Pricing options and other derivatives is one of the issues faced by researchers that attempt to nd alter-native method to include empirical features of actual nancial market. This report aim to expand theanalytical exibility of the previous pricing model (such as black-scholes model, Merton’s Jump model).We illustrate that the double exponential jump model can generate accuracy solutions for path-dependentoptions efciently. We also compare the simulation result of three models(geometric Brownian motionmodel, Merton’s Jump diffusion model and Double Exponential Jump diffusion model) with the actualdata in UK stock market.
Numercial results demonstrate that the double exponential model do betterperformance in actual stock market than other models.
Keywords : Option pricing, Double Exponential Model, Brownian Motion model, stock market
In nancial stock market, the stock price is the main source of risks for investors. The insurance com-pany, security and other nancial institutions pay more attention to predict the trend or distribution ofstock prices by analysing the historical data. The Black-Scholes Model, which based on Brownianmotion model and normal distribution has been used to predict the asset future prices in recent years.
However, the Black-Scholes cannot describe the two empirical features in actual nancial market: (1)The asymmetric features, the distribution of asset return has left skewness, higher peak and two heaviertails than normal distribution. (2) The volatility smile, which means the implied volatility should beconstant, the curve of implied volatility is not smile and the curve of the strike price is not convex inactual market if Black-Scholes model is correct. There are variety models that was proposed to incorpo-rate the two empirical phenomena in Black-Scholes model such as chaos theory, generalized hyperbolicmodel and time changed Brownian motions model (Heyde, 2000) that used to solute the questions aboutasymmetric leptokurtic features.
The stochastic volatility and GARCH model(Hull and White, 1987),constant elasticity model (Davydov and Linetssky, 2001), normal jump diffusion model and Levy pro-cess model that were proposed to eliminate the error from volatility smile in estimating asset prices. Kou(2004) proposed a new jumnp diffusion model to assess the asset price based on the previous researches,which was named as Double exponential Jump diffusion model (DEJ model). This model gives analyt-ical solution to a variety of option pricing problem, particularly for some path-dependent options.This report introduces the basic content of double exponential jump diffusion model, discusses its fea-tures and applications. Moreover, we prepare to discuss the praticalbility and rationality of its applica-tion in asset pricing in UK by simulating the double exponential jump diffusion model and compare theresult with previous models (Black-Scholes Model and Merton’s Jump diffusion model).
Kou(2004) proposes the double exponential jump diffusion model based on Black-Scholes model. Thismodel contains a continuous section that driven by Brownian motion method and a jump section whosec The authors 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.2 of 8jump size with a double exponential distribution.The dynamics of the asset prices can be determined bythe following equation:dS (t) dS ( t) =mdt +sdW (t) + d(N(t)Ґi = 1(Vi1)) (2.1)W (t) is a standard Brownian motion, N(t) is a Poisson process with rate l, the constant mand s> 0are the drift and volatility of the diffusion. Y1;Y2;Y3::::::are a series independent identically distributed(i.i.d) positive or zero random variables so that Y= l og (V )satises an asymmetric double exponentialdistribution. The density of Yis as following:fY (y ) = ph1e h1y1 (y > =0) +qh2e h2y1 (y < 0) (2.2)h 1 >1;h2 >0where p;q > = 0, q+ p= 1, q and p represent the probability of jump up and down respectively.l og(V ) = Y= x+;p (2.3)l og (V ) = Y= x;q (2.4)In order to get nite expectation of asset price S(t) , we set x1 >1. Thus 1 x1 and1 x2 are the mean of twoindependent exponential distributions respectively. Kou also assumed that all randomness N(t) W (t)and Yare all independent.Finally, Kou obtained the dynamic function to estimate the asset price by solving the stochastic differ-ential equation (2.1).S(t) = S(0 )ex p (m 1 2s2)t + ( t) N(t)
i = 1Vi (2.5)In this equation, E(Y ) = p h1 q h2,Var (Y ) = pq(1 h1 +1 h2 )2+ ( p h21 +q h22 )andE (V ) = E(e Y) = q( h2 h2+1) +p( h1 h11;h1 >1;h2 >0 (2.6)In order to make sure E(V )< Ґand E(S (t)) < Ґ, Kou set h1 >1, which means the scale of jump upnot exceed than 100%.2.2 The features of Double Exponential Model (leptokurtic feature)The distribution of asset return in stock market is not a normal distribution, it is skewed to left, itspeaks are higher than normal distribution and tails are heavier than normal distribution. Ramezani andZeng (1999) nd that the double exponential model appear better performance than Merton’s Jumpdiffusion model in actual stock market. Next, we would pay attention to estimating whether the doubleexponential model is suited to the leptokurtic and asymmetric features or not.According to the equation 2.5, the return of asset during the time interval Dtis:D (S (t)) S(t) =S(t + Dt) S(t) 1= ex p ((m 1 2s2)D t) + s(W (t + Dt) W (t)) + N(t+ Dt)Ґi = N(t)+ 1Yi)1 (2.7)3 of 8The return of distribution can be approximated if the time interval Dtis small, thus we can ignore theterms whose orders higher than Dtand obtain the following equation by applying ex 1+ x+ x2.D S(t) S(t) m t+ sZp Dt+ BY (2.8)Z is standard normal random variables and B is standard Bernoulli random variables, with P(B = 1) =l t, P (B = 0) = 1 l t. The g, which represents the density of right-side of 2.8 is the approximationof the asset return DS(t) S(t) . The gures show the plot of normal distribution andgwith same mean andvariance.g(x ) = 1 l t sp Dt j(x m t sp Dt )+ l t ph1e (s 2h 21 Dt) = 2e( (x m )h1)F (x m t s2h 1Dt sp Dt )+ qh2e (s 2h 22 Dt) = 2e(x m t) h2F( x m t+ s22 Dt sp Dt ) (2.9)The mean and variance is: E(g ) = mt+ l( p h1 q h2)D t (2.10)Var (g ) = s2D t+ pq (1 h1 +1 h2)2+ ( p h21 +1 h22 ) l t+ ( p h1 q h2)2l t( 1 l t) (2.11) FI G . 1. this is overall comparison FI G . 2. this is peak comparisonThe parameters of these four gures are: Dt= 1= 250 year ,s = 20%, m= 15%, l= 10 and the mean ofjump size is 2%, volatility is 2%. From the four gures, we can nd that the double exponential modelhas leptokurtic asymmetric feature. The double exponential distribution has higher peak and heavier lefttail than normal distribution. The leptokurtic feature also exit in real nancial market. We can see fromthe gure 5, the return distribution of FTSE 100 during the 8 years have high peak and heavy left tail.The second feature or property of double exponential model is memoryless. This unique property il-lustrates the issue and provide solution of the problem about option pricing. Therefore, the doubleexponential model provide a efcient and effective method to pricing different kinds of option.4 of 8FI G .
3. this is left tail comparison FI G .
4. this is right tail comparison2.
3 The application of double exponential modelIt is crucial for investors and nancial companies to control and manage risk by predicting option inadvance. The previous models such as Black-Scholes model that were used to estimate option pricesresulting in volatility smile and Merton’s Jump model cannot apply analytic solution for path-dependentoptions, especially American options, lookback options and barrier options (Liang and Xu, 2019).Thedouble exponential jump model not only provide an appropriate approach to describe time-varyingvolatility, explain volatility smile, but also show asymmetric leptokurtic features. Although the modelwith jump section can explain two emerical features in actual market, it is hard to calculate analyti-cal solutions for path-dependent options (Davydov and Linetsky, 2001). The following section wouldexplain and discuss the application of double exponential jump model in pricing path-dependent options.Lookback options and barrier options, which traded in exchanges and over-the-counter markets arepopular and received much attentions from investors. For lookback options, Kou and Wang(2003) solvethe pricing of lookback option by nding the relationship between the Laplace transform of lookbackoption and double exponential jump model.LookBack(T ) = E [e ( rT )(max M ;max0 6 t6 T S(t) ] S(0 ) (2.12)The barrier options are divided into eight types, which are up(down)-and-in(out) call (put) options. Wewill introduce how to pricing up-and-in call option in this report. The pricing formula is:Cal l=E [e ( eT )(S(T ) K)+1fmax(0t T )S (t) > =Hg](2.13)H >S(0 ) is barrier level in this equation. Laplace transforms are important to pricing lookback optionsand barrier options in double exponential jump model. Kou and Wang (2001) use Gaver-Stehfest algo-rithm to implement the pricing method. Then, this report will provide numerical solutions about doubleexponential jump model pricing lookback options and barrier options.The parameters of numerical result for lookback option and barrier option are: M=110, H=120,the strike price is K= 100, the expiration date is T= 1. We compare the result from double expo-nential model ( l= 3) and geometric Brownian Motion ( l= 0:01).The Monte Carlo method simulation5 of 8FI G . 5. FTSE 100result appear biased lower than double exponential jump method due to the systematic discretizationbias (Kou and Wang, 2003). Thus, we can conclude that the double exponential jump model do bet-ter performance in pricing path-dependent options than previous methods, especially for large jump size.Although double exponential jump diffusion model can describe the empirical features in stock mar-ket and propose analytical answer for path-dependent options, there are several disadvantages of doubleexponential jump diffusion model. One of the limitations of this model is that it is difcult to calcu-late and derivate the pricing formula, because there are many differential equations in pricing process.Secondly, the double exponential jump diffusion model is difcult to hedge risk in stock market. Theconventional riskless hedging method can not used in this model because the nancial market is incom-plete due to the jump section. Thus, the result of this model maybe incorrect if we not consider the riskhedging part.
Monte Carlo simulation is an appropriate method for nancial derivatives and stocks pricing. It use a se-ries of repeated random variables to estimate the option price. Different from other models, Monte Carlocan solve multi-dimensional or multi-factor problems effectively. In Normal Jump diffusion model(Merton’s Jump Diffusion model), is normal distribution (Merton, 1976). Although the double ex-ponential jump diffusion model and Merton’s Jump Diffusion model both can explain the empericalfeatures (asymmetric leptokurtic and volatility smile features), Merton’s Jump Diffusion model is failto estimate the value of path-dependent options (Liu, 2017). This report use Monte Carlo method tosimulate the UK stock market, the Merton’s jump model and double exponential model and comparethis three models.6 of 8FI G .
Numerical Result of Monte Carlo SimulationFigure 8 shows the historical trend of FTSE 100 and FTSE 250 by analysing 2000 historical trans-action data. Figure 9 shows the path of stock price through Brownian motion that be simulated basedon historical data. FI G . The historical data of FTSE 100 FTSE 250According to the result of these two gures, the Geometric Brownian Motion simulation trend of UKstock market (FTSE 100 and FTSE 250) is different from the actual stock market. In other world, theUK stock market is not suited to geometric Brownian motion model. Then we want to estimate whetherthe double exponential model and Merton’s jump model are suited for UK actual stock market or not.7 of 8FI G .
The Merton’s Jump model simulation of FTSE 100 FTSE 250The gure 10 and gure 11 show the simulation result of Double Exponential model and Merton’sJump model. These two gures indicate that the classical geometric Brownian motion model with jumppart can be a good indication of the sudden rise or fall in the actual stock market. Then we combinethe gure 5 (the historical data distribution of FTSE 100) with two gures and nd that the doubleexponential jump model can describe the asymmetric leptokurtic phenomenon in UK stock market.
Unpredictable events, varying nancial regulators and other risk factors all put forward higher require-ments in derivatives pricing models. Many nancial researchers pay attention to incorporate the asym-metric feature and volatility smile in Black-Scholes method. In this paper, we describe a appropriateapproach to pricing European options and path-dependent options. The numerical results show that thedouble exponential jump model perform better job than previous models in pricing path-dependent op-8 of 8FI G .
The double exponential Jump model simulation of FTSE 100 FTSE 250tions. Moreover, the simulation studies indicate that the double exponential jump model can describethe asymmetric and volatility smile in actual nancial market. We can also nd that the simulation dataof double exponential jump model do better job in UK stock market than geometric Brownian motionand Merton’s Jump Diffusion model. For the limitations of jump model, we can use programming tosolve the problem about the complex calculation and modify the assumption or the jump section for theriskless hedging problems.
D AV Y D OV , D. & L I N E T S K Y, V. (2001) Pricing and Hedging Path-dependent Options Under the CEV Process,MANAGEMENT SCIENCE, 47(7), 949″965.H E Y D E , C. C. (2000) A risky asset model with strong dependence through fractal activity time, Applied Probabil-ity, 36, 1234″1239.K O U , S.G. (1999) A jump diffusion model for option pricing with three properties: Leptokurtic Feature, VolatilitySmile, and Analytical Tractability, Columbia University,.K O U , S.G. (2002) A jump diffusion model for option pricing, MANAGEMENT SCIENCE,48,1086″1011.K O U , S.G. (2003) First passage times of a jump diffusion process, MANAGEMENT SCIENCE,47(7), 949″965.K O U , S.G. & W A N G, H. (2004) Option Pricing Under a Double Exponential Jump Diffusion Model, MANAGE-MENT SCIENCE, 50(9), 1178″1192.L I A N G , Y.J. & X U, X.C. (2019) Variance and Dimension Reduction Monte Carlo Method for pricing EuropeanMulti-Asset Options with Stochastic Volatilities, Sustainability,11(815).L I U , Q.W. (2017) A Jump diffusion Model for Asset pricing, University of science and Technology of China.H U L L , J.A N D WH I T E , A. (1987) The pricing of options on assets with stochastic volatilities, Journal of Financial,42 , 281″300.M E RTO N , R. C. (1976) Option pricing when underlying stock returns are discontinuous, Journal of FinancialEconomics, 3(1976), 125″144.R A M E Z A N I , C.A. & Z E N G, Y. (1999) Maximum likelihood estimation of asymmetric jump-diffusion pro-cess:Application to security prices, Department of Statistics, University of Wisconsin, Madison, WI .