The Heston Option Pricing Model Ser-Huang Poon June 27, 2011 If Black-Scholes (BS) is the correct option pricing model, then there can only be one BS implied volatility regardless of the strike price of the option. BS implied volatility smile and skew are clear evidence that market option prices are not priced according to the BS formula. Nevertheless, the BS implied volatility, despite all its shortcomings, has been proven overwhelmingly to be the best forecast of volatility. This raises the important question about the relationship between BS implied volatility and the true volatility. Heston (1993), one of the most popular stochastic volatility option pricing models, is motivated by the widespread evidence that volatility is stochastic and that the distribution of risky asset returns has tail(s) longer than that of a normal distribution. A SV model with correlated price and volatility innovations can address both empirical stylized facts. The SV option pricing model was developed with a series of contributions from Johnson and Shanno (1987), Wiggins (1987), Hull and White (1987, 1988), Scott (1987), Stein and Stein (1991) and Heston (1993). It was in Heston (1993) that a semi-closed form solution was derived based on characteristic function of the price distribution. Section 1 presents the Heston SV option pricing model starting from the square root process, and shows the derivation of the Heston equation from the characteristic function. Section 2 shows how one can simulate the SV dynamics. Section 3 considers the inclusion of a jump component. In Section 4, we simulate a series of Heston option prices from a range of parameters. Then we use these option prices as if they were the market option prices to back out the corresponding BS implied volatilities. If market Some of the materials are from Poon (2005) A Practical Guide for Forecasting Financial Market Volatility, Wiley, and Jondeau, Poon and Rockinger (2006) Financial Modeling Under Non-Gaussian Distributions, Springer-Verlag London Ltd. I would like to thank He Xue Fei for his contributions to some of the derivations. 1 option prices are priced according to the Heston dynamics, the simulations in this section will give us some insight into the relationship between BS implied volatility and the true volatility. Finally, Section 5 presents empirical …ndings on the predictive power of Heston implied volatility as a volatility forecast. 1 Stochastic Volatility Model The Heston (1993) stochastic volatility model is based on the following stock price and variance dynamics dSt = dvt = p St dt + St vt dZ1 ; p ( vt ) dt + vt dZ2 ; (1) (2) where ; ; > 0 are constant parameters. The two Brownian motions, Z1 and Z2 , are correlated, i.e. Corr [dZ1 ; dZ2 ] = dt. The dynamics of the stock price St in (1) is a geometric Brownian motion with time varying volatility. The variance vt in (2) follows a square root process (also known as the Cox-Ingersoll-Ross process). As we will show below, the parameter corresponds to the long-run average of vt , and controls the speed by which vt returns to its long-run mean. Given that the Heston model has two sources of randomness, the bivariate Ito’s lemma is used to derive the fundamental partial di¤erential equation. The steps involved in the derivation of the Heston option pricing formula are the same as those in the no-arbitrage derivation for the BlackScholes formula except that two derivative assets are required to obtain a risk-neutral portfolio. For example, we could consider shorting a call option C together with long positions in units of the underlying asset and units of a second derivative C1 written on the same underlying. C1 di¤ers from C by its maturity or strike price. Let us rewrite, using shorter notations, the system of equations (1) and (2) as dS = S dt + S dZ1 ; dv = v dt + v dZ2 : Let C (S; v; t) denote the price of a call option. We obtain from the bivariate 2 Ito’s lemma that the dynamics C may be written as dC = 2C @2C 1 + @S@v 2 @C @C @C + v + dt + + S @S @v @t 1 2 2@ S @S 2 + S v 2 2@ C v @v 2 S The dynamic of the portfolio value, W = C dW = dC 1 = 2 @C dZ1 + @S S v @C dZ2 : @v C1 , is dS dC1 2 @2C 1 2 @2C @C @C @C 2@ C + + + S + v + v S S dt S @S 2 @S@v 2 v @v 2 @S @v @t @ 2 C1 1 2 @ 2 C1 @C1 @C1 @C1 1 2 @ 2 C1 + + + S + v + dt S v S v 2 2 2 @S @S@v 2 @v @S @v @t @C @C1 @C1 @C dZ1 + v dZ2 : (3) v S S S @S @S @v @v + To obtain risk neutrality, the coe¢cients of dZ1 and dZ2 must be zero, so that the two sources of uncertainty no longer play a role in the portfolio value dynamic. This means that @C @S @C @v @C1 ; @S = + = @C1 : @v (4) (5) With these two conditions, the instantaneous change of value of the fully hedged portfolio must be equal to the return on a risk-free investment. Otherwise there will be an arbitrage opportunity. Hence, dW = r [C S C1 ] dt: If we equate (3) and (6), and substitute the values of (5), we obtain = 1 2 1 2 2@ S 2C @S 2 2C 1 @S 2 2@ S + + (6) and from (4) and @2C 1 2 @2C @C @C @C @C + + rS + v + rC @S@v 2 v @v 2 @S @v @t @v 2 2 @C1 @ C1 1 2 @ C1 @C1 @C1 @C1 + + rS + v + rC1 : v v 2 @S@v 2 @v @S @v @t @v S v S Interestingly, the two sides of the equality sign are the same in their expression, di¤ering only in the option to which they apply. Given that the same equation must hold for any type of call option of any maturity and strike price, then each side of the equality must be independent from the 3 type of option that one considers. This suggests that each side will be equal to some function, say (S; v; t) that depends on S and v. This function may be interpreted as a volatility risk premium. Replacing the parameters by their actual units, we see that the fundamental partial di¤erential equation is now 1 2 @2C @2C 1 2 @2C @C 0 = vS + vS + v 2 + rS 2 2 @S @S@v 2 @v @S @C @C rC + : (Heston eq6) +[ ( v) (S; v; t)] @v @t Heston (1993) makes the important assumption that the volatility risk premium is a linear function of vt such that (S; v; t) = v. Moreover, introducing x = ln S or S = ex , we have @C @S @2C @S 2 = = = @2C @S@v = @C @x 1 @C @C = =e x @x @S S @x @x @ @C @x @ @C @C @x = = e x +e e x @x @S @S @x @x @S @x 1 @2C 1 1 @C 1 @2C 1 @C + = + S @x S @x2 S S 2 @x S 2 @x2 @ @C @x 1 @2C = @x @v @S S @x@v x@ 2C @x2 @x @S Substitute the results into (Heston eq6) to obtain 0 = 1 @2C vt + 2 @x2 +[ ( vt ) vt @2C 1 @2C + 2 vt 2 @x@v 2 @v @C 1 vt ] + r vt @v 2 @C @x rC + @C : @t (7) The key di¤erence between (Heston eq6) and (7) is that the coe¢cient of the partial derivatives does not contain S (or x) making the PDE a lot easier to solve. In the case of a European Call option, we have the following boundary conditions: C(ST ; v; T ) = max(ST K; 0); (8) C(0; v; t) = 0; (9) @C (1; v; t) = 1: (10) @St By analogy with the Black-Scholes formula, the guessed solution of this European option is of the form C(S; v; T ) = SP1 x = e P1 4 r e KP2 rt rT e KP2 (11) where P1 and P2 are cumulative density functions in relation to the moneyness of the option at maturity. From (11), we obtain the following partial derivatives: @C @x @2C @x2 @C @v @2C @v 2 @2C @x@v @C @t @P1 @P2 ert rT K @x @x 2P @P @ @ 2 P2 1 1 rt rT ex P1 + 2ex + ex e K @x @x2 @x2 @P @P 1 2 ex ert rT K @v @v @ 2 P1 @ 2 P2 rt rT ex e K @v 2 @v 2 2 @P1 @ P1 @ 2 P2 ex + ex ert rT K @v @x@v @x@v @P @P2 1 ex rert rT KP2 ert rT K @t @t = ex P1 + ex = = = = = Substitute these partial derivatives into (7) to give ex [f (P1 )] ert rT K [f (P2 )] = 0 where f (Pi ) with i = 1; 2 is a polynomial function of Pi . (12) is f (P1 ) = f (P2 ) = 0 (12) A solution for which means f (P1 ) = 1 @ 2 P1 vt + 2 @x2 +[ ( vt vt ) @ 2 P1 1 + @x@v 2 vt + @ 2 P1 1 + r + vt @v 2 2 @P1 @P1 + =0 vt ] @v @t 2 vt @P1 @x and f (P2 ) = 1 @ 2 P2 vt + 2 @x2 +[ ( vt vt ) @P2 @ 2 P2 1 @ 2 P2 1 + 2 vt + r vt 2 @x@v 2 @v 2 @x @P2 @P2 rP2 + rP2 + =0 vt ] @v @t Alternatively, as presented in Heston equation (12), 1 @ 2 Pj vt + 2 @x2 + [a @Pj @ 2 Pj @ 2 Pj 1 + 2 vt + (r + uj vt ) 2 @x@v 2 @v @x @Pj @Pj + =0 (Heston eq12) bj vt ] @v @t vt for j = 1; 2, where u1 = 12 , u2 = 1 2, a= 5 , b1 = + , b2 = + . Indeed the cumulative probability functions Pj in (Heston eq12) are not immediately available in closed form. On the other hand, it can be shown that their characteristic functions fj (x; v; T ; ) must satisfy the same PDEs in (Heston eq12) subject to the terminal condition fj (x; v; T ; ) = ei x : Assuming that the characteristic function has the solution fj (x; v; T ; ) = eC+Dv+i x (13) where C and D are functions of (T t). Hence, the task now is to …nd suitable C and D such that fj (x; v; T ; ) satis…es @2f 1 @2f 1 @2f vt 2 + vt + 2 vt 2 2 @x @x@v 2 @v @f @f @f + (r + uj vt ) + [a bj vt ] + @x @v @t = 0 (Heston A4) From (13) @f @x @f @v 2 @ f @x@v = i f; = Df; = i Df; @2f 2 = f @x2 @2f = D2 f @v 2 @f @D @C = +v @t @t @t f Substitute these partial derivatives into (Heston A4) and obtain 1 vt 2 1 2 2 + i D+ 1 2 2 2 f+ D 2 + uj i vt i Df + 1 2 2 vt D2 f + (r + uj vt ) i f @D @C + vt f @t @t @C v + ri + aD + @t + [a bj vt ] Df + = 0 bj D + @D @t = 0 This gives rise to a system of two Ricatti equations: 1 2 2 + i D+ 1 2 2 @D @t @C ri + aD + @t D 2 + uj i 6 bj D + = 0 = 0 Solving the pair of Ricatti equations, using e.g. mathematica, gives the solution for (13) with C( ; ) = ri D( ; ) = a + (bj 2 bj i +d 2 i + d) 1 ed 1 ged 1 2 ln ged 1 g ; and g = d = bj bj q ui + d ; ui d ( bj )2 i 2 (2u ji 2 ); From the Fourier inversion theorem, we have Z e i xf ( ) 1 1 1 Re d ; F (x) = 2 i 0 Z 1 1 1 e i xf ( ) Pr [X > x] = 1 F (x) = + Re d : 2 i 0 Hence, we can write P1 and P2 in (11) as 1 1 Pj = + 2 Z 1 Re e i ln K f j;xt ;vt ;t ( i 0 ) d ; with j = 1; 2. Carr and Madan (1999) and Bakshi and Madan (2000) show how to use the fast Fourier method to evaluate the integral. 2 Simulating the SV dynamics In the previous section, we considered the simpler and straightforward case of a European call option. Invoking the put-call parity, the European put price can be deduced. There are situations, e.g. pricing path dependent options, where we have to simulate trajectories either under the risk-neutral probability or under the objective probability. In this case, we will need the system of stochastic di¤erential equations and the associated PDE. This can be done by analogy. Indeed, for dS = S dt + S dZ1 ; (14) dv = v dt + v dZ2 ; (15) 7 and instantaneous correlation Corr [dZ1 ; dZ2 ] = dt, the transition probability p(ST ; vT jS; v) must satisfy @2p 1 2 @2p @p @p @p 1 2 @2p + + + S + v + = 0: v S S v 2 2 2 @S @S@v 2 @v @S @v @t Inspection of the PDE in (Heston eq12) for the case j = 2, where the transition probability is risk neutral, shows that p 1 vt ; vt ; S = 2 If we also decompose dZ1 into S =r v = dZ1 = dZ2 + p ( 1 vt ) vt ; v = p vt : 2 dZ ~1 ; (16) where Z~1 is a Brownian motion and where dZ2 and dZ~1 are uncorrelated, then we get i p p h 1 2 dZ ~1 ; vt dt + vt dZ2 + 1 (17) dxt = r 2 p dvt = [ ( vt ) vt ] dt + vdZ2 : (18) h i Since Corr dZ~1 ; dZ2 = 0; dZ~1 and dZ2 can be simulated separately and then linked via (17) to yield a correlated process. If we introduce the notations = + and = = ( + ), equation (16) can also be written as p vt dZ2 : (19) dvt = ( vt ) dt + Using the system (17) with either (18) or (19), it is possible to simulate risk neutral trajectories using a …rst or second-order discretization scheme. The di¤erence between (14) and (17) is the replacement of by r. The di¤erence between (18) and (19) is the replacement of ; and by the risk neutral equivalent and . That is the risk neutral parameters and have absorbed , the volatility risk premium. Both (18) and (19) are risk neutral SDEs. 3 Combining stochastic volatility with jumps Bates (1996) considers a compensated jump-di¤usion process with variance following a square-root process as in Heston (1993). The full model is p dSt = ( J)St dt + vt St dZ1 + St Jt dN; (20) p dvt = ( vt )dt + vt dZ2 : (21) 8 The term J is the expected instantaneous jump size. As the price process is compensated, i.e., the jump contribution in the drift will be zero. Moreover, Cov [dZ1 ; dZ2 ] = dt; (22) Pr [dNt = 1] = dt; (23) log (1 + Jt ) N log 1 + J 1 2 2 ; 2 : (24) where Jt is the stochastic jump, expressed as a percentage, whenever a jump takes place. The last equation, (24), de…nes 1 + Jt as a log-normal jump. Hence, 1 2 1 2 E[1 + Jt ] = exp log(1 + J) + = 1 + J; 2 2 and thus E[Jt ] = J: Furthermore as E[Jt dN ] = Jdt; this result shows that the expected jump contribution is zero per time unit in equation (20) because the jump comJdt in the mean. ponent has been compensated by the introduction of Next, we proceed to solve the PDE associated with equations (20) to (24). Let us start with the univariate jump di¤usion given by dSt = a dt + b dZt + cJt dN: Let h(ST ) be some integrable function and de…ne Z (S; t) E [h(ST )jS; t] = h(sT )p(sT ; T jS; t)dsT ; sT the latter integral involving the transition probability. Given that the problem is Markovian, the Chapman-Kolmogorov equation still holds and using the argument of iterated expectations as we did for Heston’s model, it is possible to show that is a martingale. It follows that its drift must be zero, or in other terms that the expected instantaneous variation must be zero. We may use the Ito formula for di¤usions involving jump-processes to obtain1 Z @p @p @p 1 2 @2p b +a + dt + b dZ 0 = E[d ] = h(sT ) 2 2 @s @s @t @s sT i + ([p(st + ct J; t) p(st ; t )] dN dsT ; 1 This extension of Ito’s lemma to processes with discrete jumps is immediate. 9 where we suppressed the arguments (sT ; T ) …guring in the transition probabilities. We notice that there are two components to the drift term. One due to the di¤usion part and the other due to the jump part. The expectation of the Brownian motion part, dZt is nil and thus it must be that Z 1 2 @2p @p @p h(sT ) b +a + dt 2 2 @s @s @t sT i + E f[p(st + ct Jt ; t) p(st ; t )] dNt g dsT = 0: Given that this equation must hold for all h(:) functions, it must also be true for Dirac functions. It follows, using the fact that the jump size Jt and the Poisson process are independent that @p @p 1 2 @2p b +a + + E[p(st + ct Jt ; t) p(st ; t )] = 0; 2 2 @s @s @t an expression called the in…nitesimal generator of the jump di¤usion. This is the PDE that the transition probability of a jump di¤usion must obey, i.e. with an additional expectation term due to the jump component. Expanding the notations for a, b and c, Bates obtains the following PDEs for the transition probability 1 vt 2 @2p +2 @s2 @p 1 @p + r vt +( J @t 2 @s @2p @2p + 2 2 + E[p(st + ; t) @s@v @v vt ) @p + @v p(st ; t)] = 0; (25) with log(1 + J ) N 1 2 log(1 + J ) 2 ; 2 : The parameters ; ; and J are risk neutral. In the case of a European call option with maturity date T , strike price K and parameter vector , the pricing formula is C(St ; T; t; K; ) = St P1 (ST > ln K) Ke r P2 (ST > ln K); where P1 and P2 are survival functions. The di¤erential equations associated with P2 follows, as in Heston, from the construction of a non-arbitrage portfolio. The partial di¤erential equation associated with P1 follows by recognizing that C must follow the same partial di¤erential equation as P2 :2 2 The appendix in Bakshi, Cao, and Chen (1997) also provides details on the various PDEs and guidence on how to solve them. 10 A characteristic function that will solve the PDE in (25) can be guessed in the usual way given the linearity of the problem ; )+B( ; )v+i S f ( ; S; v; t) = eA( : Bates derives C(S; T; t; K; ) = S P1 K e r P2 ; Z fj;x;v;t ( )e 1 1 1 Pj = Re + 2 i 0 i ln K d ; for j = 1; 2 where fj;x;v;t ( ) E[e i XT ] 1 (1 + J )uj + 2 = exp Aj ( ; ) + Bj ( ; )vt + i S + [(1 + J )i e 2 1 2 (uj i 2 ) 1 ; and Bj ( ; ) = (r J )i 2 2 Aj ( ; ) = j = u1 = 2 s ( 2 1 log 1 + ( 2 i i j i 2 j) 1 ; 2 j) 1 e j : j 2 1 j) j j 1 uj i 2 + j (1 + e i 1 ; u2 = 2 ( 2 2 )=(1 j uj i = 1 2 ; and e ) j 2 ; ; 2 = : This formula is again relatively straightforward to implement. As before, some care is needed in the computation of the inverse Fourier transform since the integrand is an oscillating function. Bates also shows that the risk neutral counterparts to equations (20)– (24) are dSt =St = (r J )dt + dv = [ Cov [dZ1 ; dZ2 ] = Pr [dN = 1] = v]dt + dt; dt; E [J ] = J : 11 p p vdZ1 + J dN; vdZ2 ; All starred variables represent risk neutral versions of the actual variables. This system of equations may be discretized with a …ne …rst or second-order scheme as the basis for simulations. The model of Bakshi, Cao, and Chen (1997) is similar to Bates’s, but with the extension to include a square-root process for interest rate as well. 4 Heston parameters and BS implied In this Section, we analyse possible BS implied bias by simulating a series of Heston option prices using asset price S = 100, interest rate r = 0, time to maturity T = 1 year, and strike prices range from 50 to 150. There are …ve other parameters used in Heston formula; namely, , the speed of mean reversion, , the long-run volatility level, , the market price of risk, , volatility of volatility, and , the correlation between the price and the volatility processes. When = 0, the volatility process becomes risk neutral, and and become and respectively. In most simulations, and unless otherwise stated, = 0:1, = 20%, = 0, = 0:6, = 0 and the current “instantaneous” volatility, vt , is set equal to the long run level, , at 20%. Figure 1: Black-Schole Series (Skewness=0, Kurtosis=3) The …rst set of simulations presented in Figure 1 involves replicating the Black-Scholes prices as a special case. Here we set = 0. Since there is no volatility risk, = 0. This is a special case where Heston price and Black-Scholes price are identical and the BS implied volatility is the same 12 across strike prices. In this special case, BS implied volatility (at any strike price) is a perfect representation of true volatility of 20%. Figure 2: E¤ect of ( =VV, kur = Kurtosis, = 0:1) In the second set of simulations presented in Figure 2, we alter , the volatility of volatility, and keep all the other parameters the same. The e¤ect of an increase in is to increase the unconditional volatility and kurtosis of risk neutral price distribution.3 In this case where volatility is RT stochastic, true volatility is 0 vt dt and can be derived from (2) and should be bounded between the min and max points of the vol smile which in Figure 2 correspond to the OTM and ATM implied vol. Hence, we can conclude that ATM implied vol underestimates true volatility while OTM implied vol overestimates it. Assuming that is constant over time, and that and are relatively stable, a time series regression of historical ‘actual volatility’ on historical ‘implied volatility’ at a particular strike will be su¢cient to correct for these biases. This is basically the Ederington and Guan (1999) approach. However, is not likely to be stable. When is not constant, the analysis below and Figure 3 shows that ATM implied volatility is least a¤ected by changing . This explains why ATM implied volatility is the most robust and popular choice of volatility forecast. 3 It is the risk neutral distribution because , the market price of risk, is set equal to zero. 13 2 Figure 3: E¤ect of Correlation, In Figure 3, it is clear that changing correlation coe¢cient alone has no impact on ATM implied volatility. Correlation has the greatest impact on skewness of the price distribution and determines the shape of volatility smile or skew. Its impact on kurtosis is less marked when compared with , the volatility of volatility. Figure 4 highlights the impact of , the mean reversion parameter when = 0:6; = 0:2; and = 0. High rate of mean reversion , or high initial volatility vt has the e¤ect of ‡attening the vol smile as it becomes hard to distinguish a stock price with a very high volatility and a stock price with a stochastic volatility. When this is the case, there is no strike price bias in BS implied (i.e. there will not be volatility smile). When is low, this is when the problem starts. A low corresponds with volatility persistence where BS implied volatility will be sensitive to the current state of volatility level. Strike price e¤ect or the volatility smile is the most acute when initial volatility level vt is low. Similar to the situation with high vol of vol , ATM implied vol will underestimate true vol vis-a-vis OTM options. Figure 5 and 6 can be used to infer the impacts of parameter estimates above when the volatility risk premium is omitted. In the literature, we often read “... volatility risk premium is negative re‡ecting the negative correlation between the price and the volatility dynamics ...” (Buraschi and 14 Figure 4: E¤ect of ( = 0:6, = 0:2) Jackwerth (2001), Bakshi and Kapadia (2003)). All series in Figure 5 have correlation = 0:5 and all series in Figure 6 have correlation = +0:5. A negative (volatility risk premium) produces higher Heston price and higher BS implied volatility. The impact is the same whether the correlation is negative or positive. Empirical evidence indicates that Figure 6 is just as likely a scenario as Figure 5. As = + and = + , a negative has the e¤ect of reducing (slower mean reversion) and increasing (higher long run mean), and a higher option price. Hence, a “negative risk premium” is to be expected whether the price and the volatility processes are positively or negatively correlated.4 Both volatility and volatility risk premium have positive impact on option price. 5 Volatility forecast using Heston model The thick tail and non-symmetrical distribution found empirically could be a result of volatility being stochastic. The simulations results in the previous section suggest that , the volatility of volatility is the main driving force for kurtosis and skewness (if correlation not equal to zero). A high , volatility 4 This is really a misnomer, while the parameter is negative, it actually results in a higher option price. So strictly speaking the volatility risk premium is positive! 15 Figure 5: E¤ect of on negatively correlated processes ( Skewness = +0:1623; Kurtosis = 3:7494) = 0:5, mean reversion will cancel out much of the impact on kurtosis and some of that on skewness. Correlation between the price and the volatility processes, , determines the sign of the skewness. But beyond that its impact on the magnitude of skewness is much less compared with the combined e¤ect of and . Correlation has negligible impact on kurtosis. The long run volatility level, , has very little impact on skewness and kurtosis, except when is very high and is very low. So a stochastic volatility pricing model is useful and will out perform Black-Scholes only when volatility is truly stochastic (i.e. high ) and volatility is persistence (i.e. low ). The di¢culty with the Heston model is that a large combinations of parameter values can produce similar skewness and kurtosis. This contributes to model parameter instability and convergence di¢culty during estimation. Through simulation results we can predict the degree of Black-Scholes pricing bias as a result of stochastic volatility. In the case where volatility is stochastic and = 0, ATM implied volatility will be lower than actual volatility while implied volatility of far-from-the-money options (i.e. either very high or very low strikes) will be higher than actual volatility. The pattern of pricing bias will be much harder to predict if is not zero and varies through time. Some of the early work on option implied volatility focus on …nding an optimal weighting scheme to aggregate implied volatility of options across strikes. (See Bates (1996) for a comprehensive survey of 16 Figure 6: E¤ect of on positively correlated processes ( = 0:5; Skewness = 0:1623; Kurtosis = 3:7494) these weighting schemes.) Since the plot of implied volatility against strikes can take many shapes, it is not likely that a single weighting scheme will consistently remove all pricing errors. For this reason and together with the liquidity argument, ATM option implied volatility is often used for volatility forecast but not implied volatilities at other strikes. 17 References [1] Bakshi G., C. Cao and Z. Chen (1997) Empirical performance of alternative option pricing models, Journal of Finance, 52, 5, 2003-2049. [2] Bakshi G. and D. Madan (2000) Spanning and derivative-security valuation, Journal of Financial Economics, 55, pp.205-238. [3] Bates D.S. (1996) Testing option pricing models, in Maddala G.S. and C.R. Rao eds. Handbook of Statistics, v.14: Statistical methods in Finance, Elsevier, Amsterdam: North Holland, pp.567-611. [4] Carr Peter and Dilip Madan (1999) Option Pricing and the Fast Fourier Transform, Journal of Computational Finance, 2, 4, pp.61-73. [5] Johnson H. and D. Shanno (1987) Option pricing when the variance is changing, Journal of Financial and Quantitative Analysis, 22, 143-151. 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