Hi Could somebody please answer the following questions.1) Why do we have volatility smile/skew in the market.2) As the option maturity increases, the smile curve starts getting flatter. why ??Regards...

wrong forum Or, if you like this forum, a brief search of threads with "the skew" in the title shows 10 pages of links, including an obviousthread in the FAQ forum

Hi bakait,Well I tried to read Alan link about why the smile and to be honest I got headache reading the explanation. Your question is very simple to answer (it is a fundamental question though). Imagine the BS world (no smile) just one vol and that's it. Now suddendly you observe that vol (the unique one) move with the time, one day that vol is 10%, another day is 12%. Well it is easy to come out with a strategy that will make you always money. That stratedy is being long fly (long strangle and short the straddle). You short a x-amount of straddle in such a way overall you are vega neutral from the day 1. Once the vol move you can easily realize you will always be in the money. That strategy got vol gamma (volga). If vol move up you are long vega and you make money, if the vol move down you are short vega and you make money!!!Since there is not such thing as free lunch then appears convexity in the market that go against the strategy I have just explained.I am sorry if my explanation does not have world like pdf, lognormal, markovian, etc... As I said the explanation is trivial. Traders explain this in just one sentence: "convexity got value and therefore human being invented the smile to trade that value!"

QuoteTraders explain this in just one sentence: "convexity got value and therefore human being invented the smile to trade that value!"Thanks for the reply, but I would be wary of quoting traders. Some of them talk by slogans and often use words with undefined meanings to hide their limited understanding of what they are trading.If a trader gave me £1 for every time they mentioned "convexity" as a magic word to justify anything that they don't understand in the pricing I wouldn't probably need to work anymore.Even if traders can explain the smile in a single sentence, to me that sentence is completely and utterly meaningless.

Hi gc,I think what you says it is true in all asset classes except in FX where the traders do trade convexity as instrument (like call or put). However the explanation of why the smile is not from my trade quote but because of the existence of free lunch with the strategy I have just explained.Cheers,

QuoteOriginally posted by: vdiclementeSince there is not such thing as free lunch then appears convexity in the market that go against the strategy I have just explained.so you claim that there is (model independent) arbitrage, if no market smile is present ?

QuoteOriginally posted by: vdiclementeSince there is not such thing as free lunch then appears convexity in the market that go against the strategy I have just explained.That's half the story. The other half is that because the market is inefficient, there are lunches to be had, some of them cheap, some of them expensive. The smile is a consequence of market inefficiency.

I will appreciate if someone explains the concept of smile. As in the original paper let the drift coefficient equal to 0. It is pure theoretical approach which states that solution of the linear SDE simultaneously is a solution of the nonlinear SDE. If it is true one can subtract one equality from the other takes a second moment and arrives at the equality that will state that diffusion coefficients of the linear and nonlinear equations coincide.

it is a well known result that the density of the underlyings distribution is equal to the second derivative of european call option prices w.r.t. strike. black scholes is a standard way to express these prices in terms of a model parameter (implied bs vol). so the volatility smile corresponds to the density with flat smile = lognormal density due to this black scholes normalization. you could also use a normal bs model for normalization for example and get a different smile / density correspondece, in particular flat smile = normal density then. it is really arbitrary.arbitrage freeness implies negative first and positive second derivatives of call option prices in strike direction hence a decreasing convex function. this translates to restrictions on the smile shape again via the black scholes normalization.now you can ask what densities are implied by market smiles, what models may produce or fit them and what the economic interpretation may be.

QuoteOriginally posted by: pcaspersit is a well known result that the density of the underlyings distribution is equal to the second derivative of european call option prices w.r.t. strike. black scholes is a standard way to express these prices in terms of a model parameter (implied bs vol). so the volatility smile corresponds to the density with flat smile = lognormal density due to this black scholes normalization. you could also use a normal bs model for normalization for example and get a different smile / density correspondece, in particular flat smile = normal density then. it is really arbitrary.arbitrage freeness implies negative first and positive second derivatives of call option prices in strike direction hence a decreasing convex function. this translates to restrictions on the smile shape again via the black scholes normalization.now you can ask what densities are implied by market smiles, what models may produce or fit them and what the economic interpretation may be.Thank you pcaspers for lightening details related to smile. Indeed second derivative with respect to K equal to density of the underlying GBM at ( t, x; T, K ). It looks like that this property does not depends on underlying. If underlying has a density then twice differentiation of the BS formula w.r.to K leads to this property. The next step does not clear. If we have a linear SDE representing stock price how we chose nonlinear SDE for the same stock that cannot be reduced to the original linear governed stock?

QuoteOriginally posted by: bakaitHi Could somebody please answer the following questions.1) Why do we have volatility smile/skew in the market.2) As the option maturity increases, the smile curve starts getting flatter. why ??Regards...the main reason for the smile is a jump where u cannot stop loss ur delta without a gap.if i am to sell a otm option, i lose big time if spot gaps 5% down so why sell these options unless u can collect a decent premium.smile curve flattens with the termstructure cos a gap affects short dated options more.

Quotethe main reason for the smile is a jump where u cannot stop loss ur delta without a gap.if i am to sell a otm option, i lose big time if spot gaps 5% down so why sell these options unless u can collect a decent premium.thus you would use some levy like process rather than pure diffusion as your risk driver and arrive at non-lognormal distribution, thus a non-flat bs-smile. also historical data tells you that log stock returns are not normal. so it would be rather surprising having no smile.QuoteThe next step does not clear. If we have a linear SDE representing stock price how we chose nonlinear SDE for the same stock that cannot be reduced to the original linear governed stock?to what paper do you refer?

According to Derman the causes of the volatility smiles are:-crash protection/fear of crashes-expectation of changes in volatility over time (4)-support/resistance level at various strike-Dealer's books tend to be filled by the demand of zero cost collars-Inability to hedge continuously (2)-Transaction costs (3)-Leverage effect-Stochastic volatility (5)-Jumps/Crashes (6)The small numbers in brackets are my personnal ranking, my number one being that the smile is created by the dynamic interaction of several agents with different strategies (namely speculators, hedgers and arbitragers as an example).

Hi pcaspers,Yes I said there is (model independent) arbitrage, if no market smile is present. The only condition required is observe a volatility which change with time. As I explained below if you are long the fly (lond Strangle and short Straddle) you can make money if there is not smile...It seems Derman (see frenchX) put a long list of reason to create smile. I believe you dont need all of them, in fact sure just one is enough ("expectation of changes in volatility over time"). Talking about my strategy, I know many hedge fund even nowdays try to explode the "efficiency" of the market to arb it. They are long fly for huge amount, from day one you are vega neutral but you got vol gamma (volga). If the volatility move up you are long vega, if the volatility move down you are short vega. In a very efficient market the smile would adjust (risk reversal adjustment) in such a way the money you get in the long fly is paid in decay (risk reversal leaking). However the market does not move quick enough and you can find a window (couple of days) where the position pay off.

Just ask the question why do you get different shapes in different markets? That's the biggest real clue, as opposed to the reverse engineering beloved by quants! P