# Are there any new Option pricing models?

Back in the mid 90's I used the Black-Scholes Model and the Cox-Ross-Rubenstein (Binomial) Model's to price Options. That was nearly 15 years ago and I was wondering if there are any new models being used to price Options?

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For FX options, many use the Garman-Kohlhagen model (en.wikipedia.org/wiki/…), since it can handle two interest rates for two currencies versus the one interest rate that Black-Scholes can handle. – barrycarter Feb 3 '11 at 8:50

Black-Scholes itself didn't change a lot but we can now adjust it to deal with a lot more complicated factors to price more complicated contracts:

• stochastic volatility (Heston, Gatheral)
• stochastic rates (Hull)
• credit risk
• dividends

Other methods (computing intensive) have also evolved to deal with various types of contracts where BS is not very appropriate choice (e.g. Monte Carlo simulation for path-dependant options).

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To the best of my knowledge, progress has been made in making pricing models more accurate, accounting for additional variables (stochastic vol. & GARCH, local vol., jumps, etc), but the option pricing equations fundamentally involve the same GBM framework, albeit they are much more complicated.

Off the top of my head, the only example I can think of something completely original is http://papers.ssrn.com/abstract=2412761 Here, option prices grow at $t^{1/4}$ instead of $t^{1/2}$

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There seems to be a tendency to include ever more risk factors into the models. CVA is a good example: based on credit default swaps it is possible to calibrate model parameters and include the risk of default into a Monte Carlo pricer.

The book Counterparty Credit Risk: The new challenge for global financial markets (Amazon) discusses counter party risk in detail.

For the implementation perspective: Mathworks has a nice practical discussion of counterparty credit risk and CVA including example code.

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There is a whole family of GARCH option pricing models; ones with complex distributions, leverage effects, skewness parameters etc. For an example see Christoffersen and Jacobs (2004).

Some example GARCH models:

• NGARCH
• EGARCH
• TGARCH

Some distributions that can be used:

• Hyperbolic
• Normal Inverse Gaussian
• Variance Gamma
• and the generalized form Generalized Hyperbolic
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GARCH models are not for pricing options. – Bob Jansen May 27 '14 at 19:37
there is a whole literature that disagrees with you @BobJansen for example: jstor.org/stable/30046228 – berkorbay May 27 '14 at 20:25
I'd call to those GARCH-based but point taken. – Bob Jansen May 28 '14 at 6:59

Option pricing is done under the risk-neutral measure, i.e. the drift term is the risk-free interest rate. Therefore the only degree of freedom to drive the underlying is the volatility. That is why volatility modelling for all (new) option pricing models is so crucial. You can find a good, concise and current overview here:

Abstract:

This document is a short summary regarding the evolution of the volatility models from Black and Scholes to the Local-Stochastic Volatility models. We show advantages as drawbacks linked to each model, and how the community has moved from one model to another in order to overcome drawbacks.

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One I like is the Artificial Neural Network model with inputs the same as for the Black-Scholes model (hence Spot, Strike, Rate, Time to expiration, Dividend).

This is a modification of the Andreou et. al. (2008) framework who use hybrid artificial neural networks that incorporate information from the parametric models. Moreover, $Call = f(Call_{cs}, Call_{bs})$ where $f(.)$ is the hybrid network that links information from the parametric models with the prices $Call_{cs}$ (Corrado and Su, 1997) and $Call_{bs}$ (Black Scholes, 1973).

In the case I proposed, one can use the same hybrid networks $f(.)$ to incorporate information from the Black-Scholes inputs (Spot, Strike, Rate, Time to expiration, Dividend). Hence, $Call = f(Spot, Strike, Rf, T, Div)$.

This would allow for complex nonlinear relations. However, the approach may be sensitive to overfitting or other calibration issues.

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Is there a paper or explanation you could link to? – chrisaycock Jun 27 '13 at 15:35
@JohnAndrews, could you please post a paper or anything you used as reference or that you can show here as reference. It would actually lend your answer much more credibility to add references rather than saying "I know of a new model called "fractional banana split option price model", but unfortunately I cannot provide any explanation what it is nor references to someone who does know". I dont mean to be condescending so please ignore the analogy if it disturbs you. TGIF. – Matt Wolf Jun 28 '13 at 4:47
See modification – JohnAndrews Jun 28 '13 at 15:09
1. The introduction of 'volume endowed' shapes that represent support and resistance forces. Theoretical paths in , unlike in Brownian motion are either strictly monotonically increasing, flat, or monotonically decreasing and obey the equilibrium equation with an inverse relation between LHS resistance/support forces and .
2. Volatility in chart to scalar is a combination of variables such as the geometric properties of the LHS resistance/support forces, change in price, trade size, and volume.
3. The interaction of buy & sell orders within the volume is the propagator of price change.
4. The use of truncated normal distributions in contrast to a log-normal distribution.
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There are plenty of other models

You can also add all the exponential Lévy processes with or without time change and also other stochastic volatility models such as SABR.

I must add that there exist a paradigm different of the "risk neutral pricing" (mainly developped by Platen and Heath) called "Benchmark Pricing" and which is in a way (that I do not fully understand yet), more general than "Risk Neutral paradigm". The biggest problem being that calculation and determination of the benchmark protfolio doesn't seem easy to achieve in this "supermartingale framework".

Regards

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@ all: There is a paper by Fontana"Weak and strong no-arbitrage conditions for continuous financial markets" :arxiv.org/pdf/1302.7192.pdf that explains very well the link between different notions of arbitrage among which the Benchmark approach can be wrapped. Regards – TheBridge Oct 2 '13 at 13:47

Maybe you think about other model than a diffusion ?

There is an article on wilmott.com about the Korn-Kreer-Lenssen Model.

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That URL has a pay wall unfortunately. That's my problem, I think, not yours, and I appreciate your answer. You gave additional content so I have a starting point. Might you have any other links that are more accessible? – Ellie Kesselman Dec 15 '11 at 20:49