BTC & ETH Options

Understanding behaviour of Diffusion and Jump-Diffusion models on Crypto-currency underlying

Why study option prices?

Option prices are a facinating bundle of information which captures expectation of investors future state of the world, into a single number. If you crunch a lot of information into a small space (i.e. a single number in this case), then naturally retreiving the information back from that said single number is a complex and cumbersome task.
Hence, if we can meaningfully breakdown the quoted market price of an option, then a lot can be revealed about investors expectations.



What information Option prices store?

Option prices are heavily dependent on the movement of the underlying, i.e. it's current level and uncertainity in it's movement - which is the volatility of prices. For an option with certain maturity T and strike K, it's price will depend on the level of underlying S and volatility (implied) at the time of valuation. There are other factors as well such as dividend yield and interest rate how-ever the movation of this reseach focuses on modeling the underlying. It is therefore critical to understand and model the underlying process as appropriately as possible so that when options are prices, the process can generate paths in future which are representative of the behaviour from the past - roughly speaking.

What's so special about options on BTC and ETH

BTC and ETH are household digital currencies which have a lot of investor instest and their options have historically had massive open interest.




It is therefore of an interest to model the underlying process (i.e. asset process for BTC and ETH) appropriately to advance the development of option pricing Methodology. My intution is that crypto-markets have lot more "optimism" and therefore has lot more positively skewed returns, which create a higher curvature in vol smile. This can be explained and re-created by introducing jumps in asset process with a stochastic volatility process.

Put/Call Ratio

Historically, when BTC and ETH show negative returns or a "material downward jump" over a small period, the put/call ratio turns bearish how-ever only for a small duration compared to equity markets. This would indicate a optimisitc investor outlook with call activity picking up relatively quickly.




The skewed behaviour would suggest a higher expectation to increased volatility and higher returns sooner than anticipated, which would inturn suggest a need for "higher" reversion in volatility process. This would further mean that with a reversion back to mean volatility, the asset price level should as well "jump up" really quickly and generate higher and positive returns in doing so.

Choice of Models

I've tried to compare results of pricing option under Heston dynamics and Bates dymanics. Heston model assumes a pure diffusion process to model prices and assumes a CIR style mean reverting Volatility process.

Heston Model

$$ d S_t = \mu S_tdt + \sqrt V_t S_t dW_{t}^{(S)} $$ $$ d V_t = \kappa (\theta - V_t) dt + \sigma_v \sqrt V_t dW_{t}^{(V)}$$ $$ Cov(dW_{t}^{(S)} , dW_{t}^{(V)}) = \rho dt $$

Bates Model

$$ d S_t = (r - q -\lambda\mu_J) S_tdt + \sqrt V_t S_t dW_{t}^{(S)} + J_tS_tdN_t $$ $$ d V_t = \kappa (\theta - V_t) dt + \sigma_v \sqrt V_t dW_{t}^{(V)} $$ $$ Cov(dW_{t}^{(S)} , dW_{t}^{(V)}) = \rho dt $$



High volaility in the past (as of June 2022) and higher returns suggests a need for adaptive process. To build the foundation of this argument it is necessary to understand the behaviour of non jump and single jump process first. .

I calibrated the two models to market observed option prices - see Results tab for details of option date used for calibration. I then re-generated the option prices and compared them against the observed market prices. The results and conclusions are presented on Results tab whereas the code is available here.

Options Prices

Smile re-generation using Heston and Bates model on a stike v/s maturity grid

Using Heston and Bates model, calibrated appropriately, it is noticable that both models struggle to achieve gain in volatility for shorter maturities and longer strikes, as seen below

10D 17D

With increased maturity, the stochastic volatility and asset processes do get enough space on the intergral over time to try adnd attain higher implied vol on longer strikes.

It is interesting to observe that while Bates (red line) closely tracks lower strikes well on 38 days maturity and underestimates it on higher strikes, how-ever it overestimates longer stikes on 66 days maturity by sacrifing fit on lower stikes.

At the same time Heston (blue line) satisfactorily comes close to lower strikes on 38 days maturity, it fails to gain necessary increament in volatility on higher strikes. Whereas with maturity of 66 days Heston is capable of generating the curvature of the smile, this comes at a sacrife of fit on lower strikes.

38D 66D

Going beyond 100 days maturity, both models get sufficient and integral space to re-generate the curvature to a great extent. The mis-alignment to market quotes in this cases can be attributed to the attempt of calibrating entire grid (K v/s T) at the same time with some of the smaller maturities - were as seen above, the fit is sub-optimal.

101D 192D 283D

It'll be interesting to understand the efficency of calibration and smile re-generations on ATM maturity slice, which should isolate the effect of varying maturities with stikes around ATMs of each underlying forward contract. I'll be working on it next!!

Coming soon...

Revisit this page later

This is a working copy of my white-paper, may not be up-to-date with my thinking and progress.. Please reach-out to me if you're interested to know more!