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 FORUMS > Technical Forum < refresh >
 Topic Title: skew and forward volatilities Created On Fri Sep 06, 02 10:19 AM Topic View: Branch View Threaded (All Messages) Threaded (Single Messages) Linear

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niclaf
Junior Member

Posts: 1
Joined: Jul 2002

Fri Sep 06, 02 10:19 AM

I've a little problem to use well the volatility surface.
If, for example, I would price today (t=t0) a forward at-the-money Call with start on t0+1y and maturity t0+2y, I should need the forward volatility and I'm able to calculate it from the spot volatilities (t0:1y ; t0:2y), but I also want to add the skew.
The question is:
Do I have to consider the volatility corresponding to an at-the-money spot option, I mean s0, or I have to consider the volatility corresponding to an at-the-money forward option, I mean s0*exp(r*T)?

Can anyone help me?..many many thanks

Sofiane
Senior Member

Posts: 294
Joined: Jul 2002

Fri Sep 06, 02 10:31 AM

Hey,

try to think with the implied tree, i.e., with local volatility sig(S,t), I guess that you can price with local volatilities derived from the implied tree or another type of implied volatility functions. Using spot IV (of the corresponding maturity) is not false since IV have a strong information content about future realized volatility or future IV, but accuracy is not sure.

That's all folk.

Johnny
Senior Member

Posts: 2433
Joined: Oct 2001

Fri Sep 06, 02 11:37 AM

niclaf

The important point is about how you intend to hedge your position. You should price your option using the forward starting volatility from your tree, but then make sure to lock this volatility in as best you can by using vanilla options to hedge.

Sofiane

I am interested in your opinion that "IV have a strong information content about future realized volatility or future IV". All the empirical evidence I have seen from academia as well as my own practical experience says that the information content of volatility surfaces is very low. Can you say something about your evidence?

Sofiane
Senior Member

Posts: 294
Joined: Jul 2002

Fri Sep 06, 02 04:00 PM

Hi Johnny,

I was actually talking about IV and not local volatility...the information content of local volatility surface is reputed to be low but not IV even if there exist some contradictories studies about this item. The sampling methodology and estimation process to look for predictive power of IV may differ from a study to another.
Do you compute local volatilities, or not and how?

Johnny
Senior Member

Posts: 2433
Joined: Oct 2001

Fri Sep 06, 02 04:06 PM

Do you compute local volatilities, or not and how?

I'm not sure I understand the question. If you mean, do I use a volatility surface when pricing or hedging, then the answer is "no". I don't think there's much financial justification for this type of joining-the-dots exercise.

reza
Senior Member

Posts: 3015
Joined: Aug 2001

Fri Sep 06, 02 05:01 PM

niclaf,
what are you trying to do? price Exotics or interpolate for vanillas?

I never used Local Vol models, Stoch-Vol should do the right thing for forward/ skew ...
check out Alan Lewis's book ...

Sofiane
Senior Member

Posts: 294
Joined: Jul 2002

Fri Sep 06, 02 05:22 PM

Hi Johnny,

not of course for pricing and hedging since as recalled reza, a heston stoch vol model is quiet efficient. No it was for its own purpose.

sdhrolia
Junior Member

Posts: 7
Joined: Jul 2002

Thu Sep 12, 02 01:25 AM

I would tend to agree with this...use the vol that you intend to hedge with....

What is the title of Alan Lewis's book?

Clarke
Member

Posts: 39
Joined: Sep 2002

Thu Sep 12, 02 05:01 AM

IMO: Local volatility is an utterly corrupt concept(*) that is popular because it is expedient. However for problems like imputing forward skew it falls flat on face.
Stochastic volatility or anyother 'genuine' probability function will give you a more meaningful value. It has it's own pitfalls of course.
Actually the more skewed a surface and most especially the more term structure there appears to be in the skewness itself, the more you need to use a pdf than includes a jump process.

(*) skewness is a corrupt concept too. This is an effect that we observe demonstrating the shortcomings of our models in particular the assumption of log-normal returns.

Edited: Thu Sep 12, 02 at 05:03 AM by Clarke

Aaron
Senior Member

Posts: 6435
Joined: Jul 2001

Thu Sep 12, 02 06:22 PM

I would agree with your general inclination, but I do not share your reformist zeal. "Corrupt" is much too strong a word for expedient working hypotheses. I would suggest "convenient for pragmatists, but may lead idealists into serious error" instead.

-------------------------
Aaron Brown

Clarke
Member

Posts: 39
Joined: Sep 2002

Fri Sep 13, 02 02:33 AM

Given any set of data (option prices) with sufficient variables you can fit a function to them. This does not make the result meaningful. I am not suggesting that it is without information. You can use it but only if you understand the limitations of the tool. Unfortunately, I don't think that most users appreciate that; they see the great fit to observable prices and conclude 'great model'. It's a trap.

numbersix
Senior Member

Posts: 462
Joined: Jul 2001

Fri Sep 13, 02 05:50 PM

Not only is local volatility a corrupt concept, it is a non-starter for pricing (not to mention hedging) of forward start options.
Local volatility models assume that the information about the underlying process is totally captured by the present day and present spot price of the vanillas.
C(K,T) maps in theory one-to-one into sigma(S,t).
And the vanilla prices depend only on the probability distribution of stock at the maturity of the option.

Now the forward start option is essentially a stochastic volatility instrument.
So long as we haven't reached the start date of the option, its gamma is zero, hence it is insensitive to jumps.
All that matters for the pricing of the forward start option today (and before its reaches its start date) is what volatility level would prevail the day its starts, for that would determine the value of the option thereafter.

Of course a local vol model would give you a price for forward start options. However, as Clarke says, it falls flat on its face it you want it to agree with the market price of the forward start options.
Heston will perform differently, but then will it agree with the market forward skew?

I am saying (and this I now know from our experience in calibration of volatility smiles) that you need a more general model and that you need the market data on forward start options as an independent source of information for calibration of the underlying process!!

I am sorry niclaf, but this answers your question at the beginning of this thread with a much bigger question!

-------------------------
I will say first, by way of introducing myself, that I have wished to understand philosophy not as a set of problems but as a set of texts. (Stanley Cavell)

Paul
Senior Member

Posts: 5256
Joined: Jul 2001

Mon Sep 16, 02 05:52 AM

Before we vote on whether local vol is a corrupt concept, let's find out who uses it: Poll.

P

Edited: Mon Sep 16, 02 at 05:54 AM by Paul

Krysia
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Posts: 32
Joined: Nov 2001

Mon Sep 16, 02 07:24 AM

"Not only is local volatility a corrupt concept, it is a non-starter for pricing (not to mention hedging) of forward start options.
Local volatility models assume that ..."

I agree that for forward-start options the concept may not be very useful. However, you can use the local volatility surface for consistent static hedging of exotics with vanillas. I seem to remember there was a paper a couple years ago in J of Financial Engineering about the performance of static replication of barrier options with vanillas using the local volatility surface. I can dig the reference if anybody's interested. Obviously, good performance of static hedging does not tell you anything about dynamic hedging. That's where those methods have big problems.

maxim
Member

Posts: 84
Joined: Apr 2002

Mon Sep 16, 02 01:52 PM

Hi all!

I've got a question about local vols.

Suppose one calibrates the local volatility surface to the market implied vols. The pricing problem (even for vanillas) has to be solved numerically, for example, with FD method. Is there a guarantee that the model price for vanilla agrees with the market price for this vanilla? Otherwise stated, is the following relationship true?

callprice_model(localvola(*,*);K,T)=callprice_BS(impliedvola(K,T);K,T)

P.S. I think the poll should contain option "I don't know...". Many people do not have experience in trading and hedging so the poll results will be disturbed.

Edited: Mon Sep 16, 02 at 02:01 PM by maxim

alkur
Senior Member

Posts: 269
Joined: Jan 2002

Mon Sep 16, 02 04:14 PM

Maxim,
no, this equality will hold only in 'weak' sense. Usually you fit local vol model parameters so, that
the sum of (LHS-RHS)^2 [LHS, RHS from your message] is minimum for all observed strikes you have.

emergix
Member

Posts: 147
Joined: Jul 2002

Mon Sep 16, 02 04:23 PM

well, I think maxim, that you are right for a continuous model, (so a continuous set of constraints)

but for a discrete set of market price (as we get them from the market) , it is always possible to build a tree that is thiner enough to reprice exactly all your market prices.
then using the model, you can by interpolation/extrapolation compute new prices for doublet (K,T) than do not exist in the market.

I you use another interpolation/extrapolation mechanism to compute options continuously first, then want to fit the model with it, it is inconsistant

maxim
Member

Posts: 84
Joined: Apr 2002

Mon Sep 16, 02 05:07 PM

In my opinion, one can implement a la Dupire model in 3 ways:

1) One uses market prices (implied vols) and the Dupire formula (in terms of implied vols) to compute a two-dimensional set {localvola(k,t): k=K_min...K_max, t=T_min,...,T_max}. One computes partial derivatives in the Dupire formula via higher-order numerical interpolation. As neatly pointed out by a forum member (NNT) this is a numerical nightmare.

2) One calibrates a parametric form for implied vols. As a (not precise) result, one can compute the desired set of local vols using this form.

3) One can use implied trees. I'm not acquainted with the method. Does it require taking partial derivatives or just two-dimensional interpolation?

If one chooses 1) and 2) the pricing problem has to be solved numerically. The point I'm confused at is why these methods lead to vanilla prices that are different from market prices?

Once, for illustrative purposes, I implemented method 2) and priced vanillas with the FD method. I found that the local vol approach gives the prices and deltas that in general are lower than the market ones. Does it indicate that the standard FD method is not applicable? Or that one needs to calibrate local vols by minimizing
sum[market_price (K,T)-model_price(locvols(Theta);K,T)]^2
wrt to model parameters vector Theta.
This is prone to be time-consuming task.

Could please anybody clarify me the calibration of local volatility model.

numbersix
Senior Member

Posts: 462
Joined: Jul 2001

Mon Sep 16, 02 06:14 PM

Local volatility models are a theoretical dream and a numerical nightmare.

Theoretical dream:
- The whole continuum of European call prices across strikes and maturities, as of today and as of present spot, C(K,T) is given
- There exists a function v(S,t) such that the solution C(S0,0), for present spot S0 and present time 0, of the BS PDE:

dC/dt + 0.5*v^2(S,t)*S^2*d2C/dS2 + r*S*dC/dS = r*C, with initial condition the European call payoff C(S,T)=max(S-K,0),

is equal to C(K,T) for all strikes K and maturities T
- This function is given by the Dupire formula which maps C(K,T) one-to-one into v(S,t)
- Indeed it guarantees that callprice_model(localvola(*,*);K,T)=callprice_BS(impliedvola(K,T);K,T)

Numerical nightmare:
- The data C(K,T) is a discrete sample.
- Try and infer a continuum with continuous second partial derivative in K, and continous first partial derivative in T (as the Dupire formula requires).
- How do you extrapolate for large / small K, large / small T?
- How do you make sure the Dupire formula doesn't yield imaginary local volatility (in the mathematical sense, not to mention the poetical)?
- What if you vary your interpolation / extrapolation method? Will the out-of-sample values (i.e. exotic option prices or unknown vanillas whose pricing is the aim of the whole proceduure) vary? By how much? (Stability problem).
- Alternatively, try and parameterize v(S,t) via splines, or some plausible function, but again how do you extrapolate?
- Solve a minimization problem: search for the parametres, of for the knots of the spline function such that sum[market_price (K,T)-model_price(locvols(parameters);K,T)]^2 is minimum.
- Local minima, speed of convergence...
- Doesn't guarantee you match the market data perfectly.
- What if you vary your parameterization functional, or your basis spline functions? Same stability problem?
- Alternatively, work directly in discrete time and discrete space: trees or FD schemes are just numerical algorithms linking v(Si,tj) to C(Ki,Tj), so why don't we search directly for v(Si,tj), in a tree or in a FD grid, such that C(Ki,Tj) matches the market? (Implied trees, or implied FDs).
- You still need to interpolate / extrapolate the sample over your discrete grid / tree, making sure that the butterflies (the discrete proxy for second partial derivative) do not go crazy.
- Same stability problems as before, even worse due to truncation error.

So in the end, it is up to you:
Are you of the kind who have the theoretical dream first, or the numerical nightmare?

-------------------------
I will say first, by way of introducing myself, that I have wished to understand philosophy not as a set of problems but as a set of texts. (Stanley Cavell)

NotANumber
Junior Member

Posts: 6
Joined: Nov 2001

Mon Sep 16, 02 06:57 PM

"Theoretical dream? Numerical nightmare?"

I am not a number...
I am the stuff dreams are made of.