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GT_Atlanta_07052011

# GT_Atlanta_07052011 - Issues in option pricing Modeling...

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Issues in option pricing Modeling dependence Risk neutral pricing Results Merton Jump-Diffusion Revisited: A L´ evy Copula Approach Pierre Garreau 1 - Craig Nolder 5 th Annual Graduate Student Conference in Probability April 29 - May 01, 2011 1 Financial Mathematics, Florida State University 1 / 29 Merton Jump-Diffusion Revisited: A L´ evy Copula Approach

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Issues in option pricing Modeling dependence Risk neutral pricing Results 1 Issues in option pricing 2 Modeling dependence 3 Risk neutral pricing 4 Results 2 / 29 Merton Jump-Diffusion Revisited: A L´ evy Copula Approach
Issues in option pricing Modeling dependence Risk neutral pricing Results The Black-Scholes syndrom 20 40 60 80 100 120 140 160 180 200 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 SPX Black-scholes diffusion: dS t = S t ( rdt + σ dW t ) 3 / 29 Merton Jump-Diffusion Revisited: A L´ evy Copula Approach

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Issues in option pricing Modeling dependence Risk neutral pricing Results The Black-Scholes syndrom 20 40 60 80 100 120 140 160 180 200 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 SPX Black-scholes diffusion: dS t = S t ( rdt + σ dW t ) Pricing - Arbitrage Theory / Feynman-Kac: P ( t , x ) = E Q bracketleftBig f ( tildewide S t , x T ) |F t bracketrightBig Hedging - Martingale representation theorem: V t = c + integraldisplay t 0 φ s d tildewide S s , Q a . s . 3 / 29 Merton Jump-Diffusion Revisited: A L´ evy Copula Approach
Issues in option pricing Modeling dependence Risk neutral pricing Results The Black-Scholes syndrom 20 40 60 80 100 120 140 160 180 200 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 SPX 0 0.2 0.4 0.6 0.8 1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Black-scholes diffusion: dS t = S t ( rdt + σ dW t + dq t ) Pricing - Arbitrage Theory / Feynman-Kac: P ( t , x ) = E Q bracketleftBig f ( tildewide S t , x T ) |F t bracketrightBig Hedging - Martingale representation theorem: V t = c + integraldisplay t 0 φ s d tildewide S s , Q a . s . 3 / 29 Merton Jump-Diffusion Revisited: A L´ evy Copula Approach

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Issues in option pricing Modeling dependence Risk neutral pricing Results The Black-Scholes syndrom 900 1000 1100 1200 1300 1400 1500 1600 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 600 650 700 750 800 850 900 950 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 σ * = C - 1 ( K , S , T , r ) , C ( K , S , T , r ) = N ( d t , 1 ) e k t N ( d t , 2 ) 4 / 29 Merton Jump-Diffusion Revisited: A L´ evy Copula Approach
Issues in option pricing Modeling dependence Risk neutral pricing Results Risk neutral in two dimensions 0 0.05 0.1 0.15 0.2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 ր Q 1 d ˜ S 1 t / ˜ S 1 t = σ 1 dB 1 t ց Q 2 d ˜ S 2 t / ˜ S 2 t = σ 2 dB 2 t P 1 ( t , x ) = E Q 1 bracketleftBig f ( tildewide S 1 , x T ) |F t bracketrightBig P 2 ( t , x ) = E Q 2 bracketleftBig f ( tildewide S 2 , x T ) |F t bracketrightBig 5 / 29 Merton Jump-Diffusion Revisited: A L´ evy Copula Approach

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Issues in option pricing Modeling dependence Risk neutral pricing Results Risk neutral in two dimensions 0 0.05 0.1 0.15 0.2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 ր Q 1 d ˜ S 1 t / ˜ S 1 t = σ 1 dB 1 t + ρ σ 1 σ 2 dB 2 t ց Q 2 d ˜ S 2 t / ˜ S 2 t = σ 2 dB 2 t + ρ σ 1 σ 2 dB 1 t P 1 ( t , x ) = E Q 1 bracketleftBig f ( tildewide S 1 , x T ) |F t bracketrightBig P 2 ( t , x ) = E Q 2 bracketleftBig f ( tildewide S 2 , x T ) |F t bracketrightBig and Q i = E bracketleftBig d Q d P | F i t bracketrightBig , d Q d P | F t = e ( θ, W t )−bardbl θ bardbl 2 t .
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