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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 04/09/2012 for the course ECON 142 taught by Professor Mess during the Fall '11 term at UCLA.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online