Portfolios Generated by Contingent Claim Functions, with Applications to Option Pricing
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916743172063232 |
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| author | Fernholz, Ricardo T. Fernholz, Robert |
| author_facet | Fernholz, Ricardo T. Fernholz, Robert |
| contents | This paper presents a synthesis of the theories of portfolio generating functions and option pricing. The theory of portfolio generation is extended to measure the value of portfolios generated by positive C^{2,1} functions of asset prices X_1,... , X_n directly, rather than with respect to a numeraire portfolio. If a portfolio generating function satisfies a specific partial differential equation, then the value of the portfolio generated by that function will replicate the value of the function. This differential equation is a general form of the Black-Scholes equation. Similar results apply to contingent claim functions, which are portfolio generating functions that are homogeneous of degree 1. With the addition of a riskless asset, an inhomogeneous portfolio generating function V : R^{+n} x [0, T] \to R^+ can be extended to an equivalent contingent claim function \hat{V} : R^+ x R^{+n} x [0, T] \to R^+ that generates the same portfolio and is replicable if and only if V is replicable. Several examples are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_13717 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Portfolios Generated by Contingent Claim Functions, with Applications to Option Pricing Fernholz, Ricardo T. Fernholz, Robert Pricing of Securities Mathematical Finance 60H30, 91G10, 91G20 This paper presents a synthesis of the theories of portfolio generating functions and option pricing. The theory of portfolio generation is extended to measure the value of portfolios generated by positive C^{2,1} functions of asset prices X_1,... , X_n directly, rather than with respect to a numeraire portfolio. If a portfolio generating function satisfies a specific partial differential equation, then the value of the portfolio generated by that function will replicate the value of the function. This differential equation is a general form of the Black-Scholes equation. Similar results apply to contingent claim functions, which are portfolio generating functions that are homogeneous of degree 1. With the addition of a riskless asset, an inhomogeneous portfolio generating function V : R^{+n} x [0, T] \to R^+ can be extended to an equivalent contingent claim function \hat{V} : R^+ x R^{+n} x [0, T] \to R^+ that generates the same portfolio and is replicable if and only if V is replicable. Several examples are presented. |
| title | Portfolios Generated by Contingent Claim Functions, with Applications to Option Pricing |
| topic | Pricing of Securities Mathematical Finance 60H30, 91G10, 91G20 |
| url | https://arxiv.org/abs/2308.13717 |