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# Mean Reversion and Stochastic Volatility Finance Essay

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“Mean Reversion and Stochastic Volatility Finance Essay”

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

Contents

## .

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

## .

=

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

&

=

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

.

Xt

=

=

#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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