Bounds for the American perpetual put on a stock index

2001 ◽  
Vol 38 (01) ◽  
pp. 55-66 ◽  
Author(s):  
V. Paulsen
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Decomposition Technique ◽  
Early Exercise

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

Bounds for the American perpetual put on a stock index

2001 ◽  
Vol 38 (1) ◽  
pp. 55-66 ◽  
Author(s):  
V. Paulsen
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Decomposition Technique ◽  
Early Exercise

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (04) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t>0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (4) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτE (max0≤t≤τXt − c τ), where X = (Xt)t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | Xt = g* (max0≤t≤sXs)} where s ↦ g*(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g*(s) ∼ ((Δ − 1) / K Δ)1 / Δs1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (supt>0Xt) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

scholarly journals Stopping at the maximum of geometric Brownian motion when signals are received

2005 ◽  
Vol 42 (03) ◽  
pp. 826-838 ◽  
Author(s):  
X. Guo ◽  
J. Liu
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Geometric Brownian Motion ◽  
Stopping Rules ◽  
Optimal Stopping Problem ◽  
Signal Process ◽  
Optimal Stopping Time ◽  
A Free Boundary Problem ◽  
Random Times

Consider a geometric Brownian motion X t (ω) with drift. Suppose that there is an independent source that sends signals at random times τ 1 &lt; τ 2 &lt; ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward S τ , where S t = max(max0≤u≤t X u , s) for some constant s ≥ X 0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−r τ i S τ i | X 0 = x, S 0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ&gt;λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ * is governed by a threshold a * such that τ * = inf{τ n : X τ n ≤a * S τ n }. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ * = τ 1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

scholarly journals PENENTUAN NILAI VALUE at RISK PADA SAHAM IHSG MENGGUNAKAN MODEL GEOMETRIC BROWNIAN MOTION DENGAN LOMPATAN

2015 ◽  
Vol 4 (2) ◽  
pp. 67
Author(s):  
I GEDE ARYA DUTA PRATAMA ◽  
KOMANG DHARMAWAN ◽  
LUH PUTU IDA HARINI
Keyword(s):  
At Risk ◽  
Brownian Motion ◽  
Confidence Level ◽  
Value At Risk ◽  
Model Calibration ◽  
Historical Data ◽  
Likelihood Estimation ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Peak Over Threshold

The aim of this research was to measure the risk of the IHSG stock data using the Value at Risk (VaR). IHSG stock index data typically indicates a jump. However, Geometric Brownian Motion (GBM) model can not catch any of the jumps. To view the jumps, it is necessary that the model was then developed into a Geometric Brownian Motion (GBM) model with Jumps. On the GBM model with Jumps, returns the data are discontinuous. To determine the value of VaR, the value of return to perform the simulation model of GBM with Jumps is required. To represent processes that contain jumps, discontinuous Poisson process using the Peak-Over Threshold is required. To determine the parameters of model, calibration of historical data using the Maximum Likelihood Estimation (MLE) method is performed. VaR value for GBM model with Jumps with a 95% and 99% confidence level are -0,0580 and -0,0818 while VaR value for GBM model with a 95% and 99% confidence level are -0,0101 and -0,0199. VaR for GBM model with Jumps with a confidence level of 95% and 99% show greater than the model VaR for GBM.

scholarly journals Stopping at the maximum of geometric Brownian motion when signals are received

2005 ◽  
Vol 42 (3) ◽  
pp. 826-838 ◽  
Author(s):  
X. Guo ◽  
J. Liu
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Geometric Brownian Motion ◽  
Stopping Rules ◽  
Optimal Stopping Problem ◽  
Signal Process ◽  
Optimal Stopping Time ◽  
A Free Boundary Problem ◽  
Random Times

Consider a geometric Brownian motion Xt(ω) with drift. Suppose that there is an independent source that sends signals at random times τ1 < τ2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward Sτ, where St = max(max0≤u≤tXu, s) for some constant s ≥ X0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−rτiSτi | X0 = x, S0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ* is governed by a threshold a* such that τ* = inf{τn: Xτn≤a*Sτn}. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ* = τ1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

scholarly journals Efficient or Fractal Market Hypothesis? A Stock Indexes Modelling Using Geometric Brownian Motion and Geometric Fractional Brownian Motion

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2983
Author(s):  
Vasile Brătian ◽  
Ana-Maria Acu ◽  
Camelia Oprean-Stan ◽  
Emil Dinga ◽  
Gabriela-Mariana Ionescu
Keyword(s):  
Brownian Motion ◽  
Stock Market ◽  
Fractional Brownian Motion ◽  
Dynamic Equation ◽  
Geometric Brownian Motion ◽  
Stock Index ◽  
Market Indexes ◽  
Stock Indexes ◽  
Fractal Market Hypothesis ◽  
Geometric Fractional Brownian Motion

In this article, we propose a test of the dynamics of stock market indexes typical of the US and EU capital markets in order to determine which of the two fundamental hypotheses, efficient market hypothesis (EMH) or fractal market hypothesis (FMH), best describes market behavior. The article’s major goal is to show how to appropriately model return distributions for financial market indexes, specifically which geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM) dynamic equations best define the evolution of the S&P 500 and Stoxx Europe 600 stock indexes. Daily stock index data were acquired from the Thomson Reuters Eikon database during a ten-year period, from January 2011 to December 2020. The main contribution of this work is determining whether these markets are efficient (as defined by the EMH), in which case the appropriate stock indexes dynamic equation is the GBM, or fractal (as described by the FMH), in which case the appropriate stock indexes dynamic equation is the GFBM. In this paper, we consider two methods for calculating the Hurst exponent: the rescaled range method (RS) and the periodogram method (PE). To determine which of the dynamics (GBM, GFBM) is more appropriate, we employed the mean absolute percentage error (MAPE) method. The simulation results demonstrate that the GFBM is better suited for forecasting stock market indexes than the GBM when the analyzed markets display fractality. However, while these findings cannot be generalized, they are verisimilar.

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