# Stochastic flow approach to Dupire’s formula

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## Abstract

D:\TexFiles\1Research\TimeReversal\PCR5.DVI Put Call Reversal Jesper Andreasen Peter Carr Bank of America Courant Institute 1 Alie St. 251 Mercer Street London England New York, NY 10012 44-207-809-5360 (212) 260-3765 [email protected] [email protected] Initial version: February 4, 2002 Current version: April 25, 2002 File reference: pcr5.tex Assuming that the stock price process is a jump diffusion, we derive a new relation between puts and calls termed Put Call Reversal (PCR). We show how PCR gives simple new probabilistic interpretations of deltas and gammas. We also show how PCR simplifies semi-static hedging of long dated options. We thank Leif Andersen, Sebastian Jaimungal, Dilip Madan, Philip Protter, P. Sundar, and the par- ticipants of the 2002 Fields Institute workshop on Computational Methods and Applications in Finance. We are solely responsible for any errors. Put Call Reversal I Introduction There are many known results which relate put and call values to each other. These results differ in terms of the generality of the model for the underlying dynamics and in terms of whether they apply to both European and American options. For example, put call parity (PCP) implies that a put and a call have the same time value. As is well known, PCP is a completely model-free result governing European option values, but not American ones (see Merton[18]). Put Call Equivalence (PCE) implies that a put on one currency is a call on the other (Grabbe[14]). By considering a different currency more generally as a change of numeraire, PCE can be extended to options on many different underlyings (Schroder[22]). In contrast to PCP, PCE governs both European and American options (DeTemple[11]). PCE is almost model-free, but it does assume that the numeraire is never worth zero. Put Call Symmetry (PCS) is a result that relates an out-of-the-money put on one asset to an out-of-the-money call on the same asset (Bates[3]). In contrast to PCP, PCS appl

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