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

2007 ◽  
Vol 44 (04) ◽  
pp. 865-879 ◽  
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.

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

2007 ◽  
Vol 44 (4) ◽  
pp. 865-879 ◽  
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.

scholarly journals HISTORICAL BACKTESTING OF LOCAL VOLATILITY MODEL USING AUD/USD VANILLA OPTIONS

2016 ◽  
Vol 57 (3) ◽  
pp. 319-338
Author(s):  
T. G. LING ◽  
P. V. SHEVCHENKO
Keyword(s):  
Implied Volatility ◽  
Underlying Asset ◽  
Local Volatility ◽  
Market Data ◽  
Black Scholes Model ◽  
Local Volatility Model ◽  
Volatility Model ◽  
Wide Range ◽  
Black Scholes ◽  
Delta Hedging

The local volatility model is a well-known extension of the Black–Scholes constant volatility model, whereby the volatility is dependent on both time and the underlying asset. This model can be calibrated to provide a perfect fit to a wide range of implied volatility surfaces. The model is easy to calibrate and still very popular in foreign exchange option trading. In this paper, we address a question of validation of the local volatility model. Different stochastic models for the underlying asset can be calibrated to provide a good fit to the current market data, which should be recalibrated every trading date. A good fit to the current market data does not imply that the model is appropriate, and historical backtesting should be performed for validation purposes. We study delta hedging errors under the local volatility model using historical data from 2005 to 2011 for the AUD/USD implied volatility. We performed backtests for a range of option maturities and strikes using sticky delta and theoretically correct delta hedging. The results show that delta hedging errors under the standard Black–Scholes model are no worse than those of the local volatility model. Moreover, for the case of in- and at-the-money options, the hedging error for the Black–Scholes model is significantly better.

SIMPLIFIED HEDGE FOR PATH-DEPENDENT DERIVATIVES

2016 ◽  
Vol 19 (07) ◽  
pp. 1650045 ◽  
Author(s):  
CAROLE BERNARD ◽  
JUNSEN TANG
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Asian Option ◽  
Distributional Properties ◽  
Lookback Option ◽  
Volatility Model ◽  
Black Scholes ◽  
Delta Hedging ◽  
Path Dependent

Path-dependent derivatives are typically difficult to hedge. Traditional dynamic delta hedging does not perform well because of the difficulty to evaluate the Greeks and the high cost of constantly rebalancing. We propose to price and hedge path-dependent derivatives by constructing simplified alternatives that preserve certain distributional properties of their terminal payoffs, and that can be hedged by semi-static replication. The method is illustrated by a geometric Asian option and by a lookback option in the Black–Scholes setting, for which explicit forms of the simplified alternatives exist. Extensions to a Lévy market and to a Heston stochastic volatility model are discussed as well.

EXPANSION FORMULAS FOR BIVARIATE PAYOFFS WITH APPLICATION TO BEST-OF OPTIONS ON EQUITY AND INFLATION

2014 ◽  
Vol 17 (02) ◽  
pp. 1450010 ◽  
Author(s):  
EMMANUEL GOBET ◽  
JULIEN HOK
Keyword(s):  
Interest Rates ◽  
Stock Price ◽  
Local Volatility ◽  
Local Volatility Model ◽  
Volatility Model ◽  
Asset Value ◽  
Black Scholes ◽  
Approximation Formulas ◽  
Volatility Function ◽  
Log Normal

A wide class of hybrid products are evaluated with a model where one of the underlying price follows a local volatility diffusion and the other asset value a log-normal process. Because of the generality for the local volatility function, the numerical pricing is usually much time consuming. Using proxy approximations related to log-normal modeling, we derive approximation formulas of Black–Scholes type for the price, that have the advantage of giving very rapid numerical procedures. This derivation is illustrated with the best-of option between equity and inflation where the stock price follows a local volatility model and the inflation rate a Hull–White process. The approximations possibly account for Gaussian HJM (Heath-Jarrow-Morton) models for interest rates. The experiments show an excellent accuracy.

scholarly journals EXPANSION FORMULAS FOR EUROPEAN OPTIONS IN A LOCAL VOLATILITY MODEL

2010 ◽  
Vol 13 (04) ◽  
pp. 603-634 ◽  
Author(s):  
E. BENHAMOU ◽  
E. GOBET ◽  
M. MIRI
Keyword(s):  
Asymptotic Expansion ◽  
Analytical Solution ◽  
Closed Form ◽  
European Options ◽  
Local Volatility ◽  
Closed Form Solutions ◽  
Local Volatility Model ◽  
Volatility Model ◽  
Black Scholes

Because of its very general formulation, the local volatility model does not have an analytical solution for European options. In this article, we present a new methodology to derive closed form solutions for the price of any European options. The formula results from an asymptotic expansion, terms of which are Black-Scholes price and related Greeks. The accuracy of the formula depends on the payoff smoothness and it converges with very few terms.

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