Questions tagged [lognormal]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
75 views

How can I estimate value-at-risk of a long/short portfolio without making simplifying assumptions?

I have had a couple of long-standing questions about the mathematics behind a simple "vanilla" parametric VaR calculation and I'm hoping someone could clear up my confusion. Most likely I am ...
user avatar
1 vote
1 answer
130 views

Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions

Are there optimal portfolio theories than instead of the expected value they were based on the Mode of distributions? During my engineer student days I saw the Markowitz theory for portfolio selection ...
user avatar
  • 61
0 votes
1 answer
56 views

Should future price scenarios be symmetric around the current market price?

Assume a financial instrument which has a (roughly) log-normal price distribution and behaves like a random walk. I would like to generate some possible scenarios for where the price might be tomorrow....
user avatar
3 votes
1 answer
101 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 ...
user avatar
1 vote
0 answers
91 views

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}...
user avatar
1 vote
0 answers
56 views

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 ...
user avatar
4 votes
1 answer
243 views

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(...
user avatar
0 votes
1 answer
94 views

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 ...
user avatar
6 votes
2 answers
135 views

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 ...
user avatar
1 vote
0 answers
122 views

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: ...
user avatar
  • 61
0 votes
2 answers
177 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....
user avatar
  • 41
0 votes
1 answer
84 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, $\mu$ ...
user avatar
  • 121
2 votes
1 answer
370 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 $P(S_t\leq x)=1-a$ ($a:$...
user avatar
3 votes
1 answer
130 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 $$\...
user avatar
0 votes
0 answers
78 views

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_{...
user avatar
  • 356
0 votes
1 answer
86 views

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 ...
user avatar
  • 356
2 votes
2 answers
417 views

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 ...
user avatar
  • 778
1 vote
0 answers
73 views

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 ...
user avatar
  • 11
1 vote
2 answers
608 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 ...
user avatar
  • 23
1 vote
3 answers
807 views

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 ...
user avatar
2 votes
1 answer
130 views

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 ...
user avatar
  • 41
0 votes
0 answers
95 views

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. ...
user avatar
0 votes
1 answer
417 views

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 ...
user avatar
  • 13
1 vote
1 answer
258 views

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 ...
user avatar
3 votes
2 answers
1k views

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 ...
user avatar
  • 33
1 vote
2 answers
372 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 ...
user avatar
  • 587
0 votes
1 answer
356 views

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\...
user avatar
  • 181
0 votes
2 answers
250 views

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 ...
user avatar
  • 830
1 vote
0 answers
63 views

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? ...
user avatar
1 vote
1 answer
75 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 ...
user avatar
  • 507
0 votes
1 answer
86 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 ...
user avatar
  • 1
1 vote
1 answer
987 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 ...
user avatar
  • 619
3 votes
2 answers
367 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})$ $y_t∼logN(log(y_{t-1})+μ_{daily},σ^2_{daily}))$ ...
user avatar
  • 39
3 votes
1 answer
5k views

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 ...
user avatar
  • 171
2 votes
1 answer
200 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 ...
user avatar
0 votes
1 answer
258 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?...
user avatar
  • 25
0 votes
1 answer
767 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 ...
user avatar
  • 113
0 votes
3 answers
284 views

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 ...
user avatar
1 vote
0 answers
35 views

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 ...
user avatar
1 vote
2 answers
11k views

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??
user avatar
  • 2,151
3 votes
2 answers
756 views

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}...
user avatar
  • 101
2 votes
1 answer
375 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 ...
user avatar
  • 197
1 vote
1 answer
543 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 $...
user avatar
1 vote
2 answers
212 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 ...
user avatar
  • 11
2 votes
0 answers
260 views

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) : \...
user avatar
  • 73
0 votes
2 answers
414 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 ...
user avatar
0 votes
1 answer
70 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 ...
user avatar
  • 123
2 votes
1 answer
385 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 {...
user avatar
  • 123
1 vote
0 answers
74 views

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, ...
user avatar
0 votes
2 answers
297 views

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 ...
user avatar
  • 143