scholarly journals An Asymptotic Solution for Call Options on Zero-Coupon Bonds

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1940
Author(s):  
Michael J. Tomas Tomas III ◽  
Jun Yu
Keyword(s):  
Stochastic Process ◽  
Asymptotic Solution ◽  
Interest Rates ◽  
Perturbation Approach ◽  
Return Volatility ◽  
Leading Order ◽  
Bond Price ◽  
Call Options ◽  
Black Scholes Model ◽  
Black Scholes

We present an asymptotic solution for call options on zero-coupon bonds, assuming a stochastic process for the price of the bond, rather than for interest rates in general. The stochastic process for the bond price incorporates dampening of the price return volatility based on the maturity of the bond. We derive the PDE in a similar way to Black and Scholes. Using a perturbation approach, we derive an asymptotic solution for the value of a call option. The result is interesting, as the leading order terms are equivalent to the Black–Scholes model and the additional next order terms provide an adjustment to Black–Scholes that results from the stochastic process for the price of the bond. In addition, based on the asymptotic solution, we derive delta, gamma, vega and theta solutions. We present some comparison values for the solution and the Greeks.

Pricing resettable convertible bonds using an integral equation approach

2019 ◽  
Vol 31 (4) ◽  
pp. 417-443
Author(s):  
Sha Lin ◽  
Song-Ping Zhu
Keyword(s):  
Integral Equation ◽  
Convertible Bonds ◽  
Convertible Bond ◽  
Equation Approach ◽  
Fair Price ◽  
Bond Price ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Integral Equation Approach ◽  
Increasing Function

Abstract In this paper, the fair price of an American-style resettable convertible bond (CB) under the Black–Scholes model with a particular reset clause is calculated. This is a challenging problem because an unknown optimal conversion price needs to be determined together with the bond price. There is also an additional complexity that the value of the conversion ratio will change when the underlying price touches the reset price. Because of the additional reset clause, the bond price is not always a monotonically increasing function with the underlying price, which is impossible for other types of the CBs. Of course, the problem can be dealt with using the Monte-Carlo simulation. But, a partial differential equation (PDE)/integral equation approach is far superior in terms of computational efficiency. Fortunately, after establishing the PDE system governing the bond price, we are able to present an integral equation representation by applying the incomplete Fourier transform on the PDE system.

UNCERTAINTY IN PRICING TRADABLE OPTIONS

2003 ◽  
Vol 06 (02) ◽  
pp. 103-117 ◽  
Author(s):  
JORGE R. SOBEHART ◽  
SEAN C. KEENAN
Keyword(s):  
Interest Rates ◽  
Market Equilibrium ◽  
Expected Value ◽  
Options Pricing ◽  
Options Market ◽  
Pricing Model ◽  
Random Delays ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Random Fluctuations

In this paper we introduce an options pricing model consistent with the level of uncertainty observed in the options market. By assuming that the price at which an option can be traded is intrinsically uncertain, either because of the inability to hedge continuously or because of errors in the estimation of the security's volatility and interest rates, random delays in the execution of orders or information deficiencies, we show that the Black-Scholes model produces a biased estimate of the expected value of tradable options. Information deficiencies lead to a call-put relationship that reduces to the standard call-put expression on average but shows random fluctuations consistent with the concept of market equilibrium. The same information deficiencies can contribute to the volatility skew that affects the Black-Scholes model.

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.

scholarly journals Black-Scholes Model in Determining European Option Prices on Netflix,Inc.

d'CARTESIAN ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 80
Author(s):  
Desty A. Tambingon ◽  
Jullia Titaley ◽  
Tohap Manurung
Keyword(s):  
Share Price ◽  
Option Prices ◽  
European Option ◽  
Put Options ◽  
Call Options ◽  
Price Data ◽  
Black Scholes Model ◽  
Black Scholes

Research has been conducted to compare the prices of European option on the Yahoo Finance website with prices obtained from the Black-Scholes model (theoretical price). Data was taken on January 31, 2019 which included the daily share price of Netflix, Inc. (NFLX) on February 14, 2018 - January 31, 2019 to obtain volatility, and NFLX options data due on January 17, 2020. Options with prices lower than theoretical prices are said to be underpriced, so the decision taken is to buy the options. Whereas options with prices higher than theoretical prices are said to be overpriced, so it has to be reconsidered. The proportion of the underpriced call options for the total number of call options is 77.7778%, while the proportion of the underpriced put options for the total number of put options is 38.5714%.

scholarly journals Stochastic pricing formulation for hybrid equity warrants

2021 ◽  
Vol 7 (1) ◽  
pp. 398-424
Author(s):  
Teh Raihana Nazirah Roslan ◽  
◽  
Sharmila Karim ◽  
Siti Zulaiha Ibrahim ◽  
Ali Fareed Jameel ◽  
...  
Keyword(s):  
Interest Rates ◽  
Stochastic Volatility ◽  
Statistical Error ◽  
Optimization Method ◽  
Calibration Procedure ◽  
Stochastic Interest Rates ◽  
Black Scholes Model ◽  
Stochastic Interest ◽  
Black Scholes ◽  
Pricing Formula

<abstract> <p>A warrant is a financial agreement that gives the right but not the responsibility, to buy or sell a security at a specific price prior to expiration. Many researchers inadvertently utilize call option pricing models to price equity warrants, such as the Black Scholes model which had been found to hold many shortcomings. This paper investigates the pricing of equity warrants under a hybrid model of Heston stochastic volatility together with stochastic interest rates from Cox-Ingersoll-Ross model. This work contributes to exploration of the combined effects of stochastic volatility and stochastic interest rates on pricing equity warrants which fills the gap in the current literature. Analytical pricing formulas for hybrid equity warrants are firstly derived using partial differential equation approaches. Further, to implement the pricing formula to realistic contexts, a calibration procedure is performed using local optimization method to estimate all parameters involved. We then conducted an empirical application of our pricing formula, the Black Scholes model, and the Noreen Wolfson model against the real market data. The comparison between these models is presented along with the investigation of the models' accuracy using statistical error measurements. The outcomes revealed that our proposed model gives the best performance which highlights the crucial elements of both stochastic volatility and stochastic interest rates in valuation of equity warrants. We also examine the warrants' moneyness and found that 96.875% of the warrants are in-the-money which gives positive returns to investors. Thus, it is beneficial for warrant holders concerned in purchasing warrants to elect the best warrant with the most profitable and more benefits at a future date.</p> </abstract>

Strategi Pengelolaan Risiko dan Peningkatan Kualitas Pengelolaan Kas Negara

2017 ◽  
Vol 2 (2) ◽  
pp. 43-54
Author(s):  
Agung Mulyono
Keyword(s):  
Interest Rates ◽  
Financial Risk ◽  
Vector Autoregression ◽  
Secondary Market ◽  
Key Factors ◽  
Bond Price ◽  
Hedging Strategy ◽  
Foreign Loan ◽  
Term Operation ◽  
Loan Interest Rates

Cash management is  one of treasury’s main functions in which has a potential financial risk. A potential financial risk emerges when State Treasurer manages cash surplus and or/ shortages in order to maintain optimum liquidity. By applying Vector Autoregression (VAR) system on empirical data provided by Bank Indonesia and the Ministry of Finance of Indonesia, we found that currency value  flunctuation is a significant factor for repayment value of foreign loan. Interest rates and amount of government’s bond held by foreign investors are also variables impacted on government’s bond price movement in secondary market. Currency value  flunctuation and price of government’s bond in secondary market are the key factors that have to be considered by State Treasurer (BUN) in managing state’s money. Hedging strategy by using derivatif product is possible to be utilized by State Treasurer (BUN) due to it’s flexibility for short-term operation.   Abstrak Pengelolaan kas negara merupakan salah satu fungsi pokok perbendaharaan yang dalam proses pelaksanaannya menyimpan potensi berbagai risiko keuangan. Risiko keuangan, khususnya dalam investasi berpotensi muncul ketika Bendahara Umum Negara (BUN) melakukan kegiatan pengelolaan kelebihan dan/ kekurangan kas dalam rangka menjamin ketersediaan dan optimalisasi kas. Dengan menggunakan analisis Vector Autoregression (VAR) atas data empiris yang diperoleh dari Bank Indonesia dan Kementerian Keuangan Indonesia, penulis menemukan bahwa fluktuasi nilai tukar mata uang merupakan faktor yang signifikan terhadap besaran pembayaran utang luar negeri pemerintah. Tingkat suku bunga acuan dan pergerakan besaran kepemilikan SUN oleh investor asing juga merupakan variabel yang berpengaruh terhadap pergerakan harga SUN di pasar sekunder. Fluktuasi nilai tukar mata uang dan pergerakan harga SUN di pasar sekunder menjadi faktor penting dalam pelaksanaan investasi yang dilakukan BUN dalam rangka pengelolaan kelebihan dan/ kekurangan kas. Berdasarkan hasil tersebut, strategi pengelolaan risiko atau hedging dengan menggunakan produk-produk derivatif dalam pengelolaan kelebihan dan/ kekurangan kas jangka pendek – menengah sangat dimungkinkan karena sifat instrumen derivatif yang fleksibel.

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