scholarly journals A constructive method for convex solutions of a class of nonlinear Black-Scholes equations

2019 ◽  
Vol 9 (1) ◽  
pp. 654-664 ◽  
Author(s):  
Mostafa Abounouh ◽  
Hassan Al Moatassime ◽  
Aicha Driouch ◽  
Olivier Goubet
Keyword(s):  
Transaction Costs ◽  
Strong Solution ◽  
Theoretical Study ◽  
Limit Problem ◽  
Nonlinear Dependence ◽  
Constructive Method ◽  
Monotone Sequence ◽  
Mathematical Approach ◽  
Deterministic Problem ◽  
Black Scholes

Abstract In this work, we are concerned with the theoretical study of a nonlinear Black-Scholes equation resulting from market frictions. We will focus our attention on Barles and Soner’s model where the volatility is enlarged due to the presence of transaction costs. The aim of this paper is to give a constructive mathematical approach for proving the existence of convex solutions to a non degenerate fully nonlinear deterministic problem with nonlinear dependence upon the highest derivative. The existence of a strong solution to the original equation is shown by considering a monotone sequence satisfying an abstract Barenblatt equation and converging toward the solution of a limit problem.

scholarly journals CRITICAL TRANSACTION COSTS AND 1-STEP ASYMPTOTIC ARBITRAGE IN FRACTIONAL BINARY MARKETS

2015 ◽  
Vol 18 (05) ◽  
pp. 1550029 ◽  
Author(s):  
FERNANDO CORDERO ◽  
LAVINIA PEREZ-OSTAFE
Keyword(s):  
Transaction Costs ◽  
Trading Strategies ◽  
Apparent Contradiction ◽  
New Family ◽  
Black Scholes Model ◽  
Asymptotic Arbitrage ◽  
Arbitrage Opportunities ◽  
Large Financial Market ◽  
Approximating Sequence ◽  
Black Scholes

We study the arbitrage opportunities in the presence of transaction costs in a sequence of binary markets approximating the fractional Black–Scholes model. This approximating sequence was constructed by Sottinen and named fractional binary markets. Since, in the frictionless case, these markets admit arbitrage, we aim to determine the size of the transaction costs needed to eliminate the arbitrage from these models. To gain more insight, we first consider only 1-step trading strategies and we prove that arbitrage opportunities appear when the transaction costs are of order [Formula: see text]. Next, we characterize the asymptotic behavior of the smallest transaction costs [Formula: see text], called "critical" transaction costs, starting from which the arbitrage disappears. Since the fractional Black–Scholes model is arbitrage-free under arbitrarily small transaction costs, one could expect that [Formula: see text] converges to zero. However, the true behavior of [Formula: see text] is opposed to this intuition. More precisely, we show, with the help of a new family of trading strategies, that [Formula: see text] converges to one. We explain this apparent contradiction and conclude that it is appropriate to see the fractional binary markets as a large financial market and to study its asymptotic arbitrage opportunities. Finally, we construct a 1-step asymptotic arbitrage in this large market when the transaction costs are of order o(1/NH), whereas for constant transaction costs, we prove that no such opportunity exists.

scholarly journals On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile

2005 ◽  
Vol 2005 (3) ◽  
pp. 235-258 ◽  
Author(s):  
Martin Jandačka ◽  
Daniel Ševčovič
Keyword(s):  
Transaction Costs ◽  
Real Option ◽  
Time Lag ◽  
Derivative Securities ◽  
Volatility Smile ◽  
Market Data ◽  
Nonlinear Parabolic ◽  
Black Scholes ◽  
Parabolic Pde ◽  
Option Market

We analyse a model for pricing derivative securities in the presence of both transaction costs as well as the risk from a volatile portfolio. The model is based on the Black-Scholes parabolic PDE in which transaction costs are described following the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the variance of the synthesized portfolio. Transaction costs as well as the volatile portfolio risk depend on the time lag between two consecutive transactions. Minimizing their sum yields the optimal length of the hedge interval. In this model, prices of vanilla options can be computed from a solution to a fully nonlinear parabolic equation in which a diffusion coefficient representing volatility nonlinearly depends on the solution itself giving rise to explaining the volatility smile analytically. We derive a robust numerical scheme for solving the governing equation and perform extensive numerical testing of the model and compare the results to real option market data. Implied risk and volatility are introduced and computed for large option datasets. We discuss how they can be used in qualitative and quantitative analysis of option market data.

scholarly journals OPTIMAL HEDGING OF DERIVATIVES WITH TRANSACTION COSTS

2006 ◽  
Vol 09 (07) ◽  
pp. 1051-1069 ◽  
Author(s):  
ERIK AURELL ◽  
PAOLO MURATORE-GINANNESCHI
Keyword(s):  
Transaction Costs ◽  
Optimization Problem ◽  
Optimal Investment ◽  
Market Model ◽  
Black Scholes ◽  
Hamilton Jacobi Bellman Equation ◽  
Finite Time Horizon ◽  
Hamilton Jacobi Bellman ◽  
Delta Hedge ◽  
Strong Control

We investigate the optimal strategy over a finite time horizon for a portfolio of stock and bond and a derivative in an multiplicative Markovian market model with transaction costs (friction). The optimization problem is solved by a Hamilton–Jacobi–Bellman equation, which by the verification theorem has well-behaved solutions if certain conditions on a potential are satisfied. In the case at hand, these conditions simply imply arbitrage-free ("Black–Scholes") pricing of the derivative. While pricing is hence not changed by friction allow a portfolio to fluctuate around a delta hedge. In the limit of weak friction, we determine the optimal control to essentially be of two parts: a strong control, which tries to bring the stock-and-derivative portfolio towards a Black–Scholes delta hedge; and a weak control, which moves the portfolio by adding or subtracting a Black–Scholes hedge. For simplicity we assume growth-optimal investment criteria and quadratic friction.

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