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risk neutral measure - How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion

Because $\mathbb{E}\left(e^{\sigma W_t}\right) = e^{\frac{1}{2}\sigma^2T} > 1$, you need that correction to ensure that your asset grows on average at rate $\mu$ (or $r$ in the risk-neutral measure).... Read More

stochastic processes - Covariance of logarithms of geometric Brownian motion

Let $Y = \log X$, then: $$\begin{align} Y &= Y0 + (\mu-\frac{\sigma^2}{2})t + \sigma Wt \\ EYt &=Y0 + (\mu-\frac{\sigma^2}{2})t \\ EYtEYs &= Y0^2 + Y0 (\mu-\frac{\sigma^2}{2}) (t+s) + (\mu-\frac{\sig... Read More

forecasting - Predicting natural gas prices using weather data

Natural gas prices and temperature are correlated, but there are other factors here: In the long term, the intrayear shape of gas prices is a function of temperature, but that curve is usually non t... Read More

data - Discrepancy in stock volume values for different intervals for the same day on Yahoo Finance

Regular trading hours for this stock on its primary exchange (in Tokyo) are from 09:00 JST - 11:30 JST, then it closes for 1 hour, then it reopens and trades from 12:30 JST - 15:00 JST. On top of tha... Read More

risk - PD and LGD for ECL calculations needs to be time dependent?

I assume that you calculate ECL in the context of IFRS9 -correct? market practice often follows the following appraoch: estimate a TTC PD/LGD (TTC = through the cycle). This corresponds to your life... Read More

history - Which of Terry Tao's co-authors on compressed sensing consulted for Renaissance Technologies?

The is the article you are looking for: Stable Signal Recovery from Incomplete and Inaccurate Measurements Emmanuel Candes, Justin Romberg, Terence Tao It provided the basis for compressed sensing.... Read More

black scholes - What is the formula to calculate Implied Volatility Percentile

As volatility has a great influence on option prices, you'd like to sell options in high volatility environments and purchase options in moments of low volatility. But what is high/low volatility? Im... Read More

portfolio optimization - Rigorous proof that volatility target strategies actually tend to the target

Suppose that you are riskless asset with return $r{ft}$ and a risky asset with return $rt$ and conditional volatility $\sigmat(rt) := \sqrt{Vt(rt)}$. We build a portfolio using weights $(w1, w2) \in... Read More

Calculating covariance from three variances

I think what you are effectively looking at is $$\ \begin{align} \log(S{AUDCAD})&=\log(S{AUDUSD})\pm\log(S_{USDCAD})\\ \Rightarrow z&=x\pm y \end{align} $$ Thus, $$ \sigmaz^2=\mathrm{E}\left(\left(x\... Read More

How to use black scholes for spot trading?

I am going to assume that you meant to ask how we can use option prices to inform trading of underlying stocks. The Black-Scholes-Merton model says that under the physical measure, the underlying sto... Read More