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0
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1answer
24 views

Creating the histogram for the distribution of the portfolio returns

Given log returns for some stocks $A$ and $B$, which are the constituents of our hypothetical portfolio in equal weights, how does one actually come up with a distribution of the log returns of the ...
3
votes
0answers
107 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
votes
2answers
220 views

how do we know if the volatility which is quoted in market is Normal (Bachelier model) or log normal (Black 76)?

in market, many instruments are quoted in volatility, but how we can tell what kind of volatility is this? is it normal volatility, or lognormal volatility. because it affect our hedging positions so ...
4
votes
2answers
136 views

Bloomberg implied volatility smile for equities

I was wondering if someone knows how Bloomberg does their computations for the implied volatility smile for equities. As far as I understand, they use a lognormal mixture to model the stock prices. ...
1
vote
1answer
32 views

Expected value of bivariate lognormal spread

I don´t know how to derivate the Expected Value for the following problem: Suppose that the random vector (S_1, S_2) has a bivariate lognormal distribution with ...
0
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1answer
44 views

Should earnings be modelled normally or lognormally?

I am having difficulty deciding whether a company's earnings should be modelled normally or lognormally. If we consider two arguments: (i) The earnings of a company are the returns on the assets of ...
0
votes
1answer
35 views

Test Log-Normality for LIBOR forward rates under the Libor Market Model

As far as I understand, under the Libor Market Model the forward rates are assumed to have a log-normal distribution. Given that I have constructed my LMM model and now have a matrix of: k different ...
2
votes
2answers
126 views

The Distribution of Future Stock Price

In Hull, we are presented that $$\frac{\Delta S}{S_{0}}=\mu \Delta t+\sigma\sqrt{\Delta t}\cdot \varepsilon.$$ Following some algebra, $$ \begin{align*} \frac{\Delta S}{S_{0}} &=\mu \Delta ...
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0answers
41 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
vote
0answers
149 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 ...
0
votes
1answer
71 views

Is the value also log-normally distributed?

My book assumes many times that $log(1+R)$ is normally distributed, so R is log-normal. But does this also mean that the value process is log-normal? Since $V=V_0(1+R)\rightarrow V/V_0=1+R$, and since ...
0
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0answers
120 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?
3
votes
1answer
200 views

Trouble arriving at Black-Scholes Formula

I am attempting to arrive at the Black-Scholes formula for my own understanding. I can accept one can use the risk-free distribution & rate, so I am attempting to use the distrution to arrive at ...
1
vote
3answers
321 views

Risk Neutral Evaluation - Exchange/Spread Options

I have two assets, $S_1$ and $S_2$, which follow geometric Brownian motion processes. This implies that both $S_1$ and $S_2$ have a lognormal distribution. I'm trying to get the exchange option price ...
1
vote
2answers
145 views

how to extend lognormal model so that $\sigma$ is correlated to $\mu$?

Consider a log-normal model, $dx / x = \mu dt + \sigma dW$, where $W(t)$ is a Wiener process. Let's say $\mu$ and $\sigma$ change with time, slowly, so we note them by $\mu(t)$ and $\sigma(t)$. ...
3
votes
1answer
369 views

Integrating log-normal

The usual log normal model in differential form is: $dS = \mu S dt + \sigma S dX$ where $dX$ is the stochastic part, so $\frac{dS}{S} = \mu dt + \sigma dX$ (1) and we normally solve this by ...
6
votes
2answers
551 views

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

I am preparing an undergraduate QuantFinance lecture. I want to demonstrate the ideas of Ito's correction term and Ito's lemma in the most accessible manner. My idea is to take the "working horse" of ...
6
votes
1answer
5k views

Annualzing the log of daily returns riddle

Two popular ways to measure returns are Arithmetic returns and Log returns. Let's define arithmetic (simple period) returns as: P(t) - P(t-1) / P(t-1). Let's define log return as Ln( P(t)/P(t-1) ) or ...
9
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3answers
4k views

How to calculate future distribution of price using volatility?

I want to create a lognormal distribution of future stock prices. Using a monte carlo simulation I came up with the standard deviation as being $\sqrt{(days/252)}$ $*volatility*mean*$ $\log(mean)$. ...