QEF16 – A quantum jump model of option pricing
As seen in QEF15, stock market data follows q-variance and the q-distribution. Q-variance refers to the property that the expected variance of log returns x, corrected for drift, over a period T follows to good approximation the formula V(z)=^2+z^2/2 where z=xT. This property follows because we model price change as the displacement of a quantum harmonic oscillator, so a larger displacement is associated with higher energy and therefore higher volatility.
The probability distribution associated with q-variance is a Poisson-weighted sum of Gaussians known as the q-distribution. The Poisson rate parameter is =0.5 and the standard deviation for each term is _n=(1+2n).
The q-distribution can be used to calculate the price of a call option. We can also use it to calculate the q-implied volatility _q, which is the volatility to be used in the Black-Scholes model for this distribution. The answer is seen to be
_q^2=q^2 ^2+(1-q^2 ) z^2
where q0.95.
We can therefore compare three kinds of volatility. Q-implied volatility is the number suggested by quantum model. Implied volatility is the number used by traders. Payout-implied volatility is the number which corresponds to average payouts.
Plots show q-implied (left), implied (middle), and payout-implied (right) volatility as a function of x and T (top row) and z only (bottom row).To summarise, the q-distribution can be used to calculate the expected payout of a call option. The q-implied volatility (the volatility suggested by the quantum model) is quite close to both implied volatility (the volatility used by traders) and payout-implied volatility (the volatility which reflects average payout)
This suggests that markets follow a quantum distribution, and traders are using a quantum model to price options - they just don't know it
For further reading, see:
Orrell D (2025) A Quantum of Variance, and the Challenge for Finance. Wilmott 2025(138).
Orrell D (2025) A Quantum Jump Model of Option Pricing. Journal of Derivatives 33(1).
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