6

No. Implied volatility isn't a historical measure of standard deviation. Implied volatility is used to relate a market price to some model, be that Black-Scholes or something more sophisticated. Another way to phrase it, implied vol is that single vol input into a model, such that the model reproduces the market prices. Different models will have ...


5

If you're annualising your data with T it should always be the same, not changing with the length of your data. To demonstrate, annualising monthly returns, the Sharpe ratios turn out fairly similar:- Note The reason for multiplying by root 12 is that the mean return is annualised by multiplying by 12 and volatility is annualised by m = 12. 12 on the ...


5

There are sufficiently different ways to calculate the Sharpe ratio that the best advice I can give is to do whatever your boss wants. Also, if it is for a paper or research document, just make clear you document your method. My approach is usually to calculate the highest frequency Sharpe ratio I can based on the data. The higher frequency choice is to get ...


4

beta_A = correlation_A_Index * (stdd_A / stdd_Index ) The difference you see is due to correlation. The correlation between A and the index is lower than B and the index, and that's why you're seeing a lower beta. The moral of the story is that risk is subjective, and in fact you need to understand how your portfolio is correlated with these stocks in ...


4

If $$ X_1, X_2, \dots, X_{12} $$ are i.i.d. (stochastic independent identical distributed) it holds $$ var(\sum X_i) = \sum var(X_i) = \sum var(X_1) = 12var(X_1) $$. now take the square root to get the stated result.


4

The units of returns are 'per time', while the units of variance are also 'per time', thus the units of the Sharpe ratio are 'per square root time'. See section 2.2 of the Short Sharpe Course for a discussion of units, and section 3.3.2 of the same for more information on how moments of the Sharpe are affected by the sampling rate.


4

I'll try to answer according to what I've read (and I hope mostly understood). Let's assume the mean of daily returns is 1%, and the standard deviation of daily returns is 1%. Then: $$ Sharpe = \sqrt{252} \frac{mean(daily\ return)}{stddev(daily\ return)} \approx \sqrt{252} \frac{1 \%}{1 \%} = \sqrt{252}$$ Now let's assume we work with monthly returns. In ...


4

Ideally you'd want to use daily returns and just annualise it, but if you only have monthly returns then calculating the weighted variance in the following way might do it: $$ Var = \frac{\sum_{i=0}^{24}(R_i - \mu)^2}{24 + \frac{21}{31}} + \frac{\frac{21}{31} (R_{25}' - \mu)^2}{24 + \frac{21}{31}} $$ $$ Vol = \sqrt{Var} $$ Where $R_i$ is the returns of ...


3

You know that : $X \sim N(\mu,\sigma^2)$. $Z = \large\frac{X-\mu}{\sigma}$. $\text{Var}(Z) = \large\frac{1}{\sigma^2}\text{Var}(X) = \large\frac{1}{\sigma^2}\sigma^2 = 1$. So that $Z \sim N(0,1)$. However note that the pdf evaluated for X and Z have different domains. The following figure illustrate it : $X$ is plotted in a) and $Z$ in b) Their ...


3

Assume the weights of the two assets are $w$,$1-w$ respectively;the expected returns and standard deviations are denoted by $\mu$,$\sigma$ with subscripts 1,2,p(for portfolio),i.e,we have $\mu_1$,$\mu_2$,$\mu_p$,$\sigma_1$,$\sigma_2$,$\sigma_p$.The correlation coefficent is $\rho$ Then $$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho ...


3

Focusing on intuition rather than theory, $\beta$ can also be thought of as the "risk premium" of that specific asset relative to the market. In general, market risk premium links two very important aspects of the world: Consumption & Return. So if we look at the world in two states, an "Up State" & "Down State", here is what we would see: States:...


3

The simplest and most common method for finding the Tracking Error of Fund X versus a Benchmark B is to compute the standard deviation of the differences in monthly returns of the Fund and the Benchmark: $$TE=\text{STDEV}(r_{X,i}-r_{B,i})$$ A slightly more complicated method involves performing a regression of Fund X returns on the benchmark returns. Then ...


2

Intuitively put you can say that volatility is the within variation and beta is the between variation. Within means the variation that A has within its own time-series, whereas between means between A and the index.


2

Here is an example calculation according to the formula by William F. Sharpe, 1994. The OP's method of annualising the variance (as used below), is also specified by the Committee of European Securities Regulators in this document, page 5, box 1. For this example, taking 24 months of returns of risk-free proxy (US 4-week T-bills) and an example stock, (and ...


2

It seems to me that you want to use the series of option prices to estimate the Sharpe ratio given the option prices in your sample. If so, the idea is to realise that for each option price you have at different times $t_1, t_2, ...$ you could actually close the position and realise the profit or loss. So, basically if you have the option prices you just ...


2

PerformanceAnalytics in R and PortfolioAnalytics in R Here is a tutorial from UW http://faculty.washington.edu/ezivot/econ424/portfolioFunctionsPowerPoint.pdf


2

A pithy way to put it is "implied volatility is the wrong number to put in the wrong formula to get the right price." That is, implied volatility is by definition the parameter $\sigma$ to plug into the Black-Scholes option pricing formula to get the market price of a vanilla option. This is called "volatility," but in reality it isn't the same as the ...


2

Another way to skin cat: # risk-free = 0 require(quantmod) require( PerformanceAnalytics) getSymbols('DJIA', src='yahoo', from = '2009-01-01', to ='2014-12-31') price <- Cl(DJIA) simple.ret <- price/lag(price)-1 table.AnnualizedReturns(simple.ret,Rf=0)[3,] # [1] 0.7267 log.ret <- na.omit(ROC(price)) SD <- sd(log.ret)*sqrt(252) R <-...


2

This is how people usually approach calculating SR with logreturns: library(quantmod) getSymbols('DJIA', src='yahoo', from = '2009-01-01') price <- Cl(DJIA) log_ret <- log(price/lag(price,1)) mean_log_ret <- mean(log_ret, na.rm=T) sd_log_ret <- sd(log_ret, na.rm=T) rf <- 0.0025 # benchmark SR <- (252 * mean_log_ret - log(1+rf))/(sd_log_ret*...


2

What is risk? If one defines risk heuristically as deviation from expectation, then (assuming returns have finite variance) standard deviation can be considered a first approximation for risk. For most distributions the mean and variance do not fully parameterize the distribution. Some standard measures of risk for general distributions include Value at ...


2

You can calculate variance of a portfolio/basket by taking direct weighed averages of the components and then adding the relevant correlation terms * weights for each pair. Can take sqrt of the expression obtained to have Standard deviation. Exact formula for calculation goes like this : (source: benetzkorn.com)


2

Rt in your notation is "filtered" variance R(t|t). The prediction of variance R(t+1|t) adds another term which is not guaranteed to be decreasing overtime. I think another critical assumption is Ve in your equation. How do you define Ve? For price series Ve as a proxy for volatility makes sense to be time-varying, and probably exhibit some auto-correlation....


2

I would suggest check out the Wikipedia page first and use more stylized notations. In your update equation mean(t) = mean(t-1) + K(t) * ( price(t) - mean(t-1) ) you are basically saying that your state process is mean(t) and price(t) is a measurement of mean(t). This doesn't sound legit On the other hand, you could have a mean reverting process $$\text{...


2

This is only correct if the expected returns are normally distributed. Remember that z-score is in essence the quantile function or the VaR of the normal distribution. If you try to apply this to any other distribution, you are going to be sorry. Take a lognormal distributed variable$\sim \text{logN}(\mu,\sigma^2)$, the VaR in that case is $$\text{VaR}_{95\...


2

Your estimator $\hat{s_i}$ for stock $i$ is an unbiased estimator of its latent standard deviation $\sigma_i$ (which is constant for your model). When applying your "window rolling" for calculating $\hat{s_i}$, you get a time-series $ts_{\sigma_i}$ for each stock $i$. With an intercept-only OLS-regression for each time-series $ts_{\sigma_i}$, $$\hat{s_{it}}...


1

In the set of an index where all insturments are traded in the same time zone I would agree that vola pa from say weekly returns is lower than from daily returns. Besides this, the distribution of weekly returns should look "more" Gaussian than the one of daily returns. This is called aggregational Gaussianity e.g. in the paper by Rogers and Zhang. The term ...


1

It is most common to use the "square root of time" method to scale volatility (i.e. standard deviation of returns) to a year (annualize it) if needed, i.e. if the estimate is based on a sample with higher frequency (daily, weekly,..). Mathematically this requires the underlying stochastic process $(X_t)_{t\in T}$ (I've omitted some technical prerequisites ...


1

Have a look at ?dnorm, and rather use the standardized value as argument in your function, in addition to mean and sd: a_<-dnorm((0.001-0.0001)/0.4, mean=0, sd=1) Hope it helps [EDIT] Likewise from ?dsgt st<-(0.001-0.0001)/0.4 skewt<-dsgt(st, mu=0, sigma=1, lambda=0.1, p = 2, q=5, mean.cent=TRUE, var.adj=TRUE) results in skewt=0.4302996 (close ...


1

I believe a few things need to be said here. First, returns are usually calculated (END_VALUE-BEGIN_VALUE)/BEGIN_VALE. There are other ways, but this is what is usually used, and much arguments can be had on what "value" actual is. Second, data frequency should be aligned so daily standard deviation should be aligned to daily expected returns. Third, the ...


1

Severals thoughts: First: this topic is already covered here - mabye you have to collect parts. YOu can start here. A very good overview is given here too. Let us define the following notions. Let $P_t^1$ and $P_t^2$ denote the prices of zwo assets at time $t$. Then $$ r_t = \frac{P_t^1-P_{t-1}^1}{P_{t-1}^1} = \frac{P_t^1}{P_{t-1}^1} -1 $$ denotes the ...


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