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2

According to the blog post you cited above, all you have to do is simply back out Black Scholes Implied Volatilities from the prices in the first part of the website. For a given strike $X$, risk-free rate of zero, and jump-level $H$, the present value according to the blog post is: $$PV(H,X)=\mathbb{E}_\mathbf{Q}\left((S-X)^+\right)=p_\mathbf{Q}(H-X)^+ + (1-...


2

To accurately compute volatility for future events using past data and present data ( for many of the reasons Stephane has already eluded to), is a fruitless and frustrating task. Goldman, Citadel , and the Fed already know that you and all others trying to predict this will do such. Put down the complicated calculations and spend as much time as you can to ...


0

Besides a couple of ways you might try to improve your code (which I will not do here); your jump check is not working correctly: In a time step $\Delta t$, the process will jump with probability $\approx exp(-\lambda \Delta t) $. Hence, you need to compare if (unifrnd(0,1) > exp(-lambda * dt)) % jump occured % flip state else % no jump occured % ...


0

Just use the definition of conditional probability. With $\gamma_t \in \{-1,1\}$ an indicator returning 1 if $\sigma = 0.8$ and -1 otherwise, and $x_{1:T}$ the path of the Brownian motion over the time-period $[1,T]$ you have $$p(x_{1:T}, \gamma_{1:T}) = p(x_{1:T} \vert \gamma_{1:T})p(\gamma_{1:T}).$$ In practice you just simulate, until time $T$, ...


9

Great question. Let me try provide some insights and thoughts regarding your points and questions raised. It may not be a full answer but hopefully it helps to connect the contents in the paper/book with some trading intuition: From a theoretical perspective, I don't see any mistake in your thinking regarding skew decay but two questions arise on my end: ...


4

It depends what you exactly call Dupire's formula. If you take the original formula, valid under zero interest rates and dividends (or equivalently, considering undiscounted option prices on the forwards), which reads $$\sigma_L^2 = 2 \frac{ \frac{\partial C}{\partial T} }{K^2 \frac{\partial^2 C}{\partial K^2}}\,.$$ Then the formula for a put is the same, ...


1

I agree that the above mentioned eSSVI extension is a very efficient and elegant method for calibration purposes. Arbitrage free slices and interpolations can easily be created by making use of the criteria in the papers. It is also described in great detail in the thesis "Extending the SSVI model with arbitrage-free conditions" (google it). It ...


1

For the first question, there have been a number of improvements on SVI/SSVI that are much more flexible than SSVI and also come with easy-to-impose no-arb conditions. See below: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2971502 https://arxiv.org/pdf/1804.04924.pdf Paper 2 builds on Paper 1 and comes with a robust fitting procedure. One thing to ...


1

You are almost there! The annual variance is the sum of the daily variance: $$\sigma_y^2 t_y = \sum_{i=1}^{252}{\sigma_d(i)^2 t_d}$$ where $t_d=\frac{1}{252}$ and $t_y=1$ hence the $\sqrt{252}$ term you get if you assume that the volatility is expected to be the same every day. The daily variance is related to average 10min bar variances $\sigma_m$: $$\...


1

You could try to do regression analysis, where you sub-divide the day into time windows and then try to fit a seasonality by saying that x% of daily variance will accrue in the first hour of trading etc.


-1

Standard deviation method Historical price returns also on close prices. The simplest method for Parkinson's High Close method is The high Low method is statistically more efficient than the standard close method. However it assumes continuous trading and observations of high and Low prices. The method can therefore underestimate the true volativity. The ...


0

Here is the simplest implementation of an implied volatility estimation in VBA by Espen Haug, it is easily portable to Python, however, I am not sure, but I think QuantLib in Python has a built-in Implied volatility estimator. Here is the code for reference: Public Function GBlackScholesImpVolBisection(CallPutFlag As String, S As Double, _ X As ...


0

The most common methods apart from the standard deviation of returns which is the most common method of estimating volatility are, Parkinsons extreme value method Sheldon Natenberg suggests These are all Historic or realized volatility estimates NOT Implied volatility which is a whole different ball game.


1

I would like to posit a more straightforward answer, it is a mathematical illusion. Although this can be solved through formal theory because the distributions are known, doing so would create a long post. Instead, it can be quickly illustrated through a simulation. Let us assume that the data are normally distributed. The results depend on that. If they ...


2

Call option: $$\mathbb{P}\left(S_t\geq K\right)=\mathbb{P}\left(S_0e^{(rt-0.5\sigma^2t+\sigma W_t)}\geq K\right)=\\=\mathbb{P}\left(W_t\geq \frac{ln\left(\frac{K}{S_0}\right)-rt+0.5\sigma^2t}{\sigma}\right)=\\=\mathbb{P}\left(Z\geq \frac{ln\left(\frac{K}{S_0}\right)-rt+0.5\sigma^2t}{\sigma\sqrt{t}}\right)=\mathbb{P}(Z\leq d2)$$ So we have shown the well-...


2

This is largely because the variance of stock returns is high relative to their mean. The idea that stock return means are harder to estimate is old and was already known before high frequency data, or even GARCH models, were widely used. The point is made e.g. in this 85 paper by Jorion who writes: On the other hand, uncertainty in variances and covariances ...


3

The option payoff diagram for European Option will be exactly the same. The intuitive reason that option value changes with volatility is that it changes the probability of winning the jackpot. Think about it, options provide you downside protection. So, for example, when the stock is volatile, it has a higher probability of getting to a high price. If you ...


1

Have you reviewed the Black Scholes formulae? For a given spot price, volatility is back-calculated based on options premium that investors are willing to pay. Given constant volatility as you suggest, the option value will decay as it gets close to the expiration. In other words, an option at a strike price equal to spot price, will have an intrinsic value ...


2

I am reading this 2.5 months after the question was asked but I still see some confusion in the answers (or at least I am confused by them). The OP claims that the variance of asset returns is easier to estimate than the mean, but the statement is not formulated mathematically. The currently available answers do not formulate it mathematically either. This ...


2

In fact, a standard way to estimate the volatility does not use the mean at all (the mean is set to zero in the formula), because, as pointed out in @Kevin's answer, it really makes no difference, so the premise of the question is a bit fraught. It should be noted that the market mean return is extremely robust (and very close to constant, at around 4 basis ...


3

I am not sure where you're going, but GARCH models usually have two equations: (1) an equation describing the conditional expectation of the dependent variable and (2) an equation describing the conditional variance of the error term in equation (1) as a process that is perfectly anticipated 1 period ahead. When you use returns in equation (1), equation (2) ...


0

If you are looking for exact volatility and variance swap market prices then as others have said you need to go to a data provider. If you are satisfied with good/accurate approximate prices for both instruments, then the paper It takes three to smile explains how to find these prices from three quoted options.


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