LOG-SKEW-NORMAL MIXTURE MODEL FOR OPTION VALUATION

Transcription

1 LOG-SKEW-NORMAL MIXTURE MODEL FOR OPTION VALUATION J.A. Jiménez and V. Arunachalam Department of Statistics Universidad Nacional de Colombia Bogotá, Colombia ICASQF 2016 Cartagena Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 1 / 34

2 AGENDA 1 INTRODUCTION 2 LOG-SKEW-NORMAL DISTRIBUTION 3 THE SKEW NORMAL MIXTURES 4 OPTION PRICING FORMULA 5 SOME SPECIAL CASES 6 NUMERICAL RESULTS 7 CONCLUSIONS Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 2 / 34

3 INTRODUCTION Introduction There is empirical evidence that financial stock returns are not normally distributed but are characterized by skewness, leptokurticity, heavy-tailedness and other non-gaussian properties. The skewness and kurtosis of the empirical distribution function (EDF) of stock returns contribute significantly to the phenomenon of volatility smile. In recent years, there have been considerable efforts to report that the unconditional probability distributions of returns on financial stocks are not normally distributed. Specifically, these distributions tend to have heavier tails (leptokurtic) and asymmetry. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 3 / 34

4 INTRODUCTION Introduction There is empirical evidence that financial stock returns are not normally distributed but are characterized by skewness, leptokurticity, heavy-tailedness and other non-gaussian properties. The skewness and kurtosis of the empirical distribution function (EDF) of stock returns contribute significantly to the phenomenon of volatility smile. In recent years, there have been considerable efforts to report that the unconditional probability distributions of returns on financial stocks are not normally distributed. Specifically, these distributions tend to have heavier tails (leptokurtic) and asymmetry. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 3 / 34

5 INTRODUCTION Introduction There is empirical evidence that financial stock returns are not normally distributed but are characterized by skewness, leptokurticity, heavy-tailedness and other non-gaussian properties. The skewness and kurtosis of the empirical distribution function (EDF) of stock returns contribute significantly to the phenomenon of volatility smile. In recent years, there have been considerable efforts to report that the unconditional probability distributions of returns on financial stocks are not normally distributed. Specifically, these distributions tend to have heavier tails (leptokurtic) and asymmetry. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 3 / 34

6 INTRODUCTION Introduction There is empirical evidence that financial stock returns are not normally distributed but are characterized by skewness, leptokurticity, heavy-tailedness and other non-gaussian properties. The skewness and kurtosis of the empirical distribution function (EDF) of stock returns contribute significantly to the phenomenon of volatility smile. In recent years, there have been considerable efforts to report that the unconditional probability distributions of returns on financial stocks are not normally distributed. Specifically, these distributions tend to have heavier tails (leptokurtic) and asymmetry. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 3 / 34

7 INTRODUCTION Introduction [Brigo and Mercurio(2002)] - the lognormal mixture (LNMIX) diffusion for a stock price to derive an option formula for exotic derivatives. The Skew Brownian motion has been used for pricing European options ([Corns and Satchell(2007)]). The Log-Skew-Normal (LSN) distribution, which is an extension for the positive data of the Log-Normal (LN) distribution used to form data with asymmetry and kurtosis. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 4 / 34

8 INTRODUCTION Introduction [Brigo and Mercurio(2002)] - the lognormal mixture (LNMIX) diffusion for a stock price to derive an option formula for exotic derivatives. The Skew Brownian motion has been used for pricing European options ([Corns and Satchell(2007)]). The Log-Skew-Normal (LSN) distribution, which is an extension for the positive data of the Log-Normal (LN) distribution used to form data with asymmetry and kurtosis. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 4 / 34

9 INTRODUCTION Introduction [Brigo and Mercurio(2002)] - the lognormal mixture (LNMIX) diffusion for a stock price to derive an option formula for exotic derivatives. The Skew Brownian motion has been used for pricing European options ([Corns and Satchell(2007)]). The Log-Skew-Normal (LSN) distribution, which is an extension for the positive data of the Log-Normal (LN) distribution used to form data with asymmetry and kurtosis. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 4 / 34

10 OBJECTIVE Introduction In this talk By assuming that the stock distribution follows a Log-Skew-Normal mixture (LSNMIX) distribution, we calculate an explicit formula for option valuation for both European call and put options and Greek measures. We also show that some of the well-known models are obtained as special cases from the proposed model([black and Scholes(1973)], [Bahra(1997)] and [Corns and Satchell(2007)]). An example from S&P500 daily returns to price is presented to illustrate the model. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 5 / 34

11 OBJECTIVE Introduction In this talk By assuming that the stock distribution follows a Log-Skew-Normal mixture (LSNMIX) distribution, we calculate an explicit formula for option valuation for both European call and put options and Greek measures. We also show that some of the well-known models are obtained as special cases from the proposed model([black and Scholes(1973)], [Bahra(1997)] and [Corns and Satchell(2007)]). An example from S&P500 daily returns to price is presented to illustrate the model. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 5 / 34

12 OBJECTIVE Introduction In this talk By assuming that the stock distribution follows a Log-Skew-Normal mixture (LSNMIX) distribution, we calculate an explicit formula for option valuation for both European call and put options and Greek measures. We also show that some of the well-known models are obtained as special cases from the proposed model([black and Scholes(1973)], [Bahra(1997)] and [Corns and Satchell(2007)]). An example from S&P500 daily returns to price is presented to illustrate the model. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 5 / 34

13 Log-Skew-Normal distribution LOG-SKEW-NORMAL DISTRIBUTION (LSN) The LSN distribution was first introduced by [Azzalini et al.(2003)azzalini, Dal Cappello, and Kotz] and later by [Lin and Stoyanov(2009)]. A random variable Y has a LSN distribution with asymmetry parameter λ R, denoted as Y LSN (Λ 1 ), if its pdf is of the form ( ) ( ( ) 2 ln y µ f Y y; Λ1 = σy ϕ Φ λ ln y µ ) σ σ = 1 y φ SN( ln y; Λ1 ), y R +, (1) where Λ 1 = ( µ, σ, λ ), σ > 0, φ SN ( ) denotes the pdf of the skew-normal (SN) distribution, and ϕ ( ) and Φ( ) denote the pdf and cdf of a standard univariate normal variable, respectively. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 6 / 34

14 Log-Skew-Normal distribution LOG-SKEW-NORMAL DISTRIBUTION (LSN) The LSN distribution was first introduced by [Azzalini et al.(2003)azzalini, Dal Cappello, and Kotz] and later by [Lin and Stoyanov(2009)]. A random variable Y has a LSN distribution with asymmetry parameter λ R, denoted as Y LSN (Λ 1 ), if its pdf is of the form ( ) ( ( ) 2 ln y µ f Y y; Λ1 = σy ϕ Φ λ ln y µ ) σ σ = 1 y φ SN( ln y; Λ1 ), y R +, (1) where Λ 1 = ( µ, σ, λ ), σ > 0, φ SN ( ) denotes the pdf of the skew-normal (SN) distribution, and ϕ ( ) and Φ( ) denote the pdf and cdf of a standard univariate normal variable, respectively. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 6 / 34

15 Log-Skew-Normal distribution LOG-SKEW-NORMAL DISTRIBUTION If µ = 0 and σ = 1 then Y is said to have a (standard) LSN distribution, i.e., Y LSN (λ). The parameter λ controls the skewness, which is positive when λ > 0 and negative when λ < 0. The cdf of (1) is given by ( ) ( ) ln y µ ln y µ F Y (y; Λ 1 ) = Φ 2T ; 0, λ, (2) σ σ where the function T (z; α, λ) with α 0 is given as [ arctan ( λ ) z ] λ x T (z; α, λ) =sign (λ) ϕ(x, α, 1)ϕ (y)dydx, 2π and sign( ) is the signum function. α 0 Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 7 / 34

16 Log-Skew-Normal distribution Probability Density Functions SN(µ,σ,λ), µ= 0, σ= 1 λ= 0 λ= 2 λ= 5 SN(µ,σ,λ) x FIGURE: Comparison of the SN pdf with λ > 0. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 8 / 34

17 Log-Skew-Normal distribution Probability Density Functions SN(µ,σ,λ), µ= 0, σ= 1 λ= 0 λ= 2 λ= 5 SN(µ,σ,λ) x FIGURE: Comparison of the SN pdf with λ < 0. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 9 / 34

18 The Skew Normal Mixtures SKEW NORMAL MIXTURES we consider the finite mixture model suggested by [Lin et al.(2007)lin, Lee, and Yen]: We assume that if Y is SNMIX distributed then the transformation X = exp {Y} is distributed as a LSNMIX. Let us assume that f Y (y) is the weighted sum of m-component SNMIX densities, that is, f Y (y; Λ) = m ω j φ SN (y; µ j, σ j, λ j ). (3) j=1 We use the notation Y SNMIX (Λ), where Λ = (ξ 1,..., ξ m ), and ξ j = (ω j, µ j, σ j, λ j ) is the parameter vector that defines the j-th component and probability weights, ω j, satisfy the conditions m ω j =1, 0 < ω j < 1, for each j. (4) j=1 Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 10 / 34

19 The Skew Normal Mixtures The choice of a finite mixture is attractive from the application view point because of its flexibility and allows us to consider different kinds of shaped distributions. For instance, the two component SNMIX model has the advantage of numerical tractability, because it has only seven parameters. Assuming ξ j = (ω j, µ j, σ j, λ j ), with µ 1 = 1, µ 2 = 1, σ 1 = σ 2 = 1, λ 1 =.5 and λ 2 = 2. Figure 3 shows the pdf shape of the SNMIX for three values of ω. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 11 / 34

20 The Skew Normal Mixtures SNMix(µ 1,σ 1,λ 1,µ 2,σ 2,λ 2,ω) Probability Density Functions SNMix(µ 1,σ 1,λ 1,µ 2,σ 2,λ 2,ω) µ 1 = 1, σ 1 = 1, λ 1 = 0.5, µ 2 = 1, σ 2 = 1, λ 2 = 2 ω= 0.06 ω= 0.46 ω= 0.91 Normal x FIGURE: Comparison of the pdf of the SNMIX with varying ω. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 12 / 34

21 OPTION PRICING Option pricing formula Let S t be the price of the underlying stock at time t and C(t, τ; K) the price of the call option with strike price K and maturity date of T = t + τ. It is assumed that r is the annual risk-free rate. According to [Harrison and Pliska(1981)], in the absence of arbitration, the price of European call options can be written as follows: C(t, τ; K) = E [ e rτ max{s T K, 0} ] = e rτ E [max{s T K, 0}] C t (K) = e rτ (S T K) f (S T ) ds T. K Here E[.] is the expected value conditional (risk neutral) on any information that is available at time t, f (S T ) is the risk-neutral pdf (risk-neutral distribution, RND) for the underlying stock. In an arbitrage-free economy, the stock price discounted by the risk free rate becomes martingale, that is: E[e rτ S T ] =S t, τ > 0 (5) Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 13 / 34

22 Option pricing formula OPTION PRICE USING THE LSNMIX We define the distribution of the logarithm of the stock price S T using its location and scale parameters A and B, respectively, and also Λ, the parameter of the SNMIX. These parameters satisfy the following relationship: ln [S T ] =A + BY, with Y SNMIX (Λ). (6) Then, the pdf of S T is a LSNMIX. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 14 / 34

23 Option pricing formula OPTION PRICE USING THE LSNMIX PROPOSITION The price of a European call option is given by e rτ C t ( K; Λ ) = m j=1 2ω j E [S T ] Υ j (Λ, B) δ 1j ϕ (z) Φ [λ j z j ]dz m ω j K [1 F Y ( δ 2j ; λ j )], (7) j=1 where z j = z + Bσ j and F Y ( ) is given in (2), δ 1j = δ 2j + Bσ j, δ 2j = κ µ j, κ = ln K A, (8) σ j B δ 2j = 1 { ( ) } E [ST ] ln ln [Υ j (Λ, B)] 1 Bσ j K 2 Bσ j, Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 15 / 34

24 Option pricing formula OPTION PRICE USING THE LSNMIX PROPOSITION The LSNMIX European put option price is given by e rτ ( ) m P t K; Λ = ω j KF Y ( δ 2j ; λ j ) j=1 m j=1 2E [S T ] ω j Υ j (Λ, B) δ1j where δ 1j and δ 2j are given in (8) and Υ j (Λ, B) is given in (9). ϕ (z) Φ [λ j z j ]dz, Using the option valuation formula, we can obtain the put-call parity relationship by subtracting expression (7) from (10) to obtain the following equality (10) e rτ ( C t ( K; Λ ) Pt ( K; Λ )) =E [XT ] K. (11) Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 16 / 34

25 BLACK-SCHOLES Some Special Cases Assuming ξ j = ( 1 m, µ, σ, 0) for all j in (3), substituting in expressions (7) and (10) yields, respectively, e rτ C t ( K; Λ ) =E [ST ] Φ [ 1 Bσ ln [ E [ST ] K ] + 12 ] [ 1 Bσ KΦ Bσ ln [ E [ST ] K ] 1 2 =E [S T ] Φ (d 1 ) KΦ (d 2 ), (12) where d 2 = 1 Bσ ln ( E [ST ] K ) 1 Bσ, (13) 2 and d 1 = d 2 + Bσ. Note that when B = τ, these expressions coincide with the option pricing formula of [Black and Scholes(1973)]. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 17 / 34

26 [BAHRA(1997)] Some Special Cases ( ) ( ) When ξ j = ω j, µ j, σ j, 0 for all j, where µ j = B µ j 1 2 σ2 j in (3), substituting in expressions (7) and (10) yields, respectively, e rτ C t ( K; Λ ) = m j=1 where δ 1j and δ 2j are given in (8) and Υ j (Λ, B) = ω j E [S T ] Υ j (Λ, B) Φ (δ 1j) K m ω j Φ (δ 2j ), (14) j=1 m ω l exp { B 2 (µ l µ j ) }. (15) l=1 Note that when B = τ, these expressions coincide with the option pricing formula given in [Bahra(1997)]. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 18 / 34

27 Some Special Cases [CORNS AND SATCHELL(2007)] When ξ j = ( 1 m, µ, σ, λ) for all j in (3), substituting in expressions (7) and (10) yields, respectively, e rτ C t ( K; Λ ) = E [S T ] Φ (ρσb) δ 1 ϕ (z) Φ [λ (z + Bσ)]dz K [1 F Y ( δ 2 ; λ)], (16) where δ 2 = 1 { } E Bσ ln [ST ] 1 Bσ, (17) 2K Φ (ρσb) 2 and δ 1 = δ 2 + Bσ. Note that when B = τ, these expressions coincide with the option pricing formula in [Corns and Satchell(2007)]. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 19 / 34

28 Some Special Cases GREEK MEASURE-DELTA The delta of an option is defined as the rate of change of the option price with respect to the price of the underlying stock. Call = m j=1 Put = Call 1. 2ω j ϕ (z) Φ [λ j z j ]dz, Υ j (Λ, B) δ 1j Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 20 / 34

29 Some Special Cases GREEK MEASURE-GAMMA The gamma of a portfolio of options on an underlying stock is the rate of change of the portfolio s delta with respect to the price of the underlying stock, i.e., the second partial derivative of the portfolio with respect to stock price Γ Call = Γ Put = m j=1 2ω j Φ [ λ j δ 2j ] Υ j (Λ, B) ϕ (δ 1j ) Bσ j S. (18) Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 21 / 34

30 GREEK MEASURES Some Special Cases OPTION PRICING BASED ON A LOG-SKEW-NORMAL MIXTURE by J. A. JIMÉNEZ, V. ARUNACHALAM, and G. M. SERNA, Int. J. Theor. Appl. Finan. Vol.18 (5), December (2015) DOI:S X Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 22 / 34

31 Numerical Results We now present an example from real market data to model the distribution of the stock prices and compare the numerical values of a European option under the assumption that the stock movement follow a LSNMIX distribution. The market data of the S&P500 index for the daily prices were considered from January 4, 2010 to October 13, Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 23 / 34

32 Numerical Results We now present an example from real market data to model the distribution of the stock prices and compare the numerical values of a European option under the assumption that the stock movement follow a LSNMIX distribution. The market data of the S&P500 index for the daily prices were considered from January 4, 2010 to October 13, Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 23 / 34

33 Numerical Results Statistics Values Mean Stan. Dev Minimum Maximum Skewness Kurtosis JB test TABLE: Summary of the Descriptive Statistics The empirical distribution and its descriptive statistics presented in table 1 are analysed using the test proposed by Jarque & Bera (1987), in particular, the skewness and kurtosis, which confirms that the null hypothesis of a normal distribution must be rejected. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 24 / 34

34 Numerical Results Estimate MME MLE µ µ σ σ λ λ ω TABLE: Estimates for adjusting the SNMIX (Λ) Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 25 / 34

35 Numerical Results Density functions adjusted for daily returns on S&P 500 Index Histogram Normal Empirical NMix SNMix Frequency Daily returns FIGURE: Returns vs. normal distribution and SNMIX. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 26 / 34

36 Numerical Results Strike Maturity (K) τ = 0.25 τ = 0.5 τ = 0.75 τ = , , , , , , , , , , , , , , , , , , , ,5672 TABLE: Comparison prices of call option Black Scholes. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 27 / 34

37 Numerical Results Strike Maturity (K) τ = 0.25 τ = 0.5 τ = 0.75 τ = , , , , , , , , , , , , , , , , , , , ,1000 TABLE: Comparison prices of call option Corrado & Su. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 28 / 34

38 Numerical Results Strike Maturities (K) τ = 0.25 τ = 0.5 τ = 0.75 τ = , , , , , , , , , , , , , , , , ,6477 8, , ,6045 TABLE: Comparison prices of call option SNMIX. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 29 / 34

39 Numerical Results Call option BS CS SNMIX Moneyness (%) FIGURE: Call option for different strikes. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 30 / 34

40 Numerical Results Call option BS CS SNMIX Maturities FIGURE: Call option for different maturities. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 31 / 34

41 Numerical Results BS SNMIX 0.2 Implied volatility Strike FIGURE: Implied volatility. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 32 / 34

42 CONCLUSIONS Conclusions We have proposed an alternative approach for calculating the price of the call and put options when the stock return distribution follows the SNMIX distribution. We have obtained an explicit expression for the price of the European options, and from the proposed model, we deduce some well-known models, such as [Black and Scholes(1973)], [Bahra(1997)] and [Corns and Satchell(2007)]. An example from real market data is also presented to implement the proposed model, and comparisons are made with well-known models. The derivation of an expression for the implied volatilities at and around-the money and studies of its asymptotic behaviour are in progress. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 33 / 34

43 CONCLUSIONS Conclusions We have proposed an alternative approach for calculating the price of the call and put options when the stock return distribution follows the SNMIX distribution. We have obtained an explicit expression for the price of the European options, and from the proposed model, we deduce some well-known models, such as [Black and Scholes(1973)], [Bahra(1997)] and [Corns and Satchell(2007)]. An example from real market data is also presented to implement the proposed model, and comparisons are made with well-known models. The derivation of an expression for the implied volatilities at and around-the money and studies of its asymptotic behaviour are in progress. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 33 / 34

44 CONCLUSIONS Conclusions We have proposed an alternative approach for calculating the price of the call and put options when the stock return distribution follows the SNMIX distribution. We have obtained an explicit expression for the price of the European options, and from the proposed model, we deduce some well-known models, such as [Black and Scholes(1973)], [Bahra(1997)] and [Corns and Satchell(2007)]. An example from real market data is also presented to implement the proposed model, and comparisons are made with well-known models. The derivation of an expression for the implied volatilities at and around-the money and studies of its asymptotic behaviour are in progress. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 33 / 34

45 CONCLUSIONS Conclusions We have proposed an alternative approach for calculating the price of the call and put options when the stock return distribution follows the SNMIX distribution. We have obtained an explicit expression for the price of the European options, and from the proposed model, we deduce some well-known models, such as [Black and Scholes(1973)], [Bahra(1997)] and [Corns and Satchell(2007)]. An example from real market data is also presented to implement the proposed model, and comparisons are made with well-known models. The derivation of an expression for the implied volatilities at and around-the money and studies of its asymptotic behaviour are in progress. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 33 / 34

46 Conclusions THANK YOU VERY MUCH Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 34 / 34

47 Conclusions C. Harvey and A Siddique. Autoregressive conditional skewness. Journal of Financial and Quantitative Analysis, 34(4): , Robert J Ritchey. Call option valuation for discrete normal mixtures. Journal of Financial Research, 13(4): , William R. Melick and Charles P. Thomas. Recovering an asset s implied pdf from option prices: An application to crude oil during the gulf crisis. The Journal of Financial and Quantitative Analysis, 32(1):91 115, Bhupinder Bahra. Implied risk-neutral probability density functions from option prices: A central bank perspective. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 34 / 34

48 Conclusions In: Knight, John, and Satchell, Stephen (Eds.), Forecasting Volatility in the Financial Markets, third edition, pages , Elsevier Finance. Dimitris N. Politis. A heavy tailed distribution for arch residuals with application to volatility prediction. Annals of Economics and Finance, 5(2): , R. Jarrow and A. Rudd. Approximate option valuation for arbitrary stochastic processes. Journal of Financial Economics, 10(3): , Charles J. Corrado and Tie Su. Skewness and kurtosis in s&p 500 index returns implied by option prices. Journal of Financial Research, 19(2): , Charles J. Corrado and Tie Su. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 34 / 34

49 Conclusions Implied volatility skews and stock index skewness and kurtosis implied by s&p 500 index option prices. Journal of Derivatives, 4(4):8 19, Mark Rubinstein. Edgeworth binomial trees. Journal of Derivatives, 5(3):20 27, D. Brigo and Fabio Mercurio. Lognormal-mixture dynamics and calibration to market volatility smiles. International Journal of Theoretical and Applied Finance, 5(4): , T.R.A Corns and S.E. Satchell. Skew brownian motion and pricing european options. The European Journal of Finance, 13(6): , G. D Lin and J. Stoyanov. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 34 / 34

50 Conclusions The logarithmic skew-normal distributions are moment-indeterminate. Journal of Applied Probability Trust, 46(3): , F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3): , A. Azzalini, T. Dal Cappello, and S Kotz. Log-skew normal and log-skew-t distributions as models for family income data. Journal of Income Distribution, 11(3):12 20, High Seng Chai and Kent R. Bailey. Use of log-skew-normal distribution in analysis of continuous data with a discrete component at zero. Statistics in Medicine, 27: , Tsung I Lin, Jack C Lee, and Shu Y. Yen. Finite mixture modelling using the skew normal distribution. Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 34 / 34

51 Conclusions Statistica Sinica, 17: , J. M. Harrison and S. R. Pliska. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes Applications, 11(3): , Ali Hirsa and Salih N Neftci. An introduction to the mathematics of financial derivatives. Academic Press, Hans R. Stoll. The relationship between put and call option prices. The Journal of Finance, 24(5): , Dec Jiménez & Arunachalam (UNAL) Log-Skew-Normal mixture ICASQF 2016 Cartagena 34 / 34