Pricing of Path-Dependent European-Type Options Using Monte Carlo Simulation

American Option Pricing with Discrete and Continuous Time Models: An Empirical Comparison

This article proposes a closed pricing formula for European options when the return of the underlying asset follows extended normal distribution, that is, any different degrees of skewness and kurtosis relative to the normal distribution induced by the Black-Scholes model. The moment restriction is suggested, so that the pricing model under any arbitrary distribution for an underlying asset must satisfy the arbitrage-free condition. Numerical experiments and comparison of empirical performance of the proposed model with the Black-Scholes, ad hoc Black-Scholes, and Gram-Charlier distribution models are carried out. In particular, an estimation of implied parameters such as standard deviation, skewness, and kurtosis of the return on the underlying asset from the market prices of the KOSPI 200 index options is made, and in-sample and out-of-sample tests are performed. These results not only support the previous finding that the actual density of the underlying asset shows skewness to the left and high peaks, but also demonstrate that the present model has good explanatory power for option prices. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:845–871, 2005

Emphasis is placed on the valuation of plain vanilla option when the price process of underlying asset is described by the stochastic Verhulst-Gompertz Equation with network externality effects in a complete market. The method is based on the change of measure, Girsanov theorem and martingale valuation techinique. The application to an exchange option is made attempt and the valuation formula for this option like the Black-Scholes one is derived. A simple relationship similar to the put-call-parity between the exchange call option and the put option is provided. Our results will be useful to analyze and to hedge the price evolution at a sudden rise or crash of stock and commodity markets. 1 Introduction. The prices of some assets traded in the financial market have the ten- dency raised more when the prices begin to rise, it is because most traders expect that a price rises more and the purchase orders of other traders are induced as a result. Conversely, when the prices begin to fall, selling orders of other traders are induced. In the actual mar- ket, we often observe such a phenomenon called network externality price effects. These effects have been recognized as very significant concept in the field of marketing. Nerlove and Arrow(8) , Bass(2), Vidale and Wolfe(11) and Gould(4) made use of this deterministic version to study the dynamic behavior of new product and advertising policy. Their models were stimulated by a classical theory of logistic curve by Verhulst(10). Gompertz extended Verhulst's model to so-called a deterministic Verhulst-Gompertz model in order to investi- gate the growth of population. In the fields of finance, Schwartz(9) gave a stochastic version of Verhulst-Gompertz model and developed the random behavior of commodity price. His model was based on the biological problem studied by Goel and Richer-Dyn(3). The main purpose of this paper is devoted to describe the network externality effects by a generalized stochastic Verhulst-Gompertz equation and to derive pricing formulae for a plain vanilla and an exchange options when the price of underlying asset shows the network externality price effects. The organization of this paper is as follows. The next section briefly introduces the generalized stochastic Verhulst-Gompertz equation to express the network externality effects and in Section 3 we develope the martingale measure to evaluate the value of option. In Section 4 we derive a hedging strategy and a pricing formula of an exchange option that gives the holder the right which exchanges the underlying asset with the stochastic Verhulst-Gompertz equation for the asset with the geometric Brownian motion (Black-Scholes model) by means of martingale pricing method(7). If we regard the former as the spot price of energy commodity and the latter as the stock price of business firm that trades the energy commodity, such an exchang option will be very useful to hedge the sudden rise or heavy fall in price of energy and make a contribution to the stabilized price of energy. The paper concludes with a brief summary.

This paper obtains an analytic pricing formula for vulnerable European options. This formula allows the option writer to default at any time during the life of the option, and the recovery value of the option in default to be an amount equal to the writerfs asset-to-debt ratio multiplied by the economic equivalent of the option right, rather than its default-free value. Numerical results indicate that the economic equivalent of the option right, assumed as its default-free value, is inappropriate, and a default, restricted to occur only at the exercise date, seriously understates the impact of credit risk on option prices. Key words: vulnerable option, option pricing, credit risk

Multiscale stochastic elasticity of variance for options and equity linked annuity; A Mellin transform approach

In this article, we generalize the classical Edgeworth series expansion used in the option pricing literature. We obtain a closed-form pricing formula for European options by employing a generalized Hermite expansion for the risk neutral density. The main advantage of the generalized expansion is that it can be applied to heavy-tailed return distributions, a case for which the standard Edgeworth expansions are not suitable. We also show how the expansion coefficients can be inferred directly from market option prices.

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

This paper considers the introduction of stock options in an (dynamically) incomplete securities market made up of a riskless bond and the stock. The stock price follows a geometric Brownian motion with constant drift. However, there is incomplete information about the unknown stochastic volatility. The option price is determined by a uniform-price auction. Thus an option pricing formula results from the interaction of market participants relying on private information on the unknown stochastic volatility under an explicit market structure. This paper incorporates market microstructure considerations into an extended Black-Scholes model with incomplete information on the underlying volatility. It relies on the growing importance of auctionlike trading rules in financial markets.

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

In this paper, an additional factor is introduced into the Heston–Hull–White (HHW) hybrid model, which originally combines the Heston stochastic volatility model and the Hull–White stochastic interest rate model, to capture the correlation between the underlying price and the interest rate, while at the same time preserve the analytical tractability. With the analytical solution to the characteristic function of the underlying price being successfully derived, a closed‐form pricing formula for European options under the two‐factor HHW hybrid model is presented. Numerical experiments are also carried out to show the effect of the newly introduced factor. To further demonstrate the importance of introducing the correlation, we have also conducted an empirical study, comparing the performance of our model and that of the HHW model, based on European options written on S&P 500 index.

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.

The operation of digital systems are based on time slotting and it finds many applications particularly in communication networks. It is important to analyze a digital network on some digital concepts. In this paper we analyze the network buffer through discrete time queueing system in which queue is discrete in nature and it is used to analyze the performance of the communication networks buffer. Discrete time models can be used to represents such type of systems in terms of slotted time units to models any physical system. More than one state changes in discrete time models from one time unit to another. Discrete time markov chains are used to describes the behavior of the network under the specific conditions and various performance measures are evaluated such as queue length, mean waiting time and mean number in the system.

Modeling the stock price development as a geometric Brownian motion or, more generally, as a stochastic exponential of a diffusion, requires the use of specific statistical methods. For instance, the observations seldom reach us in the form of a continuous record and we are led to infer about diffusion coefficients from discrete time data. Next, often the classical assumption that the volatility is constant has to be dropped. Instead, a range of various stochastic volatility models is formed by the limiting transition from known volatility models in discrete time towards their continuous time counterparts. These are the main topics of the present survey. It is closed by a quick look beyond the usual Gaussian world in continuous time modeling by allowing a Levy process to be the driving process.

In energy markets, the use of quanto options has increased significantly in recent years. The payoff from such options are typically written on an underlying energy index and a measure of temperature. They are suited to managing the joint price and volume risk in energy markets. Using a Heath-Jarrow-Morton approach, we derive a closed-form option pricing formula for energy quanto options under the assumption that the underlying assets are lognormally distributed. Our approach encompasses several interesting cases, such as geometric Brownian motions and multifactor spot models. We also derive Delta and Gamma expressions for hedging. Further, we illustrate the use of our model by an empirical pricing exercise using NewYork Mercantile Exchange-traded natural gas futures and Chicago Mercantile Exchange-traded heating degree days futures for New York.

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.