scholarly journals Stochastic Volatility Estimation of Stock Prices using the Ensemble Kalman Filter

2021 ◽  
Vol 3 (2) ◽  
pp. 136-143
Author(s):  
Yudi Mahatma ◽  
Ibnu Hadi
Keyword(s):  
Kalman Filter ◽  
Interest Rate ◽  
Interest Rates ◽  
State Space ◽  
Stochastic Volatility ◽  
Ensemble Kalman Filter ◽  
Price Volatility ◽  
Call Option ◽  
Put Option ◽  
Black Scholes

AbstractVolatility plays important role in options trading.  In their seminal paper published in 1973, Black and Scholes assume that the stock price volatility, which is the underlying security volatility of a call option, is constant.  But thereafter, researchers found that the return volatility was not constant but conditional to the information set available at the computation time.  In this research, we improve a methodology to estimate volatility and interest rate using Ensemble Kalman Filter (EnKF).  The price of call and put option used in the observation and the forecasting step of the EnKF algorithm computed using the solution of Black-Scholes PDE.  The state-space used in this method is the augmented state space, which consists of static variables: volatility and interest rate, and dynamic variables: call and put option price. The numerical experiment shows that the EnKF algorithm is able to estimate accurately the estimated volatility and interest rates with an RMSE value of 0.0506.Keywords: stochastic volatility; call option; put option; Ensemble Kalman Filter. AbstrakVolatilitas adalah faktor penting dalam perdagangan suatu opsi.  Dalam makalahnya yang dipublikasikan tahun 1973, Black dan Scholes mengasumsikan bahwa volatilitas harga saham, yang merupakan volatilitas sekuritas yang mendasari opsi beli, adalah konstan. Akan tetapi, para peneliti menemukan bahwa volatilitas pengembalian tidaklah konstan melainkan tergantung pada kumpulan informasi yang dapat digunakan pada saat perhitungan.  Pada penelitian ini dikembangkan metodologi untuk mengestimasi volatilitas dan suku bunga menggunakan metode Ensembel Kalman Filter (EnKF).  Harga opsi beli dan opsi jual yang digunakan pada observasi dan pada tahap prakiraan pada algoritma EnKF dihitung menggunakan solusi persamaan Black-Scholes.  Ruang keadaan yang digunakan adalah ruang keadaan yang diperluas yang terdiri dari variabel statis yaitu volatilitas dan suku bunga, dan variabel dinamis yaitu harga opsi beli dan harga opsi jual. Eksperimen numerik menunjukkan bahwa algoritma ENKF dapat secara akurat mengestimasi volatiltas dan suku bunga dengan RMSE 0.0506.Kata kunci: volatilitas stokastik; opsi beli; opsi jual; Ensembel Kalman Filter.

Volatility integration of gold and crude oil prices with the interest rates in India

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Shailesh Rastogi ◽  
Adesh Doifode ◽  
Jagjeevan Kanoujiya ◽  
Satyendra Pratap Singh
Keyword(s):  
Interest Rate ◽  
Crude Oil ◽  
Interest Rates ◽  
Global Economy ◽  
Spillover Effect ◽  
Price Volatility ◽  
Volatility Spillover ◽  
Content Type ◽  
Crude Oil Prices ◽  
The Impact

PurposeCrude oil, gold and interest rates are some of the key indicators of the health of domestic as well as global economy. The purpose of the study is to find the shock volatility and price volatility effects of gold and crude oil market on interest rates in India.Design/methodology/approachThis study finds the mutual and directional association of the volatility of gold, crude oil and interest rates in India. The bi-variate GARCH models (Diagonal VEC GARCH and BEKK GARCH) are applied on the sample data of gold price, crude oil price and yield (interest rate) gathered from November 30, 2015 to November 16, 2020 (weekly basis) to investigate the volatility association including the volatility spillover effect in the three markets.FindingsThe main findings of the study focus on having a long-term conditional correlation between gold and interest rates, but there is no evidence of volatility spillover from gold and crude oil on the interest rates. The findings of the study are of great importance especially to the policymakers, as they state that the fluctuations in prices of gold and crude oil do not adversely impact the interest rates in India. Therefore, the fluctuations in prices of gold and crude may generally impact the economy, but it has nothing to do with interest rate in particular. This implies that domestic and foreign investments in the country will not be affected by gold and crude oil that are largely driven by interest rates in the country.Practical implicationsGold and crude oil are two very important commodities that have their importance not only for domestic affairs but also for international business. They veritably influence the economy including forex exchange for any nation. In addition to this, the researchers believe the findings will provide insights to policymakers, stakeholders and investors.Originality/valueGold and crude oil undoubtedly influence the exchange rates but their impact on the interest rates in an economy is not definite and remains ambiguous owing to the mixed findings of the studies. The lack of studies related to the impact of gold and crude oil on the interest rates, despite them being essentials for the health of any economy is the main motivation of this study. This study is novel as it investigates the volatility impact of crude oil and gold on interest rates and contributes to the existing literature with its findings.

scholarly journals TESTING OF THE BLACK SCHOLES AND GARCH MODELS IN LQ45 USING LONG STRADDLE STRATEGY IN 2009-2018

2021 ◽  
Vol 22 (1) ◽  
pp. 30-39
Author(s):  
Riko Hendrawan ◽  
Anggadi Sasmito
Keyword(s):  
Time Series Data ◽  
Mean Squared Error ◽  
Series Data ◽  
Call Option ◽  
Due Date ◽  
Average Percentage ◽  
Put Option ◽  
Option Contract ◽  
Black Scholes ◽  
Better Than

The purpose of this study is to examine the implementation of option contracts using Black Scholes and GARCH on the LQ45 index using the long straddle strategy. This study uses time-series data as a time frame for conducting research, using a sample of closing price data for the LQ 45 daily index for 2009-2018. For the test the model, we used the secondary data of the closing stock price index from February 28, 2009 to March 31, 2019The results of this study are seen by comparing the average percentage value of Average Mean Squared Error (AMSE) of Black Scholes and GARCH with the application of a long straddle strategy, where the smaller the percentage value, the better the model will be. Within one month of option contract due date, Black Scholes is better than GARCH, with an error value on the call option of 2.77% and the put option of 1.56%. Within two months of option contract due date, GARCH is better than Black Scholes, with an error value on the call option of 8.12% and the put option of 4.00%. Within three months of option contract due date, Black Scholes is better than GARCH, with an error value on the call option of 12.38% and on the put option of 5.50%. The long straddle strategy in the LQ45 index only reached a maximum of 60% of possible profits, with an average of around 30% possible profits.

scholarly journals A comparison of assimilation results from the ensemble Kalman Filter and a reduced-rank extended Kalman Filter

2003 ◽  
Vol 10 (6) ◽  
pp. 477-491 ◽  
Author(s):  
X. Zang ◽  
P. Malanotte-Rizzoli
Keyword(s):  
Kalman Filter ◽  
State Space ◽  
Extended Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Forecast Error ◽  
High Viscosity ◽  
Empirical Orthogonal Functions ◽  
Reduced Rank ◽  
Low Viscosity ◽  
Reduced State

Abstract. The goal of this study is to compare the performances of the ensemble Kalman filter and a reduced-rank extended Kalman filter when applied to different dynamic regimes. Data assimilation experiments are performed using an eddy-resolving quasi-geostrophic model of the wind-driven ocean circulation. By changing eddy viscosity, this model exhibits two qualitatively distinct behaviors: strongly chaotic for the low viscosity case and quasi-periodic for the high viscosity case. In the reduced-rank extended Kalman filter algorithm, the model is linearized with respect to the time-mean from a long model run without assimilation, a reduced state space is obtained from a small number (100 for the low viscosity case and 20 for the high viscosity case) of leading empirical orthogonal functions (EOFs) derived from the long model run without assimilation. Corrections to the forecasts are only made in the reduced state space at the analysis time, and it is assumed that a steady state filter exists so that a faster filter algorithm is obtained. The ensemble Kalman filter is based on estimating the state-dependent forecast error statistics using Monte Carlo methods. The ensemble Kalman filter is computationally more expensive than the reduced-rank extended Kalman filter.The results show that for strongly nonlinear case, chaotic regime, about 32 ensemble members are sufficient to accurately describe the non-stationary, inhomogeneous, and anisotropic structure of the forecast error covariance and the performance of the reduced-rank extended Kalman filter is very similar to simple optimal interpolation and the ensemble Kalman filter greatly outperforms the reduced-rank extended Kalman filter. For the high viscosity case, both the reduced-rank extended Kalman filter and the ensemble Kalman filter are able to significantly reduce the analysis error and their performances are similar. For the high viscosity case, the model has three preferred regimes, each with distinct energy levels. Therefore, the probability density of the system has a multi-modal distribution and the error of the ensemble mean for the ensemble Kalman filter using larger ensembles can be larger than with smaller ensembles.

THE ENTROPY THEORY OF STOCK OPTION PRICING

1999 ◽  
Vol 02 (03) ◽  
pp. 331-355 ◽  
Author(s):  
LES GULKO
Keyword(s):  
Option Pricing ◽  
Interest Rates ◽  
Stock Price ◽  
Price Change ◽  
Price Volatility ◽  
Stock Option ◽  
Market Uncertainty ◽  
Gamma Model ◽  
Black Scholes ◽  
Market Beliefs

An informationally efficient price keeps investors as a group in the state of maximum uncertainty about the next price change. The Entropy Pricing Theory (EPT) captures this intuition and suggests that, in informationally efficient markets, perfectly uncertain market beliefs must prevail. When the entropy functional is used to index the market uncertainty, then the entropy-maximizing market beliefs must prevail. The EPT resolves the ambiguity of asset valuation in incomplete markets, notably, the valuation of derivative securities. We use the EPT to derive a new stock option pricing model that is similar to Black–Scholes' with the lognormal distribution replaced by a gamma distribution. Unlike the Black–Scholes model, the gamma model does not restrict the dynamics of the stock price or the short-term interest rate. Option replication based on the gamma model accounts for random changes in the stock price, price volatility and interest rates.

PENENTUAN HARGA OPSI DENGAN MODEL BLACK-SCHOLES MENGGUNALKAN METODE BEDA HINGGA FORWARD TIME CENTRAL SPACE

2018 ◽  
Vol 1 (1) ◽  
pp. 45
Author(s):  
Werry Febrianti
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Option Price ◽  
Call Option ◽  
Put Option ◽  
Daily Data ◽  
Black Scholes ◽  
One Year ◽  
The Right

Option can be defined as a contract between two sides/parties said party one and party two. Party one has the right to buy or sell of stock to party two. Party two can invest by observe the put option price or call option price on a time period in the option contract. Black-Scholes option solution using finite difference method based on forward time central space (FTCS) can be used as the reference for party two in the investment determining. Option price determining by using Black-Scholes was applied on Samsung stock (SSNLF) by using finite difference method FTCS. Daily data of Samsung stock in one year was processed to obtain the volatility of the stock. Then, the call option and put option are calculated by using FTCS method after discretization on the Black-Scholes model. The value of call option was obtained as $1.457695030014260 and the put option value was obtained as $1.476925604670225.

scholarly journals PENENTUAN HARGA PREMI ASURANSI UNIT LINK DENGAN GARANSI MINIMUM

2019 ◽  
Vol 8 (3) ◽  
pp. 172
Author(s):  
NI LUH PUTU RATNA DEWI ◽  
I NYOMAN WIDANA ◽  
LUH PUTU IDA HARINI
Keyword(s):  
Interest Rate ◽  
Interest Rates ◽  
The Other ◽  
Other Hand ◽  
Black Scholes ◽  
The Interest Rate

The goal of this research is to determine the pricing of unit-linked insurance after attaching the minimum guarantees, which are guaranteed minimum maturity benefit (GMMB) and guaranteed minimum death benefit (GMDB) using the Black-Scholes-Merton Method. Before the new price is determined, the previous steps are find the value of  and  for GMMB or  and   for GMDB. The result of pricing on the cases in this research, resulted if the new price included GMMB with the interest rate 6% and management expenses 0% and 2% are changed from Rp 21.000.000,00 become Rp 21.003.000,00 and Rp 21.031.000,00. On the other hand, the new price for interest rates 14% and 20% with both management expenses are constant. Furthermore, the new price included GMDB with management expense 0% and 2% also interest rates 6%, 14%, and 20% in succession are changed from Rp 21.000.000,00 become Rp 25.132.000,00; Rp 21.031.000,00; Rp 21.002.000,00; Rp 44.521.000,00; Rp 44.520.000,00 and Rp 44.520.000,00.

RENORMALIZATION OF BLACK-SCHOLES EQUATION FOR STOCHASTICALLY FLUCTUATING INTEREST RATE

2001 ◽  
Vol 04 (04) ◽  
pp. 621-634
Author(s):  
ALEXANDER G. MUSLIMOV ◽  
NIKOLAI A. SILANT'EV
Keyword(s):  
Interest Rate ◽  
Option Pricing ◽  
Numerical Solution ◽  
Short Term ◽  
Stochastic Component ◽  
Put Option ◽  
Stochastic Fluctuations ◽  
Black Scholes ◽  
The Interest Rate ◽  
American Put

We investigate the effect of stochastic fluctuations of an interest rate on the value of a derivative. We derive the modified Black-Scholes equation that describes evolution of the value of a derivative averaged over an ensemble of stochastic fluctuations of the rate of interest and depends on the "renormalized" values of volatility and rate of interest. We present the explicit expressions for the renormalized volatility and interest rate that incorporate the corrections owing to the short-term stochastic variations of the interest rate. The stochastic component of the interest rate tends to enhance the effective volatility and reduce the effective interest rate that determine an evolution of the option pricing "smoothed out" over the stochastic variations. The results of numerical solution of the modified Black-Scholes equation with the renormalized coefficients are illustrated for an American put option on non-dividend-paying stock.

scholarly journals Estimasi Harga Multi-State European Call Option Menggunakan Model Binomial

CAUCHY ◽  
2011 ◽  
Vol 1 (4) ◽  
pp. 182
Author(s):  
Mila Kurniawaty, Endah Rokhmati ◽  
Endah Rokhmati
Keyword(s):  
Mean Square Error ◽  
Mean Square ◽  
Call Option ◽  
Expiration Date ◽  
Strike Price ◽  
Underlying Asset ◽  
Put Option ◽  
European Call Option ◽  
Black Scholes

Option merupakan kontrak yang memberikan hak kepada pemiliknya untuk membeli (call option) atau menjual (put option) sejumlah aset dasar tertentu (underlying asset) dengan harga tertentu (strike price) dalam jangka waktu tertentu (sebelum atau saat expiration date). Perkembangan option belakangan ini memunculkan banyak model pricing untuk mengestimasi harga option, salah satu model yang digunakan adalah formula Black-Scholes. Multi-state option merupakan sebuah option yang payoff-nya didasarkan pada dua atau lebih aset dasar. Ada beberapa metode yang dapat digunakan dalam mengestimasi harga call option, salah satunya masyarakat finance sering menggunakan model binomial untuk estimasi berbagai model option yang lebih luas seperti multi-state call option. Selanjutnya, dari hasil estimasi call option dengan model binomial didapatkan formula terbaik berdasarkan penghitungan eror dengan mean square error. Dari penghitungan eror didapatkan eror rata-rata dari masing-masing formula pada model binomial. Hasil eror rata-rata menunjukkan bahwa estimasi menggunakan formula 5 titik lebih baik dari pada estimasi menggunakan formula 4 titik.

THE ENTROPY THEORY OF BOND OPTION PRICING

2002 ◽  
Vol 05 (04) ◽  
pp. 355-383 ◽  
Author(s):  
LES GULKO
Keyword(s):  
Interest Rates ◽  
Term Structure ◽  
Price Change ◽  
Price Volatility ◽  
Market Uncertainty ◽  
Bond Price ◽  
Entropy Functional ◽  
Black Scholes ◽  
Option Model ◽  
Bond Option

An informationally efficient price keeps investors as a group in the state of maximum uncertainty about the next price change. The Entropy Pricing Theory (EPT) captures this intuition and suggests that, in informationally efficient markets, perfectly uncertain market beliefs must prevail. When the entropy functional is used to index collective market uncertainty, then the entropy-maximizing consensus beliefs must prevail. The EPT resolves the ambiguity of arbitrage-free valuation in incomplete markets. The EPT produces a new bond option model that is similar to Black–Scholes' with the lognormal distribution replaced by a beta distribution. Unlike alternative models, the beta model is valid for arbitrary term structure dynamics and for arbitrary credit risk of the underlying bonds. Option replication and hedging under the beta model accounts for random changes in the underlying bond price, price volatility and short-term interest rates.

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