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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2309.05054 |
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| _version_ | 1866916951201153024 |
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| author | Armstrong, John Ionescu, Andrei |
| author_facet | Armstrong, John Ionescu, Andrei |
| contents | We apply rough-path theory to study the discrete-time gamma-hedging strategy. We show that if a trader knows that the market price of a set of European options will be given by a diffusive pricing model, then the discrete-time gamma-hedging strategy will enable them to replicate other European options so long as the underlying pricing signal is sufficiently regular. This is a sure result and does not require that the underlying pricing signal has a quadratic variation corresponding to a probabilisitic pricing model. We show how to generalise this result to exotic derivatives when the gamma is defined to be the Gubinelli derivative of the delta by deriving rough-path versions of the Clark--Ocone formula which hold surely.
We illustrate our theory by proving that if the stock price process is sufficiently regular, as is the implied volatility process of a European derivative with maturity $T$ and smooth payoff $f(S_T)$ satisfying $f^{\prime \prime}>0$, one can replicate with certainty any European derivative with smooth payoff and maturity $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05054 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gamma Hedging and Rough Paths Armstrong, John Ionescu, Andrei Mathematical Finance We apply rough-path theory to study the discrete-time gamma-hedging strategy. We show that if a trader knows that the market price of a set of European options will be given by a diffusive pricing model, then the discrete-time gamma-hedging strategy will enable them to replicate other European options so long as the underlying pricing signal is sufficiently regular. This is a sure result and does not require that the underlying pricing signal has a quadratic variation corresponding to a probabilisitic pricing model. We show how to generalise this result to exotic derivatives when the gamma is defined to be the Gubinelli derivative of the delta by deriving rough-path versions of the Clark--Ocone formula which hold surely. We illustrate our theory by proving that if the stock price process is sufficiently regular, as is the implied volatility process of a European derivative with maturity $T$ and smooth payoff $f(S_T)$ satisfying $f^{\prime \prime}>0$, one can replicate with certainty any European derivative with smooth payoff and maturity $T$. |
| title | Gamma Hedging and Rough Paths |
| topic | Mathematical Finance |
| url | https://arxiv.org/abs/2309.05054 |