scholarly journals PUT OPTION PRICES AS JOINT DISTRIBUTION FUNCTIONS IN STRIKE AND MATURITY: THE BLACK–SCHOLES CASE

2009 ◽  
Vol 12 (08) ◽  
pp. 1075-1090
Author(s):  
D. MADAN ◽  
B. ROYNETTE ◽  
M. YOR
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Large Class ◽  
Joint Distribution ◽  
Distribution Functions ◽  
Standard Brownian Motion ◽  
Option Prices ◽  
Put Option ◽  
Black Scholes

For a large class of ℝ+ valued, continuous local martingales (Mtt ≥ 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 [Formula: see text], for (Bt, t ≥ 0) a standard Brownian motion.

scholarly journals PENENTUAN HARGA KONTRAK OPSI KOMODITAS EMAS MENGGUNAKAN METODE POHON BINOMIAL

2017 ◽  
Vol 6 (2) ◽  
pp. 99
Author(s):  
I GEDE RENDIAWAN ADI BRATHA ◽  
KOMANG DHARMAWAN ◽  
NI LUH PUTU SUCIPTAWATI
Keyword(s):  
Option Prices ◽  
European Option ◽  
Binomial Tree ◽  
Call Options ◽  
Option Contracts ◽  
Put Option ◽  
Black Scholes ◽  
Tree Method

Holding option contracts are considered as a new way to invest. In pricing the option contracts, an investor can apply the binomial tree method. The aim of this paper is to present how the European option contracts are calculated using binomial tree method with some different choices of strike prices. Then, the results are compared with the Black-Scholes method. The results obtained show the prices of call options contracts of European type calculated by the binomial tree method tends to be cheaper compared with the price of that calculated by the Black-Scholes method. In contrast to the put option prices, the prices calculated by the binomial tree method are slightly more expensive.

scholarly journals On the Uniqueness of Martingales with Certain Prescribed Marginals

2013 ◽  
Vol 50 (2) ◽  
pp. 557-575
Author(s):  
Michael R. Tehranchi
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Asset Price ◽  
Simple Random Walk ◽  
Option Prices ◽  
Tree Model ◽  
Binomial Tree ◽  
Black Scholes ◽  
Risk Neutral ◽  
Binomial Tree Model

This note contains two main results. (i) (Discrete time) Suppose that S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Cox-Ross-Rubinstein binomial tree model.) Then S is a geometric simple random walk. (ii) (Continuous time) Suppose that S=S0eσ X-σ2〈 X〉/2 is a continuous martingale whose marginal laws agree with a geometric Brownian motion. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Black-Scholes model with volatility σ>0.) Then there exists a Brownian motion W such that Xt=Wt+o(t1/4+ ε) as t↑∞ for any ε> 0.

On the Uniqueness of Martingales with Certain Prescribed Marginals

2013 ◽  
Vol 50 (02) ◽  
pp. 557-575
Author(s):  
Michael R. Tehranchi
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Asset Price ◽  
Simple Random Walk ◽  
Option Prices ◽  
Tree Model ◽  
Binomial Tree ◽  
Black Scholes ◽  
Risk Neutral ◽  
Binomial Tree Model

This note contains two main results. (i) (Discrete time) Suppose that S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Cox-Ross-Rubinstein binomial tree model.) Then S is a geometric simple random walk. (ii) (Continuous time) Suppose that S=S 0eσ X-σ2〈 X〉/2 is a continuous martingale whose marginal laws agree with a geometric Brownian motion. (In financial terms, let S be a risk-neutral asset price and suppose that the initial option prices agree with the Black-Scholes model with volatility σ>0.) Then there exists a Brownian motion W such that X t =W t +o(t 1/4+ ε) as t↑∞ for any ε> 0.

scholarly journals Measures of concordance determined byD4-invariant copulas

2004 ◽  
Vol 2004 (70) ◽  
pp. 3867-3875 ◽  
Author(s):  
H. H. Edwards ◽  
P. Mikusiński ◽  
M. D. Taylor
Keyword(s):  
Distribution Function ◽  
Mass Distribution ◽  
Random Vector ◽  
Joint Distribution ◽  
Distribution Functions ◽  
Joint Distribution Function ◽  
Unit Square ◽  
Special Cases ◽  
Measures Of Concordance ◽  
Measure Of Association

A continuous random vector(X,Y)uniquely determines a copulaC:[0,1]2→[0,1]such that when the distribution functions ofXandYare properly composed intoC, the joint distribution function of(X,Y)results. A copula is said to beD4-invariant if its mass distribution is invariant with respect to the symmetries of the unit square. AD4-invariant copula leads naturally to a family of measures of concordance having a particular form, and all copulas generating this family areD4-invariant. The construction examined here includes Spearman’s rho and Gini’s measure of association as special cases.

scholarly journals Some limit theorems connected with Brownian local time

2006 ◽  
Vol 2006 ◽  
pp. 1-5
Author(s):  
Raouf Ghomrasni
Keyword(s):  
Brownian Motion ◽  
Large Class ◽  
Local Time ◽  
Limit Theorems ◽  
Standard Brownian Motion ◽  
Time Process ◽  
Continuous Version ◽  
Brownian Local Time ◽  
Class Of Functions

Let B=(Bt)t≥0 be a standard Brownian motion and let (Ltx;t≥0,x∈ℝ) be a continuous version of its local time process. We show that the following limitlim⁡ε↓0(1/2ε)∫0t{F(s,Bs−ε)−F(s,Bs+ε)}ds is well defined for a large class of functions F(t,x), and moreover we connect it with the integration with respect to local time Ltx . We give an illustrative example of the nonlinearity of the integration with respect to local time in the random case.

scholarly journals FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE

2003 ◽  
Vol 06 (01) ◽  
pp. 1-32 ◽  
Author(s):  
YAOZHONG HU ◽  
BERNT ØKSENDAL
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Standard Brownian Motion ◽  
Stochastic Integration ◽  
European Option ◽  
No Arbitrage ◽  
Black Scholes ◽  
White Noise Calculus ◽  
Replicating Portfolio

The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

OPTIMAL CONSUMPTION AND PORTFOLIO IN A BLACK–SCHOLES MARKET DRIVEN BY FRACTIONAL BROWNIAN MOTION

2003 ◽  
Vol 06 (04) ◽  
pp. 519-536 ◽  
Author(s):  
YAOZHONG HU ◽  
BERNT ØKSENDAL ◽  
AGNÈS SULEM
Keyword(s):  
Mathematical Model ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Consumption Rate ◽  
Hurst Parameter ◽  
Optimal Consumption ◽  
Standard Brownian Motion ◽  
Power Type ◽  
Black Scholes ◽  
Market Driven

We present a mathematical model for a Black–Scholes market driven by fractional Brownian motion BH(t) with Hurst parameter [Formula: see text]. The interpretation of the integrals with respect to BH(t) is in the sense of Itô (Skorohod–Wick), not pathwise (which is known to lead to arbitrage). We find explicitly the optimal consumption rate and the optimal portfolio in such a market for an agent with utility functions of power type. When H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

Sign in / Sign up

Export Citation Format

Share Document