Derivatives_NYU_Lecture_4

# Derivatives_NYU_Lecture_4 - Dynamic Assets Option Pricing 4...

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1 22/04/2009 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected] 1 Dynamic Assets & Option Pricing 4 Sebastien Galy Overview 4. In continuous time - Pricing with Characteristic functions - Intro 5. In discrete time – Pricing with Utility functions - Intro 6. In discrete time – Pricing with Certainty Equivalence - Intro 7. Multiple sources of risk – Interest Rate Modeling – Intro 8. Real options - Introduction Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]

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2 22/04/2009 S. Galy 3 Continuous PDE Discrete CRR Binomial Hull and White Discretise/Grid Utility Theory MRS discounting Feynman Kac BSM Analytical Convergence Monte Carlo Sim Replication Complete Markets =>No Arbitrage=> Unique state price Incomplete Markets Risk Neutral Transformation Girsanov Theorem Change distribution to move to risk free drift 3. In continuous Time: Black Scholes and Merton Exercise - Vega 22/04/2009 4 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]
3 BSM 22/04/2009 Charles S. Tapiero, NYU Poly Tech Institute 5 12 1 1 (,) ( ) () ln( / ) 1 ; 2 f f R R f CS SNd Ke Nd CN d N d S K e SS S R SK dd d τ σ τσ στ =− ∂∂ =+ + = 11 f Rt CSt e σσ Φ Φ ⎛⎞ ⎜⎟ ⎝⎠ 2 2 1/2 /2 2 log( / ) ( )( / 2) (2 ) , = - f d T tR y de d Tt π −− ⎡⎤ +− + ∂Φ ⎢⎥ ⎣⎦ 22 2 2 2 2 2 1 2 f f d Sd e d d Ke e Ke d e υτ σπ + . Calculating the Vega The Vega can be calculated by noting that: Since We obtain:

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4 δσ δ ϑ C = X S S ) , ( t S C Vega Source: Greeks_graph.m 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 Call and lower bound 0 20 40 60 80 100 120 140 160 180 200 0 10 20 30 40 Vega Same shape as Gamma
5 4. In continuous Time: Pricing with Characteristic Functions Introduction 22/04/2009 9 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]

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8 5. In discrete Time: Pricing with Utility Functions Exercise 22/04/2009 15 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected] • Assume that you have a representative agent who can invest in a risk free bond, consume and receives a wage Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]
9 Solution 1 ) 1 ( ) 1 ( ) , ( 1 , 1 , 1 , , , 1 , 1 , 0 , , , = + + = + = + + + + = + + l n r B B n W n r B n C p l C U E Max t w t t t t t t B t t w t t t t B t t i i t i t t l n l C t B t t β Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected] The firm’s problem π tt t t n t t t t pf z k n wn rk = (, , ) r p PmK t t t = w P PmN t t t = Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]

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10 The consumer’s problem EU c l t t tt 0 0 β = (,) ck p p B p p B w p n r p kk r p B r p B t B t t t B t tk t t t t t t B t t t B t t k k ++ + = + + + + +++ 11 1 1 1 () Under the constraint that Maximize Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected] The consumer’s first order conditions L n t : w UU t n t l t =− = λλ L K t + 1 : λ βλ δ t t Er = + 1 (( ) ) L C t : t c t U p = Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]
11 The consumer’s first order conditions w p

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Derivatives_NYU_Lecture_4 - Dynamic Assets Option Pricing 4...

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