Summary and Comparison

PT pricing involves balancing trade-offs between manipulation resistance, market responsiveness, simplicity, and capital efficiency. The optimal approach depends on the protocol's risk preferences, market conditions, and asset liquidity.

Method	Formula	Pros	Cons	Preferred Environment
Linear Discount Rate	$P_t = 1 - d \times \frac{T - t}{T}$	Simple, predictable, manipulation-resistant	Underprices, unresponsive to market dynamics	Low-liquidity markets
Zero-Coupon Bond Rate	$P_t = \frac{1}{(1 + r)^T}$	Reflects compounding, predictable, manipulation-resistant	Unresponsive to market dynamics	Longer-term markets with stable yield
AMM-TWAP	${\text{TWAP}_t} = \frac{\sum P_t \times \Delta t}{\sum \Delta t}$	Market-responsive	Latency, liquidity-dependent	Active DeFi markets with deep liquidity
Hybrid Approach	$P_t=Max( \frac{\sum P_t \times \Delta t}{\sum \Delta t},1 - d \times \frac{T - t}{T}),$ $P_t=Max( \frac{\sum P_t \times \Delta t}{\sum \Delta t},\frac{1}{(1 + r)^T})$	Manipulation-resistant, market-responsive	Higher complexity	Markets requiring both stability and adaptability