17

I generally agree with @dm63's answer: A convex (concave) smile around the forward usually indicates and leptokurtic (platykurtic) implied risk-neutral probability density. Both situations can or cannot admit arbitrage. I provide you with two counterexamples to your statements. A volatility smile that is concave around the forward does not necessarily ...


11

From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit: Forward variance swap volatility (A) Forward implied volatility smile (B) I really recommend reading Bergomi's "Stochastic Volatility Modeling" which is an excellent book for equity practitioners. The topics ...


10

Some Notations It's easy to get lost so let's introduce some notations and let $$ \sigma : (t, S, K, \tau) \to \sigma(K,\tau; S, t) $$ denote the implied volatility smile prevailing at time $t$ when the spot price is $S_t=S$ for an option with strike level $K$ and time to expiry $\tau=T-t$. From here onward, we drop the $t$ argument to keep notations ...


8

It's because of the settlement days you passed when you initialized the flat volatility curve. You're creating the spot, forward and flat volatilities as: boost::shared_ptr<BlackVarianceSurface> volatilitySurface( new BlackVarianceSurface(todaysDate, calendar, maturityArray, strikeArray, ...


8

The central limit theorem guarantees, under fairly general assumptions, that the sum of returns becomes more normally distributed as the number of returns grows (technically, defining a return as $\mathrm{log}(S_{t+\Delta t}/S_t)$, $\sum_i ^n \mathrm{log}(S_{t+\Delta t i}/S_{t+\Delta t (i-1)} \to \mathcal{N}(\cdot,\cdot)$ as $ n \to \infty $). Thus, as $T$ ...


8

There are several reasons, maybe the most important and also quite intuitive one: Implied volatility more or less assumes that the stock price is driven by Brownian motion and thus moves in a continuous fashion. What we observe is that stocks can jump (usually downwards, sometimes upwards) which needs to be modelled using something like a jump process (...


8

Either you or some reference you are following is in error here. At-the-money (or at least near-the-money) options are the most liquidly traded. And trading is much more heavy in out-of-the-money than in-the-money options.


8

Stochastic-Local Vol (SLV) is an attempt to mix the strengths and weaknesses of both Stochastic Vol and Local Vol models. Below, I'll quickly summarise each model and their strengths and weaknesses, and then discuss how SLV tries to improve things. Although there are many stochastic vol models, I limit the discussion here to the Heston model to keep things ...


8

One possible reason could be jumps. Over the longer maturity, there could be more jumps so the jumps average out in a way; whereas over the short term, a jump can make a bigger difference and hence the risk of jump increases demand. This reasoning is used to justify Stochastic volatility with jumps models in some books.


7

There are lots of papers online and here are a few I would suggest math.umn riskworx G. Dimitroff, J. de Kock Nowak, Sibetz I you have matlab there is an step step example to calibrate SABR model. Since it uses the financial toolbox of matlab for a few functions I dont think you can replicate it in any other language. There must be C++ code available ...


7

You can see concavity in mean-reverting underlying assets where the option tenor is comparable to the characteristic reversion time of the asset. For a geometric brownian motion, all underlying prices are possible, so any mean reversion or other limitation on large changes that might occur in reality would ultimately appear as a skinny tail and negative ...


7

You are absolutely correct that they should be seen as approximations. While it would be nice to let h go to zero in a mathematical sense this is of course impossible in real life as the options are only traded in particular intervals. While the smallest interval may be less than 25, for historical reasons traders have gotten used to using the 25 point. ...


7

It is possible, yes, but it requires assumptions. But, philosophically speaking, this is the case as with all pricing, of any instrument. For example, given only the price of a 6Y and 7Y IRS can you correctly price the 6.5Y IRS rate? Well, yes you can, but it depends upon your assumptions about interpolation which is a subjective choice. Lets look ...


7

In practice, things are actually quite different and a bit more subtle. You really need to differentiate between the underlying being an index or e.g. a single stock. I will try to provide some insight: Index options are, in general, of European type. The market quotes prices for calls and puts and you can back out the implied vols via the usual BS formula. ...


7

Further notes: One shouldn't build an implied volatility surface just from call prices or just from put prices. One should build it from liquid instrument quotes and, if necessary, some less liquid ones. Some markets, like FX option one, quote package prices (butterfly, risk reversal, ATM straddles). Deciding how to parameterize the implied volatility ...


6

There is nothing in simple cubic spline fitting routines that would prevent arbitrage. Even with conscientious use of knot points and smoothing techniques you may end up with simple spread and local volatility arbitrage conditions. Stochastic volatility models on the other hand can explicitly constrain your solutions to prevent call/ put spread arbitrage at ...


6

First of all, may I point out two big misperceptions that you may have: Implied Volatility (IV) is the input to any vanilla option pricing model (not just Black Scholes (BS) that impacts the pricing the most. You can verify this by flipping through the different risk exposures (greeks and higher order sensitivities) and study mean volatilities in such risk ...


6

A model that reflects the volatility smile is one with dynamics that approximate pricing that yields an implied volatility smile. However, your question makes me suspect you are fuzzy on some of these pieces, so let's go through this in more detail. Implied Volatilities $\implies$ Correct Price? You mention that implied volatility in the Black-Scholes model ...


6

It's probably important that we're talking about IV of an index. From "Volatility Trading" by Euan Sinclair: In equity indexes the skew will be more pronounced than in the individual stocks that make up the index. The volatility of an index, $σ$, is related to the volatility of the components, $σ_i$, by: $$σ^2=\sum_{i=1}^N w_i^2 σ_i^2+2\sum_{i=1}^{...


5

This is merely a question of notation, you should simply read $$ \sigma(K,T) = \sigma(S_t=K, t=T) $$ For an easy to follow derivation see this excellent note from Fabrice Rouah Some intuition behind the developments: The price of a European option, for instance a call, can be written in integral form: $$ C(t, S_t, K, T) = e^{-r(T-t)} \int_0^\infty (S_T-K)^...


5

I think you would find the following paper very useful. It compares different pricing models applied to VIX options. You can use it as starting point to apply to VSTOXX options and see where it gets you. The Performance of VIX Option Pricing Models: EmpiricalEvidence Beyond Simulation The following models were tested: Whaley (1993) Grunbichler and ...


5

Would it be OK to mix put/call prices such that I only ever calculate implied volatility for in-the-money options? No. Use OTM options because they usually have narrower bid-ask spread. Ideally you calculate all IVs, and then use highest bid IV, smallest ask IV. If so, I assume this surface can then immediately be used to calculate ... Yes, then you ...


5

Intuition: You can think of the vol smile as a reflection of the risk neutral distribution (compared to the Black Scholes Gaussian density). A fat tailed distribution creates the smile: fat tail -> higher prob of exercise than Gaussian with constant stdev -> higher option price than BS with ATM vol -> higher implied vol for given strike. Skewed distributions ...


5

Regarding your second question: Remember that Black/Scholes start by postulating a stochastic model for the dynamics of the underlying asset - a geometric Brownian motion with a constant diffusion coefficient $\sigma$. This asset price process should be the same no matter what option you want to value based on it. Saying that you allow for different values ...


5

Stochastic local volatility model means $dS_t/S_t=...dt+\sigma_t L(S_t,t)dW_t$ with $\sigma_t$ the stochastic part (modeled for instance as in the Heston model, or any other dynamics deemed appropriate) and $L(S_t,t)$ the local part. The local part $L(S_t,t)$ is computed from "Dupire’s unified theory of volatility" which states that $$ σ_{\text{local}}(S,...


5

I work in a relatively illiquid and old-fashioned market (options on power), where trades are arranged via phone & broker, so the issue of low underlying liquidity is definitely there. To remedy this, all options are dealt with delta hedge, where the price level of the delta hedge is pre-agreed, so market moves during arrange a trade do not matter as ...


5

Call and a put of the same strike have the same I.V, in theory. The ONLY reason for this to differ is the limits to arbitrage on call put parity. Now this is a static strategy that has no rebalancing - so the only problem here is transaction costs in buying/shorting the stock. So if you have reason to believe that this strategy is difficult to implement, ...


4

In addition to the presented answers, I just wanted to mention that such a situation is described in Hull, page 419 (Chapter 19 Volatility Smiles, 19.8: "When a single large jump is anticipated"). This happens when probability distribution of returns is binomial. It can occur in a situation when market is expecting some announcement which will either ...


4

Implied Black-Scholes volatility is much more than just a parameter in a formula that can be fudged to produce a reasonable price. When an option position is hedged in Black-Scholes, the daily P&L is proportional to the realized minus implied variance. It follows that implied volatility corresponds to the consensual prediction of realized volatility by ...


4

Negative excess kurtosis leads to a concave vol smile. By the way, no-arbitrage arguments are of theoretical nature: implied volatilities can exhibit no-arbitrate violations in the theoretical sense for extended periods given that such arbitrate cannot be traded due to other factors, such as liquidity, spreads, transaction related costs...not saying this ...


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