an investigation of heston and svj model during financial crises2751

5/21/2018 An Investigation of Heston and SVJ model during Financial Crises2751

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LONDON SCHOOL OF ECONOMICS AND POLITICAL SCIENCE

An Investigation of Heston

and SVJ model duringFinancial CrisesFM408 Financial Engineering

MSc Finance (Full-Time)

2012-2013

Candidate Number: 82751

Word Count: 5752

The copyright of this dissertation rests with the author and no quotation from it or information derivedfrom it may be published without prior written consent of the author.


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Table of Contents

Abstract ........................................................................................................................... 2

1. Literature review and Introduction ............................................................................... 2

2. Description for the Market Data ................................................................................... 4

2.1 Data Source and its Selection .............................................................................................. 4

2.2 Normality Test of the data .................................................................................................. 6

3. Black Scholes Model ..................................................................................................... 9

4. Heston Stochastic Volatility Pricing Model .................................................................. 11

4.1 The Heston Model ............................................................................................................ 11

4.2 Calibration of the Heston Model ....................................................................................... 12

5. An Extension of Heston Model .................................................................................... 16

5.1 Add Jumps ....................................................................................................................... 165.2 Calibration of the SVJ Model ............................................................................................. 16

6. Financial Crises Analysis with the SVJ model ............................................................... 19

7. Conclusion .................................................................................................................. 22

Acknowledgement ......................................................................................................... 23

References ..................................................................................................................... 23

Appendix ........................................................................................................................ 24


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Abstract

After the crash of the stock market in the 20th century, the implied volatility surface of the stock

market exhibits a smile shape. That aroused the interests of researchers to develop more

sophisticated models beyond the basic Black-Scholes model. In this study, two stochastic models were

studiedthe Heston model and the Stochastic Volatility Jump model. They were used to analyze the

implied volatility surface for S&P 500 index options during two typical financial crises 2000 Dot

com bubble crisis and 2008 US Subprime Mortgage crisis. The results from this study show that the

parameters of these two models capture the volatility and stock crash, yet these models are still

imperfect at pricing options.

1. Literature review and Introduction

In 1973, the paper published byBlack and Scholes (1973)introduced an option pricing model

based on the assumptions that the return of the stock is log normal and the volatility isconstant across all options with different maturities. After the stock crash in 1987, the marketimplied volatility was observed to perform a smile-like shape, i.e. the implied volatility ishigher for the lower end of the strike price. This smile feature of the volatility surfacesuggests that the BlackScholes model is no longer good enough as an option pricing model.In particular, for equities, the volatility smile is noticed to be right skewed. Thus for equities,Black Scholes model would tend to under price the in the money puts while over estimatingthe out of the money calls. Figure 1.1by C.Chen (2007)reveals the crashing impact of thestock market on the implied volatility surface. We can easily see that before the crash, the

implied volatility surface roughly exhibited a table top shape. This suggests that the Black-Scholes model indeed shall give a good prediction before the crash in the 20 th century.However, things have changed after the crash, the implied volatility surface has become moreskewed (See Figure 1.1:Right). Moreover, the price for the out of the money puts (in themoney calls) shoots up drastically due to the over demand of the put options from theinvestors in the market.

Figure 1.1: Left: the Implied volatility smile curves for pre and post the crash in the 20 th century (1987);

Right: A skewed surface for low strike and short maturity options by C. Chen (2007).


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Since then, a significant amount of studies have been done on developing more sophisticatedoption pricing models. In 1987, Hull and White proposed a stochastic volatility (SV) modelto capture the smile effect in the stock, among which the Heston model is seen as the mostwidely accepted one. The Heston model corrects various restrictions imposed by the Black-

Scholes model. It allows the volatility to be stochastic; addresses the leverage effect (a pricedrop normally is accompanied by an increase in the volatility); includes the non log returncharacter of the stock; moreover, the Heston model also imposes a boundary restriction toexclude negative volatilities while at the same allows a volatility mean reverting property.Furthermore, the Heston model is also well - known for this computational advantage amongother SV models. It has a closed loop formula as was derived by Heston (1993), which could

be solved easily by using computational method whereas other SV models require complexnumerical methods. To conclude, the SV model could be regarded as an extension of the

basic Black-Scholes model. In this model, it resolves some limitations imposed by the Black-Scholes model and gives an improved representation of the implied volatility surface.

Although it is an improvement over the BS model, the SV model still does not fully capturethe smile feature. It mirrors the implied volatility surface for mid and long - term maturities

but not for the short-term maturities options. The high implied volatility for the out of themoney puts with shorter maturities would be under priced under this model. This leads to thedevelopment of the SVJ model by Bakshi, Cao and Chen (1997). The jump feature of thismore advanced model could be used to explain the smile shape of the surface better than the

previous two models. Yet empirical study shows that neither SV (Heston) nor SVJ alonecould give an accurate estimation of the market. SVJ still has its own drawbacks. It excludes

the relationship between the size of a market crash and its impact on the quantity that thevolatility could be increased. Researchers have been dedicated to develop more advancedmodels to address these issues, which is beyond the scope of this study.

This dissertation aims to find how the implied volatility surface changes during differentfinancial crises period. In particular, the 2000 Dot com bubble crisis and the 2008 USSubprime Mortgage crisis were studied in this dissertation. A time period extending from1999to 2010was selected, covering the two crises periods.

This study consists of five main chapters, starting with the raw data analysis in Chapter 2, inwhich the normality test was conducted for several selected time horizons. Followed byChapter 3 - a brief introduction on the basic Black Scholes model and how the impliedvolatility surface from the Black Scholes model could be used to benchmark other moreadvanced models. In Chapter 4, the Heston model without jumps was computed, and thecalibration method was employed to give the best fit of the market data. In the next chapter,

based on the Heston model built in Chapter 4, a stochastic volatility jump (SVJ) model wasdeveloped to address the shortcomings of the Heston model. Again, the SVJ model wascalibrated by employing a least square non - linear optimisation method in Matlab. In the lastchapter Chapter 6, I have discussed some findings by comparing the implied volatility

surfaces using the developed models. All the Matlab codes used in this dissertation wereattached in the Appendix for future reference.


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2. Description for the Market Data

The main purpose of this study is to investigate how the implied volatility surface changes inand out of the financial crisis by using the Heston and the SVJ model. To give a goodoverview of change of the implied volatility surface over several financial crises periods, the

data used in this dissertation is based on the daily closing price of S&P 500 index optionsfrom 1999to 2010, retrieved from Wharton Database. It would have been good to comparethe data of today with that of the 1980s where very few jumps could be observed in the stockmarket. However, due the limited data that I could import from Wharton Database, the timehorizon of the data investigated in this study is from1999to 2010.

2.1 Data Source and its Selection

For this study, I have chosen the S&P 500 index options data. This index has beeninvestigated in many literature, such asMadan and Chang(1996)andBakshi, Cao and Chen(1997)etc. S&P 500 index options is one of the many indices that have been studied in the

past due to its larg trading volumes and big open interest. On top of that, because S&P 500index options exhibit a European exercise style, it eliminates the complication of the model

building due to early exercise problems.

S&P 500 Index Time Series

Figure 2.1: Times Series for S&P 500 index

The underlying stock (S&P 500 index) of the options was collected using the historicaladjusted closing price from Yahoo Finance. The whole time series studied in this dissertationmainly covers two recent financial crises in the history, namely, 2002Dotcom Bubble crisis,2008the US Subprime Mortgage Crisis.Figure 2.1depicts the whole time series, from whichwe observe that the stock price experienced drastic drops during periods 2003 and end of

2008. A similar pattern could also be found for the return statistics we see the daily returnsof S&P 500 index becomes more volatile, dropping significantly to negative levels. The

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returns are also observed to have more clustering during the crisis periods, especially around2000-2001 and 2008.

S&P 500 Index Options Data:

The data source extracted from Wharton Database has the following format (Table 2.1). TheStrikeprice in the data has been scaled by a notional number of 1000. TheImpl_volatilitywascomputed by reverse-engineering the Black-Scholes formula, taking the option price as theaverage of theBest_bidandBest _offer.

Date Exdate CP_flag Strike Best_bid Best_offer Volume Impl_Volatility

19990601 19990717 C 1475 0.875 1 43 0.188725

19990601 19990619 C 1275 37.75 39.75 5 0.249485

19990601 19990619 P 675 0 0.0625 0

19990601 20000617 P 1350 138.25 142.25 0 0.275875

19990601 20000617 C 750 581.125 585.125 0 0.39934

19990601 20000318 C 1450 68.25 70.25 100 0.242048

Table 2.1: Historical Data of S&P 500 Index Options - source from Wharton Database

According to the paper published by the International Monetary Fund (IMF), the crisis dates for

Dot com and US Subprime Mortgage were defined as follows:

Crisis Name Crisis Date Period

Dot com 28/02/2000 07/06/2000

US Subprime 26/07/2007 31/12/2008Table 2.2: Crisis dates defined by IMF (2010)

Following a paper written by Mo Chaudhury (2011), I have further specified the crisis intopre, in and later crisis periods. In particular, the US Subprime Mortgage crisis in 2008 wasdivided into three sub-periods pre-crisis period starting from 01/01/2007 to 31/08/2007,early crisis period from 01/09/2007 to 14/07/2008, and lastly the later crisis period from15/07/2008until 31/12/2008.Figure 2.2(a)plots the smile curves for the 2008US SubprimeMortgage crisis in particular, we can see that the plot (green dots) before the crisis, the IV(implied volatility) exhibits a smile curve, reaching a minimum point at Moneynessequals 1.

Figure 2.2(a)also plots the implied volatility curves in periods during and after the financialcrisis. The curve during 2008 US Subprime crisis became more right skewed, gradually

losing its smile-like curve shape.

Similar results were observed for the Dot com crisis in 2000. The crisis periods were splitinto several sub periods. Pre-crisis period starts from 01/06/1999 to 27/02/2008; the actualDot com crisis period starts from 28/02/2000 to 15/06/2000; finally, the later crisis periodwas defined to be from 16/06/200031/12/2000. We see that the implied volatility curves

become right- skewed significantly during the crisis period, indicating a change of the riskappetite of the investors. People are more worried about more crashes in the stock market,

buying put options to hedge their portfolios, pushing up the prices for out-the-money puts.

After 2000 - the Dot com crisis, we see the implied volatility on 01/06/2001for example, hasrecovered its smile shape.


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Figure 2.2 (a): 2008 US Subprime Crisis: The smile curves for S&P 500 index Call options for pre, in and later

crisis periods. *Notice that the discontinuity of the points is due to the incompletion of the data

Figure 2.2 (b): 2000 Dot com Crisis: The smile curves for S&P 500 index Call options for pre, in and later crisis

periods

2.2 Normality Test of the data

The normality test was conducted on the return of the S&P 500 index. The following figuresdisplay the results from the normality tests conducted for S&P 500 stock returns studied inthis paper. During the sample period extending from 1999to 2010, we can see that the stock

return does not follow a perfect uniform normal distribution shape as depicted in the BlackScholes model - it has kurtosis and fat tails. The effect becomes more pronounced for periods

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2001- 2004and 20072008, which are the time periods covering the Doc.com and the USSubprime Mortgage crisis. During 20012004, we observe a right skewness in thedistribution, accompanied by a few more spikes occurring in the left tail. A similar patternwas observed again during the 2008 crisis. The kutosis (peakness) of the distribution

increased during the crisis periods, spreading into both tails. However, since investorsdemand more puts than the calls to hedge their portfolios in crisis, the whole distribution,therefore, is skewed towards the right, squeezing the right tail more.

This phenomenon seems counterfactual in a Black-Scholes world, where the return isassumed to be normally distributed, and the instant volatility is considered to be constant.Thus, if Black Scholes models assumption is valid, we should not observe any of the

patterns described above. However, Black Scholes does not hold in the real market, themarket data tells us that the implied volatility against the strikes shows a smile pattern(Figure 2.2), and furthermore this smile curve becomes right skewed during the crisis.

More recent studies have been focusing on stochastic volatility and jump diffusion models,which was further discussed in Chapter 3.

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Figure 2.3: Normality test on the return of the S&P500 index for period from 1999 to 2010: (a) Pre financial

period: 1999 2000; (b) Dot com Bubble: 2001 2004; (c) After Doc.com before US Subprime Mortgage crisis

period: 2004 2007; (d) In US Subprime Mortgage crisis: 2007 -2008; (e) After US crisis and during Eurozone

crisis: 2009 -2010.

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3. Black Scholes Model

The model was first published in Black and Scholes (1973), in which the stock price wasassumed to be a Geometric Brownian motion process.

(3.1) (3.2)

where is the instantaneous rate of return and is the volatility of the stock price. Then byIts lemmausing a replicated portfolio, Black and Scholes reached the following equationsfor the price for European style call options. Then the put price could be computed by usingPut-Call Parity.

(3.3)

(3.4)

(3.5)

As discussed before, the Black-Scholes model alone would not be used as a tool to price thesecurities in the market. However, we can use Black-Scholes model as a benchmark toreverse engineer the implied volatility of an option for a specific strike and maturity given itsoption price observed in the market. Therefore, the implied volatility surface computed bythis method could be used as a reference to benchmark more advanced models such as the

Heston and SVJ models.

By reverse engineering the raw data from Wharton Database and making assumptions thatthe option price would be calculated as the average of the best-bid and best-ask prices in themarket, the implied volatility surface is calculated. Figure 3.1(a), (b) shows one of theimplied volatility surface plotted by reverse engineering the call prices observed in themarket. The Matlab code used to plot the surface was written by Rodolphe Sitter (2009).However, one drawback of this Matlab function is that for maturity and moneyness pointswhere no option price data data is given, the Gaussian function would provide anextrapolated value which might be inaccurate. Thus this Matlab function should not be usedwith incomplete data sets to plot the implied volatility surface.

By comparing Figure 3.2 (a)and (b)one in the financial crisis (2008) and one out of thefinancial crisis (2005), we can see that the 2008plot is more skewed, whereas the 2005plotlooks more smile shaped. Note that the Moneyness (M) is defined as K/S0, and time tomaturity (T) is in years.


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Figure 3.1 (a): In 2008 US financial crisis period: Implied Volatility Surface on 01/10/2008

Figure 3.2 (b): Out of the crisis period: Implied Volatility Surface on 03/01/2005


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4. Heston Stochastic Volatility Pricing Model

4.1 The Heston Model

The Black-Scholes model is limited in real practice due to its restrictive assumptions. One

way to model the stock price and volatility is to adopt a stochastic volatility model. Morerecent studies have been focusing on developing these stochastic models. In 1993, StevenHeston published his first paper on the Heston model. In the paper, the stochastic volatilityHeston model was defined by the following stochastic differential equation:

(4.6) (4.7)

Unlike the Black-Scholes model, the Heston model assumes that the volatility term is

stochastic. The diffusion term in Equation 4.6 depends on the instantaneous volatility Vtinstead of a constant volatility. The Heston model also includes the mean-reversion feature,which the Black-Scholes model does not have. InEquation 4.7, denotes the mean revertingspeed, is the long-run mean, is the volatility of volatility. Lastly, dZ1tand dZ2tare twoBrownian motions, which are related to each other by Equation 4.8. The log-return of thestock and the volatility is correlated to each other by a coefficient . This additionalcoefficient resolves the limitation of the Black-Scholes model on modelling non-log normalreturn stocks.

(4.8)

Therefore, the Heston model could be regarded as an extension of the Black-Scholes model.It captures other features that a basic Black-Scholes model does not have; for example, ittakes into account of the non-normal distribution feature of the stock return, the meanreverting property of the volatility as well as the leverage effect. Beside these, one principleadvantage of Heston model is that, unlike other stochastic volatility models (SV), the Hestonmodel yields to a closed-loop form solutionno need of heavy numerical computation. In the

paper byHeston (1993), it derived the price of a European call option has the following form:

(4.9)By Put-Call Parity, we can reachEquation (3.10)for a European put option price:

(4.10)where,P1andP2are the time and state dependent conditional probabilities that a call to be in-the-money (ITM).

{

} (4.11)


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( )

Forj=1,2we have

Note that denotes the market price of the volatility, which becomes zero under the riskneutral world . Therefore, by pricing under , we can eliminate the parameter in theequation, i.e. = 0, leaving only five unknown parameters *, *, *, V0*, *

1. These

unknown parameters could be computed by calibrating them against market data, which isdiscussed in the next chapterChapter 5.

The above equations look intimidating but they can actually be solved very easily usingnumerical methods. There are several ways to reach the pricing solutions for a Heston model.

F D Rouah & G Vainberg (2007)explains two separate computational methods by using thefull valuation approach (Monte Carlo) as well as the closed loop-form approach. A third

method Fast Fourier Transform (FFT), which was proposed by Carr & Madan (1999),proved to be considerably faster than most other existing numerical methods.

I have adopted the closed-loop method based on the following considerations. Monte Carloapproach is the most mathematically clear way to simulate the process. However, theaccuracy of the model depends on the number of paths included in the modelincreasing thenumber of paths could be computationally expensive. Secondly, although the Fast FourierTransform method is the fastest, my attempts at it were unsuccessful. However, the closedform approach was promising and the results of using this model is thus reported in this

dissertation. The Matlab code HestonCallby Moeti Ncube (2010) was employed tocalculate the Heston option price.

4.2 Calibration of the Heston Model

Calibration is an important concept in the Heston model. In the previous section we havedefined six unknown parameters (, , , V0, , ), which are not observed directly in themarket. We have further addressed that under the risk neutral probability world , one

parametercould be eliminated, leaving only five parameters which are to be determined.

1Parameters with * means pricing under


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The calibration method is based on the idea that there is minimum error between pricecomputed from a mathematical model and the price, which is observed in the market.Therefore, we start from a set of guessing parameters, compute the Heston option priceindicated by these guessing parameters, and then compare the obtained Heston price with the

actual market price. By iteratively changing the parameters to minimise the errors betweenthe Heston model price and the actual market price, we can calibrate the parmeters for theHeston model, which can be used to price the securities in the market. Nevertheless, thiscalibration approach relies strongly on the assumption that the market price is traded at a fair

price. It captures all the information used to predict the unknown stock.

The Matlab function called HestonCalibration written by Moeti Ncube (2010) wasemployed to calibrate the Heston model for the entire time horizon covered in thisdissertation. HestonCalibration contains a Matlab least square non-linear optimisationfunction lsqnonlin. It calibrates the data by searching for an optimum set of parameters (V0,, , ,), which would give the best fit for the price observed in the market.

Table 4.1displays one set of calibrated parameters on September 15, 2005for SPX. The rightcolumn, shows the calibration results obtained from Gatheral (2006) for a non-jump Hestonmodel. We can see that these two sets of data are comparable.

Calibration result Gatheral (2006)

V0 0.0162 0.0174

1.2933 1.3252

0.0399 0.0354 0.4243 0.3877

-0.6833 -0.7165

Table 4.1: Heston calibration to S&P 500 index option surface on 15/09/2005

Figure 4.1(a) shows the daily calibrated surfaces by Heston model. The top surface is theimplied volatility surface obtained from the market data, and the flatter surface below is thecalibrated Heston model. We can see that the simple calibrated Heston model impliedvolatility surface does not reveal the actual market implied volatility surface perfectly.

Nevertheless, it shows a more skewed smile shape in comparison with the table top surface

plotted from the Black-Scholes model. It could be concluded that the Heston model is animprovement from the simple Black-Scholes model by introducing non-log normal return andstochastic volatility characteristics of the underlying. On top of that, it is shown in Figure4.1(a) in particular, the daily calibrated Heston model could predict the market impliedvolatility better for longer maturities T > 2 years, but for short maturities (T < 1.5 years),Heston model deviates from the actual implied volatility surface significantly. This featurefor the Heston model was discussed in more details inZhang and Shu J (2003),

Last but not least,Figure 4.1(a)is a plot of the implied volatility surface during the 2008USSubprime crisis. The actual surface lost its smile shape, and becomes skewed as discussed inChapter 2.1. The calibrated Heston surface follows this skewed pattern. However, the surfacefit is not satisfactory for options which are deeply in-the-money and out-of-the money. This


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characteristic reveals one of the shortcomings of the Heston model a simple Heston doesnot capture the jumps, which is especially considered as an important feature during thecrisis.

Figure 4.1(a): Daily Basis: calibrated Surface on 2008/11/03 by using parameters from 2008/11/03

Figure 4.1(b): 15-day Basis: calibrated Surface on 2008/11/03 by using parameters from 2008/10/15


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Figure 4.1(c): Monthly Basis: calibrated Surface on 2008/11/03 by using parameters from 2008/10/01

The remaining issue is how frequently the data should be calibrated. In Figure 4.1, thecalibrated surfaces by Heston model based on three different calibration windows are plotted.As discussed before, the daily calibrated Heston model does not mirror the actual market data

perfectly, yet it still gives a good representation for mid and long term maturities. However,

the monthly calibrated surface in Figure 4.1(c), notably deviates from the actual marketimplied volatility. By looking at the daily calibrated surface with the surface calibrated on a15-day interval inFigure 4.1(a) and(b), no significant difference was observed. Therefore, tosimplify the computation method in this report, the parameters were calibrated every 15 days.

The following table shows a list of calibrated parameters.

V0

2005/01/03 0.01971 1.38512 0.03492 0.40540 -0.62959

2005/09/15 0.01594 1.14916 0.04165 0.39069 -0.71728

2008/01/02 0.06142 0.49913 0.08582 0.43720 -0.73465

2008/08/01 0.04595 0.80769 0.07529 0.38107 -0.73625

2008/10/01 0.10939 0.67064 0.08086 0.49670 -0.76216

2008/10/15 0.30641 2.43717 0.05022 0.5000 -0.89999

2008/11/03 0.21221 1.15175 0.10446 0.50000 0.90000

2008/12/01 0.39647 2.83012 0.08625 0.50000 -0.89995

Table 4.2: Calibration results


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5. An Extension of Heston Model

5.1 Add Jumps

In this Chapter, a more advanced stochastic diffusion jump model (SVJ) was investigated to

create a better option pricing model. The development of the model was based on Bakshi,Cao, and Chen (1997), in which a stochastic jump model was modified into the equations ontop of the basic Heston model. Notice that the notations were changed from the original paperin order to be consistent with the Heston model developed in Chapter 4.

(3.6) (3.7)

All the parameters have the same meanings as that were defined in Chapter 4 except Jt

which is a jump percentage defined by Equation 3.8. The log normal of Jtfollows a normaldistribution with a mean of ln(1+J) ()J

2, and a standard deviation of J. The jump isdefined by a Poisson process, Equation 3.9, which follows the same definition in Bakshi,Cao, and Chen (1997).

[ ] ( ) (3.8)

{ (3.9)

The stochastic volatility jump model (SVJ) was coded into a Matlab function namedSJVCall. The algorithm followed to compute a SVJ call price is based on the equations

presented inBakshi, Cao, and Chen (1997).

Comparing the basic Heston model and Black-Scholes model, SVJ model completes theHeston model by including a jump feature of the stock market into the model. However, onedrawback of the SVJ model is that it assumes an independent relationship between large pricemovements (jumps) with the implied volatility indicated by the model. This seems

counterfactual, as a big price crash should results in an increase of the volatility. Thisshortcoming of the SVJ model was addressed in the SVCJ model, in which it includes anadditional parameter j compensating for the shortcoming of the SVJ model. Here in thisstudy, the SVCJ was not included.

5.2 Calibration of the SVJ Model

The calibration method used for the SVJ model follows a similar Matlab routine as was forthe Heston model. It employs a Matlab optimisation function lsqnonlinthe least square

non-linear optimisation function to search for an optimal solution which could minimise theerror between the call price from a SVJ model and that of a market call price. Again this


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calibration approach is based on the assumptions that the market is efficient and contains allthe available which could be used to price the options. The Matlab function called SVJcalibrationNplotwritten by the author would calibrate and plot the SVJ model.

Figure 5.1: Comparison between Heston model and SVJ model on 2005/09/15

Figure 5.1illustrates one example of the calibrated SVJ surface. In addition, it also plots theactual implied volatility surface (the most top surface), and the simple Heston surface (most

bottom surface) which was computed from Chapter 4. The advantage of the SVJ model wasvery obvious. By adding an additional jump diffusion term into the model, the fitting of themodel to the market data was improved notably, fixing the limitations of the Heston model in

pricing the short-term maturity options.

V0 J J J

Heston 0.0159 1.1492 0.0416 0.3907 -0.7173

SVJ 0.0675 5.3800 0.0400 0.3800 -0.5700 -0.0530 0.0967 0.1308

Gatheral 0.0174 1.3252 0.0354 0.3877 -0.7165

BCC 0.5394 0.0439 0.3038 -0.6974

Table 5.1: Calibrated parameters for SVJ and the Heston model on 2005/09/15* BCC parameters are the results obtained from Bakshi, Cao, and Chen (1997); Gatheral parameters are from Gatheral (2006).

The calibrated parameters are displayed in Table 5.1, for comparison, Table 5.1also lists thecalibrated parameters for the Heston model on the same day. To examine the results, it wasnoticed that the mean reverting speed ( = 5.3800) increased dramatically for a SVJ modelstudied in this dissertation, while the rest of the parameters remain approximately unchanged.The calibrated results for SVJ model are comparable with what was found in literature, apartfrom the mean reverting speed . This deviation is due to the limitation of the calibrationcode. Although the SVJ model is of a closed form, the equations are highly non-linear.

Besides, there are 8 parameters to calibrate. The code works by searching for a combinationof the 8 parameters that give the least error. Due to the non-linearity and high number of

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an investigation of heston and svj model during financial crises2751

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black scholes model

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financial crises2751

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financial crises analysis

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