Special Topic: The Geometric Random Walk and the Binomial Tree Model of Mathematical Finance

2021 ◽  
pp. 279-289
Author(s):  
Rabi Bhattacharya ◽  
Edward C. Waymire
Keyword(s):  
Random Walk ◽  
Mathematical Finance ◽  
Special Topic ◽  
Tree Model ◽  
Binomial Tree ◽  
Binomial Tree Model ◽  
Geometric Random Walk

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 Backward stochastic difference equations for dynamic convex risk measures on a binomial tree

2015 ◽  
Vol 52 (03) ◽  
pp. 771-785 ◽  
Author(s):  
Robert J. Elliott ◽  
Tak Kuen Siu ◽  
Samuel N. Cohen
Keyword(s):  
Difference Equations ◽  
Risk Measures ◽  
Time Consistency ◽  
Tree Model ◽  
Binomial Tree ◽  
Convex Risk Measures ◽  
Stochastic Difference Equations ◽  
Mathematical Properties ◽  
Binomial Tree Model ◽  
Nonlinear Expectations

Using backward stochastic difference equations (BSDEs), this paper studies dynamic convex risk measures for risky positions in a simple discrete-time, binomial tree model. A relationship between BSDEs and dynamic convex risk measures is developed using nonlinear expectations. The time consistency of dynamic convex risk measures is discussed in the binomial tree framework. A relationship between prices and risks is also established. Two particular cases of dynamic convex risk measures, namely risk measures with stochastic distortions and entropic risk measures, and their mathematical properties are discussed.

scholarly journals Pricing Exotic Options Using Some Lattice Procedures

2021 ◽  
Vol 41 (1) ◽  
pp. 26-40
Author(s):  
Sadia Anjum Jumana ◽  
ABM Shahadat Hossain
Keyword(s):  
Stock Price ◽  
Lattice Models ◽  
Exotic Options ◽  
Barrier Options ◽  
Tree Model ◽  
Binomial Tree ◽  
Black Scholes Model ◽  
Black Scholes ◽  
Binomial Tree Model ◽  
Trinomial Tree Model

In this work, we discuss some very simple and extremely efficient lattice models, namely, Binomial tree model (BTM) and Trinomial tree model (TTM) for valuing some types of exotic barrier options in details. For both these models, we consider the concept of random walks in the simulation of the path which is followed by the underlying stock price. Our main objective is to estimate the value of barrier options by using BTM and TTM for different time steps and compare these with the exact values obtained by the benchmark Black-Scholes model (BSM). Moreover, we analyze the convergence of these lattice models for these exotic options. All the results have been shown numerically as well as graphically. GANITJ. Bangladesh Math. Soc.41.1 (2021) 26-40

scholarly journals PENERAPAN STATIC HEDGE DALAM PENGELOLAAN RISIKO PADA OPSI TIPE BARRIER

2018 ◽  
Vol 7 (4) ◽  
pp. 357
Author(s):  
NI MADE NITA ASTUTI ◽  
KOMANG DHARMAWAN ◽  
TJOKORDA BAGUS OKA
Keyword(s):  
Numerical Approach ◽  
Option Value ◽  
Barrier Option ◽  
Tree Model ◽  
Strike Price ◽  
Underlying Asset ◽  
Binomial Tree ◽  
Static Hedging ◽  
Binomial Tree Model

The barrier option is an option whose payoff depends on whether the underlying asset touches the barrier or not during the lifetime of the option. The determination of the barrier option requires a numerical approach, one of which is the Binomial Tree model. The purpose of this study  is to determine barrier option type down and out call on a static hedging using the Binomial Tree model and compare it with the analytic value. The results show that the increases in strike price would decrease the option value. Moreover, values from 80 periods using the Binomial Tree model for the four strike prices are close to analytic with error less than or equal to 0.00182.

Replication and Pricing in the Binomial Tree Model

2021 ◽  
pp. 331-396
Author(s):  
Giuseppe Campolieti ◽  
Roman N. Makarov
Keyword(s):  
Tree Model ◽  
Binomial Tree ◽  
Binomial Tree Model

Binomial tree method for option pricing: Discrete Carr and Madan formula approach

2021 ◽  
pp. 2150024
Author(s):  
Yoshifumi Muroi ◽  
Ryota Saeki ◽  
Shintaro Suda
Keyword(s):  
Option Pricing ◽  
Broad Class ◽  
Asset Price ◽  
Option Prices ◽  
European Option ◽  
Tree Model ◽  
Binomial Tree ◽  
Tree Models ◽  
Binomial Tree Model ◽  
Tree Method

This paper suggests a new Fourier analysis approach to evaluate the option prices and its sensitivities (Greeks) using the binomial tree model. In the last half of this paper, we show that option prices are efficiently and effectively evaluated using a semi-closed form formula for European option prices. We can compute option prices in a broad class of jump-diffusion models because we calculate the characteristic function for an underlying asset price numerically. Furthermore, we also compute the price of European options in the exp-Levy model. This numerical experiment gives new insights into option pricing in the nonparametric Levy model. The option prices and sensitivities can be computed very accurately and efficiently, even in binomial tree models with jumps.

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