Journal of Statistical Mechanics: Theory and Experiment
Volume 2008
June 2008
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Josep Perelló et al J. Stat. Mech. (2008) P06010 doi:10.1088/1742-5468/2008/06/P06010
Option pricing under stochastic volatility: the exponential Ornstein–Uhlenbeck model
Josep Perelló1, Ronnie Sircar2 and Jaume Masoliver1
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| Figure 1. Risk-neutral return density (42) for t = 20 days and with terms provided by equations (37)–(40) when z0 = 0 and assuming Λ0 = 10–3 and Λ1 = 10–3 (cf equation (19)). We depart from the parameters m = 10–2 day–1/2, α = 8 × 10–3 day–1, ρ = –0.4 and k = 0.11 day–1/2 and slightly modify them in each of these plots. |
Figure 2. Normalized  call price (52) as a function of the moneyness
S/K for
T = 20 days assuming Λ0 = 10–3 and Λ1 = 10–3 (cf equation (19)) with terms provided by equations (37)–(40) when
z0 = 0. We depart from the parameters m = 10–2 day–1/2, α = 8 × 10–3 day–1, ρ = –0.4 and k = 0.11 day–1/2 and slightly modify them in each of these plots. |
| Figure 3. Implied volatility (in yearly units) as a function of the moneyness S/K for T = 20 days assuming Λ0 = 10–3 and Λ1 = 10–3 (cf equation (19)) with terms provided by equations (37)–(40) when z0 = 0. We depart from the parameters m = 10–2 day–1/2, α = 8 × 10–3 day–1, ρ = –0.4 and k = 0.11 day–1/2 and slightly modify them in each of these plots. |
| Figure 4. Delta hedging (53) divided by strike K as a function of the moneyness S/K for T = 20 days assuming Λ0 = 10–3 and Λ1 = 10–3 (cf equation (19)) with terms provided by equations (37)–(40) when z0 = 0. We depart from the parameters m = 10–2 day–1/2, α = 8 × 10–3 day–1, ρ = –0.4 and k = 0.11 day–1/2 and slightly modify them in each of these plots. |
| Figure 5. Call price (48) and implied volatility (in yearly units) as a function of the moneyness S/K for T = 20 days. The left column studies the effects of a non-zero initial volatility assuming Λ0 = 10–3 and Λ1 = 10–3. The right column shows those caused by changing the constant involved in the risk aversion function (19) when z0 = 0. The rest of the parameters are m = 10–2 day–1/2, α = 8 × 10–3 day–1, ρ = –0.4 and k = 0.11 day–1/2. |
| Figure 6. Call price as a function of the moneyness S/K for T = 10 days. Points represent the empirical call option prices on the Dow Jones index (DJX) at a precise date (2 May 2008) and with maturity on 16 May 2008. The dashed line takes a call price (52) fit having fixed the model parameters estimated from historical data and with the initial volatility assumed to be the CBOE DJIA volatility index (VDX) and the current interest rate ratio r = 2%. The curve thus provides a fit with proper risk aversion parameters Λ0 and Λ1. |
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