dorsal/arxiv
View SchemaPricing European Options in Realistic Markets
| Authors | Martin Schaden |
|---|---|
| Categories | |
| ArXiv ID | physics/0210025 |
| URL | https://arxiv.org/abs/physics/0210025 |
Abstract
We investigate the relation between the fair price for European-style vanilla options and the distribution of short-term returns on the underlying asset ignoring transaction and other costs. We compute the risk-neutral probability density conditional on the total variance of the asset's returns when the option expires. If the asset's future price has finite expectation, the option's fair value satisfies a parabolic partial differential equation of the Black-Scholes type in which the variance of the asset's returns rather than a trading time is the evolution parameter. By immunizing the portfolio against large-scale price fluctuations of the asset, the valuation of options is extended to the realistic case\cite{St99} of assets whose short-term returns have finite variance but very large, or even infinite, higher moments. A dynamic Delta-hedged portfolio that is statically insured against exceptionally large fluctuations includes at least two different options on the asset. The fair value of an option in this case is determined by a universal drift function that is common to all options on the asset. This drift is interpreted as the premium for an investment exposed to risk due to exceptionally large variations of the asset's price. It affects the option valuation like an effective cost-of-carry for the underlying in the Black-Scholes world would. The derived pricing formula for options in realistic markets is arbitrage free by construction. A simple model with constant drift qualitatively reproduces the often observed volatility -skew and -term structure.
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"abstract": "We investigate the relation between the fair price for European-style vanilla\noptions and the distribution of short-term returns on the underlying asset\nignoring transaction and other costs. We compute the risk-neutral probability\ndensity conditional on the total variance of the asset\u0027s returns when the\noption expires. If the asset\u0027s future price has finite expectation, the\noption\u0027s fair value satisfies a parabolic partial differential equation of the\nBlack-Scholes type in which the variance of the asset\u0027s returns rather than a\ntrading time is the evolution parameter. By immunizing the portfolio against\nlarge-scale price fluctuations of the asset, the valuation of options is\nextended to the realistic case\\cite{St99} of assets whose short-term returns\nhave finite variance but very large, or even infinite, higher moments. A\ndynamic Delta-hedged portfolio that is statically insured against exceptionally\nlarge fluctuations includes at least two different options on the asset. The\nfair value of an option in this case is determined by a universal drift\nfunction that is common to all options on the asset. This drift is interpreted\nas the premium for an investment exposed to risk due to exceptionally large\nvariations of the asset\u0027s price. It affects the option valuation like an\neffective cost-of-carry for the underlying in the Black-Scholes world would.\nThe derived pricing formula for options in realistic markets is arbitrage free\nby construction. A simple model with constant drift qualitatively reproduces\nthe often observed volatility -skew and -term structure.",
"arxiv_id": "physics/0210025",
"authors": [
"Martin Schaden"
],
"categories": [
"physics.soc-ph",
"physics.data-an",
"q-fin.PR"
],
"title": "Pricing European Options in Realistic Markets",
"url": "https://arxiv.org/abs/physics/0210025"
},
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