# Tag Info

7

The way market makers mark their volatility curves is by using models which 'fill in the gaps', i.e. they will make a price for a given option even if they do not believe this option is going to get a lot of volume. They are still willing to go long/short because they have a strategy to hedge their overall position (i.e. by managing their greeks and ...

6

Consider a more financially plausible model than Black-Scholes: one where the stock can suddenly go bankrupt due to fraud, and the volatility varies over time. Neither model is perfect, but the new one (call it SVJ) will be "less wrong". Mathematically, we no longer have the Black-Scholes SDE based on a single stochastic generator $W$ $$\frac{dS}{S} = ... 5 The Black-Scholes option pricing model provides a closed-form pricing formula BS(\sigma) for a European-exercise option with price P. There is no closed-form inverse for it, but because it has a closed-form vega (volatility derivative) \nu(\sigma), and the derivative is nonnegative, we can use the Newton-Raphson formula with confidence. Essentially, ... 5 It is a very simple procedure and yes, Newton-Raphson is used because it converges sufficiently quickly: You need to obviously supply an option pricing model such as BS. Plug in an initial guess for implied volatility -> calculate the the option price as a function of your initial iVol guess -> apply NR -> minimize the error term until it is sufficiently ... 4 First we must define what we mean by implied volatility. Let c_{BS}(t,S(t),K,T;\sigma) denote the price of the call option with strike price K and maturity T in the Black-Scholes model with the volatility \sigma (emphasized in the argument). Furthermore, let c_{MA}(t,S(t),K,T;\sigma) denote the corresponding price on the market. The volatility ... 4 Options on almost all Korean equities today present flat implied volatility, as well as options on some Japanese equities, especially in 60-90 days maturity. Here how the smile looks for T&D Holdings (ISIN:JP3539220008): 3 Actually, short term implied volatility is higher at high volatility periods and lower at low realized volatility periods. From a quantitative perspective, the explanation for this is usually that short-term implied volatility is more influenced by the recent realized volatility, while longer-term is more influenced by the long term average. From a market ... 3 Implied volatility represents the market expectation of "stuff happening in the future". Over long periods, all that stuff tends to cancel out a bit, so long-term vols are much more stable. In the short term, a single news item may be enough to drastically increase (or decrease) return uncertainty. From a quant point of view, we think of volatility as ... 3 Implied volatility has very little to do with any particular pricing model, especially not much with BS. BS is a translation tool between prices and volatility, with its own many model deficiencies. I won't get into such model assumptions because my point is an entirely different one. Even the smile/smirk is entirely unrelated to the Black-Scholes model and ... 3 The volatility smile is made out of implied volatilities. This means that you take as input K,S,r,T and the price of the option p and you use it to find \sigma such that$$ p = BS(K,S,r,T,\sigma)$$But p is defined by the market, so the \sigma you find are the estimated volatilities by market participants, if they believe in the Black Scholes ... 3 Implied volatility skew is simply collection of implied volatilities on the same underlying instrument for a given expiration. Term "implied volatility skew" is only loosely connected to statistical definition of skewness. Implied volatility surface is the collection of implied volatilities on the same underlying for several expirations. If BS formula were ... 2 Bisection method is rather fast but it has only linear convergence. Newton's method offers quadratic convergence but it requires the knowledge of Vega (which AFAIK is only accessible numerically with binomial model). However, the convergence of Newton's method can suffer from poor initial approximation. In this case Brent's method tends to perform better. ... 2 I assume no interest rates to clarify the approach. The Heston model is written under the risk-neutral probability as$$ \frac{dS_t}{S_t} = \sqrt{v_t}dW_t  dv_t = -\kappa(v_t-\eta)dt + \theta \sqrt{v_t}dZ_t $$with d\langle W,Z\rangle_t = \rho dt and v_0 = \sigma_0^2. Using Itô's lemma we can derive$$ \log\left(\frac{S_t}{S_0}\right) = \int_0^t ...

2

It's extremely common in the industry to have a parabolic skew of this type with some cutoff parameters. At it's simplest, such a model looks like this $$\sigma_{ATM}(t) = \sigma_0 + s(t)$$ where $s(t)$ is a vol term structure function and can be further simplified to $$s(t) = \frac{s}{\sqrt{t}}$$ if you are willing to accept the inaccuracies. ...

2

If you look at tick data, you will probably get an even better analysis. However, vix correlation tends to be negative with spx but remember that this is generally more true for when spx tanks. When spx goes up, the correlation isn't as strong. Why? People panic after a drop, therefore leading to people buying options. They don't care about black scholes ...

2

The Black-Scholes model is based on the assumption of lognormal returns of the underlying asset. There is much evidence and argument that stock market returns are not normal on a logarithmic basis, and there is no particular reason to assume a normal distribution, either. In particular an implied volatility smile is evidence of "fat tails" in the returns ...

2

The line Dim S, X, T, r, d, Price As Double is better written as Dim S As Double, X As Double, T As Double, r As Double, d As Double, Price As Double because 'as' only applies to the variable directly before it. The bisection algorithm seems to work as advertised. You could check with an online calculator such as this one. I've rewritten the ...

1

you can use this code from Uwe Wystup FX Options and Structured Products. you can find it online. it uses vega and taylor expansion (just to 1st derivative which is vega) to find vol. you have to have code for european call price, but you have it already Function VanillaVolRetriever(spot As Double, rd As Double,rf As Double, strike As Double, T As Double, ...

1

There is a market accepted standard to translate vanilla option prices to implied vols and backward which is the Black Scholes (BS) options pricing formula. There is no ambiguity here, everyone knows of the deficiencies of BS yet its what people use to translate between iVols <-> prices. The numerical difficulty I see is to make more realistic ...

1

Just couple points to ponder about: Weighting schemes are not a magic bullet and none of them in isolation will give you higher predictive power (aka. edge). Do not rely on how you weigh components. Why would you want to be highly correlated with VIX? Its a poorly constructed index and I find the whole rational behind the implementation details quite ...

1

I agree with some arguments above. One can find several explanattion to volatility smile: Against the BS framework assumption, volatility is not constant and traders don't expect it to be constant This is a matter of option supply and demand volatility smile incorporates the kurtosis seen in the underlying One can find hints on this issue in P.Wilmott's ...

1

The easiest way is to use single-expiry volatility that you would get from your volatility surface. It is usually good enough for government work (e.g. to get a sense if you are getting raped by a dealer or to understand your vega risk). A better way is to use local volatility model and the whole volatility surface up to the date of expiry. There is also a ...

1

You might want to look at "If the skew fits" article by Gregory Brown and Curt Randall from Risk magazine (April, 1999). Their parameterization has the following form: $$\sigma(S,t) = \sigma_{ATM}(t) + \\ \sigma_{skew}(t) * tanh(\gamma_{skew} (t) * {\log(S/S_{0})} - \theta_{skew}(t)) + \\ \sigma_{smile}(t) * [1 - sech (\gamma_{smile}(t) * ... 1 definition of a variance swap is  \int^{T+\Delta}_T \mathbb{E}_t[v_s] ds  where v_s is the variance and \mathbb{E}_t[v_s] is the expectation of the variance of time s at time t. therefore, pnl is:  (\int^{T+\Delta}_T \mathbb{E}_t[v_s] ds - \int^{T+\Delta}_{T} \mathbb{E}_{t-\delta}[v_s] ds)*d\delta  1 The negative slope is not just because people think there is going to be a crash, it is also just function of supply and demand on options: Many investors want to buy protection on their portfolio, and there is no natural seller so it's the speculator who end up taking that side of the bet, but obviously only if there is a sufficient premium with low ... 1 What you are observing is a natural occurence in the options market. If you think of PUT options as insurance, the premium is based on what the historical volatility of the underlying is. If option writers sold you the options at or below historical volatility they on average would loose money. So therefore, they "mark up" the historical volatility and that ... 1 Dealing with model error under stochastic volatility (in a more formal way) you could use the UVM (Uncertain Volatility Framework). Here are what i think are the most seminal references: Avellenada et al (1995) Pricing And Hedging Derivative Securities In Markets With Uncertain Volatilities http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.3736 ... 1 The mistaken belief you refer to is a common confusion between stating that the price went up because demand went up and/or supply went down (tautologically true, but meaningless statement) or because buyers entered the market and/or sellers exited the market (usually false). This has nothing to do with options or implied volatility. I see such confused ... 1 This is a common convention. If your spot is S and you're looking at options maturity in T, it is natural to look at the the strikes S_\pm=S.exp^{-\frac12\sigma^2T\pm\alpha\sigma\sqrt T} for a fixed \alpha. So your skew measure will be something like$$ \frac{\sqrt T(\sigma_{S_+} -\sigma_{S_-} )}{\log (S_+/S_-)} = \frac{\sqrt T(\sigma_{S_+} ...

1

Simple explanation of VIX formula is that it is a square root of weighted sum of out of the money SPX options. If options are more expensive (implying that market is paying for insurance from a major move) VIX will higher, if options are cheap (implying that the market does not think that there will be a lot of movement in the SPX) The weighting formula ...

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