scholarly journals A SIMPLE TIME-CONSISTENT MODEL FOR THE FORWARD DENSITY PROCESS

2013 ◽  
Vol 16 (08) ◽  
pp. 1350048
Author(s):  
HENRIK HULT ◽  
FILIP LINDSKOG ◽  
JOHAN NYKVIST
Keyword(s):  
Brownian Motion ◽  
Particle Filtering ◽  
Option Price ◽  
Time Varying ◽  
Option Prices ◽  
Price Data ◽  
Consistent Model ◽  
Forward Contract ◽  
Future Value ◽  
Density Process

In this paper, a simple model for the evolution of the forward density of the future value of an asset is proposed. The model allows for a straightforward initial calibration to option prices and has dynamics that are consistent with empirical findings from option price data. The model is constructed with the aim of being both simple and realistic, and avoid the need for frequent re-calibration. The model prices of n options and a forward contract are expressed as time-varying functions of an (n + 1)-dimensional Brownian motion and it is investigated how the Brownian trajectory can be determined from the trajectories of the price processes. An approach based on particle filtering is presented for determining the location of the driving Brownian motion from option prices observed in discrete time. A simulation study and an empirical study of call options on the S&P 500 index illustrate that the model provides a good fit to option price data.

Option pricing under the fractional stochastic volatility model

2021 ◽  
Vol 63 ◽  
pp. 123-142
Author(s):  
Yuecai Han ◽  
Zheng Li ◽  
Chunyang Liu
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

OPTION PRICING UNDER THE FRACTIONAL STOCHASTIC VOLATILITY MODEL

2021 ◽  
pp. 1-20
Author(s):  
Y. HAN ◽  
Z. LI ◽  
C. LIU
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Option Prices ◽  
European Call Option ◽  
Volatility Model ◽  
The Impact

Abstract We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented.

Analytical Approximations of American Call Options with Discrete Dividends

2018 ◽  
Vol 26 (3) ◽  
pp. 283-310
Author(s):  
Kwangil Bae
Keyword(s):  
Brownian Motion ◽  
Stock Prices ◽  
Stock Price ◽  
Computing Time ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Option Prices ◽  
Call Options ◽  
Analytical Approximations ◽  
Discrete Dividends

In this study, we assume that stock prices follow piecewise geometric Brownian motion, a variant of geometric Brownian motion except the ex-dividend date, and find pricing formulas of American call options. While piecewise geometric Brownian motion can effectively incorporate discrete dividends into stock prices without losing consistency, the process results in the lack of closed-form solutions for option prices. We aim to resolve this by providing analytical approximation formulas for American call option prices under this process. Our work differs from other studies using the same assumption in at least three respects. First, we investigate the analytical approximations of American call options and examine European call options as a special case, while most analytical approximations in the literature cover only European options. Second, we provide both the upper and the lower bounds of option prices. Third, our solutions are equal to the exact price when the size of the dividend is proportional to the stock price, while binomial tree results never match the exact option price in any circumstance. The numerical analysis therefore demonstrates the efficiency of our method. Especially, the lower bound formula is accurate, and it can be further improved by considering second order approximations although it requires more computing time.

scholarly journals Limit theorems for prices of options written on semi-Markov processes

2021 ◽  
Vol 105 (0) ◽  
pp. 3-33
Author(s):  
E. Scalas ◽  
B. Toaldo
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Option Price ◽  
Type Equation ◽  
Geometric Brownian Motion ◽  
Bernstein Function ◽  
Option Prices ◽  
Motion Time ◽  
Semi Markov Processes ◽  
Inverse Stable Subordinator

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

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 HOLDER-EXTENDIBLE EUROPEAN OPTION: CORRECTIONS AND EXTENSIONS

2015 ◽  
Vol 56 (4) ◽  
pp. 359-372 ◽  
Author(s):  
PAVEL V. SHEVCHENKO
Keyword(s):  
Brownian Motion ◽  
Interest Rate ◽  
Real Life ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Fixed Amount ◽  
Underlying Asset ◽  
Closed Form Solutions ◽  
Pricing Options ◽  
Interest Rate Volatility

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

scholarly journals On Market Completions Approach to Option Pricing

2021 ◽  
Vol 9 (3) ◽  
pp. 77-93
Author(s):  
I. Vasilev ◽  
A. Melnikov
Keyword(s):  
Option Pricing ◽  
Incomplete Markets ◽  
Option Price ◽  
Optimal Investment ◽  
Market Model ◽  
Option Prices ◽  
End Points ◽  
Hedging Strategies ◽  
Risk Neutral ◽  
Interesting Approach

Option pricing is one of the most important problems of contemporary quantitative finance. It can be solved in complete markets with non-arbitrage option price being uniquely determined via averaging with respect to a unique risk-neutral measure. In incomplete markets, an adequate option pricing is achieved by determining an interval of non-arbitrage option prices as a region of negotiation between seller and buyer of the option. End points of this interval characterise the minimum and maximum average of discounted pay-off function over the set of equivalent risk-neutral measures. By estimating these end points, one constructs super hedging strategies providing a risk-management in such contracts. The current paper analyses an interesting approach to this pricing problem, which consists of introducing the necessary amount of auxiliary assets such that the market becomes complete with option price uniquely determined. One can estimate the interval of non-arbitrage prices by taking minimal and maximal price values from various numbers calculated with the help of different completions. It is a dual characterisation of option prices in incomplete markets, and it is described here in detail for the multivariate diffusion market model. Besides that, the paper discusses how this method can be exploited in optimal investment and partial hedging problems.

scholarly journals PENENTUAN HARGA OPSI BELI ATAS SAHAM PT. ANTAM (PERSERO) MENGGUNAKAN MODEL BINOMIAL FUZZY

2018 ◽  
Vol 19 (1) ◽  
pp. 8-24
Author(s):  
Agung Prabowo ◽  
Zulfatul Mukarromah ◽  
Lisnawati Lisnawati ◽  
Pramono Sidi
Keyword(s):  
Stock Price ◽  
Market Price ◽  
Option Price ◽  
Risk Level ◽  
Binomial Model ◽  
Purchase Price ◽  
Option Prices ◽  
Movement Data ◽  
Curve Representation ◽  
Price Option

Option is a financial instrument where price depends on the underlying stock price. The pricing of options, both selling options and purchase options, may use the CRR (Cox-Ross-Rubinstein) binomial model. Only two possible parameters were used that is u if the stock price rises and d when the stock price down. One of the elements that determine option prices is volatility. In the binomial model CRR volatility is constant. In fact, the financial market price of stocks fluctuates so that volatility also fluctuates. This article discusses volatility of fluctuating stock price movements by modeling it using binomial fuzzy with triangular curve representation. The analysis is carried out in relation to the existence of three interpretations of the triangular curve representation resulting in different degrees of membership. In addition to volatility, this study added the size or risk level ρ. As an illustration, this study used stock price movement data from PT. Antam (Persero) from August 2015 until July 2016. The results of one period obtained from the purchase price option for August 2016 with the largest volatility, medium and smallest respectively were Rp.143,43, Rp.95,49, and Rp.79,00. There was calculated at the risk level of  ρ = 90%. The degree of membership for each option price varies depending on the interpretation of the triangle curve representation.   Opsi merupakan instrumen keuangan yang harganya tergantung pada harga saham yang mendasarinya. Penentuan harga opsi, baik opsi jual maupun opsi beli dapat menggunakan model binomial CRR (Cox-Ross-Rubinstein). Dalam model ini hanya dimungkinkan adanya dua parameter yaitu u apabila harga saham naik dan d pada saat harga saham turun. Salah satu unsur yang menentukan harga opsi adalah volatilitas. Dalam model binomial CRR digunakan volatilitas yang bersifat konstan. Padahal, pada pasar keuangan pergerakan harga saham mengalami fluktuasi sehingga volatilitas juga menjadi fluktuatif. Artikel ini membahas volatilitas pergerakan harga saham yang fluktuatif dengan memodelkannya menggunakan binomial fuzzy dengan representasi kurva segitiga. Analisis dilakukan terkait dengan adanya tiga interpretasi terhadap representasi kurva segitiga tersebut yang menghasilkan derajat keanggotaan yang berbeda. Selain volatilitas, dalam penelitian ini ditambahkan ukuran atau tingkat risiko ρ. Sebagai ilustrasi, digunakan data pergerakan harga saham PT. Antam (Persero) dari Agustus 2015 hingga Juli 2016. Hasil penelitian dengan perhitungan satu periode diperoleh hasil harga opsi beli untuk bulan Agustus 2016 dengan volatilitas terbesar, menengah, dan terkecil masing-masing adalah Rp.143,43, Rp.95,49, dan Rp.79,00 yang dihitung pada tingkat risiko ρ = 90%. Derajat keanggotaan untuk masing-masing harga opsi berbeda-beda tergantung pada interpretasi dari representasi kurva segitiga.

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