Lecture 10a[1]

# Lecture 10a[1] - ACTL3004: Week 10a Financial Economics for...

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ACTL3004: Week 10a Financial Economics for Insurance and Superannuation: Week 10a Black Scholes Option Pricing Model 1 / 16

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ACTL3004: Week 10a Black-Scholes Model Derivation Black-Scholes Model Derivation The price of a derivative in the Black Scholes model is e - rT E Q h ( S ( T ) - K ) + i how do we calculate this? We ﬁrst need the dynamics of S ( T ) under the Q probabilities. 2 / 16
ACTL3004: Week 10a Black-Scholes Model Derivation Our real world model is dS = μ Sdt + σ SdW where W is a P Brownian motion. To get the Q dynamics remember the rule from CMG that: P Q = ﬁnd W Q with W Q a Q Browmian Motion. In particular when we deﬁned Q we found that dW Q = dW + μ - r σ dt and so plugging this into the dynamics of S: dS = μ Sdt + σ S ± dW Q - μ - r σ dt ² = rSdt + σ SdW Q and S still follows GBM under Q . 3 / 16

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ACTL3004: Week 10a Black-Scholes Model Derivation From dS = rSdt + σ SdW Q we have S ( t ) = S ( 0 ) e ( r - 1 2 σ 2 ) t + σ W Q ( t ) In particular S ( T ) is lognormal with parameters ( ln S ( 0 ) + ( r - 1 2 σ 2 ) T , σ 2 T ) . Represent e - rT E Q h ( S ( T ) - K ) + i = e - rT E Q ± S ( T ) 1 S ( T ) > K ² - Ke - rT E Q ± 1 S ( T ) > K ² and we can consider the easier part Ke - rT E Q ± 1 S ( T ) > K ² =? 4 / 16
ACTL3004: Week 10a Black-Scholes Model Derivation To calculate e - rT E Q ± S ( T ) 1 S ( T ) > K ² we can complete the square. Letting Z = ln ( S ( T ) / S ( 0 )) ˜ N ³ rT - 1 2 σ 2 T , σ 2 T ´ we have 5 / 16

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ACTL3004: Week 10a Black-Scholes Model Derivation E Q ± S ( T ) 1 S ( T ) > K ² = Z z = ln K S ( 0 ) ( S ( 0 ) exp { z } ) · 1 σ 2 T 2 π exp - 1 2 z - ( rT - 1 2 σ 2 T ) σ T ! 2
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## Lecture 10a[1] - ACTL3004: Week 10a Financial Economics for...

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