On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile
We analyse a model for pricing derivative securities in the presence of both transaction costs as well as the risk from a volatile portfolio. The model is based on the Black-Scholes parabolic PDE in which transaction costs are described following the Hoggard, Whalley, and Wilmott approach. The risk...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2005-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/JAM.2005.235 |
| Summary: | We analyse a model for pricing derivative securities in the
presence of both transaction costs as well as the risk from a
volatile portfolio. The model is based on the Black-Scholes
parabolic PDE in which transaction costs are described following
the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the
variance of the synthesized portfolio. Transaction
costs as well as the volatile portfolio risk depend on the time
lag between two consecutive transactions. Minimizing their sum
yields the optimal length of the hedge interval. In this model,
prices of vanilla options can be computed from a solution to a
fully nonlinear parabolic equation in which a diffusion
coefficient representing volatility nonlinearly depends on the
solution itself giving rise to explaining the volatility smile
analytically. We derive a robust numerical scheme for solving the
governing equation and perform extensive numerical testing of the
model and compare the results to real option market data. Implied
risk and volatility are introduced and computed for large option
datasets. We discuss how they can be used in qualitative and
quantitative analysis of option market data. |
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| ISSN: | 1110-757X 1687-0042 |