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Questions tagged [lognormal]

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4
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440 views

How to compute the stochastic integral of log-normal process?

How do you compute the following integral: $$\int_0^t e^{\mu s + \sigma W_s} ds$$ or $$\int_0^t e^{\mu s + \sigma W_s} dW_s$$ ? Are those integrals stochastic processes of some well-know type (...
3
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0answers
40 views

Polynomial interpolation of corrected lognormal distribution

Can anyone provide a formula for a polynomial interpolation of the corrected lognormal distribution used to model returns traditionally resulting from the wrong Brownian motion generated model? ...
2
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0answers
153 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) : \...
2
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0answers
218 views

How do we know that the instaneous rate of return on this option, $\gamma$ is negative?

I am self-studying models for financial economics and encountered the following problem: I don't see how the author can conclude that $\gamma = -0.62$. Let's rearrange the second to last equation: $$\...
1
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0answers
30 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? ...
1
vote
0answers
30 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 ...
1
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0answers
63 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, ...
1
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0answers
79 views

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 ...
1
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0answers
264 views

Interest rates - Swaptions implied volatility - Volatility anchoring with Black and with normal volatilities

In a LMM+ with displacement factor a volatility anchoring technique is used, i.e. a long term volatility assumptions is applied, derived from historic time series. Should I adjust this historic ...
1
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0answers
102 views

Distribution of running maximums of a log normal process

I've been searching for quite some time and would appreciate any guidance! What I'm looking for is the distribution of running maximums for a log-normal process. If anyone is familiar with any ...
1
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0answers
569 views

Monte Carlo simulation returns not normal distributed

I am generating 100,000 paths of SPX out to 1 year using Euler discretization. I look at how S is distributed for 100,000 paths at the 1 year point and I find it is lognormally distributed. I look at ...
1
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0answers
231 views

BS Implied Volatility under Normal returns

If I use theoretical prices under a normal valuation model, and I estimate their implied volatility using BLACK SCHOLES implied volatility, do I'll get corresponding log normal volatility?
0
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0answers
46 views

Displaced diffusion LMM

In the standard LMM a rate $L_i(t)=L(t,T_{i-1},T_i)$ has under the $T_n$-forward measure ($n>i$) the dynamics \begin{equation} d{L_i}(t) = - {\sigma _i}(t){L_i}(t)\sum\limits_{j = i + 1}^n {\frac{{...
0
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0answers
48 views

Reference for pricing geometric-mean basket option

Let $(Z_1,\ldots,Z_N)$ be an $N$-dimensional Brownian motion with correlation matrix $\rho$ and consider the multivariate Black-Scholes model \begin{align} dS_i(t) \ = \ (r-q_i)\, S_i(t) \, dt \, + \,...