Narrow Results

Uncertain differential equation with jumps is a type of differential equation driven by two classes of uncertain processes, namely canonical process and renewal process. Based on uncertain differential equation with jumps, this paper proposes a stock model with jumps for uncertain financial markets. Furthermore, the European call and put option pricing formulas for the stock model are formulated and some mathematical properties of them are studied. Finally, some generalized uncertain stock models with jumps are discussed.

The option-pricing problem is an important topic in modern finance. In this paper, we propose a stock model with varying stock diffusion based on uncertainty theory. The European option pricing formulas are derived from the proposed uncertain stock model, and some mathematical properties of these formulas are investigated. Moreover, extended uncertain stock models are introduced and discussed. Finally, numerical examples are given to illustrate the proposed model.

It is well known that as the time interval between two consecutive observations shrinks to zero, a properly constructed GARCH model will weakly converge to a bivariate diffusion. Naturally the European option price under the GARCH model will also converge to its bivariate diffusion counterpart. This paper investigates the convergence speed of the GARCH option price. We show that the European option prices under the two corresponding models are equal up to an order near the square root of the length of discrete time interval.

This paper continues elements of the research direction of the work of Madan et al. [(1998) The variance gamma process and option pricing, European Finance Review2, 79–105] and gives analytical expressions for the prices of digital and European call options in the variance-gamma model under the assumption that the linear drift rate of stock log-returns can suddenly jump downwards. The time of the jump is taken to be exponentially distributed. The formulas obtained require the computation of some generalized hyperbolic functions.

In this paper we obtain expressions for truncated moment-generating functions of the normal-inverse Gaussian (NIG) process in closed forms. The result is derived without any additional martingale-type restriction on the initial probability measure. Applications to option pricing and risk measuring in the NIG model are given. The established formulas depend on values of the Humbert series.

In this paper, we provide an alternative framework for constructing an arbitrage-free European-style option surface. The main motivation for our work is that such a construction has rarely been achieved in the literature so far. The novelty of our approach is that we perform the calibration and interpolation in the put option space. To demonstrate the applicability of our technique, we extract the model-free implied volatility from S&P 500 index options. Subsequently, we compare its information content to that of the CBOE VIX index. Our empirical tests indicate that information content of the option-implied volatility values based on our method are superior to the VIX index.

In this paper the stochastic volatility model of Stein and Stein is extended to treat the long memory character of the volatility. It is proposed to model the volatility by a mean reverting Langevin equation driven by fractional Brownian motions. The risk-minimizing hedging price for European call options is obtained and its computation is discussed.

The Leland strategy of approximate hedging of the call-option under proportional transaction costs prescribes to use, at equidistant instants of portfolio revisions, the classical Black–Scholes formula but with a suitably enlarged volatility. An appropriate mathematical framework is a scheme of series, i.e. a sequence of models M_n with the transaction costs coefficients k_n depending on n, the number of the revision intervals. The enlarged volatility , in general, also depends on n. Lott investigated in detail the particular case where the transaction costs coefficients decrease as n^-1/2 and where the Leland formula yields not depending on n. He proved that the terminal value of the portfolio converges in probability to the pay-off. In the present note we show that it converges also in L² and find the first order term of asymptotics for the mean square error. The considered setting covers the case of non-uniform revision intervals. We establish the asymptotic expansion when the revision dates are where the strictly increasing scale function g : [0, 1] → [0, 1] and its inverse f are continuous with their first and second derivatives on the whole interval or g(t) = 1 - (1 - t)^β, β ≥ 1.