18 votes
Accepted

Arbitrage Free Volatility Smile

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 ...
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12 votes
Accepted

SSR definition in Bergomi in relation to sticky strike and sticky delta

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 $t$ ...
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  • 14k
11 votes

Forward implied volatility

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 ...
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  • 14k
10 votes

Why is there greater demand for OTM and ITM options than for ATM options?

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 ...
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  • 1,865
10 votes

Mixed local-stochastic volatility model in Quantlib

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, ...
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  • 2,856
8 votes
Accepted

Why does the volatility smile flatten as maturities increase?

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 $\...
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  • 1,329
8 votes
Accepted

QuantLib: Black / BSM processes and pricing via volatility surface. Different results?

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: ...
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8 votes
Accepted

Intuitive Explanation for Volatility Smile for Equity

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 ...
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  • 13.3k
8 votes

Why is volatility skew/smile for long term options flatter compare to short term options?

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 ...
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7 votes
Accepted

Volatility skew and how to capture it?

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 ...
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  • 1,551
7 votes
Accepted

SABR calibration: simple explanation and implementation

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 ...
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  • 891
7 votes

In Dupire's paper, why is $(S_t, t)$ in the $(K, T)$ space?

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 ...
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  • 14k
7 votes

Forward implied volatility

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 ...
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  • 8,057
7 votes

Implied Vol Smile: from Calls, Puts or Both?

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 ...
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  • 721
7 votes

Implied Vol Smile: from Calls, Puts or Both?

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 ...
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  • 5,038
7 votes

Bermudan Swaptions - Payer vs. Receiver (LGM)

I’m guessing you are finding that your model overvalues Bermudan receiver options and probably undervalues Bermudan payer options. The rationale for this has more to do with supply and demand than ...
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  • 14.1k
6 votes
Accepted

Volatility Surface Constituents, do's and dont's

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 ...
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6 votes
Accepted

Mixture models of Stochastic Volatility and Local Volatility

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 ...
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6 votes
Accepted

What does it mean that model can reflect the ”volatility smile”

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 ...
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  • 2,750
6 votes

Is the volatility smile a thing of the past?

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 ...
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  • 61
5 votes
Accepted

parameters in Heston model and their impact on volatility smile

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 -> ...
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  • 4,227
5 votes
Accepted

"Black-Scholes model implies flat implied volatility plots"?

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 ...
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5 votes

How should I convert FX Volatility Surface from one base currency to another?

There is no simple way and you have to make correlation assumptions. For instance say you have a volatility surface for $\text{EURUSD}$ and another volatility surface for $\text{USDJPY}$ and you ...
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5 votes

Sticky Strike or Sticky Delta

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 ...
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  • 1,581
5 votes

FX smile extrapolation

This is a big industry, but here are some alternatives(as usual, the best choice depends on purpose and desired accuracy): Fit a quadratic in delta space: $\sigma_{\Delta}=a + b \left( \Delta - \...
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5 votes

Implied Vol Smile: from Calls, Puts or Both?

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 ...
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  • 1,662
4 votes

In Dupire's paper, why is $(S_t, t)$ in the $(K, T)$ space?

The local volatility is just a $\mathbb{R}_+\times[0,T]\mapsto \mathbb{R}_+$ function where $T$ is some time horizon. It is the solution of a simple equation so it expression is written as $\sigma(K,t)...
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  • 545
4 votes

Volatility skew and how to capture it?

I wonder if the reasons these approximations are widely used - instead of a whole set of estimates for different deltas, as proposed - have to do with liquidity and market structure. Liquidity: A ...
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4 votes

Why does the volatility smile flatten as maturities increase?

If skew is too high, then you can have call/put spread arbitrage. An easy way to see put spread arbitrage would be to price a digital put when using skew. When using skew, the price of a digital put ...
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4 votes

Black-Scholes: Why the focus on volatility?

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 ...
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