This paper describes the Itô processes for the continuously compounded returns on European call and put stock options under the one-dimensional diffusion assumption and the Black Scholes pricing model. It uses the Itô processes to motivate discrete time approximations for the returns on calls and puts. Theses models are used in a simulation study to compute the probability of an option return v...

For a large class of R+ valued, continuous local martingales (Mt t ≥ 0), with M0 = 1 and M∞ = 0, the put quantity : ΠM(K, t) = E ( (K −Mt) ) turns out to be the distribution function in both variables K and t, for K ≤ 1 and t ≥ 0, of a probability γM on [0, 1]× [0,∞[. In this paper, the first in a series of three, we discuss in detail the case where Mt = Et := exp ( Bt − t 2 ) , for (Bt, t ≥ 0)...

The Black–Scholes model is widely used to value options. An important advantage of the model is that European options can be valued analytically by the Black–Scholes formula (Merton 1992; Hull 1997). The situation is quite different, however, for American put options with optimal early exercise. While considerable progress has been made, no completely satisfactory analytic solution has been fou...

Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optima...

We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a quasi-variational inequality and it is C across the optimal stopping boundary. Our proof only uses the classical theory of parabolic partial differential equations of Friedman (1964, 2006) ...

Credit default swaps (CDS) are similar to out-of-the-money put options in that both offer a low cost and effective protection against downside risk. This study investigates whether put optionimplied volatility is an important determinant of CDS spreads. Using a large sample of firms with both CDS and options data, we find that individual firms’ put option-implied volatility dominates historical...

Reduced European call and put option formulas by risk-neutral valuation are given. It is shown that the European call and put options for log-uniformjump-diffusion models are worth more than that for the Black-Scholes (diffusion) model with the common parameters. Due to the complexity of the jump-diffusion models, obtaining a closed option pricing formula like that of Black-Scholes is not tract...

Portfolio and asset pricing theory use symmetric volatility measures to evaluate risk. We propose a method to isolate the risk that the portfolio suffers a loss, separate from the risk that the portfolio reaps uncommon gains. Specifically we take advantage of the one-sided nature of option payments and show how downside risk will only affect the value of a put option (with correctly specified s...

Introduced by Kifer (2000), game options function in the same way as American options with the added feature that the writer may also choose to exercise at which time they must pay out the intrinsic option value of that moment plus a penalty. In Kyprianou (2004) an explicit formula was obtained for the value function of the perpetual put option of this type. Crucial to the calculations which le...

This paper builds the multi-period optimization models that incorporate put option contract and two supply chain structures to determine production decision for a supplier ordering manufacturer in two-stage chain. applies method of dynamic programming derive optimal policies provides an approximate algorithm evaluate myopic policies. also conducts numerical examples illustrate impacts contract,...