Analytical Option Pricing under an Asymmetrically Displaced Double Gamma Jump-Diffusion Model

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Analytical Option Pricing under an Asymmetrically Displaced Double Gamma Jump-Diffusion Model Advances in Computational Economics and Finance Univerity of Zürich, Switzerland Matthias Thul 1 Ally Quan Zhang 2 1 University of New South Wales School of Banking & Finance m.thul@unsw.edu.au 2 Swiss Finance Institute and Department of Banking & Finance University of Zürich quan.zhang@bf.uzh.ch March 5, 2014

The Paper in one Slide main contributions: novel jump-diffusion process for asset prices most general dynamics that admit closed-form solutions for European plain vanilla option prices newly introduced model parameters are highly statistically significant across various asset classes Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 2 / 21

The Paper in one Slide main contributions: novel jump-diffusion process for asset prices most general dynamics that admit closed-form solutions for European plain vanilla option prices newly introduced model parameters are highly statistically significant across various asset classes relevance: closed-form solutions are important for their superior computational speed and numerical robustness Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 2 / 21

Model Classification for European Plain Vanilla Options Analytical option price in terms of elementary and special functions Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 3 / 21

Model Classification for European Plain Vanilla Options Analytical option price in terms of elementary and special functions Quasi-Analytical option price in terms of integrals (e.g. Fourier inversion) Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 3 / 21

Model Classification for European Plain Vanilla Options Analytical option price in terms of elementary and special functions Quasi-Analytical option price in terms of integrals (e.g. Fourier inversion) Numerical trees, PDE methods or Monte Carlo simulation Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 3 / 21

Model Classification for European Plain Vanilla Options increased: ease of implementation, speed, numerical stability Analytical option price in terms of elementary and special functions Quasi-Analytical option price in terms of integrals (e.g. Fourier inversion) Numerical trees, PDE methods or Monte Carlo simulation increased: generality of supported underlying dynamics Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 3 / 21

2-Asset Market I frictionless market continuous trading in [0, T ] Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 4 / 21

2-Asset Market I frictionless market continuous trading in [0, T ] B = {B t : [0, T ]} money market account with dynamics db t = rb t dt, where B 0 = 1 and r R Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 4 / 21

2-Asset Market II S = {S t : t [0, T ]} limited liability spot asset with S 0 > 0 logarithmic yield X t = ln (S t /S 0 ) follows X t = where µ R and σ R + ( µ 1 ) 2 σ2 λ (φ Y ( i) 1) t + σw t + N t i=1 Y i, Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 5 / 21

2-Asset Market II S = {S t : t [0, T ]} limited liability spot asset with S 0 > 0 logarithmic yield X t = ln (S t /S 0 ) follows X t = where µ R and σ R + market is incomplete ( µ 1 ) 2 σ2 λ (φ Y ( i) 1) t + σw t + N t i=1 Y i, Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 5 / 21

Jump Size Distribution I Kou (2002): double exponential distribution 40 sample jump size density 35 30 25 density 20 15 10 5 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 jump size Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 6 / 21

Jump Size Distribution II extension I: asymmetrically displaced double exponential distribution 40 sample jump size density 35 30 25 density 20 15 10 5 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 jump size Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 7 / 21

Jump Size Distribution III extension II: asymmetrically displaced double gamma distribution 40 sample jump size density 35 30 25 density 20 15 10 5 0 0.15 0.10 0.05 0.00 0.05 0.10 0.15 jump size Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 8 / 21

Jump Size Distribution IV Y i follows an asymmterically displaced double gamma (AD-DG) distribution { ξ + with probability p [0, 1] Y i ξ with probability 1 p ξ + κ + Γ (δ +, η + ), ξ + κ Γ (δ, η ) gamma random variables Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 9 / 21

Jump Size Distribution IV Y i follows an asymmterically displaced double gamma (AD-DG) distribution { ξ + with probability p [0, 1] Y i ξ with probability 1 p ξ + κ + Γ (δ +, η + ), ξ + κ Γ (δ, η ) gamma random variables parameters: (i) p [0, 1] mixing weight, (ii) δ ± N shape, (iii) η + > 1, η > 0 tail decay, (iv) κ 0 κ + displacement Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 9 / 21

Sample Trajectories parameters: p = 40%, δ + = 1, δ = 1, η + = 80, η = 60, κ + = 1.50%, κ = 2.00% 1.4 sample spot price trajectories 1.3 1.2 spot price 1.1 1.0 0.9 0.8 0.7 0.00 0.20 0.40 0.60 0.80 1.00 time Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 10 / 21

Sample Trajectories parameters: p = 40%, δ + = 1, δ = 1, η + = 80, η = 60, κ + = 1.50%, κ = 2.00% 1.4 sample spot price trajectories 1.3 1.2 spot price 1.1 1.0 0.9 0.8 0.7 0.00 0.20 0.40 0.60 0.80 1.00 time Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 10 / 21

General Equilibrium Framework I continuous time pure-exchange economy based on Naik and Lee (1990) infinite horizon (T ) Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 11 / 21

General Equilibrium Framework I continuous time pure-exchange economy based on Naik and Lee (1990) infinite horizon (T ) single perishable consumption good (numéraire) single fully equity financed firm which (i) has one unit of share outstanding (ii) engages in costless production (iii) pays a continuous dividend at the stochastic rate δ Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 11 / 21

General Equilibrium Framework II exogenously specify ln (δ t /δ 0 ) = X t (AD-DG jump-diffusion) Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 12 / 21

General Equilibrium Framework II exogenously specify ln (δ t /δ 0 ) = X t (AD-DG jump-diffusion) three types of competitively traded and perfectly divisible assets (i) the stock S (ii) a continuum of zero-coupon bonds B(, T ) with unit notional (iii) a continuum of other contingent claims O(, T ) Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 12 / 21

General Equilibrium Framework III representative agent with isoelastic utility of consumption u (C t ) = C 1 α t 1 1 α, α > 0 coefficient of relative risk aversion Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 13 / 21

General Equilibrium Framework III representative agent with isoelastic utility of consumption u (C t ) = C 1 α t 1 1 α, α > 0 coefficient of relative risk aversion portfolio choice problem: maximize expected lifetime utility [ ] E P e ρv u (C v ) dv F t, subject to a dynamics budget constraint ρ R subjective rate of time preference t Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 13 / 21

Equilibrium Prices Proposition In a competitive equilibrium, the time t 0 price of an asset with instantaneous payout process ζ is given by [ 1 ] π t (ζ) = e ρt u (δ t ) E P e ρv u (δ v ) ζ v dv F t. t Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 14 / 21

Equilibrium Prices Proposition In a competitive equilibrium, the time t 0 price of an asset with instantaneous payout process ζ is given by [ 1 ] π t (ζ) = e ρt u (δ t ) E P e ρv u (δ v ) ζ v dv F t. t Corollary The time t 0 equilibrium prices stock price is given by S t = δ t ρ ψ X1 (i(α 1)). Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 14 / 21

Equilibrium Pricing Kernel Corollary The induced risk-neutral probability measure P corresponds to an Esscher transform of X with transform parameter α, i.e. dp dp = exp { αx T T ψ X1 (iα)} P-a.s.. Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 15 / 21

Equilibrium Pricing Kernel Corollary The induced risk-neutral probability measure P corresponds to an Esscher transform of X with transform parameter α, i.e. dp dp = exp { αx T T ψ X1 (iα)} P-a.s.. Proposition Under P, X is still an AD-DG jump-diffusion process with parameters γ = γ ασ 2, λ = λφ Y (iα), p = pφ Y +(iα) φ Y (iα), η ± = η ± ± α. σ, κ ± and δ ± are invariant under the measure change. Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 15 / 21

Option Pricing closed-form expressions for the conditional tail probability in terms of Hh n -functions P {X t x N t = n} unconditional tail probabilities can be expressed as sums over rapidly decaying terms with analytical error bound Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 16 / 21

Option Pricing closed-form expressions for the conditional tail probability in terms of Hh n -functions P {X t x N t = n} unconditional tail probabilities can be expressed as sums over rapidly decaying terms with analytical error bound closed-form solutions for the prices of European plain vanilla options on forwards and associated Greeks solution appears formidable but is straight-forward to implement and numerically robust Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 16 / 21

Methodology & Data efficient maximum likelihood estimation using Fang and Oosterlee (2008) Fourier cosine expansion (COS) method heuristic optimization routine to account for non-convexity of the objective function Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 17 / 21

Methodology & Data efficient maximum likelihood estimation using Fang and Oosterlee (2008) Fourier cosine expansion (COS) method heuristic optimization routine to account for non-convexity of the objective function 30 years of daily logarithmic returns: January 1, 1982 to December 31, 2011 17 assets across (i) equity indices, (ii) commodity indices, (iii) foreign exchange rates and (iv) precious metals Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 17 / 21

Estimation Results: Equity Indices Asset DAX 30 Dow Jones Hang Seng MSCI World NASDAQ Nikkei 225 S&P 500 TOPIX Industrial Composite σ 13.43% 9.46% 14.95% 8.44% 8.58% 9.46% 8.84% 9.96% (0.40%) (0.00%) (0.01%) (0.28%) (0.29%) (0.35%) (0.01%) (0.45%) λ 107.70 142.46 137.18 98.45 175.55 235.41 187.33 180.89 (12.92) (4.13) (6.15) (12.74) (11.18) (16.31) (9.27) (19.84) p 43.17% 44.32% 52.83% 38.66% 43.74% 56.99% 48.34% 58.56% (2.16%) (0.01%) (0.06%) (3.76%) (2.29%) (2.19%) (0.07%) (2.96%) δ + 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) δ 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) η + 90.20 133.71 83.36 143.31 99.05 129.92 133.35 134.80 (6.16) (3.58) (2.69) (11.59) (5.13) (6.16) (4.77) (8.67) η 92.55 115.71 67.42 129.15 104.28 123.52 119.62 120.67 (6.00) (2.73) (1.96) (9.29) (4.54) (5.89) (4.15) (7.15) κ + +0.10%*** +0.24%*** +0.00%*** +0.28%*** +0.05%*** +0.08%*** +0.03%*** +0.08%*** (0.01%) (0.00%) (0.00%) (0.05%) (0.00%) (0.00%) (0.00%) (0.00%) κ 0.33%*** 0.01%*** 0.10%*** 0.06%*** 0.28%*** 0.43%*** 0.00%*** 0.36%*** (0.02%) (0.00%) (0.00%) (0.01%) (0.02%) (0.02%) (0.00%) (0.02%) H (1) 0 0.23%*** +0.23%*** 0.10%*** +0.22%*** 0.23%*** 0.35%*** +0.03%*** 0.29%*** (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) L 22,179.29 24,081.39 20,456.56 26,513.74 22,755.20 21,662.19 23,974.51 22,527.43 N 7,566 7,568 7,418 7,822 7,569 7,387 7,569 7,387 Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 18 / 21

Estimation Results: Equity Indices Asset DAX 30 Dow Jones Hang Seng MSCI World NASDAQ Nikkei 225 S&P 500 TOPIX Industrial Composite σ 13.43% 9.46% 14.95% 8.44% 8.58% 9.46% 8.84% 9.96% (0.40%) (0.00%) (0.01%) (0.28%) (0.29%) (0.35%) (0.01%) (0.45%) λ 107.70 142.46 137.18 98.45 175.55 235.41 187.33 180.89 (12.92) (4.13) (6.15) (12.74) (11.18) (16.31) (9.27) (19.84) p 43.17% 44.32% 52.83% 38.66% 43.74% 56.99% 48.34% 58.56% (2.16%) (0.01%) (0.06%) (3.76%) (2.29%) (2.19%) (0.07%) (2.96%) δ + 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) δ 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) η + 90.20 133.71 83.36 143.31 99.05 129.92 133.35 134.80 (6.16) (3.58) (2.69) (11.59) (5.13) (6.16) (4.77) (8.67) η 92.55 115.71 67.42 129.15 104.28 123.52 119.62 120.67 (6.00) (2.73) (1.96) (9.29) (4.54) (5.89) (4.15) (7.15) κ + +0.10%*** +0.24%*** +0.00%*** +0.28%*** +0.05%*** +0.08%*** +0.03%*** +0.08%*** (0.01%) (0.00%) (0.00%) (0.05%) (0.00%) (0.00%) (0.00%) (0.00%) κ 0.33%*** 0.01%*** 0.10%*** 0.06%*** 0.28%*** 0.43%*** 0.00%*** 0.36%*** (0.02%) (0.00%) (0.00%) (0.01%) (0.02%) (0.02%) (0.00%) (0.02%) H (1) 0 0.23%*** +0.23%*** 0.10%*** +0.22%*** 0.23%*** 0.35%*** +0.03%*** 0.29%*** (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) L 22,179.29 24,081.39 20,456.56 26,513.74 22,755.20 21,662.19 23,974.51 22,527.43 N 7,566 7,568 7,418 7,822 7,569 7,387 7,569 7,387 Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 18 / 21

Estimation Results: Equity Indices Asset DAX 30 Dow Jones Hang Seng MSCI World NASDAQ Nikkei 225 S&P 500 TOPIX Industrial Composite σ 13.43% 9.46% 14.95% 8.44% 8.58% 9.46% 8.84% 9.96% (0.40%) (0.00%) (0.01%) (0.28%) (0.29%) (0.35%) (0.01%) (0.45%) λ 107.70 142.46 137.18 98.45 175.55 235.41 187.33 180.89 (12.92) (4.13) (6.15) (12.74) (11.18) (16.31) (9.27) (19.84) p 43.17% 44.32% 52.83% 38.66% 43.74% 56.99% 48.34% 58.56% (2.16%) (0.01%) (0.06%) (3.76%) (2.29%) (2.19%) (0.07%) (2.96%) δ + 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) δ 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) η + 90.20 133.71 83.36 143.31 99.05 129.92 133.35 134.80 (6.16) (3.58) (2.69) (11.59) (5.13) (6.16) (4.77) (8.67) η 92.55 115.71 67.42 129.15 104.28 123.52 119.62 120.67 (6.00) (2.73) (1.96) (9.29) (4.54) (5.89) (4.15) (7.15) κ + +0.10%*** +0.24%*** +0.00%*** +0.28%*** +0.05%*** +0.08%*** +0.03%*** +0.08%*** (0.01%) (0.00%) (0.00%) (0.05%) (0.00%) (0.00%) (0.00%) (0.00%) κ 0.33%*** 0.01%*** 0.10%*** 0.06%*** 0.28%*** 0.43%*** 0.00%*** 0.36%*** (0.02%) (0.00%) (0.00%) (0.01%) (0.02%) (0.02%) (0.00%) (0.02%) H (1) 0 0.23%*** +0.23%*** 0.10%*** +0.22%*** 0.23%*** 0.35%*** +0.03%*** 0.29%*** (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) L 22,179.29 24,081.39 20,456.56 26,513.74 22,755.20 21,662.19 23,974.51 22,527.43 N 7,566 7,568 7,418 7,822 7,569 7,387 7,569 7,387 Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 18 / 21

Estimation Results: Equity Indices Asset DAX 30 Dow Jones Hang Seng MSCI World NASDAQ Nikkei 225 S&P 500 TOPIX Industrial Composite σ 13.43% 9.46% 14.95% 8.44% 8.58% 9.46% 8.84% 9.96% (0.40%) (0.00%) (0.01%) (0.28%) (0.29%) (0.35%) (0.01%) (0.45%) λ 107.70 142.46 137.18 98.45 175.55 235.41 187.33 180.89 (12.92) (4.13) (6.15) (12.74) (11.18) (16.31) (9.27) (19.84) p 43.17% 44.32% 52.83% 38.66% 43.74% 56.99% 48.34% 58.56% (2.16%) (0.01%) (0.06%) (3.76%) (2.29%) (2.19%) (0.07%) (2.96%) δ + 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) δ 1 1 1 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) η + 90.20 133.71 83.36 143.31 99.05 129.92 133.35 134.80 (6.16) (3.58) (2.69) (11.59) (5.13) (6.16) (4.77) (8.67) η 92.55 115.71 67.42 129.15 104.28 123.52 119.62 120.67 (6.00) (2.73) (1.96) (9.29) (4.54) (5.89) (4.15) (7.15) κ + +0.10%*** +0.24%*** +0.00%*** +0.28%*** +0.05%*** +0.08%*** +0.03%*** +0.08%*** (0.01%) (0.00%) (0.00%) (0.05%) (0.00%) (0.00%) (0.00%) (0.00%) κ 0.33%*** 0.01%*** 0.10%*** 0.06%*** 0.28%*** 0.43%*** 0.00%*** 0.36%*** (0.02%) (0.00%) (0.00%) (0.01%) (0.02%) (0.02%) (0.00%) (0.02%) H (1) 0 0.23%*** +0.23%*** 0.10%*** +0.22%*** 0.23%*** 0.35%*** +0.03%*** 0.29%*** (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) (0.00%) L 22,179.29 24,081.39 20,456.56 26,513.74 22,755.20 21,662.19 23,974.51 22,527.43 N 7,566 7,568 7,418 7,822 7,569 7,387 7,569 7,387 Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 18 / 21

Estimation Results: Summary the AD-DE model (δ ± = 1) provides the best fit for all but one asset for all assets, both displacement terms are individually and jointly significant at the 1% level for all assets, the equality of the displacement terms can be rejected at the 1% level Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 19 / 21

Estimation Results: Summary the AD-DE model (δ ± = 1) provides the best fit for all but one asset for all assets, both displacement terms are individually and jointly significant at the 1% level for all assets, the equality of the displacement terms can be rejected at the 1% level reject both the DE (κ ± = 0) and the SD-DE (κ + + κ = 0) models in favor of our AD-DE dynamics Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 19 / 21

Conclusion novel jump-diffusion process for asset prices most general dynamics that admit closed-form solutions for European plain vanilla option prices newly introduced model parameters are highly statistically significant across various asset classes Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 20 / 21

References I Fang, Fang and Cornelis W. Oosterlee (2008) A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions, SIAM Journal on Scientific Computing, Vol. 31, No. 2, pp. 826 848. Kou, Steven G. (2002) A Jump-Diffusion Model for Option Pricing, Management Science, Vol. 48, No. 8, pp. 1086 1101. Naik, Vasanttilak and Moon Lee (1990) General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns, Review of Financial Studies, Vol. 3, No. 4, pp. 493 521. Ally Quan Zhang (Universität Zürich) AD-DG Jump-Diffusion Model March 5, 2014 21 / 21

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