1. Claims Pricing Model
In effect, the claim is equivalent of an American put option with a strike price specified by the insurer at the time s/he stakes in on the platform. It is a put option because a holder of the claim can exercise the option by selling some tokens to the insurer at the strike price. The fact that the option holder can sell it anytime before expiration date suggests that it is an American instead of an European option.
There are multiple methods of pricing American options. It is well-known in the mathemat- ical finance literature that American options do not have a closed-form solution. Researchers and practitioners can at best approximate the price. The most popular numerical methods to evaluate American options include the method based on the celerated Black-Scholes formula and Monte Carlo. Here, we provide an example of the insurance claim and the pricing of the claim based on either methods.
The Black-Scholes PDE describes the evolution of any derivative whose underlying asset satisfies the Black-Scholes assumptionsm and can be used to price American options. In the current context, the underlying asset is the token that the insurer stakes in and the derivative is the Claim, which can be sold by insurers in the market.
Since American options can be exercised at any time, we must add an extra condition at each point on the grid to verify if it is optimal to do so. Therefore, there is an extra verification for the put option
The Black-Scholes PDE for American option is given by:
Solving for the formula under the boundary condition for the American put option:
2. Price simulation
Suppose the underlying token’s initial price is 500 USDT. Then, the claim token with a strike price of 600 USDT may exhibit different historic path. Here, we provide seven different simulations to illustrate the possible claim and unclaim token prices. Figure 1 illustrates the Monte Carlo simulations of the underlying token price. Given the underlying token prices simulated, we can find instantaneous value of the claim token, which then can be used to calculated the path for the value of the claim.