III. PRICING FRAMEWORK

A. Pricing Model

The token event swap TES contract delivers a protection payment (the cost to bail-out an undercollateralized loan) at the time of the triggering event, defined as the token price declining below a pre-specified triggering barrier level (value of collateral falling below the value of debt). In exchange the TES protection buyer makes periodic premium payments at the TES rate up to the triggering event. The TES pricing model is based on the jump-to-default extended constant elasticity variance model by P. Carr & V. Linetsky [1] known as JDCEV where the token price is modeled as a diffusion, punctuated by a possible jump to zero:

where r, σ (Xt), and h (Xt) are risk free rate, instantaneous token volatility, and the state dependent jump-to-default intensity (hazard rate).

To capture the negative link between volatility and token price, we assume a constant elasticity of variance (CEV) specification for the instantaneous token volatility prior to extreme price events:

where β is the volatility elasticity parameter and a is the volatility scale parameter.

To capture the positive link between extreme price events and volatility, the hazard rate is an increasing affine function of the instantaneous variance of returns on the underlying token:

where b is a constant parameter governing the stateindependent part of the jump-to-default intensity and c is a constant parameter governing the sensitivity of the intensity to the local volatility σ 2 .

The TES price (aka premium rate) is obtained following Mendoza-Arriaga & Linetsky [2] as the rate % that equates the present value of the protection payoff to the present value of the premium payments.

The protection payment is the specified percentage (1−r) of the TES notional amount that the TES pays out to takeover the undercollateralized loan (r is the ”recovery rate” and 1 − r is the ”loss-given-the-triggering barrier crossing event”):

The first term in parenthesis is the payoff triggered by a jump and the second term is the payoff if the token price hits the barrier by diffusion. The present value of all premium payments made by the TES protection buyer is:

where L is the barrier, T is the horizon, N is the number of premium payments, ∆ = T /N is the time between premium payments, ti = ∆ · i, i = 1, 2, …, N is the i th periodic premium time, TL is first hitting time.

As collateral price and volatility change over time the premium charged to the borrowers (pricing) is adjusted using the TES pricing model; borrowers actually pay a floating premium rate. Premiums adjust inversely proportional to collateralization levels and proportional to level of collateral token volatility.

The basket TES is priced as a DV01 weighted average basket of TES spreads and the tranche pricing uses a Gaussian copula factor model as in Wang et al. [3].

B. Price Discovery

This section describes how market based price discovery is achieved. TES model prices offered to borrowers are scaled higher or lower to drive Solvency Ratio to a target set by custodians (such as 100%). The ”right” price should lead to a balance between loan collateral and insurance assets. The concept is similar to option market makers updating implied volatility based on order book imbalances. Three parameters in the pricing and risk models are scaled in an effort to drive Solvency Ratio closer to its target: hazard rate function parameters, b and c as in Eq. 3 as well as recovery r as in Eq. 4.

C. Stability

The stablecoin is designed to be price stable to USD using the following four pillars of stability:

1) Crypto-overcollateralized

Stability depends firstly on the level of overcollateralization which covers expected losses. The system prices the collateral in USD, and overcollateralization is defined as USD value of collateral minus number of stablecoins of debt. This system is extendable to create stablecoins that track anything with a price including other fiat currencies, baskets of fiat, low vol baskets of crypto etc.

2) Collateral is insured

Event risk and volatility of the collateral is transferred to insurers. Stability then also depends on the level of capital adequacy or solvency of the insurance. The system scales insurance pricing through implied hazard rates and recovery rates to drive Solvency Ratio to the target set by manager vote.

3) Final Reserve

The insurance asset pool represents capitalization to cover unexpected losses estimated by the stress model to a degree of certainty specified by VIG custodians. Actual losses may prove worse than estimated due to model risk. So the VIG final reserve backs the insurance pool as a lender of last resort covering these so-called stress losses.

4) Solvency target

Custodians set the target solvency giving them the power to run the insurance business from conservative to aggressive.

References:

[1] Carr, P., and Linetsky, V. A Jump to Default Extended CEV Model: An Application of Bessel Processes. Finance and Stochastics 10, 3 (2006), 303330.

[2] Mendoza-Arriaga, Vadim Linetsky, Pricing Equity Default Swaps under the Jump to Default Extended CEV Model, Finance and Stochastics, September 2011, Volume 15, Issue 3, pp 513540.

[3] Wang D., Rachev S.T., Fabozzi F.J. (2009) Pricing Tranches of a CDO and a CDS Index: Recent Advances and Future Research. In: Bol G., Rachev S.T., Wrth R. (eds) Risk Assessment. Contributions to Economics. Physica-Verlag HD

Vigor Stablecoin and DAC Social Links

telegram : https://t.me/vigorstablecoin

member client : https://vigor.ai

website : http://www.vigorstablecoin.com

github : https://github.com/vigorstablecoin

whitepaper : https://vig.ai/vigor.pdf

twitter: https://twitter.com/vigorstablecoin

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