Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

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Analytical pricing of single barrier options under local volatility models

Quantitative Finance ◽

10.1080/14697688.2015.1101483 ◽

2015 ◽

Vol 16 (6) ◽

pp. 867-886 ◽

Cited By ~ 4

Author(s):

Hideharu Funahashi ◽

Masaaki Kijima

Keyword(s):

Barrier Options ◽

Local Volatility ◽

Volatility Models

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Robustness of Delta Hedging for Path-Dependent Options in Local Volatility Models

Journal of Applied Probability ◽

10.1017/s0021900200003594 ◽

2007 ◽

Vol 44 (04) ◽

pp. 865-879 ◽

Cited By ~ 3

Author(s):

Alexander Schied ◽

Mitja Stadje

Keyword(s):

Payoff Function ◽

Hedging Strategy ◽

Local Volatility ◽

Directional Convexity ◽

Volatility Models ◽

Local Volatility Model ◽

Volatility Model ◽

Black Scholes ◽

Delta Hedging ◽

Path Dependent

We consider the performance of the delta hedging strategy obtained from a local volatility model when using as input the physical prices instead of the model price process. This hedging strategy is called robust if it yields a superhedge as soon as the local volatility model overestimates the market volatility. We show that robustness holds for a standard Black-Scholes model whenever we hedge a path-dependent derivative with a convex payoff function. In a genuine local volatility model the situation is shown to be less stable: robustness can break down for many relevant convex payoffs including average-strike Asian options, lookback puts, floating-strike forward starts, and their aggregated cliquets. Furthermore, we prove that a sufficient condition for the robustness in every local volatility model is the directional convexity of the payoff function.

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