scholarly journals A Generalization Of Lattice Specifications For Currency Options

2011 ◽  
Vol 1 (5) ◽  
Author(s):  
George M. Jabbour ◽  
Marat V. Kramin ◽  
Stephen D. Young
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Lattice Models ◽  
Geometric Brownian Motion ◽  
Font Size ◽  
Currency Options ◽  
Binomial Trees ◽  
Drift Condition ◽  
Binomial Models

<p class="MsoNormal" style="text-align: justify; margin: 0in 0.5in 0pt; mso-pagination: none;"><span style="font-family: Times New Roman; font-size: x-small;">This article revisits the topic of two-state pricing of currency options.<span style="mso-spacerun: yes;">&nbsp; </span>It examines the models developed by Cox, Ross, and Rubinstein, Rendleman and Bartter, and Trigeorgis, and presents two alternative binomial models based on the continuous and discrete time Geometric Brownian Motion processes respectively.<span style="mso-spacerun: yes;">&nbsp; </span>This work generalizes the standard binomial approach incorporating the main existing models as particular cases.<span style="mso-spacerun: yes;">&nbsp; </span>The proposed models are straightforward, flexible, accommodate any drift condition and afford additional insights into binomial trees and lattice models in general.<span style="mso-spacerun: yes;">&nbsp; </span>Further, the alternative parameterizations are free of the negative aspects associated with the Cox, Ross, and Rubinstein model.</span></p>

scholarly journals First Hitting Problems for Markov Chains That Converge to a Geometric Brownian Motion

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Mario Lefebvre ◽  
Moussa Kounta
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
Geometric Brownian Motion ◽  
Expected Number ◽  
Discrete Time Markov Chain

We consider a discrete-time Markov chain with state space {1,1+Δx,…,1+kΔx=N}. We compute explicitly the probability pj that the chain, starting from 1+jΔx, will hit N before 1, as well as the expected number dj of transitions needed to end the game. In the limit when Δx and the time Δt between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that pj and djΔt tend to the corresponding quantities for the geometric Brownian motion.

scholarly journals Asymptotics for the discrete-time average of the geometric Brownian motion and Asian options

2017 ◽  
Vol 49 (2) ◽  
pp. 446-480 ◽  
Author(s):  
Dan Pirjol ◽  
Lingjiong Zhu
Keyword(s):  
Brownian Motion ◽  
Discrete Time ◽  
Moment Generating Function ◽  
Mathematical Finance ◽  
Geometric Brownian Motion ◽  
Time Averaging ◽  
Asian Options ◽  
Time Average ◽  
Averaging Time ◽  
The Moment

Abstract The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps. We derive almost sure limit, fluctuations, large deviations, and also the asymptotics of the moment generating function of the average. Based on these results, we derive the asymptotics for the price of Asian options with discrete-time averaging in the Black–Scholes model, with both fixed and floating strike.

scholarly journals Geometric Brownian motion with affine drift and its time-integral

2021 ◽  
Vol 395 ◽  
pp. 125874
Author(s):  
Runhuan Feng ◽  
Pingping Jiang ◽  
Hans Volkmer
Keyword(s):  
Brownian Motion ◽  
Geometric Brownian Motion ◽  
Time Integral

scholarly journals HOLDER-EXTENDIBLE EUROPEAN OPTION: CORRECTIONS AND EXTENSIONS

2015 ◽  
Vol 56 (4) ◽  
pp. 359-372 ◽  
Author(s):  
PAVEL V. SHEVCHENKO
Keyword(s):  
Brownian Motion ◽  
Interest Rate ◽  
Real Life ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Fixed Amount ◽  
Underlying Asset ◽  
Closed Form Solutions ◽  
Pricing Options ◽  
Interest Rate Volatility

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

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