Option Pricing with Fourier Transform and Exponential Lévy Models [pdf]

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."

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.

>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)

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

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.