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Options Pricing & The AMM

This page explains how the Lyra Protocol MMVs price options.

Options market making fundamentally comes down to finding a market-value for **implied volatility (IV),** the expected volatility of the asset from now until expiration. When implied volatility is high (low), the cost to buy (sell) an option **increases (decreases). **

The core mechanic underpinning the Lyra AMM is to increase implied volatility (and therefore the cost of options) when demand for options is high, and decrease implied volatility when supply is high. By doing so, the AMM can converge to a market-clearing value for IV for each strike and expiry.

We'll get into more details as to how:

- Implied volatility translates into the price of an option
- How IV varies across strikes and expiries
- The AMM manages the risks associated trading options

in the following sections.

Black Scholes

What is the Black-Scholes model?

The Black-Scholes model is the foundation of options pricing. It takes in the following 5 inputs:

- Asset price (S)
- Strike price (K)
- Risk-free interest rate (r)
- Time to expiry (T)
- Implied volatility (sigma)

and outputs the price of an option. Knowing the inputs to the model is a good enough starting point to begin to reason about options - but interested readers can read more about the model and its derivation here.

The important point to notice here is that, **of the 5 inputs to the model, 4 are widely known to all market participants**. The current price of the asset, the strike/expiry of the options contract, and the risk-free rate are all 'known' inputs to the equation.

The only input that is not observable is implied volatility. This means that when market participants are trading options, what they are *really *trading is *implied volatility. *Therefore, an options AMM must converge to a market-clearing value for IV, which is the approach that Lyra takes.

How Lyra uses Black-Scholes

The Lyra AMM quotes options over a range of strikes and expiries. It does this by generating a market-driven value for implied volatility which varies by expiry and by strike, known as *tradeVol. *An option's *tradeVol *is the product of its expiry's *baseIV *and its strike's *skewRatio. *

When a trade is made, the* *relevant *tradeVol* is fed into a Black-Scholes pricing equation (which takes in the other 4 parameters), yielding the option price. This mechanic is explained in more detail in the next section, and for the full picture, check out sections 4 and 4.6 of the whitepaper.

Volatility Smiles & Updates

Volatility Smiles

One of the incorrect assumptions that the Black-Scholes model makes is that the implied volatility is constant across all strikes within the same expiry. In practice, different strikes have different implied volatilities, and this difference is driven by varying supply/demand for options on each strike. This phenomenon is known as a volatility smile.

Lyra's AMM accounts for volatility smiles by including a strike-specific adjustment to the implied volatility parameter that is used to price options, called the *skewRatio*.

The final implied volatility value that is fed into the Black Scholes pricing model is a combination of the *baseIV *of the relevant expiry, and the *skewRatio *of the relevant strike. That is:

Volatility Updates

The *standardSize* parameter is defined as the number of contracts a trader would have to buy (sell) for the AMM to increase (decrease) the baseline IV level for an expiry by one percentage point. This is the mechanism that allows the AMM to converge to a market-clearing level for IV.

For each *standardSize* bought (sold) per strike/expiry combination, the AMM will increase (decrease) the ratio of that strike's volatility to the baseline IV by the *skewRatioImpact *parameter (a constant).

Dynamic Fees

Lyra's fee function is made up of three distinct components:

- 1.A flat fee based on the option price
- 2.A flat fee for exchanging costs
- 3.A dynamic fee based on the pool's vega risk
- 4.A dynamic fee based on the difference between the expiry's baseIV and its GWAV, as well as the traded strike's skew ratio and vega.

1) and 2) are relatively simple fees that yield the spread that the AMM charges for providing liquidity.

The third fee is how Lyra manages its options exposure risk. The **AMM calculates its net vega exposure relative to the size of the pool** (the full algorithm is detailed in the whitepaper section 5.2.1) which we call its *vega utilization*. This term is multiplied by a boolean parameter which is equal to 1 if the trade increases the net vega exposure of the pool, and 0 otherwise. This creates an asymmetric spread around the Black Scholes theoretical value of the option, where trades that increase the risk of the pool are disincentivized relative to trades that hedge the pool.

The fourth fee is called the **variance fee** and is designed to charge more when markets are volatile. The variance fee will be present in all trades and scales with the traded strike's vega, skew and the difference of the baseline volatility from its GWAV. In rapidly changing markets, the AMM runs the risk of losing out to impermanent loss. The variance fee adds positive expectancy to the AMM, thereby improving the possible outcome for LPs. For more detail, see the final section of https://leaps.lyra.finance/leaps/leap-18/.

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Black Scholes

What is the Black-Scholes model?

How Lyra uses Black-Scholes

Volatility Smiles & Updates

Volatility Smiles

Volatility Updates

Dynamic Fees