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2

The key variable is indeed the derivative's elasticity $\Omega$ (aka leverage, Lambda). It is defined by $$\Omega=\frac{\frac{\partial V}{V}}{\frac{\partial S}{S}}=\frac{\partial V}{\partial S}\frac{S}{V}=\Delta\frac{S}{V}.$$ Here, $V$ corresponds to the value of the derivative and $S$ to its underlying. This number measures how much riskier the derivative ...


2

There are many forms of vega. For var swaps, you can directly differentiate its strike to get a vega that is the sensitivity against the var strike. In LV, you can get a vega by a parallel bump in the entire volatility surface (beware of arbitrage though). People seldom calculate vega as a partial derivative against the implied vol of a certain strike and ...


0

I think in best practice you should multiply with 100 (or even with a larger number, as long as the values are within 0 - 1000). The reason is for convergence of optimizer as others mentioned above. The volatility of course will be affected (scaled up to approximately 100 times. so after calculating the volatilities, you must rescale them to 1/100 Best ...


6

The simplest approach is to use two different variables $T_1$ and $T_2$ instead of the single variable $T$ that denotes Time To Maturity in the classic Black Scholes Merton formula. $T_1$, the time to maturity for interest rate computation purposes, is the calendar time in years between now and maturity. For example the term $-Ke^{-rT}N(d_2)$ in the formula ...


0

Without doing any calcs, I would guess it's because of the 2 days passed and the decrease in vol. Besides the more commonly known greeks (delta, gamma, vega), you have some other interesting second order greeks and the one that will explain what you are observing is probably the delta decay (Charm). Intuitively think of it like this: your put is in the money ...


3

The three ways to manufacture pseudo-implied vols I know of are: Find a related underlying and, even if only few options trade on it, 'borrow' its implied vols. Compute statistical vol from historical underlying prices (not strike dependent, still useful to know). Compute breakeven vol, still based on historical underlying prices, strike dependent, by ...


0

If you compute total return as: $$R_{2009-12-31 \rightarrow 2020-3-31} = \frac{P_{2020-3-31} + D_{2020-3-31}}{P_{2009-12-31}}-1$$ Where $D_{2020-3-31}$ are all the dividend paid in that period. Then you can do the following: First note that your sample has $12 \times 10 + 3$ months; I.e. 123 months. So the average monthly return is: $$\bar{r}_{monthly} = (...


1

There is a very popular parametrisation of the Implied Volatility (IV) Smile in terms of Total Implied Variance (see Gatheral & Jaquier). A condition for an Implied Volatility Surface (IVS) to be free of calendar spread arbitrage is also derived in terms of Total Implied Variance. Defining the latter by $w(k, t):=\sigma_{\mathrm{BS}}^{2}(k, t) t$, it ...


0

You can use Garch and VaR in complementary terms. I do not know of any top finance journal paper where that was done (as it is probably something not very novel). However, some field journals do have some interesting things on relating Garch and Value-at-Risk. For example this paper states: The results indicate that both stationary and fractionally ...


2

For Q1, Indeed the ratio of 2 zero coupon bonds associated with the forward is an exact lognormal process (Just apply Ito's lemma to the ratio, as you already know the dynamics of the 0 coupon bonds. You can disregard the drift term as the forward rate is a martingale in the bond forward measure.). The forward rate is then obtained by just adding a scalar, ...


2

Implied vol on the market depends on time, that's why we talk about vol surface as a function of maturity and strike. Here's an example from Wikipedia: Notice, that this nothing to do with time scaling, this is quoted on annual terms. In a simple model that Black and Scholes used there is no implied vol. There's only geometric Brownian motion (GBM) with $\...


2

Implied vol is per unit time, unit being a year here, so yes it is per annum. So for a T maturity, the variance will have diffusive impact of $\sigma^2 T$. I can think of two ways that might convince you that it is the variance that should scale with time. Remember the coefficient of the diffusion equation is $\sigma^2$, so the square version is important ...


1

Depends on what you're trying to do. Log-normal model Usually, you'd compute the Vol of Log-returns if you're trying to calibrate a Log-normal model, such as the Geometric-Brownian-Motion model for the stock price under the real-world probability measure: $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ If you need to calibrate a model such as the above and ...


0

In the same way. Just multiply the daily standard deviation for the square root of the trading days (i.e. 252).


1

As mentioned, the moneyness refers to the ratio of stock price to strike price, that is, $\frac{S}{K}.$. I reproduce the following plots using Python 3. Comparing the plot above to the plot in the paper, they exhibit fairly similar behaviors except at zero volatility (I get divide by zero error). The source codes are as follows and can be found at my ...


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