scholarly journals Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory

2009 ◽  
Vol 2009 ◽  
pp. 1-37
Author(s):  
Mou-Hsiung Chang ◽  
Roger K. Youree
Keyword(s):  
Stock Price ◽  
Payoff Function ◽  
Fourier Series Expansion ◽  
Spectral Approximation ◽  
European Option ◽  
Bank Account ◽  
Smoothness Assumption ◽  
Infinite Dimensional ◽  
Black Scholes ◽  
With Memory

This paper considers the pricing of a European option using a -market in which the stock price and the asset in the riskless bank account both have hereditary price structures described by the authors of this paper (1999). Under the smoothness assumption of the payoff function, it is shown that the infinite dimensional Black-Scholes equation possesses a unique classical solution. A spectral approximation scheme is developed using the Fourier series expansion in the space for the Black-Scholes equation. It is also shown that the th approximant resembles the classical Black-Scholes equation in finite dimensions.

scholarly journals ANALISIS SENSITIVITAS HARGA OPSI MENGGUNAKAN METODE GREEK BLACK SCHOLES

2018 ◽  
Vol 7 (2) ◽  
pp. 148
Author(s):  
DEVI NANDITA. N ◽  
KOMANG DHARMAWAN ◽  
DESAK PUTU EKA NILAKUSMAWATI
Keyword(s):  
Sensitivity Analysis ◽  
Stock Price ◽  
Price Change ◽  
European Option ◽  
Strike Price ◽  
Expiry Date ◽  
Hedging Strategies ◽  
Black Scholes

Sensitivity analysis can be used to carry out hedging strategies. The sensitivity value measures how much the price change of the option influenced by some parameters. The aim of this study is to determine the sensitivity analysis of the buying price of European option by using the Greek method on Black Scholes Formula. From this study we get the values of delta, gamma, theta, vega, and rho. The values of deltas, gamma, vega, and rho are positive, which means that the value of the option is more sensitive than the corresponding parameter. The most sensitive value of gamma is obtained when the stock price approaches the strike price and approaches the expiry date. The value of theta obtained is negative and hence the most sensitive theta value is when the value is getting smaller. While, the most sensitive value of vega is obtained when the stock price is close to the strike price and is far from the expiry date. The most sensitive value of rho is obtained when the stock price gets bigger and farther from the expiry date.

BSM model for ML-payoff function through PDTM

2018 ◽  
Vol 13 (01) ◽  
pp. 2050024
Author(s):  
S. J. Ghevariya
Keyword(s):  
Information Theory ◽  
Option Pricing ◽  
Payoff Function ◽  
Differential Transform Method ◽  
Entropy Function ◽  
European Option ◽  
Transform Method ◽  
European Option Pricing ◽  
Black Scholes ◽  
Differential Transform

This paper contributes to the valuation of Black–Scholes–Merton (BSM) European option pricing formula for Modified Log-payoff (ML-payoff) function, [Formula: see text] through Projected Differential Transform Method (PDTM). The ML-payoff function is closely related with the entropy function [Formula: see text] in Information Theory. It turns out that the present BSM formula is quite close to the celebrated plain vanilla option.

scholarly journals OPTIONS WRITTEN ON STOCKS WITH KNOWN DIVIDENDS

2004 ◽  
Vol 07 (07) ◽  
pp. 901-907
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Option Prices ◽  
Strike Price ◽  
Call Options ◽  
Alternative Approach ◽  
Black Scholes ◽  
Market Practice ◽  
The Difference ◽  
Risk Free Rate

There are two common methods for pricing European call options on a stock with known dividends. The market practice is to use the Black–Scholes formula with the stock price reduced by the present value of the dividends. An alternative approach is to increase the strike price with the dividends compounded to expiry at the risk-free rate. These methods correspond to different stock price models and thus in general give different option prices. In the present paper we generalize these methods to time- and level-dependent volatilities and to arbitrary contract functions. We show, for convex contract functions and under very general conditions on the volatility, that the method which is market practice gives the lower option price. For call options and some other common contracts we find bounds for the difference between the two prices in the case of constant volatility.

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 PENENTUAN HARGA KONTRAK OPSI KOMODITAS EMAS MENGGUNAKAN METODE POHON BINOMIAL

2017 ◽  
Vol 6 (2) ◽  
pp. 99
Author(s):  
I GEDE RENDIAWAN ADI BRATHA ◽  
KOMANG DHARMAWAN ◽  
NI LUH PUTU SUCIPTAWATI
Keyword(s):  
Option Prices ◽  
European Option ◽  
Binomial Tree ◽  
Call Options ◽  
Option Contracts ◽  
Put Option ◽  
Black Scholes ◽  
Tree Method

Holding option contracts are considered as a new way to invest. In pricing the option contracts, an investor can apply the binomial tree method. The aim of this paper is to present how the European option contracts are calculated using binomial tree method with some different choices of strike prices. Then, the results are compared with the Black-Scholes method. The results obtained show the prices of call options contracts of European type calculated by the binomial tree method tends to be cheaper compared with the price of that calculated by the Black-Scholes method. In contrast to the put option prices, the prices calculated by the binomial tree method are slightly more expensive.

SPECTRALLY ACCURATE OPTION PRICING UNDER THE TIME-FRACTIONAL BLACK–SCHOLES MODEL

2021 ◽  
pp. 1-21
Author(s):  
GERALDINE TOUR ◽  
NAWDHA THAKOOR ◽  
DÉSIRÉ YANNICK TANGMAN
Keyword(s):  
Finite Difference ◽  
Option Pricing ◽  
Time Discretization ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Spectral Approximation ◽  
Barrier Option ◽  
Order Accuracy ◽  
Black Scholes ◽  
Richardson Extrapolation Method

Abstract We propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular $L1$ finite difference approximation, which converges with order $\mathcal {O}((\Delta \tau )^{2-\alpha })$ for functions which are twice continuously differentiable. However, when using the $L1$ scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

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.

scholarly journals OPTION PRICING IN MARKETS WITH INFORMED TRADERS

2020 ◽  
Vol 23 (06) ◽  
pp. 2050037 ◽  
Author(s):  
Yuan Hu ◽  
Abootaleb Shirvani ◽  
Stoyan Stoyanov ◽  
Young Shin Kim ◽  
Frank J. Fabozzi ◽  
...  
Keyword(s):  
Option Pricing ◽  
Stock Price ◽  
Stock Return ◽  
Asset Pricing Theory ◽  
Black Scholes ◽  
Informed Traders ◽  
Pricing Theory ◽  
Option Pricing Theory ◽  
Information Intensity ◽  
Dynamic Asset Pricing

The objective of this paper is to introduce the theory of option pricing for markets with informed traders within the framework of dynamic asset pricing theory. We introduce new models for option pricing for informed traders in complete markets, where we consider traders with information on the stock price direction and stock return mean. The Black–Scholes–Merton option pricing theory is extended for markets with informed traders, where price processes are following continuous-diffusions. By doing so, the discontinuity puzzle in option pricing is resolved. Using market option data, we estimate the implied surface of the probability for a stock upturn, the implied mean stock return surface, and implied trader information intensity surface.

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