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9 votes
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Black Scholes and the Log Normal Distribution

The Black-Scholes-Merton (1973) model implies that the prices of the underlying asset at maturity $S_T$ are log-normally distributed $$ln(S_T)\sim N\big[ln(S_0)+(\mu-\frac{\sigma^2}{2})T,\;\sigma^2T\...
jthg's user avatar
  • 445
7 votes
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Stock Prices are Lognormal - Formal Definition

In reality, neither are stock prices log-normally distributed nor are returns normally distributed. More sophisticated models drop this assumption. For instance, returns are more peaked and have ...
Kevin's user avatar
  • 16k
6 votes
Accepted

Quantile normal and lognormal

Quantiles are preserved under monotonic transformations, hence the quantile for $Y$ is simply the exponential of the quantile of $X$, no need for corrections whatsoever (see here for instance). Put ...
Quantuple's user avatar
  • 14.7k
5 votes

Stock Prices are Lognormal - Formal Definition

Stock prices cannot be negative which means that they are not normally distributed due to the fact they cannot be negative as result of this stock prices behave similarly to exponential functions. To ...
Gogo78's user avatar
  • 636
5 votes

Why can future forward interest rates be assumed to be lognormally distributed in the standard market model?

A lognormal distribution has three valuable properties (I) It ensures that the rate is only allowed to be positive; (II) the changes in the interest rate are proportional to the interest rate; and (...
Dom's user avatar
  • 2,167
5 votes
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Difficulty with stochastic calculus problem

Shreve titles the question as: "Black-Scholes_Merton formula for time-varying, non-random interest rate and volatility". This model is well-known, so there is no ambiguity. The SDE should ...
Sebastian's user avatar
  • 168
4 votes

Downward sloping smile in normal model

The implied Black-Scholes skew will be downward sloping in the limit on both the left and the right. (I believe @Gordon's derivation claiming upward slope may have a sign error somewhere). Left Side ...
Brian B's user avatar
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4 votes
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Downward sloping smile in normal model

Since $S_T = S_0 + \sigma W_T$, \begin{align*} C &:= E\left((S_T-K)^+ \right)\\ &= E\left((S_0+\sigma W_T-K)^+ \right)\\ &=\int_{\frac{K-S_0}{\sigma \sqrt{T}}}^{\infty}(S_0+\sigma\sqrt{T} ...
Gordon's user avatar
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4 votes

How to compute the variance of this stochastic integral?

To compute the variance $$\text{Var}\left(\int _0^T e^{W_t} dt \right),$$ we need to compute \begin{align*} E\left( \left(\int _0^T e^{W_t} dt \right)^2 \right) &= \int_0^T\!\!\!\!\int_0^T E\left(...
Gordon's user avatar
  • 21.2k
4 votes

Downward sloping smile in normal model

Although it's a bit different story, there are VERY accurate approximation formulas for the implied volatility under normal model (so-called basis point volatility). Using them, you can obtain the ...
jChoi's user avatar
  • 1,174
4 votes

Why implicit volatility has the shape of a "smile"?

The smile is there exactly because the model is wrong. The reason it's used though (despite being wrong) is that it provides a convenient space to look at the underlying - the vol* The (undiscounted)...
will's user avatar
  • 2,581
4 votes

Shifted Log-Normal model

Let us assume we are interested in some (forward) rate $F_t=F(t,T)$ which we assume is log-normally distributed: $$\text{d}F_t=\sigma F_t\text{d}W_t$$ However, we observe market rates can in practice ...
Daneel Olivaw's user avatar
4 votes
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How to Understand Lognormal Distribution in the Following Case

The drift of $\mathrm{d}\ln(S_t)$ is indeed $r-\frac{1}{2}\sigma^2$ which is always negative if $r=0$. The extra $-\frac{1}{2}\sigma^2$ has many explanations. You could see it as a convexity ...
Kevin's user avatar
  • 16k
4 votes
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Why can future forward interest rates be assumed to be lognormally distributed in the standard market model?

SHORT STORY: forward Libor rates need not be assumed to be log-normally distributed. For example, they can be assumed to be normally distributed (and indeed, on Bloomberg, Swaption implied vols are ...
Jan Stuller's user avatar
  • 6,178
4 votes
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FX spot distribution with student-t returns

1. Theory The Student $t$ distribution does not exhibit a moment generating function $$ M_X(t)=\mathbb{E}\left(e^{tX} \right) $$ Hence, there exist no closed form solution for $M_X(t=1)=\mathbb{E}\...
Kermittfrog's user avatar
  • 6,772
4 votes
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Is this process log normally distributed?

I’m sorry to say you’re not correct in your conclusion. The basic problem is that in the second to last equation $$dP/P=D^*\sigma_y y_n dW_t$$ the $D^* $ is not constant but is a function of $y_n$ ...
dm63's user avatar
  • 17.2k
3 votes

Black Scholes and the Log Normal Distribution

BS assumes prices NOT returns are log-normally distributed. Why making that assumption? 1.log-normal is not perfect but OK to fit potential prices distribution. 2.The nature of log-normal distribution ...
Hui's user avatar
  • 402
3 votes
Accepted

Demonstration of Ito's correction term/lemma in binomial tree

Actually it is quite simple to demonstrate Ito's correction term in a binomial tree. Details can be found in my new paper (p. 8-10): von Jouanne-Diedrich, Holger: Ito, Stratonovich and Friends (April ...
vonjd's user avatar
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3 votes
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Why does MACD not use log normalization

The most likely reason I can think of is the ease of computation. Gerald Appel developed the MACD in the late 1970's, when computing resources were very limited. When doing calculations by hand, on ...
Joshua Ulrich's user avatar
3 votes

Transforming non-normally distributed interest rates for OLS regression

You do not need to transform the variables into normally distributed data in order to use them in a regression. That is not a requirement of ordinary least squares. If the error terms are normally ...
Dave Harris's user avatar
  • 4,299
3 votes

Asset return distribution

In the Black Scholes (1973) model, the stock price is assumed to follow a geometric Brownian motion $\mathrm{d}S_t=S_t\mu \mathrm{d}t + S_t \sigma \mathrm{d}W_t$. If you solve the SDE, $(S_t)$ is log-...
Kevin's user avatar
  • 16k
3 votes
Accepted

Drawing values from a lognormal distribution of a GBM

Assuming that your GBM is given by $$S_{T}=S_{0}e^{(r -{\frac {\sigma ^{2}}{2}})T+\sigma W_{T}}$$ then its mean and variance are: $${Mean=S_{0}e^{r T},}$$ $$ {Variance=S_{0}^{2}e^{2r T}\left(e^{\...
emcor's user avatar
  • 5,795
3 votes
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Probability of a stock price using implied volatility

I assume you want to real-world probability, because the risk-neutral probability is not a probability in the 'likelihood' sense. Under the real-world measure, we model the stock under the B-S model ...
Jan Stuller's user avatar
  • 6,178
3 votes

General Dynamics of a Tradable Asset under the Risk Neutral Measure

Our market has a tradeable asset $S$ and a risk-less money market account $B$, that is, the numéraire of the risk-neutral measure. We assume the following standard conditions, which are widely ...
Daneel Olivaw's user avatar
3 votes
Accepted

Covariance of the product of log normal process and normal procces

I'll give it a try, but am not yet 100% sure that it's the way to go. Ansatz: Let's find the distribution of the integral of a Brownian motion with respect to time (call it $x$) and find the ...
Kermittfrog's user avatar
  • 6,772
2 votes

Log normal price simulation

To my knowledge, adding a minus 1 does not transform a log normal variable into an normal distributed variable. The only thing that I can think of which make sense is the log normal represents a price ...
loxol's user avatar
  • 71
2 votes

Quantile with periodic investing

The random variable $$X = \sum_{k=1}^{n} A_k\exp(\mathcal{N}(t_k\mu-\sigma\sqrt{t_k}/2,\sigma)))$$ emerges as a weighted sum of individual random variables that are log-normally distributed. ...
Quantuple's user avatar
  • 14.7k
2 votes
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Simulating a stock price with Monte Carlo - Why my solution isn't equivalent to the author's

you got a typo. It should be 40.886 in your last equation. Then $\sigma$ should match. Also, If $\alpha$ means annualized log return, it should be $\mu\,t = \...
Will Gu's user avatar
  • 712
2 votes
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Clarification on this author's solution for this problem on lognormal stock distribution

Yes, your steps are valid This is a wrong use of the term "quantile". Here you need to compute a probability (through the normal cdf) and not a quantile (i.e. the value of a random variable ...
Quantuple's user avatar
  • 14.7k

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