ON THE CONSISTENCY OF THE DETERMINISTIC LOCAL VOLATILITY FUNCTION MODEL ('IMPLIED TREE')

Karl Strobl ()
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Karl Strobl: ABN AMRO Bank N.V., 250 Bishopsgate, London EC2M 4AA, UK

International Journal of Theoretical and Applied Finance (IJTAF), 2001, vol. 04, issue 03, 545-565

Abstract: We show that the frequent claim that the implied tree prices exotics consistently with an arbitrage-free market is untrue if the local volatilities are stochastic. This is a consequence of the market incompleteness under stochastic volatility. We also show that the problem cannot be mitigated by conveniently defining some 'weakly stochastic' local volatility, as this would violate the no-arbitrage condition. In the process of constructing the proof, we analyse — in the most general context — the impact of stochastic variables on the P&L of a hedged portfolio. We find that any stochastic tradeable either has quadratic variation — and therefore aΓ-like P&L on instruments with non-linear exposure to that asset — or it introduces arbitrage opportunities.

Keywords: Implied trees; stochastic local volatility; exotic options; vega hedging; skew models; market incompleteness (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1142/S0219024901001036

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