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

(1)

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

(2)

(3)

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.

(4)

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

(5)

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 mo-tion. 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

(6)

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

(8)

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

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

(9)

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

4.7. 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

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

(10)

1. Introduction

Despite the Black and Scholes (1973) model, [23], has been successful applied in the world of math-ematical 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 as-sumes. 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 pro-cesses, 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.

(11)

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 distri-bution to model the financial returns, probably we will underestimate the number and magnitude of crashes.

(12)

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 evi-dence 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 pre-dicted 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 par-ticularly clear on high frequency data, [45].

The implied volatility is a “wrong number which plugged into the wrong formula gives the right an-swer”, [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 fea-ture 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 indepen-dent 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 assump-tion of the random-walk model seems to be an adequate descripassump-tion 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

(13)

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

(14)

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.

(15)

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 math-ematical 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 in-dependent 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

(16)

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 au-thors 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≥0on a probability space(Ω, F , P) is a Lévy process if:

• The process starts at zero: L0= 0;

• The process has independent increments:

Lt− Lsis independent of{Lu}u≤s, for0 ≤ s < t;

• The process has stationary increments:

Lt− Lsis equal in distribution toLt−s, for0 ≤ s < t;

• The process is continuous in probability: ∀ > 0, lim_h→0_{P [|L}_t+h− L_t| ≥ ] = 0.

(17)

Definition 3.2.2. A stochastic process {Nt}t≥0is a Poisson process if it can be defined as:

Nt=

X

n≥1

1t≥Tn (3.1)

whereTn = Pni=1τi and{τi}i≥1are a sequence of independent exponential random variables with

parameterλ.

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≥0be a Poisson process. For anyt > 0, Ntfollows a Poisson distribution

with parameterλt:

∀n ∈ N, P [Nt= n] = exp (−λt)

(λt)n

n! (3.2)

Proof. See [31, Chapter 2.5].

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

P [Nt− Ns= n] = exp (−λ (t − s))

(λ (t − s))n

n! , n ∈ N (3.3)

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].

(18)

Proposition 3.2.3. Let {Nt}t≥0be a Poisson process with parameterλ. The number of expected jumps

over an interval[0, t] is given by:

E [Nt] = λt (3.4)

The variance of{N_t}_t≥0have 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:

fξ(x) = λe−λx, 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≥0is a compound Poisson process with intensityλ > 0 if

it can be defined as:

Yt=

Nt

X

i=1

(19)

where jump sizesJiare independent and identically distributed random variables with a given

distri-bution F and{N_t}_t≥0is a Poisson process with intensityλ, independent of {Ji}i≥1.

A compound Poisson process is a process whose jumps size is not any more one like the sim-ple 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≥0is given by:

E [Yt] = ∞

X

n=0

E [Yt|Nt= n] P [Nt= n] (3.8)

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 [Y}t|Nt= n] is the conditional expectation for Ytgiven that there has

been n events (jumps) until time t.

Proposition 3.2.5. Let {Yt}t≥0be 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: V ar [Yt] = λtEJ2

(20)

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

Lt= bt + σWt+ Nt X i=1 Ji− tλk ! (3.9)

where b ∈ R, σ ∈ R+, {Wt}t≥0is 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≥0can 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≥0is a

continuous functionψ such that:

E [exp {iuLt}] = exp {tψ (u)} , u ∈ R (3.11)

whereψ is given by:

ψ (u) = iub −u 2_σ2 2 + Z R eiux− 1 ν (dx) (3.12)

whereν 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]: ϕ_L_t(u) = exp t iub − u 2_σ2 2 + Z R eiux− 1 ν (dx) (3.13)

(21)

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:

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

whereatandbtare continuous processes with E

h RT

0 b2tdt

i < ∞.

Then, for anyC1,2functionf : [0, T ] × R → R, the process Yt= f (t, Xt) can be represented as

f (t, Xt) − f (0, X0) = Z t 0 ∂f ∂s (s, Xs) + as ∂f ∂x(s, Xs) ds +1 2 Z t 0 b2_s∂ 2_f ∂x2 (s, Xs) ds + Z t 0 bs ∂f ∂x(s, Xs) dWs + X {i≥1,Ti≤t} f XTi−+ 4Xi − f XTi− (3.15)

or in the differential notation

dYt= df (t, Xt) = ∂f ∂t (t, Xt) dt + at ∂f ∂x(t, Xt) dt + 1 2b 2 t ∂2f ∂x2 (t, Xt) dt + bt ∂f ∂x(t, Xt) dWt+f Xt−+ 4Xt − f Xt− (3.16) Proof. See [62, Chapter 4], [74, Chapter 1.11] and [31, Chapter 8.3] .

(22)

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

St = S0eLt 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

S0

= Lt. For a jump-diffusion process, the log-return will be a Lévy process

such that: Lt= µt + σWt+ Nt X i=1 Ji and X0 ≡ 0 (3.17)

where {Wt}t≥0 is a standard Brownian motion, {Nt}t≥0 is a Poisson process with rate λ,

PNt

i=1Ji

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 fea-ture 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 in-corporate some of the three properties — leptokurtic feature, “volatility smile” and analytical tractabil-ity. 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 mem-oryless 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

(23)

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: dSt St = (r − δ − λζ) dt + σdWt+ d Nt X i=1 (Vi− 1) ! (3.18) where ζ = E [V − 1] = p η1 η₁− 1+ q η₂ η₂+ 1− 1 (3.19)

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

fX(x) = pη1e−η1x1{x≥0}+ qη2eη2x1{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 η2represent 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:

X=d (

ξ+ with probability p

−ξ− with probability q (3.21)

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 andV = eX. Then, we have:

• Eξ+₌ 1 η₁ and Eξ −₌ 1 η₂ • V arξ+₌1 η1 2 andV arξ− = 1 η2 2 • E [X] = _ηp 1 − q η2 andV ar [X] = pq 1 η1 + 1 η2 2 +_ηp 12 + q η22 • E [V ] = EeX_{= p} η1 η1−1 + q η₂ η2+1

(24)

Proposition 3.3.2. The solution of the stochastic differential equation (3.18) is given by: St= S0exp ( r − δ − σ 2 2 − λζ t + σWt+ Nt X i=1 Xi ) (3.22)

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

St= S0exp r − δ − σ 2 2 − λζ t + σWt Nt Y i=1 Vi (3.23)

In Kou model, the Lévy process under Q is equal to Lt= r − δ − σ 2 2 − λζ t + σWt+ Nt X i=1 Xi (3.24)

Proposition 3.3.3. In the Kou model the characteristic exponent function under Q of the Lévy process is ψ_L_t(u) = iu r − δ − σ 2 2 − λζ −σ 2_u2 2 + λ p η1 η₁− iu + q η₂ η₂+ iu − 1 (3.25) Proof. See Appendix A.

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

c(S0, K, T ) = EQe

−rT_c(S

T, K, T ) = e−rTEQ(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 c(S0, K, T ) = e−rTEQ   S₀e r−δ−σ2₂ −λζT +σWT NT Y i=1 Vi − K !+  (3.27)

(25)

We know that the standard Brownian motion, {Wt}t≥0, has a normal distribution with mean 0 and

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 Wt∼ N (0, t) =⇒ σWt∼ N 0, σ2t , and Z ∼ N (0, 1) =⇒ σ√tZ ∼ N 0, σ2t , then σWt=dσ √ tZ. Consequently, equation (3.27) can be rewritten as

c(S0, K, T ) = e−rTEQ   S₀e r−δ−σ2₂ −λζT +σ√T Z NT Y i=1 Vi − K !+  (3.28)

Using the law of total probability,

c(S0, K, T ) = e−rT ∞ X n=0 EQ   S0e r−δ−σ2₂ −λζT +σ√T ZYNT i=1 Vi − K !+ | NT = n  P [NT = n] (3.29) Combining the equations (3.2) and (3.29), we get

c(S0, K, T ) = e−rT ∞ X n=0 EQ " S0e r−δ−σ2 2 −λζ T +σ√T ZYn i=1 Vi − K !+# e−λT(λT ) n n! = e−rT ∞ X n=0 EQ " S0e−λζTe r−δ−σ2₂ T +σ√T Z+Pn i=1Xi − K +# e−λT(λT ) n n! (3.30)

The randomness on the equation (3.30) comes from a normal random variable and n double expo-nential random variables. Following [57, pp. 24], the sum of n double expoexpo-nential 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

c(S0, K, T ) = e−rTEQ " S0e−λζTe r−δ−σ2₂ T +σ√T Z − K +# e−λT(λT ) 0 0!

(26)

+e−rT ∞ X n=1 n X k=1 Pn,kEQ " S0e−λζTe r−δ−σ2₂ T +σ√T Z+Pk i=1ξ + i _{− K} +# e−λT(λT ) n n! + e−rT ∞ X n=1 n X k=1 Qn,kEQ " S0e−λζTe r−δ−σ2₂ T +σ√T Z−Pk i=1ξ − i _{− K} +# e−λT(λT ) n n! (3.31) where the probabilities Pn,kand Qn,kare 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 i=1ξ + i or Pk i=1ξ − i .

Proposition 3.3.5. The density of the sum of two random variables Z = X + Y is

fZ(z) =

Z +∞

−∞

fZ(x, z − x) dx

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

fPk

i=1ξi(t) = ηe

−ηt(ηt)k−1

(k − 1)!, t > 0, gamma distribution with parameters k andη.

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

f_Z+Pk i=1ξi(t) = (ση) k e(ση)22 1 σ√2πe −tη Hhk−1 −t σ + ση f_Z−Pk i=1ξi(t) = (ση) k_e(ση)2 2 1 σ√2πe tη_Hh k−1 t σ + ση

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

(27)

Proposition 3.3.8. The expected values provided by [57, pp. 28] EQ aeb+c(Z+Pki=1ξi) − K + = aeb(ση)ke(ση)22 1 σ√2πIk−1 h; c − η; −1 σ; −ση −K(ση)ke(ση)22 1 σ√2πIk−1 h; −η; −1 σ; −ση , k ≥ 1, c < η and EQ aeb+c(Z−Pki=1ξi) − K + = aeb(ση)ke(ση)22 1 σ√2πIk−1 h; c + η;1 σ; −ση −K(ση)ke(ση)22 1 σ√2πIk−1 h; η; 1 σ; −ση , k ≥ 1, η > −c

Now, the calculation of the expected values of the equation (3.31) comes straightforward. Using the results of Proposition 3.3.8, with a = S0e−λζT, b =

r − δ − σ₂2T and c = 1, then the equation (3.31) becomes c(S0, K, T ) = e−λT S0e−(δ+λζ)TΦ (d+) − Ke−rTΦ (d−) +e−rT ∞ X n=1 n X k=1 Pn,k   S0e −λζT e r−δ−σ2₂ T σ√T η₁ k σ√T√2π e (σ √ T η1)2 2 I_k−1 h; 1 − η₁; − 1 σ√T; −σ √ T η₁ −K σ√T η₁ k σ√T√2π e (σ √ T η1)2 2 I_k−1 h; −η₁; − 1 σ√T; −σ √ T η₁ # e−λT(λT ) n n! +e−rT ∞ X n=1 n X k=1 Qn,k   S0e −λζT e r−δ−σ2 2 T σ√T η₂ k σ√T√2π e (σ√_{T η2})2 2 I_k−1 h; 1 + η₂; 1 σ√T; −σ √ T η₂ − K σ√T η₂ k σ√T√2π e (σ√_{T η2})2 2 I_k−1 h; η₂; 1 σ√T; −σ √ T η₂ # e−λT(λT ) n n! (3.32)

By simplifying these terms, we finally came to the formula for call options pricing under the Kou model:

c(S0, K, T ) = e−λT

S0e−(δ+λζ)TΦ (d+) − Ke−rTΦ (d−)

(28)

+S0e −δ+λζ+σ2₂ T e( σ√_{T η1})2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Pn,k σ√T η₁kIk−1 h; 1 − η₁; − 1 σ√T; −σ √ T η₁ −Ke−rTe( σ√_{T η1})2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Pn,k σ√T η₁kIk−1 h; −η₁; − 1 σ√T; −σ √ T η₁ +S0e −δ+λζ+σ2₂ T e( σ√_{T η2})2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Qn,k σ √ T η₂ k Ik−1 h; 1 + η₂; 1 σ√T; −σ √ T η₂ − Ke−rTe( σ √ T η2)2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Qn,k σ √ T η₂ k Ik−1 h; η₂; 1 σ√T; −σ √ T η₂ (3.33) where: • ζ is defined in equation (3.19); • d±= ln S0 K + r − δ ± σ₂2 − λζT σ√T ; • h = ln_SK 0 + λζ − r + δ +σ₂2 T ;

• Pn,kand Qn,kare defined in equations (A.14) and (A.15);

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

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

(29)

Using the put-call parity:

p(S0, K, T ) − c(S0, K, T ) = Ke−rT − S0 ⇐⇒ p(S0, K, T ) = Ke−rT − 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. p(S0, K, T ) = S0 e−(δ+λ+λζ)TΦ (d+) − 1 + Ke−rT 1 − e−λTΦ (d−) +S0e −δ+λζ+σ2₂ T e( σ√_{T η1})2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Pn,k σ√T η₁kIk−1 h; 1 − η₁; − 1 σ√T; −σ √ T η₁ −Ke−rTe( σ√_{T η1})2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Pn,k σ √ T η₁ k Ik−1 h; −η₁; − 1 σ√T; −σ √ T η₁ +S0e −δ+λζ+σ2 2 T e( σ√_{T η2})2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Qn,k σ √ T η₂ k Ik−1 h; 1 + η₂; 1 σ√T; −σ √ T η₂ − Ke−rTe( σ√_{T η2})2 2 1 σ√T√2π ∞ X n=1 e−λT(λT ) n n! n X k=1 Qn,k σ √ T η₂kIk−1 h; η₂; 1 σ√T; −σ √ T η₂ (3.35)

(30)

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 under-lying 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.

(31)

0 4,000 8,000 12,000 16,000 20,000 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.

0 400 800 1,200 1,600 2,000 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Series: CLOSEPRICE Sample 7/18/2010 6/30/2018 Observations 2905 Mean 1265.605 Median 272.9500 Maximum 19343.04 Minimum 0.050000 Std. Dev. 2825.545 Skewness 3.206615 Kurtosis 13.68523 Jarque-Bera 18798.23 Probability 0.000000

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

(32)

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. -.6 -.4 -.2 .0 .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.

0 200 400 600 800 1,000 1,200 -0.500 -0.375 -0.250 -0.125 0.000 0.125 0.250 0.375 Series: RETURNS Sample 7/18/2010 6/30/2018 Observations 2904 Mean 0.003846 Median 0.001426 Maximum 0.424580 Minimum -0.491528 Std. Dev. 0.056684 Skewness -0.157733 Kurtosis 14.69406 Jarque-Bera 16558.91 Probability 0.000000

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

(33)

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].

(34)

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.

(35)

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, η₂and the option parameters S0, K, T, r, σ, δ, where λ is the intensity of jumps; p represents the probability of upward jumps; η1is

the magnitude of up-jumps; η₂is the magnitude of down-jumps; S0is 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 mo-ments 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 expo-nential 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 nu-merical 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 com-ponent 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.

(36)

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 chap-ters.

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.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strikes ×104 0 2000 4000 6000 8000 10000 12000 14000 Options price

Calls on 16/12/2017, 13 days to maturity

Market Kou 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strikes ×104 0 2000 4000 6000 8000 10000 12000 14000 Options price

Market Black-Scholes

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.

(37)

0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strikes ×104 0 500 1000 1500 2000 2500 3000 Options price

Puts on 16/12/2017, 13 days to maturity

Market Kou 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Strikes ×104 0 500 1000 1500 2000 2500 3000 Options price

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.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0.4 0.6 0.8 1 1.2 1.4 1.6 Options price

×104 Calls on 16/12/2017, 377 days to maturity

Market Kou 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0.4 0.6 0.8 1 1.2 1.4 1.6 Options price

×104 Calls on 16/12/2017, 377 days to maturity

(38)

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

×104 Puts on 16/12/2017, 377 days to maturity

Market Kou 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 0.5 1 1.5 2 2.5 3 3.5 Options price

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.

0.5 1 1.5 2 Strikes ×104 0 500 1000 1500 2000 2500 Options price

Market Kou 0.5 1 1.5 2 Strikes ×104 0 500 1000 1500 2000 2500 Options price

(39)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 500 1000 1500 2000 2500 3000 3500 4000 Options price

Market Kou 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 500 1000 1500 2000 2500 3000 3500 4000 Options price

0.5 1 1.5 2 Strikes ×104 0 2000 4000 6000 8000 10000 12000 14000 Options price

Market Kou 0.5 1 1.5 2 Strikes ×104 0 2000 4000 6000 8000 10000 12000 14000 Options price

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

Market Kou 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Options price

(40)

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. 0.5 1 1.5 2 2.5 Strikes ×104 0 200 400 600 800 1000 1200 Options price

Market Kou 0.5 1 1.5 2 2.5 Strikes ×104 0 200 400 600 800 1000 1200 Options price

5000 6000 7000 8000 9000 10000 11000 12000 13000 Strikes 0 200 400 600 800 1000 1200 Options price

Market Kou 5000 6000 7000 8000 9000 10000 11000 12000 13000 Strikes 0 200 400 600 800 1000 1200 Options price

Market Kou

(41)

5000 5500 6000 6500 7000 7500 8000 8500 9000 Strikes 0 500 1000 1500 Options price

Market Kou 5000 5500 6000 6500 7000 7500 8000 8500 9000 Strikes 0 500 1000 1500 Options price

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Strikes ×104 0 500 1000 1500 Options price

Market Kou 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Strikes ×104 0 500 1000 1500 Options price

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Options price

Market Kou 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Options price

(42)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 500 1000 1500 2000 2500 3000 Options price

Market Kou 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Strikes ×104 0 500 1000 1500 2000 2500 3000 Options price

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. 0.5 1 1.5 2 2.5 Strikes ×104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Options price

Market Kou 0.5 1 1.5 2 2.5 Strikes ×104 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Options price

(43)

5000 6000 7000 8000 9000 10000 11000 12000 13000 Strikes 0 1000 2000 3000 4000 5000 6000 7000 Options price

Market Kou 5000 6000 7000 8000 9000 10000 11000 12000 13000 Strikes 0 1000 2000 3000 4000 5000 6000 7000 Options price

5000 5500 6000 6500 7000 7500 8000 8500 9000 Strikes 0 500 1000 1500 2000 2500 3000 3500 Options price

Market Kou 5000 5500 6000 6500 7000 7500 8000 8500 9000 Strikes 0 500 1000 1500 2000 2500 3000 3500 Options price

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Strikes ×104 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Options price

Market Kou 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Strikes ×104 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Options price

(44)

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.

(45)

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.

1.1 2.5 1.15 1.2 2 1 1.25 Im p li ed V ol at il it y σ (T ,M ) 1.3 0.8 Implied Volatility Surface

1.5 Moneyness M =S K 1.35 Time to Matutity T 0.6 1 _0.4 0.2 0.5