MARKET MODELS FOR EUROPEAN OPTIONS: DYNAMIC LOCAL VOLATILITY AND DYNAMIC LOCAL LE´VY MEASURE
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Thu, 15/10/2009 13:00 |
Sergey Nadtochiy (OMI) |
Mathematical Finance Internal Seminar  |
DH 1st floor SR |
| Most financial models introduced for the purpose of pricing and hedging derivatives concentrate on the dynamics of the underlying stocks, or underlying instruments on which the derivatives are written. However, as certain types of derivatives became liquid, it appeared reasonable to model their prices directly and use these market models to price or hedge exotic derivatives. This framework was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets. We discuss the characterization of arbitrage free dynamic stochastic models for the markets with infinite number of European Call options as the liquid derivatives. Subject to our assumptions on the presence of jumps in the underlying, the option prices are represented either through local volatility or through local L´evy measure. Each of the latter ones is then given dynamics through an Itˆo stochastic process in infinite dimensional space. The main thrust of our work is to characterize absence of arbitrage in this framework and address the issue of construction of the arbitrage-free models. | |||