The maximum maximum of a martingale with given n marginals
We obtain bounds on the distribution of the maximum of a continuous martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to n-marginal Skorokhod embedding problem in Obloj and Spoida (2013). It follows that their embedding maximises the...
| Main Authors: | , , , |
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| Format: | Journal article |
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2012
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| _version_ | 1797099013027135488 |
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| author | Henry-Labordere, P Obloj, J Spoida, P Touzi, N |
| author_facet | Henry-Labordere, P Obloj, J Spoida, P Touzi, N |
| author_sort | Henry-Labordere, P |
| collection | OXFORD |
| description | We obtain bounds on the distribution of the maximum of a continuous martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to n-marginal Skorokhod embedding problem in Obloj and Spoida (2013). It follows that their embedding maximises the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers (1998). This inequality, proved by elementary arguments, is obtained by following the stochastic control approach of Galichon, Henry-Labordere and Touzi (2011). |
| first_indexed | 2024-03-07T05:17:42Z |
| format | Journal article |
| id | oxford-uuid:ddc9c2e2-22f0-46fe-a371-66c8f572deea |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:17:42Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:ddc9c2e2-22f0-46fe-a371-66c8f572deea2022-03-27T09:27:32ZThe maximum maximum of a martingale with given n marginalsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ddc9c2e2-22f0-46fe-a371-66c8f572deeaSymplectic Elements at Oxford2012Henry-Labordere, PObloj, JSpoida, PTouzi, NWe obtain bounds on the distribution of the maximum of a continuous martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to n-marginal Skorokhod embedding problem in Obloj and Spoida (2013). It follows that their embedding maximises the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers (1998). This inequality, proved by elementary arguments, is obtained by following the stochastic control approach of Galichon, Henry-Labordere and Touzi (2011). |
| spellingShingle | Henry-Labordere, P Obloj, J Spoida, P Touzi, N The maximum maximum of a martingale with given n marginals |
| title | The maximum maximum of a martingale with given n marginals |
| title_full | The maximum maximum of a martingale with given n marginals |
| title_fullStr | The maximum maximum of a martingale with given n marginals |
| title_full_unstemmed | The maximum maximum of a martingale with given n marginals |
| title_short | The maximum maximum of a martingale with given n marginals |
| title_sort | maximum maximum of a martingale with given n marginals |
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