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.

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.

Arbitrage Pricing

2019 ◽  
pp. 95-118
Author(s):  
Tomas Björk
Keyword(s):  
Continuous Time ◽  
Implied Volatility ◽  
Futures Contracts ◽  
Martingale Measures ◽  
Bank Account ◽  
No Arbitrage ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Financial Derivative ◽  
Delta Hedging

The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.

A Simple Model for Option Pricing with Jumping Stochastic Volatility

1998 ◽  
Vol 01 (04) ◽  
pp. 487-505 ◽  
Author(s):  
Stefano Herzel
Keyword(s):  
Implied Volatility ◽  
Market Price ◽  
Volatility Risk ◽  
Simple Modification ◽  
Closed Form Solutions ◽  
Black Scholes Model ◽  
A Value ◽  
Black Scholes ◽  
Delta Hedging ◽  
Actual Price

This paper proposes a simple modification of the Black–Scholes model by assuming that the volatility of the stock may jump at a random time τ from a value σa to a value σb. It shows that, if the market price of volatility risk is unknown, but constant, all contingent claims can be valued from the actual price C0, of some arbitrarily chosen "basis" option. Closed form solutions for the prices of European options as well as explicit formulas for vega and delta hedging are given. All such solutions only depend on σa, σb and C0. The prices generated by the model produce a "smile"-shaped curve of the implied volatility.

Bifractional Black-Scholes Model for Pricing European Options and Compound Options

2020 ◽  
Vol 8 (4) ◽  
pp. 346-355
Author(s):  
Feng Xu
Keyword(s):  
Empirical Studies ◽  
Asset Price ◽  
European Options ◽  
Hedging Strategy ◽  
Underlying Asset ◽  
Bifractional Brownian Motion ◽  
Compound Options ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Delta Hedging

AbstractRecent empirical studies show that an underlying asset price process may have the property of long memory. In this paper, it is introduced the bifractional Brownian motion to capture the underlying asset of European options. Moreover, a bifractional Black-Scholes partial differential equation formulation for valuing European options based on Delta hedging strategy is proposed. Using the final condition and the method of variable substitution, the pricing formulas for the European options are derived. Furthermore, applying to risk-neutral principle, we obtain the pricing formulas for the compound options. Finally, the numerical experiments show that the parameter HK has a significant impact on the option value.

Option Pricing using Black Scholes Model (Case-Tata Motors and Reliance)

2015 ◽  
Vol 9 (1and2) ◽  
Author(s):  
Ms. Mamta Shah
Keyword(s):  
Stock Price ◽  
Implied Volatility ◽  
Safety Net ◽  
Volatility Smile ◽  
Model Case ◽  
Underlying Asset ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Option Strategies ◽  
Additional Income

The power of options lies in their versatility. It enables the investors to adjust position according to any situation that arises. Options can be speculative or conservative. This means investor can do everything from protecting a position from a decline to outright betting on the movement of a market or index. Options can enable the investor to buy a stock at a lower price, sell a stock at a higher price, or create additional income against a long or short stock position. One can also uses option strategies to profit from a movement in the price of the underlying asset regardless of market direction. the responsible act and safe thing to do. Options provide the same kind of safety net for trades and investments already committed, which is known as hedging. The research paper is based on Black Scholes Model. The study includes the Implied Volatility Test and Volatility Smile Test. This study also includes the solver available in MS Excel. This study is based on stock price of Reliance and Tata Motors.

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.

Options

2019 ◽  
pp. 231-248
Author(s):  
Alan N. Rechtschaffen
Keyword(s):  
Option Pricing ◽  
Strike Price ◽  
Over The Counter ◽  
Index Options ◽  
Underlying Asset ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Delta Hedging ◽  
Option Strategies ◽  
The Right

An option is a derivative that derives its value from another underlying asset, instrument, or index. Options “transfer the right but not the obligation to buy or sell the underlying asset, instrument or index on or before the option's exercise date at a specified price (the strike price).” A contract that gives a purchaser such a right is inherently an option even if it called something else. Options can trade over the counter or on an exchange. Regulatory jurisdiction will be defined by the underlying asset negotiated under the terms of the option, by the location where the options are traded, and by the counterparties to an option transaction. This chapter discusses the characteristics of options, how options work, the Black-Scholes model and option pricing, delta hedging, and option strategies.

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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