https://panoptic.xyz/assets/js/a1d9312c.626aae1a.js

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mathematical foundations, pricing mechanics, and applicability to decentralized markets.",authors:"A"},p=void 0,r={permalink:"/research/perpetual-option-pricing-model-comparison",source:"@site/research/2025-08-04-pricing-models-perp-options/pricing-models-perp-options.md",title:"How to Price Perpetual Options: Five Models Compared",description:"This paper compares perpetual option pricing frameworks with Panoptic\u2019s oracle-free, DeFi-native model on Uniswap v3/v4, detailing mathematical foundations, pricing mechanics, and applicability to decentralized markets.",date:"2025-08-04T00:00:00.000Z",formattedDate:"August 4, 2025",tags:[{label:"Pricing Model",permalink:"/research/tags/pricing-model"},{label:"Black Scholes",permalink:"/research/tags/black-scholes"},{label:"American Option",permalink:"/research/tags/american-option"}],readingTime:15.656666666666666,hasTruncateMarker:!1,authors:[{name:"Amy 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Explanation",id:"plain-language-explanation-2",level:3},{value:"Model IV: Gapeev &amp; Rodosthenous \u2013 Compound Perpetual American Options",id:"model-iv-gapeev--rodosthenous--compound-perpetual-american-options",level:2},{value:"Quantitative Explanation",id:"quantitative-explanation-3",level:3},{value:"Plain-Language Explanation",id:"plain-language-explanation-3",level:3},{value:"Model V: Nonlinear Volatility in Perpetual American Put Options (Grossinho, Faghan, &amp;

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1oles pricing depending on path risk."),(0,m.yg)("p",null,"In fact, Panoptic doesn\u2019t pre-price an option, it charges dynamically as market activity unfolds. When the price lingers near the strike, more fees accrue. If the market stays away, no cost is incurred. This turns options into a metered service rather than a prepaid contract. In short: users pay for ",(0,m.yg)("em",{parentName:"p"},"realized risk"),", not theoretical exposure."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Key distinction:"),(0,m.yg)("br",{parentName:"p"}),"\n","Unlike oracle-driven models, Panoptic infers implied volatility and pricing directly from liquidity and volume. This makes it not only oracle-free but also inherently composable with any Uniswap v3 and v4-compatible asset."),(0,m.yg)("h3",{id:"plain-language-explanation"},"Plain-Language Explanation"),(0,m.yg)("p",null,"Panoptic doesn\u2019t try to predict how much an option is worth using a math formula. Instead, it lets the option's cost emerge from what actually happens on-chain."),(0,m.yg)("p",null,"Here\u2019s the idea: when a trader wants to go long an option, they do it by entering a position that can cause losses to an LP, someone who provides liquidity in a specific price range on Uniswap v3 and v4. The closer the market price is to that range, the more risk the LP faces. So, Panoptic charges the trader a fee that depends on how long the price stays in that \u201cdanger zone.\u201d The longer the price remains near the strike, the more fees accumulate. This is similar to how traditional options lose value over time (called theta), but in Panoptic, it\u2019s not based on a formula, it\u2019s based on real trading activity and liquidity movement."),(0,m.yg)("p",null,"This fee is charged continuously, block by block, and only while the price is inside the strike range. If the price moves away, the trader stops paying. 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1,"\u03b8"))))),": the Black\u2013Scholes theta, representing time decay  "),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("msub",{parentName:"mrow"},(0,m.yg)("mi",{parentName:"msub"},"V"),(0,m.yg)("mtext",{parentName:"msub"},"mkt"))),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"V_{\\text{mkt}}")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,m.yg)("span",{parentName:"span",className:"mord"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.22222em"}},"V"),(0,m.yg)("span",{parentName:"span",className:"msupsub"},(0,m.yg)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.3361em"}}

1,(0,m.yg)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord text mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},"mkt")))))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.yg)("span",{parentName:"span"})))))))))),": the actual market price of the perpetual option")),(0,m.yg)("p",null,"This equation represents the rate of change of the trader\u2019s profit over time when their position is hedged against movements in the underlying asset. The term ",(0,m.yg)("span",{parentName:"p",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"\u03b8")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\theta")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}}

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1,(0,m.yg)("span",{parentName:"span",style:{top:"-2.655em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal mtight"},"d"),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal mtight"},"t")))),(0,m.yg)("span",{parentName:"span",style:{top:"-3.23em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,m.yg)("span",{parentName:"span",style:{top:"-3.4159em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal mtight"},"d"),(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.22222em"}},"V"),(0,m.yg)("span",{parentName:"span",className:"msupsub"},(0,m.yg)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.3448em"}}

1,(0,m.yg)("span",{parentName:"span",style:{top:"-2.3488em",marginLeft:"-0.2222em",marginRight:"0.0714em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"2.5em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size3 size1 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord text mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},"mkt")))))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.1512em"}},(0,m.yg)("span",{parentName:"span"})))))))))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.345em"}},(0,m.yg)("span",{parentName:"span"}))))),(0,m.yg)("span",{parentName:"span",className:"mclose nulldelimiter"}))))))," captures the market-driven price change of the perpetual option. Together, they determine the trader\u2019s net return per unit time in a funding-based system like Paradigm\u2019s."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Liquidation:"),(0,m.yg)("br",{parentName:"p"}),"\n","Positions are margin-bound. If market losses or theta payments reduce collateral below the maintenance threshold, liquidation occurs just as in perp markets."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Interpretation:"),(0,m.yg)("br",{parentName:"p"}),"\n","This structure emulates option holding costs in a perpetual fashion. The holder pays the equivalent of daily theta decay to keep the position open, just as a perp trader pays funding. While elegant, this design assumes oracle accuracy, volatility stability, and a liquid hedging infrastructure which are requirements more natural to centralized venues than DeFi."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Key difference from Panoptic:"),(0,m.yg)("br",{parentName:"p"}),"\n","Panoptic embeds the same intuition \u2014 theta-based pricing \u2014 but derives it naturally from Uniswap v3 and v4 activity, without external data. Paradigm models decay; Panoptic observes and charges it."),(0,m.yg)("h3",{id:"plain-language-explanation-1"},"Plain-Language Explanation"),(0,m.yg)("p",null,"Imagine you want to hold an option forever, but instead of paying everything upfront, you just pay a small \u201crent\u201d every block. This rent reflects how much the option would normally decay over time in a standard pricing model like Black-Scholes."),(0,m.yg)("p",null,"Paradigm\u2019s everlasting options are built exactly around this idea. They are inspired by perpetual futures contracts: just as a perp tracks an asset through a funding rate, the everlasting option tracks the value of a traditional option by charging a ",(0,m.yg)("em",{parentName:"p"},"funding rate")," based on its time decay, or theta."),(0,m.yg)("p",null,"Each second, the protocol calculates what a normal short-term option would be worth and then charges or credits the trader accordingly. If the market price drifts too far from this \u201ctheoretical\u201d value, arbitrageurs step in buying low and selling high which helps pull prices back in line."),(0,m.yg)("p",null,"But to make this work, the system depends heavily on oracles. It needs to know the live price of the asset and estimate volatility accurately. If the oracle is wrong, the whole pricing breaks down. This structure fits very well in centralized environments, where you can compute Black-Scholes precisely, but is more fragile in DeFi where data is fragmented or noisy."),(0,m.yg)("p",null,"Paradigm starts from a pricing model and builds a mechanism to match it pulling the market toward a Black-Scholes anchor. Panoptic does the opposite: it lets pricing emerge from how traders interact with the protocol, using streaming fees from LP positions. Paradigm needs an oracle; Panoptic doesn\u2019t."),(0,m.yg)("h2",{id:"model-iii-sidani--probabilistic-expiry-via-option-mixtures"},"Model III: Sidani \u2013 Probabilistic Expiry via Option Mixtures"),(0,m.yg)("h3",{id:"quantitative-explanation-2"},"Quantitative Explanation"),(0,m.yg)("p",null,"Sidani proposes a novel pricing model for perpetual options by treating them as an infinite series (or integral) of standard European options, each with a random expiry. 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1,(0,m.yg)("span",{parentName:"span",style:{top:"-2.655em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal mtight"},"\u03bb")))),(0,m.yg)("span",{parentName:"span",style:{top:"-3.23em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,m.yg)("span",{parentName:"span",style:{top:"-3.394em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},"1"))))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.345em"}},(0,m.yg)("span",{parentName:"span"}))))),(0,m.yg)("span",{parentName:"span",className:"mclose nulldelimiter"})))))),"."),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"\u03b4")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\delta")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03785em"}},"\u03b4")))))," discounts future values and can be interpreted as a funding adjustment or time preference.")),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Connection to Funding Rate:"),(0,m.yg)("br",{parentName:"p"}),"\n","While this model does not directly use a funding rate like Paradigm, the decay of weight over time (",(0,m.yg)("span",{parentName:"p",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"\u03bb")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\lambda")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal"},"\u03bb"))))),") acts as a proxy for the economic cost of holding exposure; shorter expiries dominate because longer-term exposure is discounted."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Conclusion:"),(0,m.yg)("br",{parentName:"p"}),"\n","This formulation offers a mathematically clean, expectation-based pricing for perpetual options. It does not rely on path-dependence or external price oracles (unless ",(0,m.yg)("span",{parentName:"p",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("msub",{parentName:"mrow"},(0,m.yg)("mi",{parentName:"msub"},"V"),(0,m.yg)("mtext",{parentName:"msub"},"BSM"))),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"V_{\\text{BSM}}")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,m.yg)("span",{parentName:"span",className:"mord"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.22222em"}},"V"),(0,m.yg)("span",{parentName:"span",className:"msupsub"},(0,m.yg)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.3283em"}}

1,(0,m.yg)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"-0.2222em",marginRight:"0.05em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord text mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},"BSM")))))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.yg)("span",{parentName:"span"}))))))))))," needs to be fed via oracle). The model is intuitive, composable, and compatible with both analytical and Monte Carlo methods."),(0,m.yg)("h3",{id:"plain-language-explanation-2"},"Plain-Language Explanation"),(0,m.yg)("p",null,"You can think of the Sidani model as a blend between uncertainty and flexibility. Since you don\u2019t know when the option might mature, the model prices in a whole distribution of possible expiries. This adds a kind of \u201cweighted imagination\u201d, you imagine every possible European option across time and combine them. The result is a smooth, continuous valuation that reflects both short-term and long-term risk. It\u2019s as if the market constantly rolls the dice on expiry and charges you based on the average cost of all those possible outcomes."),(0,m.yg)("p",null,"Sidani\u2019s model answers a tricky question: \u201cHow do you price something that never expires?\u201d Instead of trying to force-fit a formula, it says: what if we imagine the expiry is unknown like a spinning wheel that could land on 1 day, 10 days, or 100 days, but with shorter expiries more likely?"),(0,m.yg)("p",null,"Then, we price a normal option (using the Black-Scholes model) for every possible expiry, and take the average weighted by how likely that expiry is. If short expiries are more likely, they get more weight. If we use an exponential curve, the price looks a lot like what you\u2019d expect for a perpetual option."),(0,m.yg)("p",null,"This approach is clean, flexible, and easy to simulate. You can run a Monte Carlo with expiry drawn randomly, or integrate it analytically if you know your distribution."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Key difference from Panoptic:"),(0,m.yg)("br",{parentName:"p"}),"\n","Sidani mixes snapshots of standard options with different time horizons. Panoptic tracks the real path of the price over time and charges based on how long it stays risky. One uses a probabilistic mixture; the other uses a real-time stream."),(0,m.yg)("h2",{id:"model-iv-gapeev--rodosthenous--compound-perpetual-american-options"},"Model IV: Gapeev & Rodosthenous \u2013 Compound Perpetual American Options"),(0,m.yg)("h3",{id:"quantitative-explanation-3"},"Quantitative Explanation"),(0,m.yg)("p",null,"This model studies the valuation of ",(0,m.yg)("strong",{parentName:"p"},"compound perpetual American options"),", i.e., options whose underlying assets are themselves perpetual American options. These contracts involve a nested exercise structure, leading to a multi-layer optimal stopping problem under continuous-time diffusion models."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Contextual Note:"),(0,m.yg)("br",{parentName:"p"}),"\n","Although compound options differ from the other models in this paper: in that their underlying is itself an option rather than a spot asset, we include the Gapeev & Rodosthenous model to expand the conceptual boundaries of perpetual option pricing. By introducing perpetuality at multiple layers of optionality, this framework showcases how the perpetual structure can be embedded recursively. It provides a valuable theoretical benchmark and highlights the potential complexity and richness of derivative instruments when expiry is removed entirely."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Underlying Framework:")),(0,m.yg)("ul",null,(0,m.yg)("li",{parentName:"ul"},"The underlying asset ",(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("msub",{parentName:"mrow"},(0,m.yg)("mi",{parentName:"msub"},"S"),(0,m.yg)("mi",{parentName:"msub"},"t"))),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"S_t")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,m.yg)("span",{parentName:"span",className:"mord"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05764em"}},"S"),(0,m.yg)("span",{parentName:"span",className:"msupsub"},(0,m.yg)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.2806em"}}

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1,(0,m.yg)("span",{parentName:"span",style:{top:"-2.55em",marginLeft:"-0.0715em",marginRight:"0.05em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"2.7em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},"2")))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.15em"}},(0,m.yg)("span",{parentName:"span"})))))),(0,m.yg)("span",{parentName:"span",className:"mclose"},")")))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.8943em"}},(0,m.yg)("span",{parentName:"span"}))))),(0,m.yg)("span",{parentName:"span",className:"mclose nulldelimiter"}))))))),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Optimality Conditions:")),(0,m.yg)("ul",null,(0,m.yg)("li",{parentName:"ul"},"Martingale verification (Snell envelope)"),(0,m.yg)("li",{parentName:"ul"},"Value matching and smooth-pasting conditions"),(0,m.yg)("li",{parentName:"ul"},"Analytical solution of free-boundary PDEs")),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Interpretation:"),(0,m.yg)("br",{parentName:"p"}),"\n","Pricing a compound perpetual option requires identifying the optimal stopping time for both the outer and inner options. These solutions reflect both continuation and early exercise regions and give closed-form expressions for value and boundaries. This approach offers a rare, explicit benchmark for multi-layer optionality."),(0,m.yg)("h3",{id:"plain-language-explanation-3"},"Plain-Language Explanation"),(0,m.yg)("p",null,"Imagine an option \u2014 but instead of giving you the right to buy an asset, it gives you the right to buy another option. That\u2019s a ",(0,m.yg)("em",{parentName:"p"},"compound option"),". It\u2019s like financial nesting dolls: one option layered inside another. And in this case, both the inner and outer options are ",(0,m.yg)("strong",{parentName:"p"},"perpetual")," and ",(0,m.yg)("strong",{parentName:"p"},"American-style"),", meaning they can be exercised at any time and never expire."),(0,m.yg)("p",null,"Let\u2019s walk through a concrete example: a ",(0,m.yg)("em",{parentName:"p"},"call-on-call"),"."),(0,m.yg)("p",null,"You pay a premium today for the right (at any point in the future) to purchase a perpetual call option on a stock. That inner call option also doesn\u2019t expire and gives you the right to buy the stock. So you\u2019re essentially buying ",(0,m.yg)("em",{parentName:"p"},"optionality on optionality"),"."),(0,m.yg)("p",null,"Now here's the tricky part: you want to exercise the outer option only if the inner one is worth it. That means you need to know:"),(0,m.yg)("ul",null,(0,m.yg)("li",{parentName:"ul"},"When is the inner option \u201cin the money\u201d enough?"),(0,m.yg)("li",{parentName:"ul"},"And how much value would be left after paying the outer strike?")),(0,m.yg)("p",null,"The paper solves this problem by finding the optimal moment \u2014 or price level \u2014 at which it makes sense to pull the trigger. These price levels are called ",(0,m.yg)("em",{parentName:"p"},"critical boundaries"),", and the math uses martingale theory and optimal stopping to calculate them."),(0,m.yg)("p",null,"The result is a set of clear formulas:"),(0,m.yg)("ul",null,(0,m.yg)("li",{parentName:"ul"},"If the asset price is below a certain boundary, you wait."),(0,m.yg)("li",{parentName:"ul"},"If it's above, you exercise the outer option, claim the inner one, and then you're in the game.")),(0,m.yg)("p",null,"Each compound structure \u2014 whether it\u2019s a call-on-put, chooser, or put-on-call \u2014 has its own logic and boundary. But the idea is always the same: wait until it\u2019s worth unlocking the next layer."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Key difference from Panoptic:"),(0,m.yg)("br",{parentName:"p"}),"\n","While Panoptic relies on continuous fee accrual driven by path-dependent Uniswap activity, the Gapeev & Rodosthenous model assumes no trading activity and instead solves for closed-form optimal exercise boundaries in a frictionless market. Panoptic options are paid as-you-go and can be held indefinitely, while this compound model requires an explicit stopping decision at well-defined price levels. Unlike Panoptic\u2019s emergent pricing from market mechanics, this model is fully analytical and assumes complete market observability."),(0,m.yg)("h2",{id:"model-v-nonlinear-volatility-in-perpetual-american-put-options-grossinho-faghan--\u0161ev\u

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Model")),(0,m.yg)("ul",null,(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"S")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"S")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05764em"}},"S")))))," is the spot price of the underlying asset.  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1,"2"))))))))))),(0,m.yg)("span",{parentName:"span",style:{top:"-3.23em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"frac-line",style:{borderBottomWidth:"0.04em"}})),(0,m.yg)("span",{parentName:"span",style:{top:"-3.394em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"3em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size6 size3 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal mtight"},"d"),(0,m.yg)("span",{parentName:"span",className:"msupsub"},(0,m.yg)("span",{parentName:"span",className:"vlist-t"},(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.8913em"}},(0,m.yg)("span",{parentName:"span",style:{top:"-2.931em",marginRight:"0.0714em"}},(0,m.yg)("span",{parentName:"span",className:"pstrut",style:{height:"2.5em"}}),(0,m.yg)("span",{parentName:"span",className:"sizing reset-size3 size1 mtight"},(0,m.yg)("span",{parentName:"span",className:"mord mtight"},"2")))))))),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal mtight",style:{marginRight:"0.22222em"}},"V"))))),(0,m.yg)("span",{parentName:"span",className:"vlist-s"},"\u200b")),(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.345em"}},(0,m.yg)("span",{parentName:"span"}))))),(0,m.yg)("span",{parentName:"span",className:"mclose nulldelimiter"})))))),"."),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"\u03c1")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\rho")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal"},"\u03c1")))))," is the feedback parameter that scales how volatility responds to the option\u2019s convexity (gamma)."),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"r")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"r")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.02778em"}},"r")))))," is the risk-free interest rate. "),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"E")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"E")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05764em"}}

1,"E")))))," is the strike price of the put option.  "),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("span",{parentName:"li",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("msub",{parentName:"mrow"},(0,m.yg)("mi",{parentName:"msub"},"\u03c3"),(0,m.yg)("mn",{parentName:"msub"},"0"))),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\sigma_0")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,m.yg)("span",{parentName:"span",className:"mord"},(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.03588em"}},"\u03c3"),(0,m.yg)("span",{parentName:"span",className:"msupsub"},(0,m.yg)("span",{parentName:"span",className:"vlist-t vlist-t2"},(0,m.yg)("span",{parentName:"span",className:"vlist-r"},(0,m.yg)("span",{parentName:"span",className:"vlist",style:{height:"0.3011em"}}

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This might reflect increased trading friction, hedging demand, or liquidity sensitivity."),(0,m.yg)("p",null,"So instead of assuming fixed volatility, the model lets it grow when the option is \u201cmore dangerous.\u201d"),(0,m.yg)("p",null,"You still get a critical price ",(0,m.yg)("span",{parentName:"p",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"\u03c1")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\rho")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal"},"\u03c1"))))),": below it, you\u2019d exercise the put immediately. Above it, you wait. But now this boundary depends on more complex market dynamics."),(0,m.yg)("p",null,"If there\u2019s no feedback (i.e., ",(0,m.yg)("span",{parentName:"p",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"\u03bb"),(0,m.yg)("mo",{parentName:"mrow"},"="),(0,m.yg)("mn",{parentName:"mrow"},"0")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\lambda = 0")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal"},"\u03bb"),(0,m.yg)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.yg)("span",{parentName:"span",className:"mrel"},"="),(0,m.yg)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6444em"}}),(0,m.yg)("span",{parentName:"span",className:"mord"},"0"))))),"), the solution becomes the classical Merton perpetual put. But when feedback is present, the exercise region can shift wider or narrower depending on how reactive volatility is."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Why it matters:"),(0,m.yg)("br",{parentName:"p"}),"\n","This model adds realism by accounting for nonlinear effects. It is useful in illiquid markets or protocols with reflexive price behavior."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Relation to Other Models:"),(0,m.yg)("br",{parentName:"p"}),"\n","While Paradigm and Sidani use static volatility and external pricing, this model is internally dynamic. It doesn\u2019t require an oracle; but it also doesn\u2019t yet fit natively in DeFi. Still, it shows how feedback pricing could inspire future mechanism design in protocols."),(0,m.yg)("p",null,(0,m.yg)("strong",{parentName:"p"},"Important Question: Why is most perpetual option research focused on American-style options?")),(0,m.yg)("p",null,"Because American-style options are inherently tied to optimal stopping problems, they remain meaningful even when there is no expiry. European options with 

1no expiry don\u2019t make much sense. In fact, if you can only exercise at ",(0,m.yg)("span",{parentName:"p",className:"math math-inline"},(0,m.yg)("span",{parentName:"span",className:"katex"},(0,m.yg)("span",{parentName:"span",className:"katex-mathml"},(0,m.yg)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.yg)("semantics",{parentName:"math"},(0,m.yg)("mrow",{parentName:"semantics"},(0,m.yg)("mi",{parentName:"mrow"},"T"),(0,m.yg)("mo",{parentName:"mrow"},"="),(0,m.yg)("mi",{parentName:"mrow",mathvariant:"normal"},"\u221e")),(0,m.yg)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"T = \\infty")))),(0,m.yg)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.yg)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.13889em"}},"T"),(0,m.yg)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}}),(0,m.yg)("span",{parentName:"span",className:"mrel"},"="),(0,m.yg)("span",{parentName:"span",className:"mspace",style:{marginRight:"0.2778em"}})),(0,m.yg)("span",{parentName:"span",className:"base"},(0,m.yg)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.yg)("span",{parentName:"span",className:"mord"},"\u221e"))))),", you never actually receive a payoff.  "),(0,m.yg)("p",null,"Thus, a perpetual European option is often considered trivial or degenerate. American options, by contrast, give the holder the right to choose ",(0,m.yg)("em",{parentName:"p"},"when")," to exercise. When the expiry is removed, the problem becomes:"),(0,m.yg)("blockquote",null,(0,m.yg)("p",{parentName:"blockquote"},"\u201cWhat is the optimal price level at which to exercise, regardless of how long it takes to reach it?\u201d")),(0,m.yg)("p",null,"This setup leads to mathematically rich ",(0,m.yg)("em",{parentName:"p"},"free-boundary problems"),", which can be analyzed and solved using tools from ",(0,m.yg)("em",{parentName:"p"},"optimal stopping theory"),"."),(0,m.yg)("h2",{id:"summary-comparative-table"},"Summary Comparative Table"),(0,m.yg)("table",null,(0,m.yg)("thead",{parentName:"table"},(0,m.yg)("tr",{parentName:"thead"},(0,m.yg)("th",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"th"},"Feature")),(0,m.yg)("th",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"th"},"Panoptic")),(0,m.yg)("th",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"th"},"Paradigm")),(0,m.yg)("th",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"th"},"Sidani")),(0,m.yg)("th",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"th"},"Gapeev & Rodosthenous")),(0,m.yg)("th",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"th"},"Grossinho et al.")))),(0,m.yg)("tbody",{parentName:"table"},(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Core Idea")),(0,m.yg)("td",{parentName:"tr",align:null},"Streaming premium via theta accumulation over time on Uniswap v3 and v4"),(0,m.yg)("td",{parentName:"tr",align:null},"Funding-based perpetual option with Black-Scholes anchor"),(0,m.yg)("td",{parentName:"tr",align:null},"Probabilistic expiry: mixture of European options"),(0,m.yg)("td",{parentName:"tr",align:null},"Compound perpetual American options via optimal stopping"),(0,m.yg)("td",{parentName:"tr",align:null},"Perpetual American put with volatility driven by option gamma")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Expiry")),(0,m.yg)("td",{parentName:"tr",align:null},"None (truly perpetual)"),(0,m.yg)("td",{parentName:"tr",align:null},"None"),(0,m.yg)("td",{parentName:"tr",align:null},"Randomized (Exp or Geo)"),(0,m.yg)("td",{parentName:"tr",align:null},"None"),(0,m.yg)("td",{parentName:"tr",align:null},"None")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Pricing Method")),(0,m.yg)("td",{parentName:"tr",align:null},"Time-integrated theta, path-dependent"),(0,m.yg)("td",{parentName:"tr",align:null},"Theta as funding; price tracks oracle BSM value"),(0,m.yg)("td",{parentName:"tr",align:null},"Integral over BSM prices with time distribution weights"),(0,m.yg)("td",{parentName:"tr",align:null},"Two-layer optimal stopping; closed-form with GBM"),(0,m.yg)("td",{parentName:"tr",align:null}

1,"Free-boundary PDE with nonlinear volatility function")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Oracle Needed?")),(0,m.yg)("td",{parentName:"tr",align:null},"No"),(0,m.yg)("td",{parentName:"tr",align:null},"Yes (mark price)"),(0,m.yg)("td",{parentName:"tr",align:null},"Optional (only if BSM used)"),(0,m.yg)("td",{parentName:"tr",align:null},"No"),(0,m.yg)("td",{parentName:"tr",align:null},"No")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Hedging Model")),(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("a",{parentName:"td",href:"https://panoptic.xyz/research/options-market-making#market-making-in-panoptic"},"Delta hedging")," with perps, gamma hedging with options"),(0,m.yg)("td",{parentName:"tr",align:null},"Delta/gamma hedging via mark price and funding"),(0,m.yg)("td",{parentName:"tr",align:null},"Theoretical hedge depends on expiry weights"),(0,m.yg)("td",{parentName:"tr",align:null},"Requires boundary-aware, layered exercise logic"),(0,m.yg)("td",{parentName:"tr",align:null},"Volatility feedback complicates standard hedging")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Path Dependency")),(0,m.yg)("td",{parentName:"tr",align:null},"Yes (accumulates based on time spent near strike)"),(0,m.yg)("td",{parentName:"tr",align:null},"No (price snapshot)"),(0,m.yg)("td",{parentName:"tr",align:null},"No (distribution-driven)"),(0,m.yg)("td",{parentName:"tr",align:null},"No (exercise boundary is static)"),(0,m.yg)("td",{parentName:"tr",align:null},"No (but volatility is state-dependent)")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"On-Chain Compatibility")),(0,m.yg)("td",{parentName:"tr",align:null},"Fully native (built on AMM tick data)"),(0,m.yg)("td",{parentName:"tr",align:null},"Difficult; oracle and off-chain needed"),(0,m.yg)("td",{parentName:"tr",align:null},"Medium; numerical methods needed"),(0,m.yg)("td",{parentName:"tr",align:null},"Difficult; compound logic not natively modular"),(0,m.yg)("td",{parentName:"tr",align:null},"Challenging; requires nonlinear PDE solver")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Greeks View")),(0,m.yg)("td",{parentName:"tr",align:null},"Theta-based revenue, gamma bounded by LP range"),(0,m.yg)("td",{parentName:"tr",align:null},"Classic delta/gamma model; theta funds flow"),(0,m.yg)("td",{parentName:"tr",align:null},"Average Greeks across expiries"),(0,m.yg)("td",{parentName:"tr",align:null},"Multiple Greeks layered; sensitive to boundary rules"),(0,m.yg)("td",{parentName:"tr",align:null},"Volatility is a function of gamma; nonlinearity impacts all Greeks")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Innovation")),(0,m.yg)("td",{parentName:"tr",align:null},"Oracle-free, real-liquidity derived perpetual options"),(0,m.yg)("td",{parentName:"tr",align:null},"Replicates option decay via perps-style funding"),(0,m.yg)("td",{parentName:"tr",align:null},"Elegant link between perpetual options and vanilla blends"),(0,m.yg)("td",{parentName:"tr",align:null},"First closed-form perpetual compound American options"),(0,m.yg)("td",{parentName:"tr",align:null},"Introduces feedback-sensitive volatility into perpetual puts")),(0,m.yg)("tr",{parentName:"tbody"},(0,m.yg)("td",{parentName:"tr",align:null},(0,m.yg)("strong",{parentName:"td"},"Main Limitation")),(0,m.yg)("td",{parentName:"tr",align:null},"No closed-form price; requires simulation; high path variance"),(0,m.yg)("td",{parentName:"tr",align:null},"Dependent on oracle accuracy; off-chain components"),(0,m.yg)("td",{parentName:"tr",align:null},"Ignores real-time path; more theoretical than practical"),(0,m.yg)("td",{parentName:"tr",align:null},"Complex to implement and nest in composable DeFi"),(0,m.yg)("td",{parentName:"tr",align:null},"Hard to solve analytically; parameter calibration is complex")))),(0,m.yg)("h2",{id:"conclusion"},"Conclusion"),(0,m.yg)("p",null,"Perpetual options represent more than an innovation in pricing, they signal a shift in how 

1we design financial primitives. Without expiry, the boundaries of flexibility, composability, and on-chain autonomy expand. The challenge is no longer how to replicate Wall Street on-chain, but how to invent new forms of risk, flow, and strategy that are native to decentralized markets. Pricing becomes a question of incentive alignment, liquidity behavior, and protocol dynamics. As the ecosystem evolves, perpetual options may power the next generation of credit, leverage, insurance, and structured products; all live, modular, and governed by code. The future of derivatives is not just continuous. It is programmable."),(0,m.yg)("h2",{id:"references"},"References"),(0,m.yg)("ul",null,(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("p",{parentName:"li"},(0,m.yg)("strong",{parentName:"p"},"Panoptic Protocol Team"),(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("em",{parentName:"p"},"Panoptic Whitepaper v1.3.1"),". Published in 2023.",(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("a",{parentName:"p",href:"https://docs.panoptic.xyz/"},"https://docs.panoptic.xyz/"))),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("p",{parentName:"li"},(0,m.yg)("strong",{parentName:"p"},"Dan Robinson and Sam Bankman-Fried"),(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("em",{parentName:"p"},"Everlasting Options"),". Paradigm Research, 2021.",(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("a",{parentName:"p",href:"https://www.paradigm.xyz/2021/05/everlasting-options"},"https://www.paradigm.xyz/2021/05/everlasting-options"))),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("p",{parentName:"li"},(0,m.yg)("strong",{parentName:"p"},"Sidani, W."),(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("em",{parentName:"p"},"Pricing Perpetual Options Using a Distribution over Expiry"),".",(0,m.yg)("br",{parentName:"p"}),"\n","(Private draft/notes, circulated in DeFi research community), 2023.\n",(0,m.yg)("a",{parentName:"p",href:"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5065014"},"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5065014"))),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("p",{parentName:"li"},(0,m.yg)("strong",{parentName:"p"},"Gapeev, P. V., and Rodosthenous, N."),(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("em",{parentName:"p"},"Perpetual American Compound Option Valuation"),".",(0,m.yg)("br",{parentName:"p"}),"\n","Quantitative Finance, Volume 13, Issue 1, 2013, pp. 129\u2013139.",(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("a",{parentName:"p",href:"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1698264"},"https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1698264")))),(0,m.yg)("ul",null,(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("p",{parentName:"li"},(0,m.yg)("strong",{parentName:"p"},"Grossinho, M. R., Faghan, M., and \u0160ev\u010dovi\u010d, D."),(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("em",{parentName:"p"},"Pricing Perpetual Put Options by the Black\u2013Scholes Equation with a Nonlinear Volatility Function"),".",(0,m.yg)("br",{parentName:"p"}),"\n","arXiv preprint arXiv:1611.00885, 2017.",(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("a",{parentName:"p",href:"https://arxiv.org/abs/1611.00885"},"https://arxiv.org/abs/1611.00885"))),(0,m.yg)("li",{parentName:"ul"},(0,m.yg)("p",{parentName:"li"},(0,m.yg)("strong",{parentName:"p"},"Ekstr\xf6m, E., and Lu, B."),(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("em",{parentName:"p"},"Perpetual American Options with Stochastic Volatility"),".",(0,m.yg)("br",{parentName:"p"}),"\n","Mathematical Finance, Volume 31, Issue 2, 2021, pp. 757\u2013785.",(0,m.yg)("br",{parentName:"p"}),"\n",(0,m.yg)("a",{parentName:"p",href:"https://doi.org/10.1111/mafi.12271"},"https://doi.org/10.1111/mafi.12271")))),(0,m.yg)("p",null,(0,m.yg)("em",{parentName:"p"},"Join the growing community of Panoptimists and be the first to hear our latest updates by following us on our ",(0,m.yg)("a",{parentName:"em",href:"https://linktr.ee/panopticxyz"},"social media platforms"),". To learn more about Panoptic and all things DeFi options, check out our ",(0,m.yg)("a",{parentName:"em",href:"/docs/intro"},"docs")," and head to ",(0,m.yg)("a",{parentName:"em",href:"https://panoptic.xyz/"},"our website"),".")),(0,m.yg)("div",{className:"footnotes"},(0,m.yg)("hr",{parentName:"div"}),(0,m.yg)("ol",{parentName:"div"},(0,m.yg)("li",{parentName:"ol",id:"fn-1-009908"},"A ",(0,m.yg)("em",{parentName:"li"},"European option")," can be exercised only at expiry, while an ",(0,m.yg)("em",{parentName:"li"},"American option")," can be exercised at any time before or at expiry.",(0,m.yg)("a",{parentName:"li",href:"#fnref-1-009908",className:"footnote-backref"},"\u21a9")))))}y.isMDXComponent=!0}}]);