Pricing Blending

@gradientdev @0xBobdbldr August 10, 2023

Link to code: https://github.com/IVX-FI/ivx-diem/blob/c01b385948c67b04ccc54f65fc08d58cbdada189/src/libraries/IVXPricer.sol#L74

Black Scholes Model

https://github.com/IVX-FI/ivx-diem/blob/main/src/libraries/BlackScholes.sol

Binomial model

https://github.com/IVX-FI/ivx-diem/blob/main/src/libraries/Binomial.sol

IVX Dual Model

Link to premium calculation: https://github.com/IVX-FI/ivx-diem/blob/c01b385948c67b04ccc54f65fc08d58cbdada189/src/options/IVXDiemToken.sol#L181

The Black-Scholes model is based on a partial differential equation (PDE) that describes the dynamics of the option price over time. The solution to this equation involves the calculation of an option price using the parameters mentioned in the previous part. (here we already broke down the BS model) As the time to expiry approaches zero, the PDE used in the Black-Scholes model becomes increasingly difficult to solve. This is due to the option price becoming more sensitive to changes in the underlying asset price and volatility, which results in a breakdown of the Black-Scholes model’s assumptions and becomes increasingly unsuitable due to its reliance on certain assumptions, including constant volatility for example, which may not hold, especially for crypto markets and altcoins. This will result in inaccurate pricing of options with a short time to expiry that can get chaotic below 1 day. The Black-Scholes model assumes that the underlying asset price changes continuously over time, which may not hold in most altcoins. As such, a more conservative pricing model may be utilized to value options with a shorter time to expiry. Such a model typically prioritizes the convergence to the option’s true value as the expiry date approaches.

The binomial model is an appropriate choice for such a conservative model, as it is a discrete-time model that is better equipped to handle changing market dynamics as the option approaches expiration. To gauge the convergence of the conservative model to the binomial model, we introduce the IVX Dual model, a dynamic options pricing assumption that takes into account the stability of the Black-Scholes model over longer durations and the dynamism of the Binomial model over closer durations, and we will introduce Alpha (α), a specific metric that controls this convergence.

Price = α · Binomial Model Price + (1 − α) · Black Scholes Price (1)

▷ So let T be the expiry time, and t be the current time, so that the time to expiry is T − t.

Alpha (α) denotes the rate at which the conservative model approaches the binomial model as the time to expiry decreases, with full convergence to the Binomial model achieved as alpha approaches 100% , and full convergence to the Black-Scholes model is achieved when alpha approaches 0%.

The selection of an appropriate pricing model is critical and should depend on the specific characteristics of the option being valued and the time to expiry. While the Black-Scholes model may be appropriate for options with a longer time to expiry, the utilization of a more conservative model, such as the binomial model, may be necessary for options with a shorter time to expiry, for that reason the IVX Dual Model will be a suitable solution for a never-ending problem.

contract Alpha {
  uint public binomialCutoff;
  uint public blackScholesCutoff;

  constructor(uint _binomialCutoff, uint _blackScholesCutoff) {
   require(_blackScholesCutoff > _binomialCutoff, "Black Scholes cutoff must be greater than Binomial cutoff");
   binomialCutoff = _binomialCutoff;
   blackScholesCutoff = _blackScholesCutoff;
  }

  function calculateAlpha(uint secondsToExpiry) public {
   if (secondsToExpiry <= binomialCutoff) {
    return 1e18;
   }

   if (secondsToExpiry >= blackScholesCutoff) {
    return 0;
   }
   alpha = (blackScholesCutoff - secondsToExpiry) * 1e18 / (blackScholesCutoff - binomialCutoff);
    return alpha;
   }
}