In the information-based pricing framework of Brody, Hughston and Macrina, the market filtration $\{ \mathcal F_t\}_{t\geq 0}$ is generated by an information process $\{ ξ_t\}_{t\geq0}$ defined in such a way that at some fixed time $T$ an $\mathcal F_T$-measurable random variable $X_T$ is "revealed". A cash flow $H_T$ is taken to depend on the market factor $X_T$, and one considers the valuation of a financial asset that delivers $H_T$ at $T$. The value $S_t$ of the asset at any time $t\in[0,T)$ is the discounted conditional expectation of $H_T$ with respect to $\mathcal F_t$, where the expectation is under the risk neutral measure and the interest rate is constant. Then $S_{T^-} = H_T$, and $S_t = 0$ for $t\geq T$. In the general situation one has a countable number of cash flows, and each cash flow can depend on a vector of market factors, each associated with an information process. In the present work, we construct a new class of models for the market filtration based on the variance-gamma process. The information process is obtained by subordinating a particular type of Brownian random bridge with a gamma process. The filtration is taken to be generated by the information process together with the gamma bridge associated with the gamma subordinator. We show that the resulting extended information process has the Markov property and hence can be used to price a variety of different financial assets, several examples of which are discussed in detail.
This work extends the variance reduction method for the pricing of possibly path-dependent derivatives, which was developed in (Genin and Tankov, 2016) for exponential Lévy models, to affine stochastic volatility models (Keller-Ressel, 2011). We begin by proving a pathwise large deviations principle for affine stochastic volatility models. We then apply a time-dependent Esscher transform to the affine process and use Varadhan's lemma, in the fashion of (Guasoni and Robertson, 2008) and (Robertson, 2010), to approximate the problem of finding the Esscher measure that minimises the variance of t…
We unify and extend a number of approaches related to constructing multivariate Variance-Gamma (V.G.) models for option pricing. An overarching model is derived by subordinating multivariate Brownian motion to a subordinator from the Thorin (1977) class of generalised Gamma convolution subordinators. A class of models due to Grigelionis (2007), which contains the well-known Madan-Seneta V.G. model, is of this type, but our multivariate generalization is considerably wider, allowing in particular for processes with infinite variation and a variety of dependencies between the underlying processe…
In this chapter, we consider volatility swap, variance swap and VIX future pricing under different stochastic volatility models and jump diffusion models which are commonly used in financial market. We use convexity correction approximation technique and Laplace transform method to evaluate volatility strikes and estimate VIX future prices. In empirical study, we use Markov chain Monte Carlo algorithm for model calibration based on S&P 500 historical data, evaluate the effect of adding jumps into asset price processes on volatility derivatives pricing, and compare the performance of different …
In this paper, we consider the problem of pricing discretely-sampled variance swaps based on a hybrid model of stochastic volatility and stochastic interest rate with regime-switching. Our modelling framework extends the Heston stochastic volatility model by including the CIR stochastic interest rate and model parameters that switch according to a continuous-time observable Markov chain process. A semi-closed form pricing formula for variance swaps is derived. The pricing formula is assessed through numerical implementations, and the impact of including regime-switching on pricing variance swa…
This paper presents a multinomial method for option pricing when the underlying asset follows an exponential Variance Gamma process. The continuous time Variance Gamma process is approximated by a discrete time Markov chain with the same firsts four cumulants. This approach is particularly convenient for pricing American and Bermudan options, which can be exercised at any time up to expiration date. Numerical computations of European and American options are presented, and compared with results obtained with finite differences methods and with the Black Scholes model.
Using Maple, we compute a new exact series solution of a modified Black-Scholes equation, recently proposed, for the case of the Aunt Michaela option with a maturity condition of gamma type. We show that the modified Black-Scholes equation with the Aunt Michaela option is exactly solvable in terms of associated Laguerre polynomials or equivalently, in terms of Whittaker M functions. Finally, we make some numerical experiments with the analytical solutions
We prove that the variance swap rate (fair strike) equals the price of a co-terminal European-style contract when the underlying is an exponential Markov process, time-changed by an arbitrary continuous stochastic clock, which has arbitrary correlation with the driving Markov process, provided that the payoff function $G$ of the European contract satisfies an ordinary integro-differential equation, which depends only on the dynamics of the Markov process, not on the clock. We present examples of Markov processes where the function $G$ that prices the variance swap can be computed explicitly. I…
We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference method and the simulation method can be used for pricing under this model, but their speed is not satisfactory. So there is a need for fast but accurate approximation methods. In the case of Black-Merton-Scholes model, there are fast approximation methods, but they cannot be utiliz…
The present article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in Heath and Platen (2002) and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Lévy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. Carriere (1996), Longstaff and Schwartz (2001)). We exemplify the usefulness of o…
Paper is based on "The cost of illiquidity and its effects on hedging", L. C. G. Rogers and Surbjeet Singh, 2010. We generalize its thesis to constant elasticity model, which own previously used Black-Schoels model as a special case. The Goal of this article is to find optimal hedging strategy of European call/put option in illiquid environment. We understand illiquidity as a non linear transaction cost function depending only on rate of change of our portfolio. In case this function is quadratic, optimal policy is given by system of 3 PDE. In addition we show, that for small $ε$ costs of sell…
Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model
Samuel Curtis Johnson Graduate School of Management, Cornell University（康奈尔大学塞缪尔柯蒂斯约翰逊管理学院）；Division of Finance, Nanyang Business School, Nanyang Technological University（南洋理工大学南洋商学院金融系）
In liquid option markets, W-shaped implied volatility curves have occasionally be observed. We show that such shapes can be reproduced in a mixture of two variance-gamma models. This is in contrast to lognormal models, where at least three different distributions have to be mixed in order to produce a W-shape, as recently shown by Glasserman and Pirjol.
We extend the short rate model of Turfus and Romero-Bermúdez [2021] to facilitate accurate arbitrage-free analytic pricing of SOFR, SONIA or ESTR caplets, i.e. options on backward-looking compounded rates payments, in a manner consistent with the smile and skew levels observed in the market. These caplet pricing formulae and corresponding LIBOR or term-rate caplet results are translated into effective variance (implied volatility) formulae, which are seen to be of a particularly simple form. They show that the model is essentially equivalent to imposing on a Hull-White model an effective varia…
In this paper the zero vanna implied volatility approximation for the price of freshly minted volatility swaps is generalised to seasoned volatility swaps. We also derive how volatility swaps can be hedged using a strip of vanilla options with weights that are directly related to trading intuition. Additionally, we derive first and second order hedges for volatility swaps using only variance swaps. As dynamically trading variance swaps is in general cheaper and operationally less cumbersome compared to dynamically rebalancing a continuous strip of options, our result makes the hedging of volat…
The purpose of this article is to introduce a new Lévy process, termed Variance Gamma++ process, to model the dynamic of assets in illiquid markets. Such a process has the mathematical tractability of the Variance Gamma process and is obtained applying the self-decomposability of the gamma law. Compared to the Variance Gamma model, it has an additional parameter representing the measure of the trading activity. We give a full characterization of the Variance Gamma++ process in terms of its characteristic triplet, characteristic function and transition density. In addition, we provide efficient…
This study deals with the pricing and hedging of single-tranche collateralized debt obligations (STCDOs). We specify an affine two-factor model in which a catastrophic risk component is incorporated. Apart from being analytically tractable, this model has the feature that it captures the dynamics of super-senior tranches, thanks to the catastrophic component. We estimate the factor model based on the iTraxx Europe data with six tranches and four different maturities, using a quasi-maximum likelihood (QML) approach in conjunction with the Kalman filter. We derive the model-based variance-minimi…
Derivatives on the Chicago Board Options Exchange volatility index (VIX) have gained significant popularity over the last decade. The pricing of VIX derivatives involves evaluating the square root of the expected realised variance which cannot be computed by direct Monte Carlo methods. Least squares Monte Carlo methods can be used but the sign of the error is difficult to determine. In this paper, we propose new model independent upper and lower pricing bounds for VIX derivatives. In particular, we first present a general stochastic duality result on payoffs involving concave functions. This i…
The weak variance-alpha-gamma process is a multivariate Lévy process constructed by weakly subordinating Brownian motion, possibly with correlated components with an alpha-gamma subordinator. It generalises the variance-alpha-gamma process of Semeraro constructed by traditional subordination. We compare three calibration methods for the weak variance-alpha-gamma process, method of moments, maximum likelihood estimation (MLE) and digital moment estimation (DME). We derive a condition for Fourier invertibility needed to apply MLE and show in our simulations that MLE produces a better fit when th…
Implied volatility is at the very core of modern finance, notwithstanding standard option pricing models continue to derive option prices starting from the joint dynamics of the underlying asset price and the spot volatility. These models often cause difficulties: no closed formulas for prices, demanding calibration techniques, unclear maps between spot and implied volatility. Inspired by the practice of using implied volatility as quoting system for option prices, models for the joint dynamics of the underlying asset price and the implied volatility have been proposed to replace standard opti…
In this work, we study the value of an Asian option in the case of exponential Levy markets. More specifically, we are interested in the NIG (normal inverse Gaussian) the VG (variance gamma) models. The exponential Levy models produce incomplete markets. There are therefore an infinite number of equivalent martingale measures. We are interested in two methods of constructing of the risk-neutral measures. The first is based on the Esscher transform, and the other consists of bringing a risk-neutral correction on the dynamics of the trajectories. It turns out, according to the numerical results …
This paper investigates the hedging performance of pegged foreign exchange market in a regime switching (RS) model introduced in a recent paper by Drapeau, Wang and Wang (2019). We compare two prices, an exact solution and first order approximation and provide the bounds for the error. We provide exact RS delta, approximated RS delta as well as mean variance hedging strategies for this specific model and compare their performance. To improve the efficiency of the pricing and calibration procedure, the Fourier approach of this regime-switching model is developed in our work. It turns out that: …
Proof that under simple assumptions, such as constraints of Put-Call Parity, the probability measure for the valuation of a European option has the mean derived from the forward price which can, but does not have to be the risk-neutral one, under any general probability distribution, bypassing the Black-Scholes-Merton dynamic hedging argument, and without the requirement of complete markets and other strong assumptions. We confirm that the heuristics used by traders for centuries are both more robust, more consistent, and more rigorous than held in the economics literature. We also show that o…
In this paper, we relax the power parameter of instantaneous variance and develop a new stochastic volatility plus jumps model that generalize the Heston model and 3/2 model as special cases. This model has two distinctive features. First, we do not restrict the new parameter, letting the data speak as to its direction. The Generalized Methods of Moments suggests that the newly added parameter is to create varying volatility fluctuation in different period discovered in financial market. Moreover, upward and downward jumps are separately modeled to accommodate the market data. Our model is nov…
This paper describes another extension of the Local Variance Gamma model originally proposed by P. Carr in 2008, and then further elaborated on by Carr and Nadtochiy, 2017 (CN2017), and Carr and Itkin, 2018 (CI2018). As compared with the latest version of the model developed in CI2018 and called the ELVG (the Expanded Local Variance Gamma model), here we provide two innovations. First, in all previous papers the model was constructed based on a Gamma time-changed {\it arithmetic} Brownian motion: with no drift in CI2017, and with drift in CI2018, and the local variance to be a function of the …
The paper proposes an expanded version of the Local Variance Gamma model of Carr and Nadtochiy by adding drift to the governing underlying process. Still in this new model it is possible to derive an ordinary differential equation for the option price which plays a role of Dupire's equation for the standard local volatility model. It is shown how calibration of multiple smiles (the whole local volatility surface) can be done in such a case. Further, assuming the local variance to be a piecewise linear function of strike and piecewise constant function of time this ODE is solved in closed form …
