Interest rate derivatives for the fractional Cox-Ingersoll-Ross model

In practical applications, there is the phenomenon of volatility smirk in Classic B–S option pricing formula. In order to eliminate this phenomenon, we add an independent compound Poisson process to geometric Brownian motion, and get the corresponding option price formula. This article has done two tasks: Firstly, we estimate the volatility of classic B–S option pricing formula with stock and warrant market data, and find that volatility smirk is obvious. Secondly, we estimate the volatility of our option pricing formula with stock and warrant market data, and find that our pricing formula eliminates the smirk, and is better than classical B–S formula.

Option pricing: Stock price, stock velocity and the acceleration Lagrangian

Using physical probabilistic measure of price process and the principle of fair premium , we deal with pricing formula of option on Foreign currency option under the assumption that foreign option price process driven by fractional Brownian motion process ,we obtain the pricing formula of foreign option.

A closed-form pricing formula for European options under the Heston model with stochastic interest rate

In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance and Stochastics 16 (3), 403–422, doi: https://doi.org/10.1007/s00780-012-0177-0 ] for Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ] SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models — models utilizing a variance process postulated by Heston [(1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327–343, doi: https://doi.org/10.1093/rfs/6.2.327 ]. In particular, we inspect in detail the approximation formula for the Bates [(1996), Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1), 69–107, doi: https://doi.org/10.1093/rfs/9.1.69 ] model with log-normal jump sizes and we provide a numerical comparison with the industry standard — Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behavior under a specific SVJ model.

By means of classical Ito calculus, we decompose option prices as the sum of the classical Black–Scholes formula, with volatility parameter equal to the root-mean-square future average volatility, plus a term due to correlation and a term due to the volatility of the volatility. This decomposition allows us to develop first- and second-order approximation formulas for option prices and implied volatilities in the Heston volatility framework, as well as to study their accuracy for short maturities. Numerical examples are given.

The LIBOR (London Inter-bank Offered Rate) is a benchmark interest rate widely used as an underlying interest rate for many interest-rate derivatives. The main purpose of this thesis is to build a cross-currency LIBOR market model with uncertainty modeled by a random field and to derive pricing formulas for interest-rate derivatives. The LIBOR market model (LMM) was one of the first attempts to build an arbitrage-free mathematical model to model the dynamics of LIBOR rates to price interest-rate derivatives. The uncertainty in the LMM is driven by a finite-dimensional Brownian motion. We will first investigate the random field LIBOR market model (RFLMM) which is a generalization of the LMM to random fields and random field forward rate models. The first contribution of this thesis is we give a new proof to derive a pricing formula for bond options in the random field setting using a change of measure technique using a Girsanov transform of random fields. Moreover, we derive exact closed-form pricing formulas for interest-rate caps and dual strike caps using the RFLMM. The first contribution of this thesis is we give a new proof to derive a pricing formula for bond options in the random field setting using a change of measure technique using a Girsanov transform of random fields. Moreover, we derive exact closed-form pricing formulas interest-rate caps and dual strike caps using the RFLMM. Next, we develop an arbitrage-free cross-currency random field forward rate model for bonds. The trade between multiple economies is linked using forward exchange rates. We derive a relationship between foreign and domestic forward measures to ensure pricing formulas derived in this setting for cross-currency interest-rate derivatives are arbitrage-free. The most important contribution of this thesis is the extension of the RFLMM to the cross-currency setting. Domestic LIBOR rates, foreign LIBOR rates, forward exchange rates for all maturity times cannot be modeled as lognormal random variables simultaneously. We find approximate closed-form pricing formulas for interest-rate caps written on foreign LIBOR rates expressed in domestic currency. Further, we derive pricing formulas for cross-currency swaps under different conditions. Finally, we investigate the pricing of basket caps written on LIBOR rates in multiple foreign economies.

Empirical Study of the Effect of Including Skewness and Kurtosis in Black Scholes Option Pricing Formula on S&P CNX Nifty Index Options

In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H > 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H < 1 / 2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.

A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility with Rough Market Price of Volatility Risk

An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching

Pricing Options with Complex Fourier Series

A general framework is formulated to price various forms of European style multi‐asset barrier options and occupation time derivatives with one state variable having the barrier feature. Based on the lognormal assumption of asset price processes, the splitting direction technique is developed for deriving the joint density functions of multi‐variate terminal asset prices with provision for single or double barriers on one of the state variables. A systematic procedure is illustrated whereby multi‐asset option price formulas can be deduced in a systematic manner as extensions from those of their one‐asset counterparts. The formulation has been applied successfully to derive the analytic price formulas of multi‐asset options with external two‐sided barriers and sequential barriers, multi‐asset step options and delayed barrier options. The successful numerical implementation of these price formulas is demonstrated. Corresponding author; e‐mail: maykwok@ust.hk; fax: (852) 2358‐1643

American basket option is a contract containing multiple underlying assets, and its payoff is correlated with average prices or weighted average prices of these assets on or before the expiration date. The type of option entitles a holder the right to trade at the strike price within a specified date, and this right can be waived. Therefore, there is a certain price to be paid for acquiring this right, which produces the problem of option pricing. A lot of literature shows that basket option price is usually cheaper than option portfolios on individual underlying assets. Based on this advantage, basket option becomes popular among investors. Consequently, this paper predominantly explores four types of American basket option pricing in uncertain financial environment. Specifically they are American arithmetic basket call option, American arithmetic basket put option, American geometric basket call option and American geometric basket put option. Assuming that these stocks prices follow corresponding uncertain differential equations, we derive corresponding option pricing formulas. Some numerical examples are taken to illustrate the feasibility of pricing formulas. Simultaneously, this paper discusses the relationship between option price and some parameters.

In this paper, we propose two new representation formulas for the conditional marginal probability density of the multi-factor Heston model. The two formulas express the marginal density as a convolution with suitable Gaussian kernels whose variances are related to the conditional moments of price returns. Via asymptotic expansion of the non-Gaussian function in the convolutions, we derive explicit formulas for European-style option prices and implied volatility. The European option prices can be expressed as Black–Scholes style terms plus corrections at higher orders in the volatilities of volatilities, given by the Black–Scholes Greeks. The explicit formula for the implied volatility clearly identifies the effect of the higher moments of the price on the implied volatility surface. Further, we derive the relationship between the VIX index and the variances of the two Gaussian kernels. As a byproduct, we provide an explanation of the bias between the VIX and the volatility of total returns, which offers support to recently proposed methods to compute the variance risk premium. Via a series of numerical exercises, we analyse the accuracy of our pricing formula under different parameter settings for the one- and two-factor models applied to index options on the S&P500 and show that our approximation substantially reduces the computational time compared to that of alternative closed-form solution methods. In addition, we propose a simple approach to calibrate the parameters of the multi-factor Heston model based on the VIX index, and we apply the approach to the double and triple Heston models.