scholarly journals A numerical scheme based on semi-static hedging strategy

2014 ◽  
Vol 20 (4) ◽  
Author(s):  
Yuri Imamura ◽  
Yuta Ishigaki ◽  
Toshiki Okumura
Keyword(s):  
Diffusion Process ◽  
Numerical Scheme ◽  
Numerical Results ◽  
Barrier Options ◽  
Barrier Option ◽  
Hedging Strategy ◽  
Underlying Process ◽  
Static Hedging ◽  
Black Scholes ◽  
Path Dependent

AbstractIn the present paper, we introduce a numerical scheme for the price of a barrier option when the price of the underlying follows a diffusion process. The numerical scheme is based on an extension of a static hedging formula of barrier options. To get the static hedging formula, the underlying process needs to have a symmetry. We introduce a way to “symmetrize” a given diffusion process. Then the pricing of a barrier option is reduced to that of plain options under the symmetrized process. To show how our symmetrization scheme works, we will present some numerical results of path-independent Euler–Maruyama approximation applied to our scheme, comparing them with the path-dependent Euler–Maruyama scheme when the model is of the type Black–Scholes, CEV, Heston, and (λ)-SABR, respectively. The results show the effectiveness of our scheme.

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.

SEMI-STATIC HEDGING OF BARRIER OPTIONS UNDER POISSON JUMPS

2011 ◽  
Vol 14 (07) ◽  
pp. 1091-1111 ◽  
Author(s):  
PETER CARR
Keyword(s):  
Barrier Options ◽  
Barrier Option ◽  
Poisson Jumps ◽  
Classical Dynamic ◽  
Continuous Processes ◽  
Static Hedging ◽  
Dynamic Replication ◽  
Independent Process ◽  
The Difference ◽  
Riskless Asset

We show that the payoff to barrier options can be replicated when the underlying price process is driven by the difference of two independent Poisson processes. The replicating strategy employs simple semi-static positions in co-terminal standard options. We note that classical dynamic replication using just the underlying asset and a riskless asset is not possible in this context. When the underlying of the barrier option has no carrying cost, we show that the same semi-static trading strategy continues to replicate even when the two jump arrival rates are generalized into positive even functions of distance to the barrier and when the clock speed is randomized into a positive continuous independent process. Since the even function and the positive process need no further specification, our replicating strategies are also semi-robust. Finally, we show that previous results obtained for continuous processes arise as limits of our analysis.

scholarly journals Pricing of Barrier Options on Underlying Assets with Jump-Diffusion Dynamics: A Mellin Transform Approach

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1271
Author(s):  
Marianito R. Rodrigo
Keyword(s):  
Mellin Transform ◽  
Jump Diffusion ◽  
Futures Contracts ◽  
Barrier Options ◽  
Barrier Option ◽  
Underlying Asset ◽  
Diffusion Dynamics ◽  
Path Dependent ◽  
The Right ◽  
Right To Buy

A barrier option is an exotic path-dependent option contract where the right to buy or sell is activated or extinguished when the underlying asset reaches a certain barrier price during the lifetime of the contract. In this article we use a Mellin transform approach to derive exact pricing formulas for barrier options with general payoffs and exponential barriers on underlying assets that have jump-diffusion dynamics. With the same approach we also price barrier options on underlying futures contracts.

PRICING TWO-ASSET BARRIER OPTIONS UNDER STOCHASTIC CORRELATION VIA PERTURBATION

2015 ◽  
Vol 18 (03) ◽  
pp. 1550018 ◽  
Author(s):  
MARCOS ESCOBAR ◽  
BARBARA GÖTZ ◽  
DANIELA NEYKOVA ◽  
RUDI ZAGST
Keyword(s):  
Perturbation Theory ◽  
Stochastic Volatility ◽  
Correlation Structure ◽  
Barrier Options ◽  
Correlation Effects ◽  
Stochastic Correlation ◽  
Black Scholes ◽  
Leading Term ◽  
Path Dependent ◽  
Correction Terms

The correlation structure is crucial when pricing multi-asset products, in particular barrier options. In this work, we price two-asset path-dependent derivatives by means of perturbation theory in the context of a bi-dimensional asset model with stochastic correlation and volatilities. To our best knowledge, this is the first attempt at pricing barriers with stochastic correlation. It turns out that the leading term of the approximation corresponds to a constant covariance Black–Scholes type price with correction terms adjusting for stochastic volatility and stochastic correlation effects. The practicability of the presented method is illustrated by some numerical implementations.

scholarly journals Semi-Analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab® codes)

2018 ◽  
Vol 9 (1) ◽  
pp. 42-67 ◽  
Author(s):  
C. Guardasoni
Keyword(s):  
Analytical Method ◽  
Boundary Integral ◽  
Time Dependent ◽  
Barrier Options ◽  
Barrier Option ◽  
Boundary Integral Methods ◽  
Integral Methods ◽  
Interest Rate Volatility ◽  
Black Scholes ◽  
Barrier Option Pricing

Abstract A Semi-Analytical method for pricing of Barrier Options (SABO) is presented. The method is based on the foundations of Boundary Integral Methods which is recast here for the application to barrier option pricing in the Black-Scholes model with time-dependent interest rate, volatility and dividend yield. The validity of the numerical method is illustrated by several numerical examples and comparisons.

scholarly journals Option pricing: a yet simpler approach

2021 ◽  
Author(s):  
Jarno Talponen ◽  
Minna Turunen
Keyword(s):  
Main Tool ◽  
Random Processes ◽  
Pricing Model ◽  
Static Hedging ◽  
Merton Model ◽  
Black Scholes ◽  
Path Dependent ◽  
Digital Options ◽  
Infinite State ◽  
Drift Parameter

AbstractWe provide a lean, non-technical exposition on the pricing of path-dependent and European-style derivatives in the Cox–Ross–Rubinstein (CRR) pricing model. The main tool used in this paper for simplifying the reasoning is applying static hedging arguments. In applying the static hedging principle, we consider Arrow–Debreu securities and digital options, or backward random processes. In the last case, the CRR model is extended to an infinite state space which leads to an interesting new phenomenon not present in the classical CRR model. At the end, we discuss the paradox involving the drift parameter $$\mu $$ μ in the Black–Scholes–Merton model pricing. We provide sensitivity analysis and an approximation of the speed of convergence for the asymptotically vanishing effect of drift in prices.

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 Investigation of the barrier options pricing models

2009 ◽  
Vol 50 ◽  
Author(s):  
Rita Palivonaitė ◽  
Eimutis Valakevičius
Keyword(s):  
Option Pricing ◽  
Adaptive Mesh ◽  
Options Pricing ◽  
Barrier Options ◽  
Barrier Option ◽  
Pricing Models ◽  
Black Scholes ◽  
Barrier Option Pricing ◽  
Critical Regions ◽  
Options Pricing Models

In the article three methods of barrier option pricing are analysed and compared: Black–Scholes, trinomial ant adaptive mesh algorithm. Investigation with Lithuanian firm’s stock showed, that to get better results it is offered to adapt higer resolution mesh on critical regions of trinomial tree.

scholarly journals Pricing Parisian Option under a Stochastic Volatility Model

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Min-Ku Lee ◽  
Kyu-Hwan Jang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Barrier Options ◽  
Barrier Option ◽  
Partial Differential ◽  
Model Based ◽  
Volatility Model ◽  
Black Scholes

We study the pricing of a Parisian option under a stochastic volatility model. Based on the manipulation problem that barrier options might create near barriers, the Parisian option has been designed as an extended barrier option. A stochastic volatility correction to the Black-Scholes price of the Parisian option is obtained in a partial differential equation form and the solution is characterized numerically.

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