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6

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 ...

5

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

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 $$P(S(T)<K) = Q^{B}(S(T)<K) = ... 4 I did not look at the data, but recall that beta is a parameter in the following equation:$$ 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)}. $$Thus if assets A and B exchange roles, then only the denominator changes. In your ... 3 There are many variants proposed; some useful, some not so much. As an investor, the most important thing is to compare the exact same ratio, calculated in the exact same way, for each prospect. As the prospect/fund the most important thing is to be clear about the statistic you are reporting so your investors make well informed decisions. So let's start ... 2 If you are happy with the OptionStrategist probability values, why don't just do it exactly how they do it: function Covered() { form=document.callreturn; p=form.price.value; q=form.strike.value; t=form.days.value/365; v=form.volatility.value/100; vt=v*Math.sqrt(t); lnpq=Math.log(q/p); d1=lnpq / vt; ... 1 This is a basic utility exercise. I would guess the additional assumption you are missing to solve the exercise is that the player would be willing to accept the fair game but nothing worse. To make this a fair game, the maximal amount which could be paid to enter the game, would result in zero expected loss of utility (nobody would accept anything worse). ... 1 The delta, gamma, theta and vega exposure is just the sum of the individual positions, thus you sum up the greeks of your two puts, simple as that. Regarding implied volatility you cannot just average implied vols and say this is the implied vol of my structure (multi asset position). You can assign your own volatility expectations and compare that with ... 1 I think that you may be looking for$$ \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} &=& ...

1

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 ...

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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