On some potential applications of the heat equation with a repulsive point interaction to derivative pricing
In this note we first investigate in detail the “heat equation” with the free Laplacian replaced by the one with a repulsive point interaction centred at the origin in the case where the initial condition is given by any function proportional to e^x/2 χ(−∞,0](x). The solution is expressed in terms...
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Sapienza Università Editrice
2011-01-01
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| Series: | Rendiconti di Matematica e delle Sue Applicazioni |
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| Online Access: | https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2011(1-2)/35-52.pdf |
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doaj-992b7c4236054972b04869c5498a02a72021-08-15T14:34:11ZengSapienza Università EditriceRendiconti di Matematica e delle Sue Applicazioni1120-71832532-33502011-01-01311-23552On some potential applications of the heat equation with a repulsive point interaction to derivative pricingSilvestro Fassari0Fabio Rinaldi1Università degli Studi Guglielmo MarconiUniversità degli Studi Guglielmo MarconiIn this note we first investigate in detail the “heat equation” with the free Laplacian replaced by the one with a repulsive point interaction centred at the origin in the case where the initial condition is given by any function proportional to e^x/2 χ(−∞,0](x). The solution is expressed in terms of the cumulative function of the normal distribution in view of its direct application to derivative pricing. In the second part of the paper, with reference to the quantum mechanical approach to option pricing proposed in the last decade, we use the results in order to solve explicitly the Black-Scholes equation with a perturbing term given by a point interaction of the type λ · δ(ln( s/E )), s being the price of the underlying asset and E the exercise price of the option.https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2011(1-2)/35-52.pdfpoint interactionsheat equationheat kernel |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Silvestro Fassari Fabio Rinaldi |
| spellingShingle |
Silvestro Fassari Fabio Rinaldi On some potential applications of the heat equation with a repulsive point interaction to derivative pricing Rendiconti di Matematica e delle Sue Applicazioni point interactions heat equation heat kernel |
| author_facet |
Silvestro Fassari Fabio Rinaldi |
| author_sort |
Silvestro Fassari |
| title |
On some potential applications of the heat equation with a repulsive point interaction to derivative pricing |
| title_short |
On some potential applications of the heat equation with a repulsive point interaction to derivative pricing |
| title_full |
On some potential applications of the heat equation with a repulsive point interaction to derivative pricing |
| title_fullStr |
On some potential applications of the heat equation with a repulsive point interaction to derivative pricing |
| title_full_unstemmed |
On some potential applications of the heat equation with a repulsive point interaction to derivative pricing |
| title_sort |
on some potential applications of the heat equation with a repulsive point interaction to derivative pricing |
| publisher |
Sapienza Università Editrice |
| series |
Rendiconti di Matematica e delle Sue Applicazioni |
| issn |
1120-7183 2532-3350 |
| publishDate |
2011-01-01 |
| description |
In this note we first investigate in detail the “heat equation” with
the free Laplacian replaced by the one with a repulsive point interaction centred at the origin in the case where the initial condition is given by any function proportional to e^x/2 χ(−∞,0](x). The solution is expressed in terms of the cumulative function of the normal distribution in view of its direct application to derivative pricing. In the second part of the paper, with reference to the quantum mechanical approach to option pricing proposed in the last decade, we use the results in order to solve explicitly the Black-Scholes equation with a perturbing term given by a point interaction of the type λ · δ(ln( s/E )), s being the price of the underlying asset and E the exercise price of the option. |
| topic |
point interactions heat equation heat kernel |
| url |
https://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2011(1-2)/35-52.pdf |
| work_keys_str_mv |
AT silvestrofassari onsomepotentialapplicationsoftheheatequationwitharepulsivepointinteractiontoderivativepricing AT fabiorinaldi onsomepotentialapplicationsoftheheatequationwitharepulsivepointinteractiontoderivativepricing |
| _version_ |
1721206322217615360 |