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20 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 ...
LocalVolatility's user avatar
16 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 $t$ ...
Quantuple's user avatar
  • 14.7k
14 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 ...
Quantuple's user avatar
  • 14.7k
12 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, ...
StackG's user avatar
  • 3,026
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 ...
q.t.f.'s user avatar
  • 1,885
9 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 ...
SI7's user avatar
  • 843
9 votes

Is it possible to have only one volatility surface for american options (that fits both calls and puts)?

Usually, there is only one vol surface (I have never seen or heard of anyone using two). Almost certainly the most advanced commercially available vol surfaces are built by voladynamics. They also ...
AKdemy's user avatar
  • 8,924
8 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 ...
Attack68's user avatar
  • 10.5k
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 ...
Magic is in the chain's user avatar
8 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 ...
ir7's user avatar
  • 5,043
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 ...
Quantuple's user avatar
  • 14.7k
7 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 ...
Antoine Conze's user avatar
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 ...
dm63's user avatar
  • 17.1k
6 votes

How are the BKM risk-neutral moments derived?

Let's focus on the volatility contract price. Generalisation to cubic and quartic contracts is straightforward. Following the paper's notations, the evaluation date is $t$ and the (European) ...
Quantuple's user avatar
  • 14.7k
6 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 - \...
Magic is in the chain's user avatar
6 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 ...
Arshdeep's user avatar
  • 2,175
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 ...
kurtosis's user avatar
  • 2,900
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 ...
Adam N.'s user avatar
  • 61
5 votes

Why does the volatility smile flatten as maturities increase?

A wee bit late to the party but still worth posting an answer I think, especially since this question appears to be the only one asking why the IV flattens as $T \rightarrow \infty$. Furthermore, I am ...
Frido's user avatar
  • 1,854
5 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 ...
FinanceGuyThatCantCode's user avatar
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 ...
LocalVolatility's user avatar
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 ...
Antoine Conze's user avatar
5 votes

Forward implied volatility

The procedure outlined by @attack68 is correct for estimating forward vol assuming you are in a world where volatility is deterministic and uncorrelated with the underlying. If these assumptions are ...
dm63's user avatar
  • 17.1k
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 ...
ZRH's user avatar
  • 1,671
5 votes

Why SVI does not fit well short-maturity options?

The issue has much more to do with the SVI parameterization per se, and not with any arbitrage constraint. The fact that Heston as $T \to \infty$ becomes close to SVI is not very useful either to ...
jherek's user avatar
  • 1,394
5 votes

Forward starting options concepts

Forward-Start Options are very interesting securities, you can find a lot about them on the internet. It turns out that there is an explicit pricing formula for them in Black-Scholes, the nicest ...
StackG's user avatar
  • 3,026
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 ...
Antoine Savine's user avatar
4 votes

Downward sloping smile in normal model

The implied Black-Scholes skew will be downward sloping in the limit on both the left and the right. (I believe @Gordon's derivation claiming upward slope may have a sign error somewhere). Left Side ...
Brian B's user avatar
  • 14.9k
4 votes
Accepted

Downward sloping smile in normal model

Since $S_T = S_0 + \sigma W_T$, \begin{align*} C &:= E\left((S_T-K)^+ \right)\\ &= E\left((S_0+\sigma W_T-K)^+ \right)\\ &=\int_{\frac{K-S_0}{\sigma \sqrt{T}}}^{\infty}(S_0+\sigma\sqrt{T} ...
Gordon's user avatar
  • 21.1k
4 votes

Intuitive Explanation for Volatility Smile for Equity

Too many people think implied vol is the "volatility" that you think it is. Normally we talk about "volatility" as the measure of how unpredictable the stock movement will be in its magnitude. True. ...
Andrew's user avatar
  • 41

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