risk_neutral_pricing

risk_neutral_pricing - Economics 141A – Fall, 2010 Risk...

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Unformatted text preview: Economics 141A – Fall, 2010 Risk Neutral Pricing of Derivative Contracts last update: 11/15/10 Synopsis of Previous Results For an asset price process S ( t ) whose dynamics are given by geometric Brownian motion we established that: • The physical probability P and the associated Brownian motion W ( t ) can for the purpose of pricing by expected value be replaced by the risk-neutral probability Q and its associated Brownian motion W Q ( t ). • The discounted stock price process ˆ S ( t ) has dynamics d ˆ S ( t ) = σ ˆ S ( t ) dW Q ( t ) i.e. ˆ S ( t ) is a Q-martingale. Pricing of a Derivative Security The goal is to price a derivative security on S ( t ), viz. a contingent claim contract on the underlying stock with payoff random variable V ( T ) at maturity T . The two most natural kinds of payoff functions are of the form Φ( T,S ( T )) or Φ( S ( T )). The latter is usually called a simple claim. We seek a price process of the contract of the form F ( t,S ( t )) , for 0 ≤ t < T . We require (this is the no-arbitrage condition) that (i)the wealth process X ( t ) of the replicating portfolio for the claims contract has value equal to the value of the price process for all ≤ < t < T and (ii) the wealth process has value at T equal to the contract payoff. In symbols, these conditions are: • X ( t ) = F ( t,S ( t )) a.s. for all 0 ≤ t < T • X ( T ) = V ( T ) The portfolio process π ( t ) is a stochastic process (a function of time and state) which specifies how much wealth is to held in stock at time t . Note that π ( t ) < 0 indicates a short position in the stock....
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This note was uploaded on 04/09/2012 for the course ECON 142 taught by Professor Mess during the Fall '11 term at UCLA.

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risk_neutral_pricing - Economics 141A – Fall, 2010 Risk...

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