10

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(...


9

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 ...


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

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 ...


5

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)/...


4

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 ...


4

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 ...


4

The correct answer is "arithmetic mean, because Bill Sharpe says so". He invented the thing, and he's pretty clear on which one he was looking at. If you use the geometric mean, which is lower the higher the volatility in the returns, and then you divide by standard deviation, you have essentially discounted your result TWICE for volatility.


4

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}{\...


4

Use the call put parity : $$C(t,F_{t,T},T,\sigma,K)-P(t,F_{t,T},T,\sigma,K)=F_{t,T}-K$$ where $F_{t,T}$ is the forward rate(underlying), $K$ is the strike, $t$ the valuation date, $\sigma$ the model volatility, $T$ is the maturity. Differentiate the equation with respect to $\sigma$, and you will get the result wanted.


3

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 ...


3

Due to the leap year 366 days need to be used here to match UST conventions (which is ACT/ACT). In this case it doesn't matter whether your interest period extends to only 1 day after the 29th of February or, e.g., 200. In fact if you look at the daycount description of the bill it says: "the day count basis for price and yield calculations is 365 ...


2

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 ...


2

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 ...


2

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


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

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 ...


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 \...


2

It depends on the ratio you are looking at. Most of them are scaled by $\sqrt{12}$, but the Treynor index is a bit different and is scaled by $12$. Sharpe and Information ratios are both ratios of average returns to standard deviations. They are annualized by assuming that the monthly returns are IID. Hence, average monthly return is scaled up by 12 and ...


2

Assume we start at $t=0$ with $P_0$, there are $t=1...N$ subsequent periods, and at each period-end $t$ an (entirely arbitrary) portion $c$ of our portfolio $P_t$ is churned and $(1-c)$ remains untouched. $P$ grows over each period by a factor $(1+g)$: $P_t = P_{t-1}(1+g)$. We can partition $P_t$ into sub-portfolios, each with its own churn history, as in: ...


2

Note that with $H(\cdot)$ the Heaviside function $$\frac{d}{ds} H(s-K) = \delta(s-K)$$ but $$\frac{d}{dK} H(s-K) = \color{red}{-}\delta(s-K)$$ You can also use the Leibniz integral rule to write that $$ \frac{d}{dK} \int_K^\infty \phi_{S_T}(T,s) ds = -\phi(S_T,K) $$


1

No, if you want to calculate the annualized sharpe ratio you should 1) make sure that your risk free rate is in monthly terms (so if it's 3% annual you need to put .03/12 in cell KX32) 2) only multiply the result by the square root of 12 (not 36). To calculate the annualized sharpe ratio, you multiply the monthly ratio by the square root of the ...


1

RETURN Firstly, return is based upon the amount gained over a period of time. So your calculation for a percentage return should actually be be: (Sale price - Cost basis)/Cost Basis. TOTAL RETURN A "total return" price series or index is a transformation of the original traded price timeseries to a timeseries that can be used to estimate/calculate a ...


1

I think they messed up with the dataset. The dates are weird and the rolling volatilities do not match. They suddenly take 2 year history instead of 5. May I please ask why did you not take full columns for 1Y and 3Y stress volatilities? (the percentile() starts somewhere in middle of the column) Thank you. EDIT: You should use full dataset to calculate ...


1

At the end of each year you have wealth $X_t$ in an investment account which grows at the rate $r$. At the beginning of each year you withdraw the amount $W$ and keep it in cash to pay for your expenses through the coming year. For your investment to last $n$ years you need to start with $X=W+W\frac{1}{1+r}+\cdots+W\frac{1}{(1+r)^n}$. In this way you will ...


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 excess ...


1

Another way to look at it is that: $$\begin{align} \beta &= \frac{cov(R_p,R_M)}{var(R_M)}\\ &= \rho(R_p,R_M)\frac{\sigma(R_p)}{\sigma(R_M)} \end{align}$$ In other words, the beta is the product of the correlation between your portfolio and the market and the ratio of their volatility. You can then see why the inverse beta is not what you expected.


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 ...


Only top voted, non community-wiki answers of a minimum length are eligible