We investigate the limiting distribution of geometric Brownian motion conditional on its running maximum taking large values. The Freidlin-Wentzell large deviations theory predicts that the conditional distribution of the sample paths converge weakly to a deterministic exponential curve. We complement this result by showing that the conditional sample paths in fact converge in strong sense, and obtain quantitative bounds on the rate of convergence. As an application of our results to financial mathematics, we obtain new closed form asymptotic formulae for the fair price of barrier options with general path dependent payoff in the short maturity limit, with quantitative error estimates. We provide exact formulae for Asian and lookback style payoffs.
This paper proposes a Monte Carlo technique for pricing the forward yield to maturity, when the volatility of the zero-coupon bond is known. We make the assumption of deterministic default intensity (Hazard Rate Function). We make no assumption on the volatility of the yield. We actually calculate the initial value of the forward yield, we calculate the volatility of the yield, and we write the diffusion of the yield. As direct application we price options on Constant Maturity Treasury (CMT) in the Hull and White Model for the short interest rate. Tests results with Caps and Floors on 10 years…
A number of Bermudan option pricing methods that are applicable to options on multiple assets are studied in this thesis, one of the dominating questions being the natural scaling needed to extrapolate from Bermudan to American (both approximate and ``exact'') option prices.
Recent developments on financial markets have revealed the limits of Brownian motion pricing models when they are applied to actual markets. Lévy processes, that admit jumps over time, have been found more useful for applications. Thus, we suggest a Lévy model based on Forward-Backward Stochastic Differential Equations (FBSDEs) for option pricing in a Lévy-type market. We show the existence and uniqueness of a solution to FBSDEs driven by a Lévy process. This result is important from the mathematical point of view, and also, provides a much more realistic approach to option pricing.
In this short paper, in order to price occupation-time options, such as (double-barrier) step options and quantile options, we derive various joint distributions of a mixed-exponential jump-diffusion process and its occupation times of intervals.
This study deals with the problem of pricing compound options when the underlying asset follows a mixed fractional Brownian motion with jumps. An analytic formula for compound options is derived under the risk neutral measure. Then, these results are applied to value extendible options. Moreover, some special cases of the formula are discussed and numerical results are provided.
Barrieu, Rouault, and Yor [J. Appl. Probab. 41 (2004)] determined asymptotics for the logarithm of the distribution function of the Hartman-Watson distribution. We determine the asymptotics of the density. This refinement can be applied to the pricing of Asian options in the Black-Scholes model.
The purpose of this paper is to analyze the problem of option pricing when the short rate follows subdiffusive fractional Merton model. We incorporate the stochastic nature of the short rate in our option valuation model and derive explicit formula for call and put option and discuss the corresponding fractional Black-Scholes equation. We present some properties of this pricing model for the cases of $α$ and $H$. Moreover, the numerical simulations illustrate that our model is flexible and easy to implement.
This paper considers options pricing when the assumption of normality is replaced with that of the symmetry of the underlying distribution. Such a market affords many equivalent martingale measures (EMM). However we argue (as in the discrete-time setting of Klebaner and Landsman, 2007) that an EMM that keeps distributions within the same family is a "natural" choice. We obtain Black-Scholes type option pricing formulae for symmetric Variance-Gamma and symmetric Normal Inverse Gaussian models.
In this paper, we investigate a numerical algorithm for the pricing of swing options, relying on the so-called optimal quantization method. The numerical procedure is described in details and numerous simulations are provided to assert its efficiency. In particular, we carry out a comparison with the Longstaff-Schwartz algorithm.
We introduce a stacking version of the Monte Carlo algorithm in the context of option pricing. Introduced recently for aeronautic computations, this simple technique, in the spirit of current machine learning ideas, learns control variates by approximating Monte Carlo draws with some specified function. We describe the method from first principles and suggest appropriate fits, and show its efficiency to evaluate European and Asian Call options in constant and stochastic volatility models.
In previous works Avellaneda et al. pioneered the pricing and hedging of index options - products highly sensitive to implied volatility and correlation assumptions - with large deviations methods, assuming local volatility dynamics for all components of the index. We here present an extension applicable to non-Markovian dynamics and in particular the case of rough volatility dynamics.
The mixed fractional Brownian motion ($mfBm$) has become quite popular in finance, since it allows one to model long-range dependence and self-similarity while remaining, for certain values of the Hurst parameter, arbitrage-free. In the present paper, we propose approximate closed-form solutions for pricing arithmetic Asian options on an underlying described by the $mfBm$. Specifically, we consider both arithmetic Asian options and arithmetic Asian power options, and we obtain analytical formulas for pricing them based on a convenient approximation of the strike price. Both the standard $mfBm$…
This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develops a semi-group expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.
Pricing of European basket call option with n-assets and a bond is discussed in this paper, where all prices of n-assets and the bond are driven by Exponential Ornstein-Uhlenbeck processes. The close-form of European basket option pricing formula is derived. Utilizing with 1-order differential approximate numerical solution of stochastic differential equation (Milstein method), a simulation example of European basket option pricing with 3 assets is also given.
We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we explain -- by simple methods from chaos expansion -- why Bachelier's model yields good short-time approximations of prices and volatilities.
We study an optimal control problem related to swing option pricing in a general non-Markovian setting in continuous time. As a main result we show that the value process solves a first-order non-linear backward stochastic partial differential equation. Based on this result we can characterize the set of optimal controls and derive a dual minimization problem.
We develop an arbitrage-free random field LIBOR market model to price cross-currency derivatives. The uncertainty of the forward LIBOR rates of our cross-currency model is driven by a two time parameter random field instead of a finite dimensional Brownian motion. To demonstrate the applications of this model, we develop an approximate closed-form pricing formula for Quanto caps and cross-currency swaps. Further, we derive an exact pricing formula for an exchange rate option in the random field setting.
