Finance is one of the most rapidly changing and fastest growing areas in the business world and new mathematical models are essential to implement and price these new financial instruments. The classic Black-Scholes model, the Jump Diffusion model, Mean-Reversion Jump-Diffusion, Finite Differ ence Method and Monte-Carlo valuation arewidely used in option pricing. A meaningful number of empirical evidences in literature have tested the effectiveness of the models using historical data. This dissertation aims at analysing how do the prices of options react under mean reversion and stochastic volatility. Chapter 1 Introduction Anoptionisacontractwhichgivesits owner,therightbutnotthe obligation,tosellorbuy an underlying asset at a specified price which is known as the A¿A½strike priceA¿A½ before or at a fixed expiration date (at maturity). An important key to success in the market of options, is paying the the right price for an option, when an individual buys one, or when he writes (sells) an option. Nearly all operators on the option exchanges and most option professionals use modelsof option pricing, like A¿A½Option Master R .A¿A½ every time, for the determination whether options are overpriced or underpriced according to such pricing models. Option pricing is so important because when an individual pays too much for an option, even if the commodity futures price or the underlying stock , on which the option is purchased moves in the right direction, the individual will most probably not get enough payment for his potential risk. On the contrary, when one buys an underpriced option, he will obtain more bene fits compared to the risk when the underlying stock, commodity or index moves in the right direction. Options can be written on a variety of things: Interest rates, stock indexes, stock price. Financial options can be useful in helping banks and corporations to be more effective and efficient in meeting their risk management objectives. When financial derivatives are 1 used without a plan or improperly used, they can cause serious damage by pulling the organization in a wrong direction so that itA¿A½s ill prepared for the future, or by causing serious losses. Financial derivatives, make it possible for companies to initiate productive activities that might otherwise be done. There are many different forms of options, which is related to when you can exercise them. Among the different types of options, thereA¿A½s European option where the holder exercise only on the expiration date and is the most common type. ThereA¿A½s American option where the holder can exercise at any time before the expiration date. The payoffof an option takesinto account what the holder will get at expiration, and not what they paid for the option. An option is known to be A¿A½in the moneyA¿A½, if, it was able to be exercised immediately, so the payoff would be positive. An option is A¿A½at the moneyA¿A½, if, the underlying asset is at the strike price and, an option is known to be A¿A½out of the moneyA¿A½ if the holder wouldnA¿A½t execute immediately. For both call and put options, the greater the volatility of the underlying stock, the more likely is the option to move into the money. Financial options are important tools that help organizations to reach their specific risk management objectives. It is important that a user understands the type of the optionsA¿A½s intended function and all the safety precautions to be taken before the option is used. Bachelier (1991) proposed a formula, in particular, for the price of an option, based on the idea that the fluctuations follow a Brownian process. Bachelier (1991) is the among the first person who made a contribution which was of utmost importance to the issue of valuing stock options was the Bachelier whose approach was based on one important assumption A¿A½the expected profit of investments in options is zeroA¿A½ Hjortsberg(2007). Option pricing was then workedby Black&Scholes (1973), who introduced their sem 2 inal work on the theory of option pricing in 1973. Over the last decades, due to the work of Black&Scholes(1973),option valuationproblemhasgainedalotof attention. SinceBlack and Scholes published their seminar, in 1973 on option pricing, there has been huge investigations, theoretically and empirically on option pricing. Over the past few decades, there has been the development of options to provide the basis for corporate hedging and for liability/asset management of financial institutions and corporate hedging. The theory of option pricing has a long history, only until Black and Scholes, in the year 1973, presented their first equilibrium option pricing model. Besides, Robert Merton extended the Black- Scholes model, in the same year, in several important ways. The Black-Scholes formula, since its invention, has been extensively used by traders for the purpose of determining the price of an option. Despite this, the famous formula was questioned after the crash in 1987. Following the Black and Scholes option pricing model in the year 1973, a numbers of other approaches were developed, such as, the Jump Diffusion model which was suggestedby Merton (1976). Cox&Rubinstein (1979) introduced the numerical method FiniteDifferencesandCarlo(1964)forthepricingofthe derivativedirectedbysolvingthe underlying partial differential equation. Documentation from many empirical studies have shown that geometric Brownian motion (GBM) models are not adequate, in relation to the descriptive power, and even to the mispricing that they might generate. The contributions made to the actual volume deal with many generalizations of the basic geometric Brownian motion and here emphasisisplacedonthefactthatinterest ratesandreturnsof various assetpricesmay exhibita jumping behaviour. So possible superpositions of jump and diffusion processes have been studied, which is known as the jump-diffusion processes, the purpose of which is not the studyofaLevy drivingprocess,butto rather emphasizeonthe specific aspectsofthesubclass of Jump-diffusion. Jump-diffusion models also have some intuitive appeal because they let interest rates and prices change continuously most of the time, however, they take 3 into account the fact that now and then, larger jumps may occur and these large jumps cannotbe suitably modeledby the pure diffusion-typeprocesses. Most of the financial theory is based upon random walk of asset prices, yields and returns. Still, over the last twenty years, the theory has been extended to have a closer view of the departures from a random walk. This theory was being used for various purposes which includes capital adequacy, investment strategy, and the pricing and hedging of options. However, the random walk is a special case for a wider range of models, which includes mean reversion. It is recommended that some classes of mean reversion models reducethe capital seeminglyrequiredformany classesof insurance business,relativelyto theresults obtainedfrom random walks. This stimulatedthe actuarial interestinthe mean reversion class of models. My project will focus on Black Scholes, mean reversion and the mathematical implementation. In the next section we havePreliminaries anda deeper view on the Black& Scholes (1973) model accompanied with its drawbacks. In the next chapter, mean reversion is discussed 4 Chapter 2 Preliminaries 2.1 Introduction to Functional Programming under the Mathematica. environment Functional programming is a programming model that treats calculations as the evaluation of mathematical functions and avoids mutable and state data. It places emphasis on the applications of functions, in cont rast with the style of imperative programming that places emphasis on state changes. The functional programming languages places emphasison pattern-matchingandrules.Itis particularlyusefulfor mathematicalwork,inwhich the notion of A¿A½functionA¿A½ is well a well established concept. Mathematicar,isapowerful computational softwareprogram thatis usedin engineering, scienti fic, and mathematical fields and other areas of technical computing. The person who firsthadtheideatocreatesucha softwareis StephenWolframandthis softwareis developed by the A¿A½Wolfram Research of Champaign, llinoisA¿A½. Nearly any workflow includes computing results, and this is what Mathematicar, does. That is, from building a hedge-fund trading website, to a devel oping embedded image-recognition algorithms or the teaching of calculus. Mathematicar,is well-known as the worldA¿A½s latest application for 5 2.1 Introduction to Functional Programming under the Mathematica. environment computations, but itA¿A½s much more. It is the sole development platform that fully integrates computation into complete workflows, that moves you seamlessly from starting ideas, all the way to the arranged individual or enterprise solutions. Mathematica. is a computer language of high level, which has a large number of operations and in-built functions and this means that for various purposes, programming in the traditional senseisnot necessary. Alternatively,an individualjustusea seriesof built-in functions and this has various advantages of which the main advantage is the speed of programming, that is not having to do the basics over again, saves a lot of time. The built-in functions are very flexible and optimised and are also well joined with each other and their properties: numerics, derivatives, series and integrals, are included into the kernel. The proposed model, which will be implemented in Mathematica c enables the prices of options to be consistent simultaneously consistent with the volatility smile and the observed future prices of the option market. The proposed model is suf ficiently flexible so that financial analyst can perform scheme analysis with it and is as follows: Figure 2.1: The proposed model where #(t)isthe deterministic functionthatrepresentsthe equilibriummeanlevelofthe volatility against time, the constant . isthe meanreversionspeedofthe asset,the function a(t) is the equilibrium mean level of the volatility against time, the constant b is the mean reversion speed of the volatility. #, a constant, is the volatility coefficient of the volatility 6 2.1 Introduction to Functional Programming under the Mathematica. environment model, the process vt is the volatility of the underlying asset, that follows the Heston (1993) model, and Wt are correlatedWienerprocesses witha correlation coefficient #. The proposedmodelreducestothe Heston(1993)modelifthe meanreversionspeed, is equal to zero. With the dynamic of the underlying asset, it is possible to acquire the characteristic function for the log-asset value Xt, which is implemented in Mathematica c as follows: Figure 2.2: The Characteristic Function implemented in Mathematica c

The equations of B1, C1, D1 and other equations which leads to the characteristic function will be shown in appendix. After founding the characteristic function, The European options can be valued using Fourier inversion. Carr and Madan argues the Fast FourierTransform (FFT) to compute the vanilla call and put options that arebased on the characteristic function of the log-asset value. The payoffof the plain vanilla call option is max(ST-K, 0) where T is the optionA¿A½s maturity and K is the strike price. Let . denote the log of the strike price K and let qT (s) be the risk-neutral density of the log-asset price st

=

ln ST , and CT

(#)

be the desired valueofaT-maturity call option witha strike price exp(#). As per Carr (1999), CT (#), the modified call price is as follows: cT

(#)

=

exp(##)CT

(#)

(2.1) for some constants #> 0 7 Figure 2.3: FourierTransform .c and then call prices can numerically be obtained by using the inverse transform, implemented in Mathematica. itself: Figure 2.4: InverseTransform c

The values assigned to the various constants and variables to carry out the test in Mathematicar, is as follows: . = 0.25, n = 128, . = 1.5, a = 4.0339, . = 10, a = 0.5328, b = 3.33, . = 0.04, . = 0.9, r = 0.05, S0 = 1.3 v0 = 0.18, T = 1, and K, whichisthe strike price, variesandtheresultsofthe variationis showninthe table below: 8 2.1 Introduction to Functional Programming under the Mathematica. environment Table 2.1: Call option prices: FFT vs. Monte Carlo Strike Price FFT Monte Carlo %Difference 0.3747 1.06172 1.0658 -0.00408 0.4559 0.984477 0.9889 -0.00443 0.5549 0.890306 0.8958 -0.005494 0.6752 0.775873 0.7798 -0.003927 0.8217 0.636518 0.6413 -0.004782 1 0.466914 0.4719 -0.004986 1.2170 0.260889 0.2660 -0.005111 1.4810 0.0553335 0.0583 -0.0029665 1.8023 0.000997007 0.0011 -0.000102993 NIntegrate The function NIntegrate in Mathematica. is a general numerical integrator which can handle a large range of one-dimensional and multidimensional integrals. NIntegrate[f, x, a, b] gives a numerical approximation to the integral

b f(x)dx (2.3) a Module Mathematica normally assumes that all the variables areglobal which means that every time an individual uses a name like x, Mathematica. will assume that the individual is referring to the same object. When one write programs, he might not want all the variables to be global. For example, he may want to use the name x as reference to two different variables in two different programs. So, in this case, he need the x in each program to be treated as a local variable. He can set up the local variables in Mathematica. using modules and within each module, the individual can give a list of the variables that are 9 2.2 Stochastic Calculus to be treated as local to the module. 2.2 Stochastic Calculus Stochastic calculusisacategoryof mathematicsthatisbasedon stochasticprocesseswhich comprises of a consistent theory of integration that are defined for integrals of stochastic processes withrespectto stochasticprocesses. Stochastic calculusis usedto model systems that have random behaviour. The main flavours. Stochastic calculus is the language of risk management and option pricing at fundamentally every big financial firm, and is also the strength of a large body of academic research on an corporate finance, asset pricing, and investor behaviour. 2.3 Call and Put Option Acall option is a financial contract between two parties which consists of the buyer and the sellerwherethebuyeroftheoptionhastherighttobuythe underlying assetatafixed price (strike price/exercise price) or at any time before the option expires. The buyer of a call option purchases the underlying asset with the hope that the price of the underlying asset will increase in the future while the seller of the call options expects that the price of the underlying asset is not more than the strike price. The buyer of the option makes a profit if the value of the asset is greater than the strike price,that is the buyer of the option buys the asset at the exercise price. On the other hand, if the value of the underlying asset is less than the strike price, then the option is not exercised and expires worthless. The diagram below shows the cash payoff on a call option at expiration. 10 2.3CallandPutOption Figure 2.5: Payoff on call option 11 2.3 Call and Put Option Aput option is a financial contract between two parties that gives the buyer of the option the right to sell an underlying asset at a fixed price or at any time before the option expires. Here, the buyer of the put option has the right, but not the obligation, to sell the underlying asset at the strike price and if he or she decides to sell at any time before the expiration date, the seller of the put option (put writer) is obliged to buy at the price stated by the buyer. The buyer of the put option makes an investment by purchasing the underlying asset with the hope that the price of the underlying will decrease in the future when he will sell the underlying asset while the individual who is going to buy the put option expects that the price of the asset is less than the strike price. On the other hand, if the value of the underlying asset is more than the strike price, then the option is not exercised and expires worthless. The diagram below shows the cash payoff on a put option at expiration. Figure 2.6: Payoff on put option 12 2.3 Call and Put Option It has been seen that banks and other financial intermediaries reacted to new situation by developing many financial risk management products which were designed to control risk in a better way. The first product was the simple foreign-exchange forwards caused one party to sell, and the other party to buy, at an agreed date in the future, at a fixed amount of currency. Through entering in a foreign exchange forwardcontract, the customers could balance the risk that large changes in foreign exchange rates would ruin the viability of the economy projects and hence options were first intended to be used to hedge certain risks effectively, and in fact, that was the key that opened up their explosive development. Options also help to ameliorate the market ef ficiency because risks can be classified and be sold to those parties who are willing to accept the risks at the least cost. The use of options breaks risks into pieces so that they be managed independently. Some corporations can keep risks that they can manage most comfortably and they have the choice to transfer those risks that they do not want to manage, to other companies that are willingto accept them.Fromamarket oriented perspective, financial optionsofferfree trading of financial risks. Similarly to the prices of most things, optionsA¿A½s prices can be sensitive to market demand, but there are wise techniques one can use to determine the fair option value. There are many factors than can affect opt ion pricing. Before proceeding in the world of trading option, the investors should have a good knowledge of the different factors that are used to determine the value of an option. The factors are cash dividends, interest rates, volatility, time value or time to expiration, the intrinsic value and the current stock price. There are many options pricing models that use the mentioned parameters to determine the fair valueofanoption.Ofthe severaloptionpricingmodels,the Black-ScholesmodelBlack& Scholes (1973) is the most widely used. In many different ways, options are just similar to any other investment in which one needs to understand what determines the fair price of an option in order to be able to use them. 13 2.4 Black-Scholes European optionsare securitiesthatgivethe ownertherighttobuyan indexorastock at a certain date at a certain price. It is an option that may only be exercised at the end of its life, at its maturity. An American is an option that may be exercised at any time during its life. An American option allows the holder of the option to exercise the option at any time before and including the maturity date, which t his increases the value of the option to the holder of the option, as compared to a European option, where the option can only be exercised at maturity. ABermudaoptionisanoptionthatgivesthebuyertherighttoexerciseatasetnumber of times. This is an intermediate between a European option and an American option. 2.4 Black-Scholes TheBlack&Scholes(1973)formulaiswidelyusedto calculatepricesofoptions. However, in the year 1973, a turning point occurred in the evolution of trading of options, when the professors Black&Scholes (1973) wrote the paper A¿A½Pricingof Options and Corporate Liabilities A¿A½. Based on the assumption that a risk-free rate of interest existed, this was viewed as a fundamental effort at expressing option pricing and corporate bonds. Thou it is used mostly in institutional portfolio management departments, it is still used today for predicting what options should be worth. Since its introduction in 1973, it has been been tested against the option miss-pricing, by many. The precision of the generated price of Black-Scholes depends very much on the accuracy of the parameter inputs, such as, time, interest rate, strike and exercise price are known precisely, and hence they are relatively accurate and easily determined. The A¿A½Black-ScholesA¿A½ model which is also known as the A¿A½Black-Scholes-MertonA¿A½ model 14 2.4 Black-Scholes is one of the most important concepts in modern financial theory which contains certain derivative investmentinstruments.Itwas developedbyBlackand Scholes.Itisoneofthe best ways of determining fair prices of options and it is still widely used today. The Black-Scholes formula is as follows: C

=

SN(d1)

–

Ke..rtN(d2) (2.4) where: C=Theoretical call premium, S=Current Stock price, t=time until option expiration, K=option striking price, r=risk-free interest rate, N=Cumulative standardnormal distribution, e=exponential term(2.7183), s=standarddeviationof stockreturns, ln=natural log arithm. s 2 ln S K

+

r

+

2 tv t (2.5) v d2

=

d1

–

s t (2.6) Merton (1976) is an American economist who is known for his work on risk management and finance theory and especially for his contribution in assessing stock optionA¿A½s value and other derivatives. MertonA¿A½s work on the valuation of option is perhaps the most influential even thou his research covers many areas of economics and finance theory. When Black and Scholes published their formula, prior to 1973, which determines the value of stock options, was very difficult and risky because of the nature of options, which essentially areagreementsthatgivethe investorstherighttoeithersellorbuyanassetatafixedtime in the future. The challenge of an option is to prognosticate its value at a distant time. Before the introduction of the Black-Scholes formula, those investors investing in options determined a risk premium in order to hedge against major financial losses. It was shown 15 2.4 Black-Scholes by the Black-Scholes formula that the risk premiums are not needful for investment in stock options because those premiums are already calculated in the prices of stocks. In order to generalize the Black-Scholes formula, Robert C. Merton used his knowledge in mathematics, by modifying certain assumptions and restrictions which was set by Black and Scholes, such as the unlikely assumption that no dividends will be not be paid by the stock. By modifying this formula, Merton permitted it to be applied to other financial issues, such as student loans and mortgages. Scholes, Black and MertonA¿A½s assumptions are as follows: 1. There is no dividend during the life of the derivative 2. Options can be exercised only upon expiration 3. There are no arbitrage opportunities 4. The trading of security is continuous in time 5. There are no taxes or transaction costs, all securities are perfectly divisible 6. Stock returns are normally distributed and hence volatility is constant over time 7. Interest rates remain constant 2.4.1 Black-Scholes, Partial Differential Equation (PDE) A partial differential equation (PDE) is a differential equation that has unknown multi-variable functions and their partial derivatives. Partial differential equations are used for the formulation of problems that involve functions of various variables, and are either resolved by hand or used to create a relevant computer model. As stated above,The equationof Black&Scholes (1973)isa partialdifferential equation, that describes the price of an option over time. An idea of upmost importance behind the equation is that an individual can perfectly hedge an option by selling and buying an 16 2.4 Black-Scholes underlying asset in the right way and hence eliminating risk. The hedging in turn means thatthereisonlyone correctpriceforanoption,asreturnedbytheBlack&Scholes(1973) formula which is as follows: @V 1 @V #2S2 @2V

++

rS

–

rV =0 (2.7) @t 2 @S2 @S HereIpresent an analytical solution for the Black&Scholes (1973) PDE, @V 1 @V #2S2 @2V rf

=+

+

rS ;V

=

V (t, S) (2.8) @t 2 @S2 @S over the domain 0 <S< 8, 0

=

t

=

T , with a terminal condition V (T;S)= (S) , by the reduction of this parabolic PDE to the heat equation of physics. The substitution u

=

exp[-rt]f is made, which is stimulated by the fact that it is the portfolio value which discounted by the interest rate r that is a martingale. The product rule is used on V

=

exp rtu, and the the PDE that the function u should satisfy is derived: @u @u 1 u #2S2 @2 0=

+

rx

+

(2.9) @t @S 2 @S2 Now, we substitute y with log S, and s with T

–

t. These changes of the variables can be stimulated by observing that:

A¿A½

The underlyingprocess whichis describedby the variableSisa GBM (Geometric Brownian Motion), in order for log S describes a Brownian motion, with a possible drift. Then some sortof diffusion equation shouldbe satisfiedby log S.

A¿A½

From the terminal state of the system, the evolution of the system is backwards. Actually, the boundary condition is given as the terminal state, and the coefficient of @u is positive in equation 2.6 and in order to get the heat equation, we have to make @t the useofa substitutiontoreverse time. Since 17 2.4 Black-Scholes @u @u @u @u dy 1 @u

=

–

,

==

,

(2.10) @s @t @S @y dx [email protected]/* */ and @2us 1 @u 1 @u 1 @2u

=(

)=

–

+

(2.11) @S2 @S [email protected]/* */ S2 @y S2 @y2 we then substitute in equation 2.6, which results in: @u 1 @u 1 #2 @2u 0=

–

+(r

–

#2)+ (2.12) @s 2 @y 2 @y2 Withrespect toy, the first partial derivative does not cancel because we didnA¿A½t take into account the drift of the Brownian motion. In order to cancel the drift, the use of substitutions is made: z

=

y +(r

–

1 #2)#;p

=

s. (2.13) 2 Under the new coordinate system (z, #), we have the relations amongst vector fields: s @z

=

s @y

,

s @T

=

-(r

–

1 2 #2) s @y

+

s @s

,

(2.14) which leads to the following transformation of the equation 2.9: 0

=

–

@u @r

–

(r

–

1 2 #2) @u @z

+

1 2 #2 @2u @z2; (2.15) or @u @r

=

1 2 #2 @2u @z2

,

u

=

u(#, z) (2.16) which is one form of the diffusion equation. The domain is on -8

<

z

<

8 and 18 2.4 Black-Scholes 0

=

p

=

T ;and the initial condition is: ..rT (e u(0;z)= e z)

:=

u0(z) (2.17) The original function f can berecoveredby rt 1 f(t, x)= eu(T

–

t, log x +(r

–

#2)#) (2.18) 2 The fundamental solution of the PDE in equation 2.13 is: 1 Z G. (Z)= v exp

–

(2.19) 2#2# 2##2p and the solution u with the initial condition u0 is given by the convolution: ..rT

.

…

e(Z

–

#)2 u(#, Z)= u0

*

G. (Z)= v (e

#)

exp(-)d#. (2.20) 2#2# 2##2p

.

in terms of the original function f which is as follows: e..r.

.

…

(log x +(r

–

1 #2)p

–

#2 2 f(t, S)= v (e

.

)

exp -d#. (2.21) 2#2# 2##2p

.

19 2.4 Black-Scholes Numerical Experiment on Option Pricing The Black-Scholes Option Price Calculator (BetaVersion) is used to generate the Call Price and the Put Price that will be figure in the table below. The volatility rate is taken as 50 0.5 for , the interest is taken as 0.5 for 50 and theTimeTo Expirationis taken as1year 100 100 Table 2.2: Prices of Call and Put Option under varying Stock and Price Stock Price Strike Price Call Price Put Price 50 50 21.264 1.590 90 70 48.423 0.880 100 90 47.397 1.985 60 110 9.423 16.142 80 130 16.261 15.110 170 150 82.098 3.077 90 170 13.191 26.301 100 190 14.455 29.696 20 2.4 Black-Scholes 2.4.2 Shortcomings of the Black

&

Scholes (1973) model In the year 1973, a turning point occurred in the evolution of trading of options, when the professors Black&Scholes (1973) wrote the paper A¿A½Pricingof Options and Corporate Liabilities A¿A½. Based on the assumption that a risk-free rate of interest existed, this was viewed as a fundamental effort at expressing option pricing and corporate bonds. Thou it is used mostly in institutional portfolio management departments, it is still used today for predicting what options should be worth. Since its introduction in 1973, it has been been tested against the option miss-pricing, by many. The precision of the generated price of Black-Scholes depends very much on the accuracy of the parameter inputs, such as, time, interest rate, strike and exercise price are known precisely, and hence they are relatively accurate and easily determine. But, most individual traders have since long recognized that there are flaws in the model, in the following stated ways: 1. It was first published in 1973, when the trading of the public options was in its early stages. Puts were not publicly traded and calls were only available on a handful of listed companies. Furthermore, the population of the trading of options was extremely limited, which means that the assumptions used for the Black-Scholes model cannot be applied in the more complex modern options industry. 2. Dividends are uncalculated in the equation, which means that no dividends are applicable. Option traders know that dividends play a big role in returns and cannot be ignored. In the modern industry, anyone trading in options has to consider the impact of the dividend yield on the overall return. In comparing values of two or more underlying assets, dividend yield is often the determining factor in making the decision of which one 21 2.4 Black-Scholes has a better value. 3. Some of the assumptions are questionable under todayA¿A½s market and economic conditions. For example, the model assumes that the income and valuation have to be compared to an assumed rate of risk free intere st. It is questionable if such a rate exists today. 4. The assumption of European style expiration (that is positions can be only exercised onthelast tradingday).Withafew exceptionsof some index options, mostofthe publicly traded options can be exercised at any time before expiration and thus this changes the calculation. So, the Black-Scholes model makes one assumption that is totally flawed. The assumption follows that the implied volatility on the analysis date willremain unchanged until expiration. Every trader knows that this is inaccurate. 5. Back then, the internet was not yet created. Without the ability to fragmentize numbers automatically and easily, the Black-Scholes model depended on calculations being made manually. With hundreds of more opt ions to trade and with more detailed, faster, and more widely used formulas for the tracking of values, the entire options market is a different thing today than it was in 1973. Even the open levels of contracts have changed, increasing in billions since 1973 to a such a volume today that cannot be imagined in the past. This also directly affects valuation. The use of vega, gamma and delta are more reliable measurements of option pricing and implied volatility than the more unclear Black-Scholes model with its impractical vari ables. Black-Scholes contain numerous problems. Since its first publishment, more theories have been added to expand the Black-Scholes model in order to make it more applicable to the practical market conditions. Even so, its unlikely that an accurate market model will be produced in the near future. The Black-Scholes formula contains too many variables, and 22 2.5 The Jump Diffusion Model with the use of more variables, this results in a less reliable formula. These include the assumption of European expiration, lack of dividend, risk-free interest rate, and unchanging implied volatility. When one variable is used in the formula, it is rather troubling to a certain degree and when two or t hree variables are used, this increases the inaccuracy. The solution to the option pricing problem, should be restricted to studies of implied volatility and hence the volatility of the underlying , market conditions, and the closeness between strike of the option and current market value of stock, are the true factors that values an option. 2.5 The Jump Diffusion Model Afirst approach in further developing the basic Black and Scholes model adding the inclusion of jumps happens to be that of Merton (1976). The introduction of jumps in the Black and Scholes model has the implicatio n that the derivative prices are no longerresolvedby the principle of the absence of arbitrage only, this pricing problem was solved my Merton, following the assumption that the jump risk was not systematic. But this was in later looked through a critical point of view, that such an assumption is equal to the the presenceofamarket portfolio,thatdoesnotpresentajumping behaviourandthat containsthe underlying asset.In empirical studies,itwas do cumentedthata combinationof stochastic volatility and jumps leads to better fits and enables to avoid the implied volatility skews. The models of stochastic volatility are teated elsewhere and hence the limitation is set to stochastic volatility on affiliation with the jump-diffusion modeling. This is also because empirical documentation gives the evidence for a jump-type action in the volatility and in the correlation between jumps in jumps in prices and volatility. Dueto some limitationsof the Black& Scholes (1973) modelin the modelingof the 23 2.5 The Jump Diffusion Model distribution of logarithmic stock returns in considerable shorter periods, Merton (1976) improved the Black&Scholes (1973) modelby adding the possibilityof jumps, which occurs correspondingtoaPoissonprocess, whichis independentfromtheBrownian motion. Dynamics of the stock prices in the Merton jump-diffusion model can be described by this equation: #t+#Bt+Xt St

=

S0 exp(2.22) where Xt isa compound Poissonprocess: Nt

.

Xt

=

Yi (2.23) i=1 Yi,i = 1,2,…, are independent and identically distributed normal random variables. Nt is a Poisson process Processes Bt, Nt and variables Yt,i = 1,2,… are independent. The assumption of the standardBlack&Scholes (1973) of log-normal stock diffusion with a constant volatility is, flawed, as we have already seen in the paper. In general, it is necessary to use different volatilities for different maturities (T) and option strikes (k), to equate the formula of Black&Scholes (1973) with quoted prices of European calls and puts options. This event is most of the time referred to as the volatility smile or skew which depends on the figure of the mapping of implied volatility as of function ofTand K.Volatility skewispresentin all major stock index markets today. Usually, the steepness of the skew diminishes with the increasing option maturities. The presence of the skew is regularly associated with the fear of large downwardmarket movements. The extensions of the Black& Scholes (1973) model that capture the presence of the volatility smiles, can be grouped into three approaches. Firstly, it can be grouped in the stochastic volatility approach (Hull& White 1987), in which, the stockA¿A½s volatility is as 24 2.5 The Jump Diffusion Model sumedtobea meanrevertingdiffusionprocess, whichis usually correlatedwiththe stock process itself. A variety of volatility smiles and skews can be generated in this model, depending on the parameters and the correlation of the volatility process. It has been shown by empirical evidence from time-series analysis which confirms the presence of stochastic volatility in stock prices. However, often, unrealistically high negative correlation between volatilityand stock indexisrequired,inorderto cause implied Black-Scholes volatility skews in a stochastic volati lity that are uniform with those that are observed in traded options. Furthermore, while looking from a computational perspective, the models of stochastic volatility are not easy to handle as they are multi-factor models, which means that, an individual would nor mally need multi-dimensional lattices to evaluate, for example, American options. It has been noticed that stochastic volatility do not allow for absolute option hedging by dynamic positions in the money market account and in the stock. Firstly generated by Merton (1976), a new approach, generates volatility smiles and skews by adding discontinuous jumps to the Black-Scholes diffusion dynamics. By appropriately choosing the parameters of the jump process, a multitude of volatility skews and smiles can be generated by this s o-called jump diffusion model. Similar to stochastic volatility models, the jump-diffusion models are very challenging to handle numerically and this results in bonds and stocks forming an incomplete market. 2.5.1 The Black-Scholes vs the Jump-Diffusion model During the year 2011,in its second half,it was seen that financial marketsreacted strongly to the signs of the intensifying Eurozone crisis. During this disruptive time, the stock prices acted in a very volatile way, as a reaction to the inflow of all the information on the problems of debts of the various Eurozone members and even to the emergency solutions applied. In order to capture such a quick, sometimes discontinuous movements of 25 2.6 Finite Difference the stock prices, The jump-diffusion model of stock prices seems to be appropriate. The jump-diffusion model, on contrary to the classical Black-Scholes model, doesnA¿A½t solve the normalityofthe logarithmicreturnsanditisusuallyfitsthe distributionofthe logarithmic returns in shorter periods, in a better way, compared to the Black-Scholes model. 2.6 Finite Difference The methods of finite difference (also known as finite element methods), for option pricing, are numerical methods that are used in mathematical finance to valuate options. Schwartz (1977) first applied Finite difference methods to option pricing. It is known that finite differences methods may solve problems of derivative pricing, and in general, as complex as those problems solved by the tree app roaches. Finite difference methods are usually applied only when other approaches proves to be inappropriate but at the same time, in terms of the number of underlying variables, this approach is limited, and as for problems with multiple dimensions, the Monte Carlo methods areusually preferred, for option pricing. Finite difference methods are used for the pricing of options by the approximation of the (continuous time) differential equations which describes the evolution of an option price through time by a set of (discrete-time) differential equations. To calculate the price of the option, the discrete differential equation can then be solved iteratively. The method of finite difference value a derivative by solving a differential equation that is satisfied by the derivative. The differential equation is first converter into a set of difference equations, and then the difference equations are solved iteratively (Hull& White 1987) It is started from the Black-Scholes partial differential equation: @[email protected]/* */ +1=2#2 (2.24) 26 2.6 Finite Difference The methodsof finitedifferencecreatea mathematicalrelationshipthat connects every point on the solution domain, like a chain. From the first links, that are boundary conditions, it is discovered what every other point in the domain has to be. The most popular methods of finite difference used in computational finance are: Implicit Euler, Explicit Euler, and the Crank-Nicolson method. The use of these three methods has both advantages and disadvantages and easiest of which to implement is the Explicit Euler method. Crank-Nicolson and Implicit Euler are implicit methods, that generallyrequirea linear equations system to be solved at each step of time, that can be intensive, computationally, on a fine network.Adisadvantage for the useof Explicit Euleris the unstability for certain choices of domain discretisation. The equation of Black&Scholes (1973) has been used as the standardpricing formula fordifferenttypesofoptions.The assumptionsusedintheBlack&Scholes(1973) formula do not eternally hold and the original equation has been generalised to receive different types of options, which means that an exact solution for the Black&Scholes (1973) equation cannot be always found and one must consequently resort to approximate methods, which are: 1. Monte Carlo and quasi Monte Carlo methods (Boyle 1977) 2. Reductionofthe equationtoasimplerform(Wilmott1994) 3. Reductionto other formsby meansof Fourier transforms(Carr1999) 4. Binomial and trinomial methods by Cox in 1985 Since each of mentioned methods contain disadvantages and advantages, one prefers to approximate the PDE that models the option prices by means of finite difference methods. The advantages of using finite difference methods are: 1. They have a sound mathematical and theoretical basis 2. They are easily programmed on a digital computer 27 2.7 Geometric Brownian Motion (GBM) 3. The different methods have a long history which goes back till the eighteenth century 4. The various finite difference methods are flexible and may be applied to many types of problems of pricing. The disadvantages of using the finite difference methods are: 1. Under certain circumstances, higher-order partial differential equations can degenerate into lower-order partial differential equations (with exponentially decaying volatilities or large factorsof driftin the B lack&Scholes (1973) equation) 2. It has been seen that a number of finite difference schemes are too complicated or elaborate. The numerical schemes that, in nature, are non-linear, while the equivalent partial differential equations are linear and they deserve special attention since they can be A¿A½overkillA¿A½ in oneA¿A½s opinion. 3. New techniques are needed for proving the stability of schemes that approximate the partial differential equations that have nonlinear, non-constant and discontinuous coefficients. 2.7 Geometric Brownian Motion (GBM) AGeometric Brownian Motion which is also known as the exponential brownian motion is a time-continuous stochastic process where the logarithm of the quantity varying randomly follows a Brownian motion which is also called the A¿A½Wiener processA¿A½ with drift. It is an important example of the stochastic processes satisfying the stochastic differential equation (SDE) and in particular, its use in mathematical finance is to model stock prices in the Black&Scholes (1973) model. Many current engineering economic analyses have depended on an explicit or an im 28 2.7 Geometric Brownian Motion (GBM) plicit assumption that some quantity that changes with time with uncertainties that follow a GBM process. The GBM process, which also sometimes called a lognormal growth pro-cess,hasbeen acceptedbymanyasavalid modelforthegrowthinthepriceof stocks over time. Under this model, the formulaof Black&Scholes (1973) for the pricingof European call and put options, even as their variations for some of the morecomplicated derivatives, provide simple analytical evaluation of asymmetric risks. Many recent examples of the GBM models have appeared in the analysis of real options, where the value of some underlying asset is assumed to be evolving like a stock price. The GBM assumption is stated exp licitly, in some cases. In other cases, when optionsare evaluatedbythe Black&Scholes (1973) formula,theGBMis implicitly evaluated St,astochasticprocess,issaidto followaGBMifit satisfiesthe followingSDE (Stochastic Differential Equation): dSt

=

#Stdt

+

#StdWt (2.25) where Wt isaA¿A½WienerprocessA¿A½orBrownian motionand #, the percentage drift and #, the percentage volatility, are both constants. Hypergeometric Function A generalized hypergeometric function pFq(a1,

:::,

ap; b1,

:::,

bq; x) is a function which may be defined in terms of a hypergeometric series, which means, a series for which the ratio of successive terms may be written as: Ck+1=Ck

=

P (k)=Q(k)

=

((k+a1)(k+a2):::(k+ap)=(k+b1)(k+b2):::(k+bq)(k+1))x (2.26) In-the-money, At-the-money and Out-of-the-money 29 2.7 Geometric Brownian Motion (GBM) If the strike price is more than the market price of the underlying asset, a put option is in-the-money. Acall option is A¿A½in-the-moneyA¿A½ if the market price of the underlying asset is greater than the strike price. An option is at-the-money when the strike price is equal to the price of the underlying security. If the market price of the underlying asset is less than the strike price, a call option is out-of-the-money. Aput option is out-of-the-money if the market price of the underlying security is greater than the strike price. Derivative A derivative is a security and its price depends on or derived from one or more underlying assets. The derivative is itself a mere contract between two or more parties. The valueof the derivativeis obtainedby fluctuationsin the underlying asset. Some examples of underlying assets are: bonds, stocks, commodities, market indices, interest rates and currencies. Most derivatives are characterized by high leverage. Security Asecurityisaninstrumentthatrepresents ownership (stocks),therightsto ownership (derivatives) ora debt agreement (bonds) Underlying asset An underlying asset is a term from derivatives trading. For example, in Microsoft stock option, Microsoft stock is the underlying asset. In case of gold options, gold is the underlying asset. Price movements of the underlying assets determine the price movement of options. Stochastic Volatility 30 2.7 Geometric Brownian Motion (GBM) Models of stochastic volatility areused in mathematical finance field for the evaluation of derivative securities, such as options. The name stochastic volatility is derives from the modelA¿A½s treatment of the volatility of the underlying security as a random process, which is governedby state variablessuchasthe tendencyof volatilitytorevertto somelong-run mean value, the price level ofthe underlying security, and the variance of the process of the volatility itself. Standardized futures contract Afutures contract is known to be a standardized contract between two parties to exchangea speci fied assetof standardized quantityand standardized qualityforapricethat is agreed today. Log-Return The advantage of looking at log return is that one can see relative changes in the variable and comparedirectly with other variables whose values may have very different base values Strike Price In options, the strike price, which is also known as the exercise price, is the fixed price at which the owner of the option can purchase, in case of a call option, or sell, in case of a put option, the underlying security or commodity. Characteristic Function The characteristic functionofareal-valued random variable definesitsprobabilitydensity function.Ifa random variableintroduceaprobability density function,thenthechar acteristic functionis the FourierTransformof theprobability density function. Wiener Process 31 2.7 Geometric Brownian Motion (GBM) Inthe worldof mathematics,theWienerprocessifatime-continuous stochasticprocess which is named in honor of NorbertWiener. It is often called the Brownian motion. Is is one of the best known stochastic process and occurs often in applied and pure mathematics, quantitative finance, economics and physics. Levy process Intheprobabilitytheory,theLevyprocess, whichis named afterthePaulLevy,aFrench mathematician, is a stochastic process with independent and stationary increments. A Levy process represents the motion of a point where its successive displacements are independent and random, and statistically identical over different time intervals of the same length. Systematic Risk In economics and finance, a systematic risk is the vulnerability to events which affects the aggregate outcomes such as total economy-wide resource holdings, market returns or aggregate income. Interest rates , wars and recession all represent sources of systematic risk because they affect the entire market and cannot be diversified to avoid the risk. The Poisson Process In the theory of probability, a Poisson process is a stochastic process that counts the numberof eventsandthe timesthat these events occurinagiven intervalof time.Theduration between each pair of consecutive events is known to have an exponential distribu tion with

as parameter and each of these inter-arrival times is assumed to be independent of the other inter-arrival times. Volatility Smile or Skew In the world of finance, a volatility smile is the pattern in which out-of-and in-themoney options areobserved to have bigger implied volatilities than at-the-money options. 32 2.7 Geometric Brownian Motion (GBM) Agraph of the strike price vs. the implied volatility for a given expiry will form an upturned curve, just like the shape of a smile. Figure 2.7:Volatility Smile 20. The Deterministic Model In mathematics,a deterministic modelisasystemin whichno randomnessis involved in the process of the development of the future states of the system and will thus always produce the same output from a given initial state or starting condition. Map Function (Mathematica) Mathematica.chas many powerful functions for working with lists. It is frequently desirable to map a function into each individual element in a list. While listable functions do this by default, one can use Map to do this with functions that cannot be listed. Discretisation In mathematics, discretisation infers the process of transferring continuous equations and models into discrete counterparts. 33 2.7 Geometric Brownian Motion (GBM) Explicit Euler, Implicit Euler, Crank-Nicolson method Explicit Euler involves the calculation of the state of a system at a later point in time, from the state of the system at the actual time. Implicit euler, derivesa solution,by solving an equation that involves both the current state of the system and the later one. The Crank-Nicolson methodisafinitedifference methodthatisusedforthe numerical solving of partial differential equations and it s second order in time, and is numerically stable. The Log-normal distribution Alog-normal distributionisacontinuousprobability distributionofarandom variable whereits logarithm is normally distributed. For example, if X isarandom variable following a normal distribution, then Y

=

exp(X) contains a log-normal distribution. Similarly, if now Y is a random variable following a normal distribution, then X

=

log(Y

)

has a normal distribution. Therefore, the log-normal distribution, is a distribution of random variable which takes only real values that are positive. 34 Chapter 3 Mean Reversion Model In the history of commodity pricing, the most common way is to model the logarithmic price through a mean-reverting process (Buhler 2009). Similar to the model of BlackScholes-Merton, the process of mean-reversion is based on the exponential treatment of the stochastic spot price (Merton 1976) (Black&Scholes 1973). If for example, these models are used for electricity, they may catch the mean-reversion of electricity prices, but they will not be able to account for non-negligible and huge observed spikes in the market. It is necessaryto extendtheBlack&Scholes(1973)modelbyajump component,tobeableto catch the behavior of spikes of the electricity spot price dynamics. This model was applied to the English andWelsh electricity marketby Cartea&Figueroa (2005) and findsit gives aproper adjustments to the abnormalities of the electricity markets. Roncoroni&Geman (2006) has discussed and tried to try to fix the deficiency of this model, that is to say that it uses only one unrealistically high mean-reversion rate, both for the jump process and for the diffusion. Nevertheless, a single rate of mean-reversion for these two visible features only has limited use because the price of elasticity does not exhibit classical jumps but instead exhibit spikes and these spikes have the tendency to revert fast, which leads ti a high rate of mean-reversion following a spike . The mean-reversion rate is in fact much lower, during 35 normal times. As a consequence, the use of a single mean-reversion factor causes a too slowremovalof intenseprice movements (spikes)andalsoatoorapidreturntoa seasonal trend of periods without intense events. Asolution exists. This problem can be solved by the separation of the mean-reversion factors for the A¿A½normalA¿A½ and the A¿A½extremeA¿A½ process. An applicable approach for this purpose was described by Benth (2005) where he predicts the parameters for the diffusion process taken from historical data and considers a constant volatility over time. But this approach resulted in several drawbacks. Firstly, the spike processA¿A½s parameters are not estimated from the time series, but are based on the opinions of expert. Secondly, this approach has neglected the fact that the volatility in the electricity markets is stochastic over time. Deng (2006) then compares Merton (1976) Jump-Diffusion model with the stochastic volatility and constant and derives prices for the dissimilar energy derivatives using Fourier transform and hence shows that the stochastic volatility is important. Escribano et al. (Villaplana 2003) supply huge empirical tests on an ample range of markets and then makes the conclusion that it is important to include stochastic volatility and jumps. The model of an asset follows a mean reversion process if the prices of assets tend to fall after hitting a maximum. Similarly, the price will rise after hitting a minimum. Let us consider a deterministic model, in which, cash and world stocks must have identical returns and the cash rates must deterministically follow the existing current path of forward interest rates. Everyone can know that stockreturns or interest rates havereachedalow or high point without breaching the unimportant market and arbitrage free conditions. This could be consequently viewed as a form of mean reversion, if the current forwardinterest rate curve is smoothly downward or upwardsloping. Mean-Reverting Stochastic Process dSt

=

a(L

–

St)dt +stochastictermA¿A½ (3.1) 36 The mean-reversion stochastic process has a drift term that takes the variable being modeled back to the equilibrium level. The result at the end is that the variable will have the tendency to oscillate around this equilibrium point and each time the A¿A½stochastic termA¿A½ pushes the variable away from the equilibrium level, the deterministic term will act in a way such that the variable will go back to the equilibrium level. The stochastic term is #t St dWt where #t is a volatility (standarddeviation), #> 0, the constantListhe A¿A½long-termmeanA¿A½oftheprocess,towhichitrevertsinsometime, a> 0 is a measurementof the strengthof meanreversion and Wt isa standardWienerprocess, v Wt

N(0;t) (3.2) Lets consideranaturalgascalloptionthatfallsand becomes worthlessincasetheprice of natural gas increases above a certain price at anytime during the lifetime of the option. The use of the process of a Geometric Brownian Motion (GBM) price to model natural gas prices gives such price paths that as a result, gives a much higher probability of ending up to the barrier level during the life of the option, than a mean reverting process does. The option pricing models that involves the use of a mean reverting process ensures that prices is drawn towards the mean reversion level, which assigns less chance of touching the barrier during the option life. The graphs below shows simulated paths of price and a resulting histogram for for example, natural gas, using Geometric Brownian Motion vs. Geometric Brownian Motion with mean reversion. The greater prices that are produced by the GBM method can be seen, clearly. In this example, both processes of price produce the same result for a European exercise option, but awfully dissimilar option prices for a barrier option. 37 Figure 3.1: Geometric Brownian Motion: Sample Price Paths Figure 3.2: GeometricBrownian MotionWith Mean Reversion: Sample Price Paths 38 Drawback of the Mean-Reverting Model 1. L, the long-term mean, stays fixed over time: It needs to be readjusted on a continuous basis, to ensure that the curves resulting are mar

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