scholarly journals A VALUATION FORMULA FOR MULTI-ASSET, MULTI-PERIOD BINARIES IN A BLACK–SCHOLES ECONOMY

2009 ◽  
Vol 50 (4) ◽  
pp. 475-485
Author(s):  
MAX SKIPPER ◽  
PETER BUCHEN
Keyword(s):  
Closed Form ◽  
Present Value ◽  
Simple Analytic Expression ◽  
Black Scholes ◽  
Analytic Expression

AbstractWe present a new valuation formula for a generic, multi-period binary option in a multi-asset Black–Scholes economy. The payoff of this so-called M-binary is the most general possible, subject to the condition that a simple analytic expression exists for the present value. Portfolios of M-binaries can be used to statically replicate many European exotics for which there exist closed-form Black–Scholes prices.

Alternative Formulas to Compute Implied Standard Deviation

2009 ◽  
Vol 12 (02) ◽  
pp. 159-176 ◽  
Author(s):  
James S. Ang ◽  
Gwoduan David Jou ◽  
Tsong-Yue Lai
Keyword(s):  
Standard Deviation ◽  
Closed Form ◽  
Asset Price ◽  
Closed Form Solution ◽  
Form Solution ◽  
Call Option ◽  
Present Value ◽  
Underlying Asset ◽  
The Third ◽  
Exercise Price

We assume that the call option's value is correctly priced by Black and Scholes' option pricing model in this paper. This paper derives an exact closed-form solution for implied standard deviation under the condition that the underlying asset price equals the present value of the exercise price. The exact closed-form solution provides the true implied standard deviation and has no estimate error. This paper also develops three alternative formulas to estimate the implied standard deviation if this condition is violated. Application of the Taylor expansion on a single call option value derives the first formula. The accuracy of this formula depends on the deviation between the underlying asset price and the present value of the exercise price. Use of the Taylor formula on two call option prices with different exercise prices is used to develop the second formula, which can be used even though the underlying asset price deviates significantly from the present value of the exercise price. Extension of the second formula's approach to third options value derives the third formula. A merit of the third formula is to circumvent a required parameter used in the second formula. Simulations demonstrate that the implied standard deviations calculated by the second and third formulas provide accurate estimates of the true implied standard deviations.

scholarly journals ON THE PROFIT AND LOSS DISTRIBUTION OF DYNAMIC HEDGING STRATEGIES

1999 ◽  
Vol 02 (02) ◽  
pp. 131-152 ◽  
Author(s):  
SERGEI ESIPOV ◽  
IGOR VAYSBURD
Keyword(s):  
Monte Carlo ◽  
Probability Distribution ◽  
Present Value ◽  
Hedging Strategy ◽  
Loss Distribution ◽  
Time Dynamics ◽  
Hedging Strategies ◽  
Black Scholes ◽  
Risk Neutral ◽  
Stop Loss

Hedging a derivative security with non-risk-neutral number of shares leads to portfolio profit or loss. Unlike in the Black–Scholes world, the net present value of all future cash flows till maturity is no longer deterministic, and basis risk may be present at any time. The key object of our analysis is probability distribution of future P & L conditioned on the present value of the underlying. We consider time dynamics of this probability distribution for an arbitrary hedging strategy. We assume log-normal process for the value of the underlying asset and use convolution formula to relate conditional probability distribution of P & L at any two successive time moments. It leads to a simple PDE on the probability measure parameterized by a hedging strategy. For risk-neutral replication the P & L probability distribution collapses to a delta-function at the Black–Scholes price of the contingent claim. Therefore, our approach is consistent with the Black–Scholes one and can be viewed as its generalization. We further analyze the PDE and derive formulae for hedging strategies targeting various objectives, such as minimizing variance or optimizing distribution quantiles. The developed method of computing the profit and loss distribution for a given hedging scheme is applied to the classical example of hedging a European call option using the "stop-loss" strategy. This strategy refers to holding 1 or 0 shares of the underlying security depending on the market value of such security. It is shown that the "stop-loss" strategy can lead to a loss even for an infinite frequency of re-balancing. The analytical method allows one to compute profit and loss distributions without relying on simulations. To demonstrate the strength of the method we reproduce the Monte Carlo results on "stop-loss" strategy given in Hull's book, and improve the precision beyond the limits of regular Monte-Carlo simulations.

scholarly journals Estimating the Risk of Satellite Collisions in Densely Populated Orbit Shells

2021 ◽  
Author(s):  
William Lee ◽  
Paul Martin ◽  
Ann Smith ◽  
Giancarlo Antonucci ◽  
Georgia Brennan ◽  
...  
Keyword(s):  
Time Dependence ◽  
Low Earth Orbit ◽  
Collision Probability ◽  
Earth Orbit ◽  
Limited Information ◽  
Asymptotic Approach ◽  
Optimal Point ◽  
Single Collision ◽  
Simple Analytic Expression ◽  
Analytic Expression

Low Earth Orbit is becoming crowded with satellites. Updating estimates of collision probabilities is important as new deployments are authorised but is difficult because only limited information is given. This report investigates developing analytic estimates of collision probabilities. A survey of approaches reported in the literature is carried out. A collision involving a satellite from the Iridium cluster is reviewed. A simple analytic expression for the collision probability between two satellites is derived using the smallness of several dimensionless ratios appearing in the problem. Single collision probabilities are then extended to orbital planes populated by n satellites with the aim of finding the optimal point at which to traverse such an orbit. This report demonstrates that analytic estimates relevant to the problem can be made. Further work should focus on: making these estimates rigorous by using a formal asymptotic approach, considering multiple orbital planes and introducing time dependence

scholarly journals Pricing Basket Options by Polynomial Approximations

2021 ◽  
Author(s):  
Pablo Olivares ◽  
Alexander Alvarez
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Closed Form ◽  
Computational Effort ◽  
Basket Options ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Power Moments ◽  
Exponential Power ◽  
Closed Form Approximation

We propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model. The method is based on Taylor and Chebyshev expansions and involves mixed exponential-power moments of a Gaussian distribution. Our numerical results show that both approaches are comparable in accuracy to a standard Monte Carlo method, with a lesser computational effort

COMPUTATIONAL ANALYSIS OF STATIONARY WAITING-TIME DISTRIBUTIONS OF GIX/R/1 AND GIX/D/1 QUEUES

2005 ◽  
Vol 19 (1) ◽  
pp. 121-140 ◽  
Author(s):  
Mohan L. Chaudhry ◽  
Dae W. Choi ◽  
Kyung C. Chae
Keyword(s):  
Closed Form ◽  
Waiting Time ◽  
Time Distribution ◽  
Computational Analysis ◽  
Rational Functions ◽  
Waiting Time Distribution ◽  
Generalized Distributions ◽  
Analytic Expression ◽  
Stationary Waiting Time ◽  
Stieltjes Transforms

In this article, we obtain, in a unified way, a closed-form analytic expression, in terms of roots of the so-called characteristic equation of the stationary waiting-time distribution for the GIX/R/1 queue, where R denotes the class of distributions whose Laplace–Stieltjes transforms are rational functions (ratios of a polynomial of degree at most n to a polynomial of degree n). The analysis is not restricted to generalized distributions with phases such as Coxian-n (Cn) but also covers nonphase-type distributions such as deterministic (D). In the latter case, we get approximate results. Numerical results are presented only for (1) the first two moments of waiting time and (2) the probability that waiting time is zero. It is expected that the results obtained from the present study should prove to be useful not only for practitioners but also for queuing theorists who would like to test the accuracies of inequalities, bounds, or approximations.

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