
Option Pricing with Fourier Transform and Exponential Lévy Models [pdf] - Cieplak
http://maxmatsuda.com/Papers/2004/Matsuda%20Intro%20FT%20Pricing.pdf
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doodlebugging
Being a geophysicist, I got sucked into this by the mention of the Fourier
Transform.

I found this paper to be very wordy and littered with mathematics that was at
times impenetrable. Thank God for the Appendix to help decipher some of the
non-geophysical industry symbology. I'm familiar with the derivations of the
equations used but it's been several decades since I had taken differential
equations so there is a lot left for me to absorb.

I understand this is an old paper but I am cursed with an eye for detail and
some of the little things ended up grabbing my attention more than they should
have. I found myself wondering if, prior to publication, anyone had bothered
reading the full text with an eye for identifying simple spelling errors or
whether they had used a spell checker or other tool to maintain consistency of
spelling of uncommon words or terms. I think not. Just in my own reading these
things popped out at me:

p. 92 in the sentence just after Figure 6.1 sock is used instead of stock.

p. 153 just after Eq 9.8 is defined Meron is used instead of Merton.

p. 166 the page has all but one mention of Black-Sholes spelled as Black-
Shoels.

p. 244 the third sentence uses Bronwian instead of Brownian.

Also, they made a typical math funny on pages 190-191. On p.190 the second
sentence reads:

>Because the CF of VG process cannot be obtained by simply substituting 0 α =
in (11.23), we need to do this step-by-step.

So now they promise me some interesting step-by-step derivations in their
algebra. Instead I get the standard upper-level math statement on p. 191 after
Eq. 11.28:

>After tedious algebra:

You're almost to the appendix and you find the first mention that some of the
math got hairy. Fun stuff.

I expected the last page to read "This page intentionally left blank."

~~~
mlevental
>I found this paper to be very wordy and littered with mathematics that was at
times impenetrable.

don't understand this comment. if someone asked you to read a QFT paper would
you claim it were "littered" with impenetrable mathematics? or vice versa: if
a cond mat physicist read a geophysics paper would they be justified in
claiming the same? you're reading a paper outside of your domain of expertise;
expect to be challenged.

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cosmic_ape
>> The answer is that these three models are special cases of more general
exponential Lévy models. Options cannot be priced with general exponential
Lévy models using the traditional approach of the use of the risk-neutral
density of the terminal stock price because it is not available.

Does this mean there is no hedging strategy in these general exponential
models? My understanding is the Black-Scholes gives the price, such that if
the price was different, there would be an arbitrage strategy (under some
assumptions on the variance). And this arbitrage strategy is used for hedging.

~~~
conistonwater
That's right, in the Black Scholes model for any option you can construct
essentially a portfolio of money market+stock that _replicates_ the option. So
if you sell the option and buy and track the replicating portfolio, there is
no hedging error. But you can't hedge perfectly when there are jumps, there is
no replicating portfolio, only ways to minimize the hedging error.

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Rainymood
From the abstract

>Merton jump-diffusion model (1976) which is an exponential Lévy model with
finite arrival rate of jumps,

This is funny because my graduate thesis is about models based on Hawkes
processes, that is, jump with a path-dependent and self-exciting intensity
process. Instead of taking the arrival time of jumps (often the parameter
lambda) to be constant, in a Hawkes process a jump increases the probability
of another jump (positive feedback), leading to clusters of jumps we often see
in crises.

I love how this paper might seem "magic" and voodoo and most importantly
"true" to people who don't know much about mathematical finance. The point is
that the models here are most likely wrong and have GLARING flaws in them, yet
are still used to price REAL things in REAL life. All models are wrong, some
are less wrong than others. The point is, how wrong do our models have to be
before we get some really bad consequences (2008 financial crisis, anyone?)

P.S. Please rewrite this in LaTeX, posting a 250 page document written in Word
makes me 90% less likely to read it (compared to something written in LaTeX)

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llamaz
I'm not sure whose voting up such a technical paper. I could understand if the
subject was control theory or robotics, but economics with engineering-level
math seems out of most of our purviews.

I'm not complaining - I'm happy to see more math on HN. I'm just wondering
about HN's demographics.

~~~
skolos
Especially since the result is not new (it was written in 2004). Even though
the topic is within my interests I'm confused that this is on the front page
of HN.

~~~
yold__
I guess it's interesting to see an alternative to Black Scholes. Fourier
transform (FFT) is a common topic in advanced algos.

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FabHK
Interesting that this got voted up. Anyway, here a few notes:

1\. Mostly, the goal is not to "price options". There's a liquid market for
basic calls/puts, and those prices are used to calibrate a model and then
interpolate/extrapolate as well as price more exotic things. So the goal is to
"fit the market".

2\. Black Scholes is a well defined bijection between a Call price C(K,T) and
a BS vol sigma: C(K,T) = BS(K,T,F,df,sigma). However, prices are such that
calls at different strike K have different BS vol, thus the vol can't be a
description of the underlying stock price. BS is just "the wrong formula to
plug in the wrong number (BS vol) to get the right price". In particular, you
can't evolve a stock (in Monte Carlo, going forward, or in a PDE, going
backward) using BS vol and reprice all options correctly.

2b. But already, BS gives you a means to interpolate and hedge.

3\. The next huge step forward was local vol (Dupire), LV. Instead of assuming
fixed vol, it assume that the vol is a deterministic function of stock price S
and time t. Now you can evolve a stock (in MC or PDE) and reprice vanilla
options correctly, by and large. However, two problems remained:

4a. Forward smile. Take prices as they look today, fit a local vol model, and
evolve it forward 2 years. You've hit all the 2yr option prices correctly, and
you'll hit all the 3yr option prices. However, the 1yr options IN 2 YEARS will
look all wrong (in particular, the smile will have decayed unrealistically).

4b. Very short term smile. A gaussian will basically never go more than 3 std
devs from its mean, right. So, short term out of the money options should be
really worthless. But they aren't, because stock prices in the real world do
jump (or move 10 std devs). So, we require enormously high "lognormal" BS or
local vols to reproduce observed option prices correctly.

4a. is solved with stochastic vol models, SV. Mix SV and LV and you reprice
options perfectly, and go a few years forward, and your forward smile still
looks reasonable.

4b. is solved incorporating jumps, JD (jump diffusion).

Mix SV, JD, LV and you get a nice model that fits the market, and evolves
reasonably.

5\. Most exotic products you price have additional features that preclude
closed form pricing. If there's path dependency, you often just use Monte
Carlo. If there's calculability, you try and use PDEs. If there's both, you
have to use advanced methods: either carry state variables with you in the
PDE, or use Longstaff-Schwartz like Monte Carlo methods.

6\. However, in the last decade or so, after the financial crisis, all the
fancy stuff receded in the background, and there was more focus on the basics:
rates. Different counter parties have different credit risk, different
currencies have different credit, giving rise to cross-currency basis,
different LIBOR maturities are at different levels, giving rise to intra-
currency basis, etc. All that stuff needs to be captured properly.

~~~
keldaris
Naive question from a computational physicist with no quant experience - if
you forgo closed form pricing anyway, there doesn't seem to be any obstacle to
including all of the things you mention in your last paragraph (and more) and
just building more sophisticated numerical models to fit the market. There's
certainly no shortage of data or computational power at the scale we're
talking about. Given that, why has the "fancy stuff" receded in the
background?

~~~
FabHK
Two things, really.

The models needed to be updated to incorporate all the rates stuff. That takes
time. So the focus wasn't so much on innovation on the product front, but on
getting all existing models and products to play along with the new reality.
(That's a huge undertaking, btw... you need the market data, someone
responsible for marking it, put it in the databases, have it flow through the
infrastructure, take it into account in the models, etc.)

Second, previously there were many fancy products that allowed you to trade,
say, vol, mean reversion, correlation, etc.

However, trading this rates stuff is fairly straightforward (in particular,
you don't need optionality/convexity; linear products (such as forwards or
swaps) are enough).

Maybe I shouldn't say "recede" \- but in the earlier decade the focus was more
on fancy exotic products (with huge margins), while in the more recent decade
the focus was on simpler products, but modelling them really precisely.

(There was also some focus on systems and ops and front-to-back processing.)

------
kltutor
Does anyone know if techniques like these are applicable to betting in
baseball games?

