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15

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 ...


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, ...


7

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.


7

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 ...


7

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 ...


6

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)^...


6

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. ...


6

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 ...


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

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 ...


6

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 ...


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

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 ...


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 ...


4

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 $T$ ...


4

The choice of a model depends on what inputs you have, the complexity allowed (e.g. calculation time restrictions) and what you want to infer from it. The development of the LMM adressed the mathematical difficulty of finding a joint model for all Libor forwards and was a great achievement in the late 90'. But at that time the distribution of the Libors was ...


4

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 ...


4

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 distribution with positive (negative) excess kurtosis. Here is an example for a concave implied vol. smile. It is often observed in markets where a single ...


4

Neither situation is necessarily an arbitrage. Negative smile is consistent with a 'thin-tailed' density function , just as positive smile is consistent with a fat tailed density function . It's true that an extreme amount of negative smile could cause the implied density to be negative in places I.e an arbitrage.


4

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 want to build a volatility surface for $\text{EURJPY}$. You start from the observation that a call with maturity $T$ and strike $K$ on $\text{EURJPY}$ with ...


3

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 ...


3

First of all, a Bermudan Swaption does not have to be of American type. Consider a "9NC2 Bermudan" (9 non call 2), basically a Bermudan swaption with final maturity in 9 years which is not exercisable for the first 2 years. I have not worked at an exotic rates desk in a while (many years to be more precise) but from what I remember you need to use a ...


3

First note that the price of binary call is related to the price of an ordinary call in any model by $$ BinC(T,K) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[1_{S_T>K}] = - \frac{\partial}{\partial K}e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)_+] = - \frac{\partial}{\partial K}C(T,K) $$ Now the volatility smile is implicitly defined by $$ C(T,K) = C_{BS}(T,K,\Sigma(...


3

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 ...


3

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 market participant willing to trade e.g. a 10 delta option for no economic reason other than skew will find, for many products, that the edge evident from a fitted ...


3

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 $\sigma(K,t)$ but here $K$ is essentially a notation to denote a strike value as the Dupire equation relates the function $\sigma$ to vanilla market prices at a given ...


3

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} x-K) \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx\\ &=(S_0-K)\Phi\left(\frac{S_0-K}{\sigma \sqrt{T}}\right)+\frac{\sigma\sqrt{T}}{\sqrt{2\pi}}e^{-\frac{(S_0-K)...


3

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 For the left side it is sufficient to note that the lognormal model has no density below zero while the normal model has strictly positive density in that ...


3

In black-scholes world, correlation between volatility and spot is zero. From the above details you can estimate how the implied volatility for a given option (note options have FIXED strikes) might change for a given move in spot. If when stock goes up, the option's implied vol goes down, this would be a violation of the black-scholes model (which scenario ...


3

@q.t.f. 's answer is 100% correct. As an OMM, I wanted to add some reasoning behind this. The practice of trading ATM options has been established for over a century now, and before formal mathematical methods were developed, traders have developed many heuristics for pricing ( proportional to vol ) and hedging ( delta = 1/2 ) . Typically in a newly ...


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