt(2 * pi * T) ); a1 = exp( -(delta + lambda * zeta + sigma^2/2) * T ) * d1; a2 = exp( -(delta + lambda * zeta + sigma^2/2) * T ) * d2;

% calcule parcel 1 sum_p1 = 0; % according to Kou (2001: 17) only the ﬁrst 10 to 15 terms are needed for most applications for n = 1 : 15 sum = 0; for k = 1 : n sum = sum + P(n, k, p, q, eta1, eta2) * (sigma * sqrt(T) * eta1)^k * I(k - 1, h, 1 - eta1, -1/( sigma * sqrt(T) ), -sigma * eta1 * sqrt(T) ); end sum_p1 = sum_p1 + Pi(n, lambda, T) * sum; end p1 = S0 * a1 * sum_p1;

% calcule parcel 2 sum_p2 = 0; % according to Kou (2001: 17) only the ﬁrst 10 to 15 terms are needed for most applications

101 for n = 1 : 15 sum = 0; for k = 1 : n sum = sum + P(n, k, p, q, eta1, eta2) * (sigma * sqrt(T) * eta1)^k * I(k - 1, h, - eta1, -1/( sigma * sqrt(T) ), -sigma * eta1 * sqrt(T) ); end sum_p2 = sum_p2 + Pi(n, lambda, T) * sum; end p2 = K * exp( -r * T ) * d1 * sum_p2;

% calcule parcel 3 sum_p3 = 0; % according to Kou (2001: 17) only the ﬁrst 10 to 15 terms are needed for most applications for n = 1 : 15 sum = 0; for k = 1 : n sum = sum + Q(n, k, p, q, eta1, eta2) * (sigma * sqrt(T) * eta2)^k * I(k - 1, h, 1 + eta2, 1/( sigma * sqrt(T) ), -sigma * eta2 * sqrt(T) ); end sum_p3 = sum_p3 + Pi(n, lambda, T) * sum; end p3 = S0 * a2 * sum_p3;

% calcule parcel 4 sum_p4 = 0; % according to Kou (2001: 17) only the ﬁrst 10 to 15 terms are needed for most applications for n = 1 : 15 sum = 0; for k = 1 : n sum = sum + Q(n, k, p, q, eta1, eta2) * (sigma * sqrt(T) * eta2)^k * I(k - 1, h, eta2, 1/( sigma * sqrt(T) ), -sigma * eta2 * sqrt(T) ); end sum_p4 = sum_p4 + Pi(n, lambda, T) * sum; end p4 = K * exp( -r * T ) * d2 * sum_p4;

102 % calcule parcel 5 p5 = Pi(0, lambda, T) * ( S0 * exp(-(delta + lambda * zeta) * T) * normcdf(b_plus) - K * exp(-r * T) * normcdf(b_minus) );

% ﬁnal calculation out = p1 - p2 + p3 - p4 + p5; end

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% The Kou Model — European Put price function out = Kou_European_Put(S0, K, T, r, delta, sigma, lambda, p, eta1, eta2) out = Kou_European_Call(S0, K, T, r, delta, sigma, lambda, p, eta1, eta2) + K * exp(-r * T) - S0; end

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% The Kou Model — P Functions function out = P(n, k, p, q, eta1, eta2) if k == n out = p^n; else sum = 0; for i = k : n - 1 sum = sum + p^i * q^(n - i) * nchoosek(n - k - 1, i - k) * nchoosek(n, i) * (eta1 / (eta1 + eta2))^(i - k) * (eta2 / (eta1 + eta2))^(n - i); end out = sum; end end

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% The Kou Model — Q Functions function out = Q(n, k, p, q, eta1, eta2) if k == n

103 out = q^n; else sum = 0; for i = k : n - 1 sum = sum + q^i * p^(n - i) * nchoosek(n - k - 1, i - k) * nchoosek(n, i) * (eta2 / (eta1 + eta2))^(i - k) * (eta1 / (eta1 + eta2))^(n - i); end out = sum; end end

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% The Kou Model — Pi Functions function out = Pi (n, lambda, T) out = ( exp(-lambda * T) * (lambda * T)^n ) / ( factorial(n) ); end

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% The Kou Model — I Functions function out = I(n, h, alpha, beta, delta) sum = 0; for i = 1 : n + 1 sum = sum + (beta / alpha)^(n - (i-1)) * Hh(i-1, beta * h - delta); end out = sum; if beta > 0 && alpha ~= 0 out = -(exp(alpha * h) / alpha) * sum + (beta / alpha)^(n + 1) * (sqrt(2 * pi) / beta) * exp(alpha * delta / beta + alpha^2/(2 * beta^2)) * normcdf(-beta * h + delta + alpha / beta, 0, 1); else if beta < 0 && alpha < 0 out = -(exp(alpha * h) / alpha) * sum - (beta / alpha)^(n + 1) * (sqrt(2 * pi) / beta) * exp(alpha * delta / beta + alpha^2/(2 * beta^2) ) * normcdf(beta * h - delta - alpha / beta, 0, 1); else if beta > 0 && alpha == 0

104 out = (1/beta) * Hh(n + 1, beta * h - delta); else if beta == 0 && alpha < 0 out = Hh(n, -delta) * exp(alpha * h); else out = 0; end end end end

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% The Kou Model — Hh Functions (by recursion) function out = Hh(n, x) if n == -1 out = sqrt(2 * pi) * normpdf(x, 0, 1); end if n == 0 out = sqrt(2 * pi) * normcdf(-x, 0, 1); end if n > 0 v(1) = sqrt(2 * pi) * normpdf(x, 0, 1); v(2) = sqrt(2 * pi) * normcdf(-x, 0, 1); for i = 3 : n + 2 v(i) = (1 / (i - 2)) * ( v(i - 2) - x * v(i - 1) ); end out = v(n + 2); end end

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