Derivatives_NYU_Lecture_2

Derivatives_NYU_Lecture_2 - Dynamic Assets & Option Pricing...

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24/04/2009 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected] 1 Dynamic Assets & Option Pricing 2 Sebastien Galy
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Overview 1. In discrete time : Binomial Pricing (quick review) 2. In continuous time – Taylor to Ito (with applications) 3. In continuous time - Black, Scholes and Merton pricing (spot is the only risk) 4. In continuous time - Pricing with Characteristic functions 5. In discrete time – Pricing with Utility functions 6. In discrete time – Pricing with Certainty Equivalence Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]
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1. In discrete time: Binomial Tree Quick Reminder
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Reminder Binomial Trees A binomial tree is a discretization akin to the ticks in US stocks (minimum tick size for a trade). There is no such ticks in OTC markets. Technique: A bond can be replicated by Spot and Option position to form a risk free assets => Same payoff in up and down state => 1. find delta (relative weight) The replicating portfolio earns the risk free rate (it is after all a replicated bond) => PV of portfolio =…. .(put the delta in) . .et voila As we know that we are in complete markets/no arbitrage ( unique price Ù system solvable Ù 2 states and 2 instruments), we know that we have no risk or equivalently all risks can be removed (resold). Hence, we discount at risk free rate. Equating this equation with the previous one gives the risk neutral probabilities (or state prices – just need to discount probability at the risk free rate). Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]
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Generalization (continued) Consider the portfolio that is LONG Δ shares and SHORT 1 derivative Figure 9.2 The portfolio is riskless when S 0 u Δ –ƒ u = S 0 d Δ d or d S u S d u 0 0 ƒ ƒ = Δ S 0 u Δ u S 0 d Δ d S 0
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Generalization (continued) Value of the portfolio at time T is S 0 u Δ –ƒ u Value of the portfolio today is ( S 0 u Δ u )e rT Another expression for the portfolio value today is S 0 Δ Hence ƒ = S 0 Δ –( S 0 u Δ u )e rT
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Generalization (continued) Substituting for Δ we obtain ƒ=[ p ƒ u + (1– p d ]e rT where the so- called state-price probability is: d u d p rT = e
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Risk-Neutral Valuation The variables p & (1– p ) can be interpreted as the risk-neutral probabilities of up & down movements The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate Figure 9.2 S 0 u ƒ u S 0 d ƒ d S 0 ƒ p ( 1 ) T d u ]e ?p ) ? ( [ p ? f 1 + =
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Delta Delta ( Δ ) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock The value of Δ varies from node to node. Dynamic hedging needed!
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Convergence to BSM Pick where Δ t is the length of a time step (measured in years), and where volatility is measured in years.
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This note was uploaded on 04/05/2010 for the course FINANCE AN FRE6311 taught by Professor Galy,sebastien during the Spring '09 term at NYU Poly.

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Derivatives_NYU_Lecture_2 - Dynamic Assets & Option Pricing...

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