# Tag Info

8

In addition to John's answer and just to make things clear: The arithmetic mean is given by $$\mu_a = \frac{1}{n} \sum_{i=1}^n x_i$$ The geometric mean is given by $$\mu_g = \sqrt[n]{\prod_{i=1}^n (1+x_i)} -1$$ And we have that $$\mu_g \leq \mu_a$$ So not only would the geometric sharp ratio would be taking into account the "actual" return of the ...

7

If you use a risk-neutral pricing model and consider the probability there, then you get the probability with respect to a risk neutral measure, in addition that probability depends on the chosen numeraire. For example, in Black-Scholes model taking the risk-neutral measure with respect to the bank account $B$ gives $$P(S(T)<K) = Q^{B}(S(T)<K) = \Phi(... 6 I guess what they are trying to say here is that, assume you have two time series X and Y which are exactly the same i.e. X=Y, the correlation is :$$\rho_{X,Y}= \frac{Cov(X,Y)}{\sigma_X \sigma_Y}\overset{X=Y}{=}\frac{Cov(X,X)}{\sigma_X \sigma_X}=\frac{\sigma_X^2}{\sigma_X^2}=1$$Now assume a time series Z=2 \cdot X, you have:$$\sigma_Z=2 \...

6

I'm not sure it makes sense to think of one as more correct than another. However, they do have significant differences. It may help to distinguish between ex-post evaluation of a strategy and ex-ante prediction of what the strategy's performance will be. For simplicity, let's assume the log returns of the strategy are approximately i.i.d. univariate ...

4

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ...

4

There are many variants proposed; some useful, some not so much. As an investor, the most important thing is to compare the exact same ratio, calculated in the exact same way, for each prospect. As the prospect/fund the most important thing is to be clear about the statistic you are reporting so your investors make well informed decisions. So let's start ...

3

You can use refined methodologies but if you just need a rough estimation of liquidity, you can simply use an average of daily volume over N days. In practice, for equities, people tend to use N = 20 or 30. Once you have the average daily volume (say 100,000 shares), you compare it to your holding (say 50,000 shares) to determine the the size of your ...

3

I did not look at the data, but recall that beta is a parameter in the following equation: $$r_A = \alpha + \beta r_B + \epsilon$$ relating two returns (random variables, samples) $r_A$ and $r_B$. To calculate beta you peform $$\beta = \frac{cov(r_A,r_B)}{var(r_B)}.$$ Thus if assets $A$ and $B$ exchange roles, then only the denominator changes. In your ...

3

In the dot.com era the Internet was considered a-winner-takes-it-all market, new tech start-ups (like Netscape, Amazon.com and the famous Pets.com) was measured by how much the capital they where able to chew through, the logic being that the more they spend the more aggressive they were (at least in the investors' eyes), conquering this new market known as ...

3

If you are happy with the OptionStrategist probability values, why don't just do it exactly how they do it: function Covered() { form=document.callreturn; p=form.price.value; q=form.strike.value; t=form.days.value/365; v=form.volatility.value/100; vt=v*Math.sqrt(t); lnpq=Math.log(q/p); d1=lnpq / vt; y=Math.floor(1/(1+.2316419*Math.abs(d1))*100000)/...

3

I think that you may be looking for $$\mathbb{P}(S_T<K) = \frac{\partial P}{\partial K}(K) = 1 + \frac{\partial C}{\partial K}(K)$$ where $P(K)$ and $C(K)$ are the european put and call undiscounted price functions for the maturity $T$. The proof goes (roughly) as this: $$\begin{eqnarray} \frac{\partial P}{\partial K} &=& \frac{\partial}{\... 2 You can certainly calculate the probability of changes in variation but I have not come across a model that only looks at an isolated iVol and its associated term and then deriving a directional probability. However, what you can do, and what options traders do all the time is to look at changes in skew which involves a range of implied data points. In Fx ... 2 You wrote:$$d[5] = (DJIR[5] - \mu) * Covariance$$but you left out half of it (the inverse and the transposed vector on the right side). The correct formula is$$d[5] = (DJIR[5] - \mu)^2 / Var[DJIR]The covariance "matrix" becomes the variance in a 1-dimensional case (in other words x_i and y_i are both equal to DJIR[i] in this case) and the "matrix ... 2 I am trying to fill in what Richard left for the second part: \begin{align*} \exp(-r(T-t))E\, N'(d_2) &= \frac{1}{\sqrt{2\pi}}\exp(-r(T-t))E\, \exp\left(-\frac{1}{2}d_2^2\right) \\ &=\frac{1}{\sqrt{2\pi}}\exp(-r(T-t))E\, \exp\left(-\frac{1}{2}\big(d_1-\sigma\sqrt{T-t}\,\big)^2\right) \\ &=\frac{1}{\sqrt{2\pi}}\exp(-r(T-t))E\\ &\qquad\qquad \... 1 The numerator is S N'(d_1) = S \frac{1}{\sqrt{2 \pi}} \exp(-1/2 d_1^2) = \\ S \frac{1}{\sqrt{2 \pi}} \exp\left(- \frac12 \left(\log(S/E)+ (r + \frac12 \sigma^2(T-t)) \right)^2 / \sigma^2 (T-t) \right) $$the denominator is:$$ \exp(-r (T-t)) E N'(d_2) = \\ E \frac{1}{\sqrt{2 \pi}} \exp\left(- \frac12 \left(\log(S/E)+(r- \frac12 \sigma^2(T-t)) \...

1

I think one of the main liquidity measures is the one from Pastor and Stambaugh (2003). You can use it for both individual stocks or indexes. Just run the following intra-month regression with daily data: $r^e_{i,d+1,t} = \theta_{i,t}+\phi_{i,t}r_{i,d,t}+\gamma_{i,t}sign(r^e_{i,d,t}) \times v_{i,d,t}+\epsilon_{i,d+1,t}$. Where $r^e_{i,d+1,t}$ is the ...

1

I would consider Amihud (2002) as a good first approximation with that level of data.

1

I think you are confusing the goal with the means. The calculation of the PE is not the goal, the true goal is assessing whether a particular stock is an interesting investment opportunity (cheap) under an investment thesis (set of hypotheses). Therefore, there is an infinite number of ways to calculate PE ratios, as a results of a set of different ...

1

I strongly recommend reading an undergraduate finance textbook like Investments by Bodie, Kane, and Marcus. Your methodology may be limited by your data. For example, using forward P/E requires next fiscal year's EPS estimates. NTM (next twelve months) requires quarterly EPS estimates. If you do not have estimates, the best method is TTM (trailing twelve ...

1

"Burn rate" is a measure of "spend rate" relative to cash on hand. So if you have $10 million dollars, and you spend$1 million dollars a month, you will "burn through" your cash in ten months, at which time your company will either "take off," get new financing, or go under. Strategies that rely on "burn rate" are risky ones. Nevertheless, they are ...

1

The delta, gamma, theta and vega exposure is just the sum of the individual positions, thus you sum up the greeks of your two puts, simple as that. Regarding implied volatility you cannot just average implied vols and say this is the implied vol of my structure (multi asset position). You can assign your own volatility expectations and compare that with ...

1

Isn't the option's delta a close approximation for the probability the option will be in the money?

1

Not sure about all of the complicated math and programming above, but I can tell you that, if you want to calculate for 1 Standard Deviation from the current stock price X days away, the following calculation will give you a +/- value from the current stock price. 1 StdDev Move = (Stock Price X Implied Volatility X the Square Root of 'how many days') all ...

1

The arithmetic average of +100% in Year 1 and -100% in Year 2 is 0%, but I we all know the result is not a 0% return. So arithmetic returns are absurd to use in any real life context. Maybe in another universe they can serve some purpose.

1

The only correct way is using log returns. To keep everything consistent, take a arithmetic mean of log returns. Then calculate it net of the risk free (how do you subtract properly using geometric returns?). Then divide (how do you do this properly using geometric returns?) by the standard deviation (how would you calculate this properly with geometric ...

1

For any real world applications, the difference between the arithmetic and geometric Sharpe ratios is likely to 'fall under the noise floor', i.e. be smaller, typically much smaller, than a standard error. This is even under the generous assumptions of stationarity and absence of omitted variables.

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