19
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 ...
- 5,959
15
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 $t$ ...
- 14.5k
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 ...
- 14.5k
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, ...
- 2,996
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 ...
- 1,875
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 ...
- 8,143
8
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 ...
- 2,543
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 ...
- 9,215
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 ...
- 6,204
8
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 ...
- 823
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 ...
- 5,008
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 ...
- 14.5k
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 ...
- 16.6k
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 ...
- 5,632
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
- \...
- 6,204
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 ...
- 2,880
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 ...
- 61
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 ...
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 ...
- 5,959
5
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) ...
- 14.5k
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 ...
- 5,632
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 ...
- 16.6k
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 ...
- 1,651
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 ...
- 1,369
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 ...
- 1,875
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 ...
- 2,996
4
votes
Downward sloping smile in normal model
Although it's a bit different story, there are VERY accurate approximation formulas for the implied volatility under normal model (so-called basis point volatility). Using them, you can obtain the ...
- 1,143
4
votes
At-the-money and volatility smile
The shape of the implied volatility smile is linked to the higher moments of the underlying return distribution though there is no one-to-one relationship. A convex (concave) smile usually indicates a ...
- 5,959
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
...
- 14.7k
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} ...
- 21k
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