Portfolio Based Risk Pricing: Pricing Long Term Put Options with Gjr-Garch (1,1)/Jump Diffusion Process
15 Pages Posted: 2 Jan 1999
Abstract
This article proposes a portfolio-based pricing method to evaluate risk and systematically consider risk premium. The risk premium is charged to satisfy risk management and return on risk capital requirements. The P&L distributions are priced based on Value-at-Risk and return on capital approach. The pricing generating process is explicitly presented using an example of Standard and Poor 500 index (SPX) put options.
While the complete list of risk factors is extensive, here we focus on three major concerns an option underwriter has to consider: (1) non-log-normal distribution of the underlying asset, such as stochastic volatility and/or jumps, (2) transaction costs in dynamic hedging, (3) risk premium for unhedgeable remaining basis risk. With these complications, the final profit and loss to the option underwriter is no longer certain. The underwriter has to develop a formal procedure which would allow him/her to relate the P&L distribution at maturity to a (dollar) value of the derivative security.
We can exactly fit the implied volatility "smile" or "smirk" curve of the one-year SPX options (liquid Exchange market) in term of slope and volatility values. It is then used to price five-year out-of-the-money put (non-liquid over-the-counter market). In total, the implied volatility is about 10% to 40% higher than the historical one. This numbers are consistent with the behavior of S&P500 market.
JEL Classification: G12, G13
Suggested Citation: Suggested Citation
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