Questions tagged [lognormal]

A continuous probability distribution of a random variable whose logarithm is normally distributed.

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Asian option analytical approximation

I'm trying to approximate the price of an Asian option via the Black-Scholes formula by considering the discrete arithmetic average as a log-normal distribution. $$ A_{T}(n):=\frac{1}{n} \sum_{i=1}^{n}...
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Which Model Should I Use for Pricing USD Interest Rate Caps (7, 10, 30 year maturities) on 1Month Rates?

I am trying to price USD interest rate caps on 1M rates (e.g., LIBOR, SOFR, etc.). The caps are designed to limit the exposure on non-callable USD Pay Float / Receive fixed positions in interest rate ...
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Difficulty with stochastic calculus problem

I'm currently working through Shreve's Volume II, and I'm having some difficulty on Exercise 5.4 of Chapter 5. The problem statement is: Consider a stock whose price differential is $$ dS(t) = r(t) S(...
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Is this process log normally distributed?

I came across a question that I guess $P$ is lognormally distributed. where $y_n$ is log-normally distributed. Am I right on the guessing? Here is the full solution if interested.( my guessing comes ...
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Covariance of the product of log normal process and normal procces

I tried to compute the following covariance : $$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$ where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt $ My idea was to ...
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Equivalence of Call Option on $S_T$ and Put Option on $\frac{1}{S_T}$ in FX Markets

Part 1: I am trying to price an option in the FX world. It naturally pays in the domestic currency, but in this case the payout currency must be the foreign currency. For example, consider the payoff: ...
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126 views

VaR using normal vol vS lognormal

We are using a vendor's software to calculate the Parametric VaR (using RiskMetrics approach) that take as input the volatility figure of the risk factors. The volatility used so far was the lognormal....
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77 views

FX spot distribution with student-t returns

If I am modelling my returns as $\sim N(0, \sigma^2)$, then I can evolve my spot distribution as: $$S_{t} = S_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t + \sigma dW_{t}}$$ where $S_{0}$ is the spot, $\mu$ ...
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293 views

VaR and Expected Shortfall for Geometric Brownian Motion

Given that $dS_t=\mu S_tdt+\sigma S_tdW_t$ ,a risk free rate r and defining Value at Risk and Expected Shortfall as $VaR_{t,a}=S_0e^{rt}-x$ where $x$ is the amount such that $P(S_t\leq x)=1-a$ ($a:$...
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118 views

General Dynamics of a Tradable Asset under the Risk Neutral Measure

Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then $$\...
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How to prove that a series of random variables $Z_j = 1$ or $-1$ occurring at risk-neutral probability, converges to normal, using the CLT?

Context When pricing options with trees, it is convenient to prove that the asset value at expiry $S_t$ be of log-normal distribution: $$\log{S_t} = \log{S_0} + \mu T + \sigma \sqrt{\frac{T}{n}} \sum_{...
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Log-normal risk-neutral price derivation from binomial trees, not clear about step in derivation process

At page 64 of the book Concepts and practice of mathematical finance, 2nd edition by M. Joshi, paragraph 3.7.2 (Trees and option pricing - A log-normal model - The risk-neutral world behaviour) a ...
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Why can future forward interest rates be assumed to be lognormally distributed in the standard market model?

This seems to be the underlying assumption that allows us to use the standard market model/Black's framework in order to value interest rate derivatives, but I haven't found any understandable ...
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Formula for coskewness and cokurtosis of LogN to project linear returns

I want to find the coskewness and cokurtosis of the multivariate LogN(mu, sigma) distribution from the moments of a normally distributed multivariate distribution (ie: log returns). These higher order ...
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392 views

Ito's lemma and Lognormal Property

What would be the difference between: \begin{align} dS = udt + \sigma dz \end{align} and \begin{align} dS=u*S*dt + \sigma*S*dzdS \end{align} Is that the former is in absolute terms and the latter is ...
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Probability of a stock price using implied volatility

I have attempted to use the fact of having implied volatility, but have not been able to come up with a viable way to calculate the probability, any ideas? Suppose that a stock $S_t$ follows a ...
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What is the industry standard model for pricing Swaptions during this time of negative interest rates, normal model or shifted log-normal model?

I have referred to the some of the well known papers but none of them has a clear answer for my question. I know that both of these models have some disadvantages but, what is the industry standard ...
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Geometric brownian motion and probabilities

A stock's price movement is described by the equations $dS_t=0.02S_tdt+0.25S_tdW_t$ and $S_0=100$. An investor buys a call option on said stock with a strike price $K=95$ which expires in $T=2$ years. ...
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Generate Monte Carlo simulation of multivariate lognormal or weibull distributions in R

I intend to perform a Monte Carlo simulation of asset returns in R. I am currently using the rmvnorm function in the mvtnorm R ...
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lognormal assumption of Black Scholes

I have recently started learning about option pricing and the Black Scholes formula, where stock prices are assumed to be lognormally distributed and returns normally distributed. While trying to do ...
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Stock Prices are Lognormal - Formal Definition

I'm struggling with what the exact meaning of "stock prices are lognormal" (and its use to show normality of returns). My assumption was that given ${S_t}$ are stock prices and returns are ...
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279 views

How to Understand Lognormal Distribution in the Following Case

I got a question and corresponding solution, but have some difficulties in understand the lognormal distribution part of it, so I really appreciate your advice: Question: assume zero interest rate ...
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Drawing values from a lognormal distribution of a GBM

I'm looking at a GBM with parameters $$ r=0.05 \\ \sigma=0.2 \\ K=130\\ T=0.25\\ S_0 = 100 $$ This is a process that is lognormally distributed with mean and variance given by $ \mu = S_0e^{r T+0.5\...
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Why are put and call options worth the same despite that put has no upside whereas call has unlimited upsides?

The following is an interview question. All Black-Scholes assumptions hold. Assume no dividends. Consider a standard European call and a standard European put on the same stock. Assume that each ...
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Realized vol/var log-normal approximation

It is not clear to me what is a better approximation (based on empirical evidence or otherwise), a log-normal approximation for realized volatility or log-normal approximation for realized variance? ...
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65 views

Asset return distribution

What is the basis for assumption that asset prices follow a log normal distribution? Then how is it transformed to say that asset return follows a normal distribution? How this relationship between ...
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85 views

Transforming non-normally distributed interest rates for OLS regression

I am studying the effects of short- and long-term interest rates on bank risk-taking in the Euro zone countries. To analyse the effects, I will use, amongst other, an OLS regression. However I have ...
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829 views

How does the Black Scholes Model Incorporate Log Prices Into Model?

I am still not understanding the link between log prices and how that is incorporated into the BS model. I understand why log(S) is assumed because it makes math easier and it prevents ending prices ...
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306 views

Distribution of simple returns vs logreturns

I understand that stock prices are conditionally modeled using a log normal distribution by the relationship $ y_t/y_{t−1}∼logN(μ_{daily},σ^2_{daily})$ $y_t∼logN(log(y_{t-1})+μ_{daily},σ^2_{daily}))$ ...
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Shifted Log-Normal model

I am trying to understand how the shifted log-normal model works, in which we shift a log-normal model by a factor before the simulation so that interest rates don't turn negative during the ...
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176 views

Does GBM stock price model have E[S(t)] unaffected by volatility?

Many an author claims that, if you model stock prices through GBM, $E[S(t)]=e^{\mu t}$, and the expectation is thus not related to volatility. I keep running around in circles on this one. First ...
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237 views

ATMF FX straddle delta

I am trying to price an ATMF FX (say Usdidr) straddle - the fxdelta for call and put leg are quite different with put fxdelta being higher than call delta. (Absolute values) Why would this be the case?...
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614 views

Log Differences vs Percentage returns [closed]

When working with a single TimeSeries of Foreign Exchange price data (EUR/USD : OHLC) on a minute by minute level, is it better to use the % difference of the close vs the lognormal difference of the ...
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Does the Ito correction term in GBM result in 'real money', or is it illusory?

There are two ways to think about investment returns and randomness. First is sort of like 'bank interest', with randomness. Suppose we invest 100 units of currency. Suppose each year there is a ...
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Squaring lognormal compounding with linear addition of normal returns

Let’s say we start with $100 and invest it for 20 years in stocks and want to predict its terminal value as a random variable (RV). And let’s assume average yearly returns are 10% and volatility is ...
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Black Scholes and the Log Normal Distribution

Why does the Black Scholes Equation imply the returns are log-normally distributed?? How can we tell that the returns of the underlying asset wouldnt be normally distributed??
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About SDE of Geometric Brownian Motion

It's known that most of the financial assets are subject to Geometric Brownian Motion, which satisfies the following equations: $\frac{dS}{S}=\mu dt + \sigma dX$ (1) $S_t = S_0 e^{(\mu + \frac{1}{2}...
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345 views

Why does MACD not use log normalization

Today I wondered why the MACD oscillator uses the differences of two averages instead of the log of their quotient just like it's done for volatility estimation. With this kind of log normalization ...
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516 views

Pricing of a derivative using Risk Neutral Valuation.

I am new to option pricing and following problem came up that I don't understand how to handle. A derivative will pay out dollar amount equal to $$\frac1T\ln \frac{S_T}{S_0}$$ at maturity, where $...
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191 views

Price is Log-normal distributed, yet the return is non-normal

I have a price series. The natural logarithm of the price shows good normality. As shown in the standardized normal probability plot below: However, by viewing the standardized normal probability ...
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Problem of negative local volatility:

Consider the displaced log-normal process: $$dS(t) = \lambda(t)(a(t)+b(t)S(t))dW(t), S(0) = S_0>0, $$ where $W(t)$ is a one-dimensional Brownian motion. We suppose that $(\forall t \ge 0) : \...
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378 views

Log normal price simulation

I'm trying to figure out a spreadsheet I have which simulates 50000 returns in excel using the following function: LOGNORM.INV(RAND(),0,0.35)-1 Question: How ...
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65 views

Quantile with periodic investing

Short Version Can I get a quantile of such an expression? \begin{equation} \sum_{k=1}^{n} A_k\exp(\mathcal{N}(t_k\mu-\sigma\sqrt{t_k}/2,\sigma))) \end{equation} I know I can do it for one part of ...
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365 views

Quantile normal and lognormal

Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows: \begin{equation} \Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...
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Process for a portfolio of stocks where each share follows a log-normal process

Given a portfolio of shares $I = \sum{w_iS_i}$ for some fixed weights $w_i$ where each stok $S_i$ has a log-normal distribution, what is the process / distribution followed by the portfolio? That is, ...
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Why implicit volatility has the shape of a "smile"? [duplicate]

Two of the conditions for an asset price to have a lognormal distribution are: The volatility of the asset is constant. The price of the asset changes smoothly with no jumps. In practice, neither of ...
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504 views

Perform scipy Kolmogorov-Smirnov Test for lognormal distribution in GBM

I am simulating asset prices for n days using GMB with Euler scheme, calculate returns and then perform Kolmogorov-Smirnov test on simulated returns. Code for simulating GBM : ...
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246 views

How can I prove that the sum of two log-normal variable is not log-normal?

I am looking for an analytical proof, that the sum of two log normal random variables is not log-normal. Couldn't find it anywhere, does somebody know where to find it or know how to do it?
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152 views

What is the distribution of percentage return in general?

In finance, we often assume that the log-returns $\ln(1+R(t))$ follow a normal distribution. Since $\ln(1+R(t)) \approx R(t)$ when $R(t)$ is small, \begin{equation*} dS/S \sim \text{Normal}. \end{...
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Maximum likelihood for lognormal mixture

I have a collection of historical data that I want to fit to the following model $$ y_{t+1} - y_t = \alpha + (\rho + \sigma_2 Z_{t+1} )y_t + \sigma_1 Z_{t+1} $$ where everything except the y's are ...