Derivatives NYU_Lecture 3

Derivatives NYU_Lecture 3 - Dynamic Assets & Option...

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1 14/04/2009 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – sgaly@poly.edu 1 Dynamic Assets & Option Pricing 3 Sebastien Galy
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2 Overview 3. In continuous time - Black, Scholes and Merton pricing (spot is the only risk) – Put-call Parity reminder – Greeks – Exercise – Calculate N(d1) – Girsanov Theorem: Diffusion process transformed to RN 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 – sgaly@poly.edu
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3 14/04/2009 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
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4 3. In continuous Time: The Black, Scholes and Merton Model (BSM) 14/04/2009 4 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – sgaly@poly.edu
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5 Put-Call Parity Reminder => Either European call or put needs to be priced
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6 Terminal payoffs: Put-Call parity S ) ( K S S 1 K ) ( K S
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7 Terminal payoffs: Put-Call parity K S S 1 K Leveraged position (K is the principal payment) unleveraged above S
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8 Put Call parity •C –P = S -K / (1+ r ) T • C-P=? • Maturity payoff: S T <K S T >K -(K-S T )0 - P 0 S T -K C S T -K S T –K S - K/(1+r) T
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9 Greeks
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10 14/04/2009 Charles S. Tapiero, NYU Poly Tech Institute 10 The Well Known BS Equation 12 (, ) ( ) ( ) ( ) rt WF S t S t d K e d == Φ Φ 2 1/2 /2 2 1 21 () ( 2) , log( ( )/ ) ( )( / 2) , , y u ye d u St K T t r d Tt dd T t π σ −− −∞ Φ= ⎡⎤ +− + = ⎢⎥ ⎣⎦ =−
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11 X S S ) , ( t S C Delta Long spot No cash All Cash no spot
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12 X S S ) , ( t S C Delta S C δ = Δ S C = Δ
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13 20 40 60 80 100 120 140 160 180 200 10 20 30 40 50 60 70 80 90 100 Call and lower bound 0 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1 Delta Source: Greeks_graph.m
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14 S δ γ Δ = X S S ) , ( t S C Gamma
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15 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 0.005 0.01 0.015 0.02 Gamma Source: Greeks_graph.m
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16 δσ δ ϑ C = X S S ) , ( t S C Vega
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17 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
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18 14/04/2009 18 BSM Greeks as a function of S… 0 20 40 60 80 100 120 140 160 180 200 0 0.5 1 Delta 0 20 40 60 80 100 120 140 160 180 200 0 20 40 Lambda 0 20 40 60 80 100 120 140 160 180 200 0 0.02 0.04 Gamma Source: Greeks_graph.m
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19 Exercise Client question
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20 Problem: What are the odds that EURUSD will increase to 1.40 in 3 months? 14/04/2009 Charles S. Tapiero, NYU Poly Tech Institute 20
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21 Problem: What are the odds that EURUSD will increase to 1.40 in ?
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Derivatives NYU_Lecture 3 - Dynamic Assets &amp; Option...

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