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Find a formula for the price of a derivative paying $\max(S_T^2-K,0)$ [duplicate]

Develop a formula for the price of a derivative paying $$\max(S_T^2-K,0)$$ in the Black Scholes model. What I tried: \begin{align*} e^{-rT}\mathbb{E}^\mathbb{Q^0}[\max\{S_T^2-KS_T,0\}] = e^{-rT}\left(...
DivertingPie's user avatar
3 votes
1 answer
276 views

Geometric Brownian Motion as the limit of a Binomial Tree?

Consider the price of a stock whose drift and volatility parameters are $\mu, \sigma$ respectively, over the time interval $[0, t]$. Suppose we use an $n$-stage binomial tree to model the price ...
Bumblebee's user avatar
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0 answers
40 views

Optimal strategy assuming Geometric Brownian motion?

Let's assume that some stock price follows a geometric Brownian motion with known parameters $\log X_t-\log X_0 \sim (\mu t, \sigma^2 t).$ I'm trying to model the size of the order book (normalized) ...
Ameer Jewdaki's user avatar
2 votes
0 answers
149 views

Bound on path length of a stock price

Consider a time series $(S_i)$ representing a stock price (say close prices of one minute candles). Let $\Delta$ be a quantization step (could be the price step in the strike prices of the ...
TryingHardToBecomeAGoodPrSlvr's user avatar
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1 answer
85 views

Volatility is not becoming sigma in simulated GBM [closed]

I am trying to simulate GBM using values for sigma and then after calculating the price, calculate the log of returns and the associated volatility. I was expecting the calculated volatility of log ...
Arthur's user avatar
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0 answers
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Change of numeraire : quotient

Let's consider $X_1(t)$ a geometric brownian motion (with variable volatility) and $X_2(t)$ a Brownian bridge : $dX_1(t) = \mu X_1(t) dt + \sigma_1(t) X_1(t) dW(t)$ $dX_2(t) = \frac{b - X_2(t)}{T - t} ...
Logan's user avatar
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1 vote
1 answer
155 views

Connection between the $\sigma$ parameters of the spot price and the forward price

It is well known, that under the Black-Scholes framework: $$F\left(t,T\right)=\exp\left(r\left(T-t\right)\right)S\left(t\right),$$ where $S\left(t\right)$ is the spot price of an asset at time $t$, $F\...
Kapes Mate's user avatar
0 votes
1 answer
205 views

Present value of an FX Forward contract at each simulation and time point node of a Monte Carlo simulation

Recently I started dealing with the xVA and the associated EPE and ENE concepts. In a numerical example of an FX Forward, after simulating the underlying FX spot $S_t$ (units of domestic per unit of ...
Whitebeard13's user avatar
0 votes
1 answer
116 views

Distribution of Geometric Brownian with time-dependant volatility

The process $S(t) =\exp\left(\mu.t + \int_0^t\sigma(s) \text{d}W(s) - \int_0^t \frac{1}{2}\sigma^2(s)\text{d}s\right)$ where $\sigma(s) = 0.03s$ is log-normally distributed, but i'm not sure about the ...
Logan's user avatar
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Ito Process: How to calculate expected return?

On page 300 of Hull's Options, Futures and Other Derivatives It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected drift rate and ...
user546106's user avatar
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How can we simulate daily return based on multi-factor model?

This is an interesting question for simulation. The question is a bit lengthy but I'm trying my best to make it super clear here. Now I have some multi-factor model, say some US barra risk model from ...
xxxtttsss666's user avatar
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29 views

High/Low range under GBM - Analytic solution?

Does anybody know of an analytical solution to the expected high / low range for an asset that follows a GBM process over sampling frequency dt? I have ran numerical simulations and find that the ...
Newquant's user avatar
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Reinvesting the dividends of a dividend paying stock

Suppose we have a dividend paying stock that has the following dynamics: $$dS_t=S_t((\mu-q)dt+\sigma dW_t)$$ With a continuous dividend yield $q$. What is the portfolio $Y_t$ that results out of ...
Mr. Ivan's user avatar
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137 views

Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$

Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$....
Mr. Ivan's user avatar
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1 answer
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Solving the SDE for GBM [closed]

Let's assume that we have the following stochastic differential equation: $dX_t = \mu X_t dt + \sigma X_tdW_t$ and that we have to prove that this is its solution: $X_t = X_0 \exp\left(\left(\mu -{\...
Alessandro's user avatar
1 vote
2 answers
561 views

Why do we adjust the drift in the geometric brownian motion

I am building a monte carlo based on the GMB, and I am having a hard time understanding why we subtract 1/2 variance from the drift. If I have a drift of 12% and a volatility of 50%, that would give ...
John's user avatar
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1 vote
1 answer
266 views

Geometric Brownian motion and semi-martingality

I am trying to learn by myself stochastic calculus, and I have a question regarding semi-martingale stochastic process and geometric Brownian motion (GBM). We know that a stochastic process $S_t$ is ...
XY0's user avatar
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0 answers
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Pricing equation with two correlated states

Consider the following asset pricing setting for a perpetual defaultable coupon bond with price $P(V,c)$, where $V$ is the value of the underlying asset and $c$ is a poisson payment that occurs with ...
Luca Gi's user avatar
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122 views

multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by \begin{equation} dS(t) = S(t)\big((\mu - ...
yrual's user avatar
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1 vote
1 answer
141 views

Confusion about the formula for gain process in a financial market

In this wikipedia page, we consider the following financial market The formulas for the stocks are given here And the gain process of a portfolio $\pi$ is defined such that From what I understand, ...
yrual's user avatar
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Dynamics of independent Geometric Brownian Motions under risk-neutral measure Q

Suppose I have two Geometric Brownian motions and a bank account: $$dB_t=rB_tdt$$ $$ dS=S(\alpha dt + \sigma dW_t) $$ $$ dY = Y(\beta dt + \delta dV_t) $$ Where $dW_t$ and $dV_t$ are independent ...
zjo892's user avatar
  • 31
2 votes
1 answer
56 views

Distribution of discrete Geometric average and Stock Price

If we have $$S_t = S_0 e^{(r-\frac{1}{2} \sigma ^2) +\sigma W_t}$$ and a discrete geometric average of stock prices $$G_n = (\prod_{i=1}^{n} S_{t_i})^{\frac{1}{n}} $$ where the monitoring points are ...
nachofest's user avatar
1 vote
1 answer
347 views

If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns? [closed]

Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask ...
nearhome's user avatar
2 votes
0 answers
46 views

Bessel Correction and Geometric Brownian Motion

Does it make sense to use bessel's correction for standard deviation and variance when fitting the drift and volatility parameters of geometric brownian motion to historical return data for a security....
user3163829's user avatar
0 votes
1 answer
128 views

Floating Strike Geometric Averaged Asian Option Pricing

How can I use the risk neutral evaluation to price an asian option with floating strike using the continuous geometric average? I have tried searching for ways to do it but have found almost nothing.
nachofest's user avatar
1 vote
0 answers
49 views

References for path-dependent GBMs or continuous time analog of discrete time filters

Consider a path-dependent GBM model for a stock price: $$dS_t = \mu(t, S_.)S_tdt + \sigma(t, S_.) S_t dB_t,$$ where $\mu, \sigma : [0,\infty)\times C_{[0,\infty)}\to \mathbb{R}$ are previsible path-...
Nap D. Lover's user avatar
4 votes
1 answer
112 views

Discounted expectation of generic $\mathbb{C}^2$ function

Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$. It holds that the discounted expectation is $$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right]...
Luca Gi's user avatar
  • 327
1 vote
0 answers
33 views

Mean of diffusion term not zero using NORMINV? [closed]

maybe this is a question considered too basic for all of you but im new so please excuse: I wanted to buid a simulation in excel using the usual suspect(STANDNORMINV(RAND()) and i tried to calculate ...
BussiHasi's user avatar
1 vote
0 answers
181 views

how to estimate Geometric Brownian Motion parameters on long timeseries [closed]

I'm working on a 50-years financial timeseries and I would like to simulate GBM paths from it. The first thing I'm supposed to do is to estimate the drift $\mu$ and the volatility $\sigma$ parameters. ...
randomWalk's user avatar
1 vote
0 answers
70 views

Quantile function for fractional Brownian motion (fBm)

If anyone could help me to understand if it is possible calculate the quantile function for fBm? I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised ...
Serg Gini's user avatar
2 votes
1 answer
376 views

Cholesky decomposition reduces volatility of simulated Wiener Process / Brownian Motions

I am trying to simulate $n$ correlated geometric brownian motions (GBM) given a specified correlation matrix $\Sigma$ by following this procedure which uses Cholesky decomposition. However, when I ...
Landscape's user avatar
  • 548
5 votes
2 answers
1k views

Dynamics of FX rate

I've see a couple of places where a FX rate, denoted $X$, such as EURUSD (quoted as "the number of USD needed to buy 1 EUR") is modeled with a diffusion process / Geometric Brownian Motion ...
Landscape's user avatar
  • 548
0 votes
1 answer
818 views

What are common ways to realistically simulate the stock market using historical market data?

I am currently using the FinRL library to try to automate Trading using Reinforcement Learning. However, I wanted to understand how FinRL simulates the stock market using historical data. I read here ...
julian2000P's user avatar
0 votes
0 answers
182 views

What is the Kurtosis of Returns in Geometric Brownian Motion?

Suppose that $dS_t=S_t(\mu\mathop{dt}+\sigma\mathop{dW_t})$ which has solution $$S_t=S_0\exp\left(t\left(\mu+\frac{\sigma^2}{2}\right)+\sigma W_t\right),$$ such that $W_t$ is a Wiener process, $\mu$ ...
UNOwen's user avatar
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1 vote
1 answer
410 views

American option under Ornstein-Uhlenbeck stock price

I came across with the following problem: For the Ornstein-Uhlenbeck process $(X_t, 0\leq t\leq T)$ with initial condition $X_0 = x$, find the stopping time $\tau$ that maximizes $\mathbb{E}[e^{-r\...
Aguazz's user avatar
  • 11
1 vote
0 answers
133 views

Value of trading strategy

A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process. The problem is to find the probability of the ...
Simplexity's user avatar
0 votes
0 answers
154 views

Probability the stock price (following geometric Brownian motion) hits the upper boundary U before there is a retracement from the max by amount R?

I am looking for the probability that the stock price/Geometric Brownian Motion hits the upper boundary U, before there is a retracement (from the maximum price) that exceeds amount R. In other words,...
BillB's user avatar
  • 1
2 votes
1 answer
142 views

Modelling the instantaneous funding spread as a log-normal process

Let us consider a stochastic market model with a fixed short (risk-free) rate $r\in\mathbb{R}$. A trader can obtain unsecured funding at a rate $f_t:=r+s_t$ where $s_t$ is its stochastic funding ...
Daneel Olivaw's user avatar
3 votes
1 answer
411 views

Estimating volatility of a geometric Brownian motion at different sample rates

I have troubles estimating volatility (= standard deviation of log returns) when the data is re-sampled at different sample frequencies. Problem I have generated a time series data using a geometric ...
WolfgangP's user avatar
  • 285
0 votes
1 answer
199 views

Understanding the expected value of the average

I've been looking into Asian Options pricing. Part of the process is about looking for the expected value of the average of a time series undergoing e.g. geometric brownian motion. I came across this ...
Experience111's user avatar
2 votes
2 answers
917 views

Why would exchange rates follow a geometric brownian motion?

I'm reading Shreve's Stochastic Calculus for Finance. On page 382, he begins talking about exchange rates: Finally, there is an exchange rate $Q(t)$, which gives units of domestic currency per unit ...
user60304's user avatar
1 vote
0 answers
36 views

From the perspective of a company, when is the right time to start paying dividends?

I am trying to understand geometric Brownian motion as it relates to the present discounted value of future dividend payments. I am supposing that a company has a revenue stream $f(t)$. This is just $...
user60304's user avatar
0 votes
1 answer
508 views

GBM - How to make make annualized dividend reflected in one quarter

I want to simulate the price path of a stock for one quarter using geometric Brownian motion. The stock has a continuous dividend yield of 5% based on the annual dividend yield. However, historically ...
Chell's user avatar
  • 3
0 votes
1 answer
551 views

Drift rate in Geometric Brownian Motion

I have two questions regarding the drift term in the geometric Brownian motion that I cannot find any clear answers to online. When would we use risk-free rate as drift and when would we use the ...
Chell's user avatar
  • 3
0 votes
0 answers
109 views

Performance of dollar cost averaging

If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested? For example, modelling a stock as GBM without contributions, $ ...
Zaz's user avatar
  • 101
1 vote
1 answer
339 views

How to simulate correlated stock prices (not returns)

Suppose we have two stocks following GBMs. Drift and volatility are calculated based on historical data. Furthermore the stocks are assumed to be correlated (i.e. they move together, if stock 1 goes ...
Willart's user avatar
  • 73
4 votes
0 answers
62 views

How to compute this current value using no arbitrage condition?

Suppose $X_t$ is a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. $X_0$ is known. You have a machine that produces something worth $X_t$ at random times $t$ generated by a Poisson ...
cxxu96's user avatar
  • 141
0 votes
1 answer
213 views

GBM drift when simulating correlation betwenn GBM with Cholesky Decomposition

I am currently trying to simulate correlated GBM paths and I found the Cholesky Composition for it. From my understanding, the Cholesky Decomposition can be used to create correlated random variables ...
Merwin's user avatar
  • 21
0 votes
0 answers
22 views

Interpretation deferment rate

Geometric Brownian motion related to house prices can be written as $dH_t=(r-g)H_tdt+\sigma H_tdW_t$. How should I interpret $g$? The literature calls this a deferment rate. But I don't understand how ...
Cardinal's user avatar
  • 205
0 votes
1 answer
272 views

Distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions

I am trying to simulate the distribution of Geometric Brownian Motion drawdowns from samples of multivariate Normal and Laplace distributions under the same covariance structure. Drawdowns are defined ...
Bryan Franco's user avatar