Corridor Volatility Risk and Expected Returns

The World Price of Jump and Volatility Risk

Option prices in the market embed a probability distribution over the option’s payoff, which can be fairly easily extracted. But in this risk-neutral distribution, the market’s true expectations about future returns are distorted by risk preferences, in particular, aversion to volatility risk. To separate the returns forecasts from risk aversion, one needs another way to estimate the market’s probability estimates, which the author calls “actuarial probabilities.” This difficult problem becomes even harder in the swaptions market, because the risk exposure for a swaption depends on two different time horizons. For example, a 2 × 5 swaption has two years (“expiry”) of optionality culminating with the option to enter into a swap with a maturity (“tail”) of five years. The volatility surface is three-dimensional, with two time variables and the strike interest rate (the “volatility cube”), and it is difficult to estimate statistically using available data on long-lived contracts. The author develops a procedure based on extracting a small number of common factors of the volatility process using principal components. Since all swaptions are exposed to these factors, with different loadings, modeling the evolution of the principal components, rather than the specific swaption volatilities, allows a more efficient use of the data. Applying this approach to about 10 years of swaption data, the article reports several interesting results, including the fact that three principal components capture a large proportion of the variation in actuarial volatilities across the volatility cube, while a couple extra PCs are needed to achieve the same level of explanatory power for market implied volatilities. One clear result is that the market price of volatility risk is time-varying, and it moves rather quickly, with rates of mean-reversion that seem much faster than for actuarial volatility, particularly for short expiry–long tail swaptions. Another interesting result is that the volatility risk premium reverses sign, being largely positive (market IV > actuarial IV) for contracts with short expiry, but becoming negative for long expiry contracts.

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

In the financial world, the thing that is interesting for investors today is to predict a reasonable Asian Call Option price that does not cause the seller or buyer of the option to experience losses. The constant interest assumption is more often used by previous research as an assumption in predicting the price of Asian Call Options. This is not in line with the fact that interest rate movements are not constant. The use of incorrect assumptions will affect the accuracy of prediction results. In overcoming this problem, this study aims to predict the price of Asian Call Options with non-constant interest rates using Monte Carlo simulation. The non-constant interest rate model used is Cox-Ingersoll-Ross (CIR) because this model produces positive interest rates. This study begins with a screening of Normally distributed interest rate data. The data is used to determine the parameters of the CIR interest rate model. Next, an Asian Call Option price simulation with non-constant interest rates was carried out and ended by analyzing a graph of the Asian Call Option price simulation results with non-constant interest rates. The simulated price result of the Asian Call Option with non-constant interest rates converged at the value of $170.82. Based on the price chart of Asian Call Options with different strike prices, the larger the strike price causes, the larger the Asian Call Option price, which corresponds to the price characteristics of the Asian Call Option. In addition, by extending the maturity period, the price of Asian Call Options with non-constant interest rates is greater according to the price characteristics of Asian Call Options.

Volumetric Risk Hedging Strategies and Basis Risk Premium for Solar Power

The Indonesian stock market, especially PT Bank Central Asia (BCA) stock experiences high volatility. Therefore, this research is very important to provide a deeper understanding of the pricing of put and call options as a risk management instrument. This study aims to analyze the pricing of put and call options on PT Bank Central Asia (BCA) shares using the Black-Scholes model and identify factors that affect stock price fluctuations on option prices. This study aims to understand the changes in put and call option prices on PT Bank Central Asia (BCA) shares caused by fluctuations in stock prices. This study uses historical data of PT Bank Central Asia (BCA) shares, volatility, and current interest rates as the basis for analysis. The Black-Scholes method is used as a framework to analyze the pricing of put and call options by considering volatility and interest rates. The results of this study analyze the price changes of put and call options in the face of stock price fluctuations and the factors that affect option prices so as to provide insight into risk management and investment decision making. This research is expected to provide practical guidance for investors and risk managers in making decisions related to put and call options on PT Bank Central Asia (BCA) shares.

The Information Content of The Implied Volatility Surface: Can Option Prices Predict Jumps?

The Pricing of Volatility Risk in the US Equity Market

We introduced a new method to compute the European Call (and Put) Option price under the assumption of multifractional Brownian motion (mBm). The reason why we need a procedure for estimating the Option price is due to the absence of a closed formula for this process. To compute the Option price, we first simulated the logarithmic price under mBm and, by using a discount factor, we computed the option’s pay-off. Then, we fitted the best probability distribution associated to the discounted pay-off, computing the European Call Option price as its average.

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

The binomial formula given by Cox, Ross and Rubinstein (1979) is a tool for valuating the call option price. It is well known that the price from binomial formula converges to the price from Black-Scholes formula which was given by Black, Scholes and Merton (1973) as the number of periods (n) converges to infinity. In 1988, Boyle introduced the trinomial formula which is another tool for calculating call option price. He considered the trinomial formula in the case that the rising rate of a stock price is u = e λσ√T n and the falling rate of the stock price is d = u −1 , where T is maturity time, σ is volatility and λ > 1. After that, Entit et al. (2013) gave examples which show that the call option price from trinomial formula is closed to the call option price from the Black-Scholes formula. In this thesis, we give the rigorous proof of this conjecture by showing that the trinomial formula converges to the Black-Scholes formula. Moreover, we prove that the rate of this convergence is 1 √ n .

We develop a new high-performance spectral collocation method for the computation of American put and call option prices. The proposed algorithm involves a carefully posed Jacobi-Newton iteration for the optimal exercise boundary, aided by Gauss-Legendre quadrature and Chebyshev polynomial interpolation on a certain transformation of the boundary. The resulting scheme is straightforward to implement and converges at a speed several orders of magnitude faster than existing approaches. Computational effort depends on required accuracy; at precision levels similar to, say, those computed by a finite-difference grid with several hundred steps, the computational throughput of the algorithm in the Black-Scholes model is typically close to 100 000 option prices per second per CPU. For benchmarking purposes, Black-Scholes American option prices can generally be computed to ten or eleven significant digits in less than one-tenth of a second.

Investment is a saving activity with the aim of overcoming price increases or often called inflation. Investments can be in the form of gold, property, silver or stock investments. Stock investment is considered more profitable than just saving at a bank. Currency values are declining due to inflation. This results in a tendency to invest in shares. Stock investment carries a great risk. Therefore, in 2004 stock options began to trade. Stock options are contracts that give the holder the right to buy / sell shares at the agreed time, at a certain price. Stock option prices tend to be cheaper than stock prices. Therefore, determining the right price of stock options is needed. In this study, we will focus on the European type of buying options, the right to buy shares at an agreed price at maturity. The purpose of this study is the completion of the Black-Scholes model of European type option prices at a constant market, assuming stock movements meet the stochastic differential equation, fixed risk-free interest rates, companies distributing dividends, no taxes, no transaction costs, and free market arbitration. The results of this research are in the form of differential equations and the settlement of the Black-Scholes model of European type call option prices, and a case study used by stock option contracts with a maturity of January 4, 2010, PT Aqua Golden Mississippi Tbk.

Identifying Volatility Risk Premia from Fixed Income Asian Options

We use a novel pricing model to filter times series of diffusive volatility and jump intensity from S&P 500 index options. These two measures capture the ex-ante risk assessed by investors. We find that both components of risk vary substantially over time, are quite persistent, and correlate with each other and with the stock index. Using a simple general equilibrium model with a representative investor, we translate the filtered measures of ex-ante risk into an ex-ante risk premium. We find that the average premium that compensates the investor for the risks implicit in option prices, 10.1 percent, is about twice the premium required to compensate the same investor for the realized volatility, 5.8 percent. Moreover, the ex-ante equity premium that we uncover is highly volatile, with values between 2 and 32 percent. The component of the premium that corresponds to the jump risk varies between 0 and 12 percent.