& 26 AB>ก F G& 205? A?>H<I ก$J

Transcription

1 ก ก ก 21! 2556 ก$%& Beyond Black-Scholes : The Heston Stochastic Volatility Option Pricing Model :;$?AK< \K & 26 AB>ก F G& 205? A?>H<I ก$J GK>H< LA <

2 Beyond Black-Scholes The Heston Stochastic Volatility Option Pricing Model Derivative An Financial Innovation A derivative is an innovation of financial instrument which its value can derive from underlying asset. The typical derivative instrument. Option contracts Forward contracts Futures contracts Derivative of SET50 is the financial instrument that its value depends on SET50 index.

3 Derivative of SET50 Thailand Future Exchange or TFEX is the exchange market for derivative products in Thailand. Future and Forward contract is the obligation to buy/sell the SET50 index at the specific price and time. Option is the Right to buy/sell the SET50 index at the specific strike price and expiration time. European Option American Option Derivative An Financial Innovation

4 Derivative An Financial Innovation Derivative An Financial Innovation Since the option price is gain(loss) limitation For Call, If S<K, then investor loss equal to option price. For put, if S>K, then investor loss equal to option price. To arrive with the effective investing strategies, investor require to obtain the precise option price. So, Option Pricing Model play the critical role to quantify the option value.

5 Option Pricing Model The novel theoretical option pricing model was developed by Black-Merton-Schole(1973)(BS- Model). BS model provided two wisdom of knowledge. One is that BS-model provided closed form solution for option price. Second, BS-model also provide the closed form solution for Greek s hedging parameters delta, gamma, roh,vega, theta. BS Model - Assumption Stock price follow GBM. The short selling of securities with full use of processds is permitted There are no transaction cost or taxes. All securities are perfectly divisible. There are no dividiends duing the life of the derivative There are no riskless arbitrage opportunities. Security trading is continuous. Volatility is constant

6 BS Model-Solution BS Model-Solution Implied Volatilities is the volatilities implied by the option price observed in the market

7 BS Model-Greek Delta Hedging : is the rate of change of the option price with respect to underlying asset price. Gamma Hedging : is the rate of change of the delta parameter or how fast that delta change it value. Vega Hedging : is the rate of change of portfolio value with respect to volatility Rho Hedging : is the rate of change of portfolio value with respect to risk free rate Theta Hedging : is the rate of change of the value of portfolio with respect to the passage of time or time decay of portfolio. Investor can manage risk by Heding Delta and Gamma BS-Assumed that implied volatility is constant Implied Volatility One of the flawed assumption of BS mode is the constant implied volatility The volatility smile also can be observed empirically in many market. For example, Rubinstein(1985), Taylor and Xu(1994) Bakshi, Cao and Chen(1997), Dumas, Fleming and Whaley(1998). The result showed the volatility smile by the u-shaped relationship between level of moneynessand volatility. Dumas, Fleming and Whaley(1998) They concluded that the constant volatility assumption of BS model is flawed. volatility seems to exhibit the mean reversion properties(scott1986,stein1989).

8 Implied Volatility-SET50 Call Put Patakkinang et al(2012) study the volatility smile in Thai market on SET50 option. They showed that both put and call exhibit the volatility smile using the sample of SET50 option expiring between Mar 2551 to Dec So, this result showed that the violation of BS assumption lead to the mis-pricing To reconfirm the smile in SET50 option Data cover from May 2011 October 2013 Put Call T=40 days Put Call T=60 days

9 The affect of smile The result from volatility smile lead to the rejection of the constant volatility of BS-model. Patakkinang et al(2012) also document that BS provides biasness to option pricing DOTM DITM CALL Over Price Under Price PUT Under Price Over Price Delta cannot Neutralized portfolio and hence the investor need to concern the Vega. Therefore, the risk management concerning BS model show a improper result. Investor should concern BS model carefully Heston Stochastic Volatility Model Heston(1993) assumed that Stock price follow GBM Volatility process follow CIR(1985)

10 Heston Stochastic Volatility Model So, there are two source of risk Risk from underlying stock price Risk from stochastic volatility Heston set up the riskless portfolio with additional option to manage the stochastic volatility risk. First, Heston set up the PDE (eq6.) Heston Stochastic Volatility Model

11 Heston Stochastic Volatility Model Heston Stochastic Volatility Model Obtain Heston s Price

12 Implication of Heston s OPM Introduction to Excel

13 Introduction to Excel The excel calculation sheet can download from free source or from CD attached with Option Pricing Model & Volatility using Excel-VBA by Fabrice Douglas Rouah and Gregory Vainberg Parameter Estimation

14 Parameter Estimation Parameter Estimation The estimated parameter value will be in green box Then we can use such parameter to calculate for Heston Option Pricing Model The Heston price is in the Heston price column

15 BS Model and Risk Management BS Model and Risk Management

16 BS Model and Risk Management Heston Model and Risk Management For Heston s model, there are two source of risk Risk from underlying stock price Risk from stochastic volatility Therefore, we need to an option for each of risk to be hedged. Intuitively, we do need at least two option available.

17 Heston Model and Risk Management Heston Model and Risk Management

18 Heston Model and Risk Management Heston Model and Risk Management

19 Heston Model and Risk Management Conclusion The empirical data from international and from SET, we found that volatility exhibit a stochastic property. This evidence lead to the rejection of constant volatility used by BS model. So, BS yield at a biased pricing. Moreover, with a constant volatility assumption, investor will under estimate the risk from stochastic volatility. Thus, the short fall of BS seem to provide both price biasness and lead to improper hedging strategies.

20 Conclusion For Heston(1993) model that incorporate the stochastic volatility in pricing process should yield a more accurate price and a more proper risk management strategy. Further Study Another stylize fact that we can observe the in market is that the returns are non-normal. As the result the log-normal random walk process would be rejected. Bate(1996) suggested model of option pricing that incorporate the jump-diffusion process to the underlying asset movement. The contribution is to suggest new form of risk called Jump risk

21 Back up

22 Beyond Black-Scholes : The Heston Stochastic Volatility Option Pricing Model 1. Introduction : Index option In Thai Market. A derivative is an innovation of financial instrument which its value can derive from underlying asset. The important feature of derivative is that investors can use it as alternative investment and use to hedge the investment risk. Although, there are many type of derivative -- future contract, forward contract, swaps contract, option and etc, Investor can use it to hedge in different ways depend on their objective. One of the most liquid derivative in almost market is the index derivatives--the INDEX future and INDEX option. In Thailand, the SET50 future start to trade in exchange market called Thailand Future Exchange(TFEX) since And the SET50 option starts to trade in Stock Exchange of Thailand (SET) since However, there are the different between both derivatives. The future and forward contract is the obligation to buy or sell the underlying asset at the specific price and time. Hence, the investors who participates in the future contract need to be realized all of principle gain or loss. For example, if the index rises beyond the agreed price, the buyer can realize the profit at the end of contract. However, the future contract can use to log-in the future price for specific type of hedging for example, for the consumption goods. Unlike future contract, option is the right for holder to buy or sell its underlying asset at the specific amount, strike price and time. This means that the option holders have the right to exercise it right.,0 If its underlying asset goes beyond the strike price, the option holders will exercise the option. If there is not the case, the holder will not exercise and incurred the realized loss at the premium of each option price. Hence, investing in option can be more effective to hedge investment risk in index security. Moreover, with index option, investor can insure realized gain(loss) within the specific range. To arrive with the effective investing strategies, investor requires obtaining the precise option price. So, Option Pricing Model plays the critical role to quantify the option value. The Pricing model offer theoretical price which should to reflect the market price and can forecast the future movement of the variables(khanthavit, 2007). To obtain the option price, the novel theoretical option pricing model developed by Black-Merton-Schole(1973)(BS-Model) is the most popular model.(black and Scholes, 1973) This is because BS model provided two wisdom of knowledge. One is

23 that BS-model provided closed form solution for option price. So both practitioner and academic can obtain the theoretical option price easily. Second, BS-model also provide the closed form solution for Greek:s hedging parameters;delta, gamma, roh,vega, theta. 2. BS-Model The BS model grounds on a set of strong assumption. The model assumed that underlying asset returns are log-normal distributed and followed Geometric Brownian Motion (GBM) with known mean and constant variance. Moreover, the model assumed risk-free asset exist, no transaction cost and continuous trading. So, the European call and put option value can be written as,,,,,!"#!$ %" & ',!! %!( )( *!"#!$, +('( &! ) +)! )' ()!( max,0 /,0 According to the above condition and put-call parity, BS-Model arrived at the closed form solution for both put and call option. SN 2 N 3 / N 3 N 2 2 log log d1 and d2 are the probabilities that options will be expired in-the-money in both risk neutral and physical measures respectively. Moreover, the investor can construct the riskless portfolio using a unit of option and a delta unit of underlying asset. The delta of an option or a portfolio of options is the sensitivity of the option to the underlying asset value. < 1 =,Δ?? So, the fully hedge portfolio will grow at risk-free rate. dπrπdt

24 3.Volatility Smile Although BS model assumed that the volatility is constant, it cannot be observed directly. The solution to this problem is to apply the implied volatility. The implied volatility can obtain through the market data. Investors can observe the historical market data of option price, risk-free rate, and time to expiration. As the result, they can plug-in those market data into the BS-model and obtained the implied volatility.,,,,,&!)*! & %" However, many previous studied showed that the constant variance assumption is flawed and the variance exhibit the stochastic properties. (Geske, 1979, Scott, 1987, Johnson and Shanno, 1987),(Rubinstein, 1985, Wiggins, 1987). Moreover, volatility seems to exhibit the mean reversion properties(scott, 1987, Stein and Stein, 1991). The volatility smile also can be observed empirically in many markets. For example, (Rubinstein, 1985), (Taylor, 1994)Tompkins(1998). (Bakshi et al., 1997) published the empirical result of their study on S&P500 option during 1988 to The result showed the volatility smile by the u-shaped relationship between level of moneyness and volatility. (Dumas et al., 1998) showed the cross-sectional result of volatility smile on S&P500. Therefore, they concluded that the constant volatility assumption of BS model is flawed. Beber A. studied the implied volatility of Italian Stock Market option using MIBO30 which is the most liquid traded stock on that market. The sample between exhibit the u-shape relationship between volatility and moneyness on both short and long maturity option(beber, 2001). The more recent research also ascertains the consistence of this finding. For example, in 2003, a researcher documented an evidence of volatility smile on more liquid market,s&p100 option, using a sample period of 1986 to 2000 including crisis period(jones, 2003). In 2002, Yakoob provided the empirical result of smile volatility on a study of option index on S&P500 and S&P100 using a data between 2000 and 2001(Yakoob and Economics, 2002). More recently, there is study on Nikkei 225 option on a period after Asian crisis. They provide the evidence that Nikkei 225:s volatility exhibit smile(fukuta and Ma). Another recent studied found the stochastic relationship of volatility using ultrahigh frequency data. They documented asymmetry volatility smile on ASX200 between bull and bear

25 market. The sample period of 2006 E 2008 cover the more recent financial crisis. In sum, they found the volatility smile on ASX200 for put option in both bull and bear market while the call option exhibit the smile only on bear market. (Larkin et al., 2012) However, there are few evidences on the emerging market. Pena,Rubio and Serna studied the option on Spanish IBEX-35 index which is the most liquid market in Spain. The sample of 1994 to 1998 showed the biasness between volatility and moneyness(pena et al., 1999). The result on this market also ascertained by(fiorentini et al., 2002). They use daily data sample pre-crisis period between Jan and April 1996 to confirm the volatility smile in Spanish market. Singh document the volatility and moneyness bias on NSE index option of Indian market(singh, 2013). This can showed the consistence to the previous study. Patakkinang et al study the volatility smile in Thai market on SET50 option. They showed that both put and call exhibit the volatility smile using the sample of SET50 option expiring between Mar 2551 to Dec As the result of stochastic volatility, the BS model lead to mispricing of the option and hence the wrong hedging parameter. Besides, they suggested to hedge the volatility risk using Vega along with the delta hedging to alleviate the investment impact from this mispricing(patakkinang Tanasak, 2012). Therefore, the implied volatility was founded not to be constant lead to the violation of BS model:s assumption We use the data of SET50 Option between May 2553 E Oct 2556 to extend the result of previous studies. The result showed the volatility smile in both Put and Call option on SET50 at 40 and 60 days before expiration. (a.) (b.)

26 Figure1. (a.) The relationship between volatility and moneyness on Put Option on SET50 at 40 days to expiration(t=40). (b.) The relationship between volatility and moneyness on Call Option on SET50 at 40 days to expiration(t= =40) (a.) (b.) Figure2. (a.) The relationship between volatility and moneyness on Put Option on SET50 at 60 days to expiration (T=60). (b.) The relationship between volatility and moneyness on Call Option on SET50 at 60 days to expiration (T=60) The affect from the smile suggested that the stochastic volatility caused the pricing obtained by BS should contain biasness. Besides, (Patakkinang Tanasak, 2012) argued that BS model yield overprice on call option at deep-out-the-money and put option at deep-in-the-money. On the other sides, BS suggested the underprice on put option at deep-out-the-money and call option at deep-in-the-money. Moreover, the hedging using Greek that obtain from BS also based on the BS-pricing fomula. Therefore, the hedging parameter also affect from this stochastic volatility. Therefore, the investor should apply BS-model carefully. 4.The Heston Model. From the observed stylized fact that the returns are not normally distributed. Besides, it exhibits the skewness and kurtosis including ng fat tail. In addition, the volatility is changing overtime. However, many empirical showed that the volatility depicts the mean reversion process;the volatility changing around its long run mean. From this observation, various researcher construct the volatility model which it volatility drive by their own stochastic process. This include the models of Hull and White (1987), Scott (1987), Wiggins (1987), and Stein and Stein (1991)(Heston1993)

27 In the same way, Heston(1993) models their volatility based their stochastic process under Cox- Ross-Ingersoll(1985) or CIR process and the stock price followed GBM as in BS model(heston, 1993). Moreover, the Heston model allows Weiner:s process for both stock and volatility to correlate through the correlation coefficient. This mean that Heston suggested that there are two source of risk in the pricing process. One is the risk from stochastic movement of underlying asset. Two is the risk from the stochastic volatility. D% 2, % EF% D% 3, G H I 2,, 3, JK κ > 0 the mean reversion speed for variance θ > 0 the mean reversion level for the variance σ > 0 the volatility of the variance ν 0 > 0 the initial level of variance at time zero The Heston model suggested that the value of option is the function of stock price, time to expiration, strike price, risk-free rate, volatility of variance, volatility of underlying asset, mean reversion speed, mean reversion level for the variance, volatility risk premium and correlation between two process. L,,,M N ;P,Q,R,S,T,UV In order to arrive with the closed form formula, Heston based his model on arbitrage free argument like BS to construct the riskless portfolio. Since, the pricing process contains two source of risk. So, the riskless portfolio needs at least two option to hedge each risk. ΠΔWX The risk portfolio contains a unit of option, delta fraction of underlying asset and a W fraction of another option. Therefore, the motion of this portfolio is dπδdwx Then heston set up the Heston:s PDE (eq6) as follow.?x? 1 X 2 Y3?3? 3 KY?3 X??Y 1 X 2 Y3?3?Y 3 X?X? ZEFY[,Y,?X?Y 0 Like BS-Model, Heston model provided the closed form solution. The pricing solution of call option is in the same form of BS. Where / 2! / 3 are the probability of option to expire in the money. / 2 \ / 3

28 ?/ ]? 1 / ] 2 Y?3? 3 KY? 3 / ]??Y 1 / ] 2 Y3?3?Y 3 ]%?/ ]? b ] %?/ ]?Y 0 1 ) ^ 1,2; 2 2 ; ; EF,b 2 E[K;b 2 E[ This in the money probability function / ] can be recovered from the characteristic function which given by ' _ `; ;% via IInversion theoremj suggested by Gil-Pelaez (1951). Where Re is the real part of the function and c 1 is the imagin part of the function. Heston assumed that the characteristic function of log-stock price is in the log linear form. Therefore, the parameter requires to estimate heston price are shown 5.Option and Risk Management 5.1 BS and Risk Management One of the novel of BS-Model is that it can provide a formula to constructt fully hedge portfolio using and an option. The delta hedging is the sensitivity of option value to changing in underlying asset. a = ; = de df The Value at Risk (VaR) is an estimate of how large of lose in portfolio that could happen given the degree of confidence. So, VaR technique usually applies to quantify the levell of maximum risk in portfolio. The value of VaR for the portfolio that contain an option can be given as,) ; g "!( 1 h???? 1 2 3?3? 3i??

29 g 1 ( jh???? 1 2 3?3? 3i?? k % g l? 3?( m 3 Mn o g o pqr s t u 3 v So, the level of VaR can be eliminate by changing the value of Delta in portfolio. Moreover, with BS formula, it can easily obtain the Delta parameter from closed form solution =(""w 2! =+ w 2 1 Since, BS model based on the flawed assumption of constant volatility, therefore, the Greek parameters were calculated from the constant volatility through time. Thus, this flawed calculation can lead to the improper hedging parameter. 5.1 Heston and Risk Management Heston model based on the two source of risk in the pricing process. Therefore, to hedge the risk, portfolio need at least each option on each risk source;risk of underlying asset and risk of volatility. D% 2, % EF% D% 3, G H I 2,, 3, JK,% g "!( 2 e ( v LD% v V( x LEF% D% x V 1 2 ( vv 3 ( xx % 3 ( vx % yl( v ( x EF% V 1 2 ( vv 3 ( xx % 3 ( vx %z( v D% v ( x D% x % g ( v ( 3 %( x ( 3 %27( v ( ( x ( K8 { r}~ ;{ Q Q~ƒ Unlike BS model, Heston cannot provide the close form solution to each Greek:s parameter. But we intuitively know that Delta is the sensitivity of option value to changing in the value of underlying asset and Vega is the sensitivity of option value to changing in the value of volatility. So obtain the value of Delta and Vega we can calculate numerically.

30 $?,%?% ' ' lim ˆ "?,%? $?,%?% ; "?,%? '' 2 ˆ=ˆ= 2= %ˆ=%%ˆ=% 2=% Next, we will set up a riskless portfolio using two option and quantify the optimal hedging ratio. Let the riskless portfolio is, < 1 = The motion of portfolio is < = = ==D% v ll( v ( x EF% V 2 3 ( vv 3 ( xx % 3 ( vx %m v D% v hl( v ( x EF% V 1 2 ( vv 3 ( xx % 3 ( vx %i x D% x v D% v x D% x Since the portfolio is riskless, then the stochastic part would be zero < 1 = < Contain only deterministic part < ( v ll( v ( x EF% V 2 3 ( vv 3 ( xx % 3 ( vx %m ll( v ( x EF% V 2 3 ( vv 3 ( xx % 3 ( vx %m The stochastic part equation are IZeroJ =D% +( v D% v v D% v ` x D% x + x D% x ` Therefore, we can obtain the optimal hedging ratio. ( x x $ ) = ( x ( v ( v 1 ( x x ( v x =!)')+)! ) $ )(( )%! )'!"#!$ )(& +(!)')+)! ) $ )(( )%! )'%)""#

31 6. Conclusion Since the Options on index are introduced as the instrument for various objective, one of it purpose is for hedging risk. For that purpose, investors do need to price the option accurately. However, many found the flawed on novel BS model:s assumption of constant volatility. The empirical data from both international and Thai market, we found that volatility exhibit a smile shape- -stochastic property. This evidence leads to the rejection of constant volatility used by BS model. Hence, BS is a model that yields at a biased pricing. As suggested by (Patakkinang Tanasak, 2012)they showed that BS model yield overprice on call option at deep-out-the-money and put option at deep-in-the-money. Moreover, BS underprices on put option at deep-out-the-money and call option at deep-in-the-money. This is because with a constant volatility assumption, investor will under estimate the risk from stochastic volatility. As the result, the short fall of BS seem to provide both price biasness and lead to improper hedging strategies. Heston(1993) model that incorporate the stochastic volatility in pricing process should yield a more accurate price and a more proper risk management strategy. Therefore, lead to more effective on hedging strategies. Although, heston model can provide a more accurate on pricing, the model does not provide a closed form formula for The Greek;Hedging parameter like BS. To solve this problem, we showed that numerical method can apply to find the Greek for both Delta and Vega parameters. However, Another stylize fact that we can observe the in market is that the returns are nonnormal. As the result the log-normal random walk process would be rejected. Therefore, any model that relied on the log-normal assumption would affect from the non-normal distribution. For example BS and Heston model. Therefore, Bate(1996) suggested model of option pricing that incorporate the jump-diffusion process to the underlying asset movement. This is to incorporate the risk involving in jump which is called jump-risk. Reference BAKSHI, G., CAO, C. & CHEN, Z. (1997) Empirical Performance of Alternative Option Pricing Models. The Journal of Finance, 52, BEBER, A. (2001) Determinants of the implied volatility function on the Italian Stock Market ALEA Tech Reports, Tech Report Nr.10. BLACK, F. & SCHOLES, M. (1973) The Pricing of Options and Corporate Liabilities. The Journal of Political Economy, 81, DUMAS, B., FLEMING, J. & WHALEY, R. E. (1998) Implied Volatility Functions: Empirical Tests. The Journal of Finance, 53,

32 FIORENTINI, G., LENON, A. & RUBIO, G. (2002) Estimation and empirical performance of Heston's stochastic volatility model: the case of a thinly traded market. Journal of Empirical Finance, 9, FUKUTA, Y. & MA, W. Implied volatility smiles in the Nikkei 225 options. Applied Financial Economics, 23, GESKE, R. (1979) The valuation of compound options. Journal of Financial Economics, 7, HESTON, S. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, JOHNSON, H. & SHANNO, D. (1987) Option Pricing when the Variance is Changing. The Journal of Financial and Quantitative Analysis, 22, JONES, C. S. (2003) The dynamics of stochastic volatility: evidence from underlying and options markets. Journal of Econometrics, 116, KHANTHAVIT, A. (2007) กQNRSTUQSVWXYZZ[\Q]R^_`abRก\QcSdNQeQRRfgWbShR^dWgSi SET50. NQj^QSXk[WjXkgQกQN eol`qokgjmqnvny]lกqnzwogi UcQXkpjQ]WjqNNUmQnVN. LARKIN, J., BROOKSBY, A., LIN, C. T. & ZURBRUEGG, R. (2012) Implied volatility smiles, option mispricing and net buying pressure: evidence around the global financial crisis. Accounting & Finance, 52, PATAKKINANG TANASAK, T. L., CHALAKARN SASIPA AND THARAVANIJ PIYAPAS (2012) Volatility Smile and Forecasting Performance of Implied Volatility from SET50 Option. NIDA Development Journal, 52, 34. PENA, I., RUBIO, G. & SERNA, G. (1999) Why do we smile? On the determinants of the implied volatility function. Journal of Banking & Finance, 23, RUBINSTEIN, M. (1985) Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, The Journal of Finance, 40, SCOTT, L. O. (1987) Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application. The Journal of Financial and Quantitative Analysis, 22, SINGH, V. K. (2013) Empirical Performance of Option Pricing Models: Evidence from India International Journal of Economics and Finance, 5. STEIN, E. M. & STEIN, J. C. (1991) Stock price distributions with stochastic volatility: an analytic approach. Review of Financial Studies, 4, TAYLOR, S. J. (1994) MODELING STOCHASTIC VOLATILITY: A REVIEW AND COMPARATIVE STUDY. Mathematical Finance, 4, WIGGINS, J. B. (1987) Option values under stochastic volatility: Theory and empirical estimates. Journal of Financial Economics, 19, YAKOOB, M. Y. & ECONOMICS, D. U. D. O. (2002) An Empirical Analysis of Option Valuation Techniques Using Stock Index Options.