With the democratization of trading quickly under way, the markets have experienced an influx of retail traders with small accounts looking to speculate and test their luck in the market.
Such a situation lends itself perfectly towards facilitating rising demand for alternative assets such as various derivative instruments, leveraged products, cryptocurrencies, etc.
Another driver of speculative activities that results in rising demand for derivatives is the increasingly loose monetary policy which has been a trend since Volker left the Federal Reserve in the late 80’s. Having said this, if doves continue to operate at the helm of the Fed, it is likely that the option volume will continue to rise in the future as it has over the last 30 years.
Assuming that demand for options will continue to increase, it is almost certain that the future of the derivatives market will include a wider range of underlying assets.
Furthermore, the burgeoning of option trading will give rise to more data that can be used by quantitative traders to devise strategies with higher statistical significance.
However, abnormal events caused by derivatives, such as the gamma-squeeze that occurred with Gamestop’s stock and a handful of others may force regulators to step in and introduce new restrictions that would hinder the proliferation of trading volume in the option market.
Quantitative Overview:
The Black-Scholes model has been widely employed since the 70’s and 80’s. Currently, research focused on option pricing aims at building upon this framework and relaxing some of the assumptions it requires.
- Figure 1: Black-Scholes Model for European Call
- C=Nd1S0-Nd2K-rt
- d1=ln S0K +r+22tt
- d2=d1-σt
- Where St is the stock price at time t, K is the strike price, is the historical volatility, r is the interest rate, and t is the time until expiration. N() represents the CDF of a normal distribution.
One of the Black-Scholes assumptions is that volatility is constant across strike prices. However, from the “volatility skew/smirk” that we see in option chains, it is obvious that this is a false assumption.
To correct for this, paradigms such as the “Heston Model” were developed in the mid 90’s which actually included volatility dynamics in the option pricing model.
Most of these paradigms model volatility dynamics using interest rate models such as a CIR model that have a lower bound at 0 and a mean-reverting process.
Models like the Heston do a fairly good job at replicating the volatility skew; however, there is still significant room for improvement.
Today, research on option pricing also attempts to incorporate jumps in volatility into the volatility’s dynamics using mathematical methods such as the Poisson process. These models are surprisingly accurate and do a good job at replicating the volatility skew/smirk.
- Figure 2: Heston Model
- Stock Price Dynamics: dSt= μStdt+ tStdWtS
- Volatility Dynamics (CIR Model): dt=θ-tdt+ ξtdWt
- Where is the stock’s drift, is the long-run volatility, determines the speed at which volatility reverts to its mean or long-run equilibrium , is the variance of t
- Figure 3: Poisson Process
- PX=x= xe-λx!
- Where x = number of occurrences, λ = mean number of occurrences in the interval, e = Euler’s Constant
- PX=x= xe-λx!
Going forward, it will be interesting to see the ways in which such models are modified to price newly-devised derivatives on alternative assets and financial instruments.
Due to the incredibly deep intuition that is afforded to users of these option pricing models, they should not be too difficult to modify to account for new types of derivatives that are made in the future.
