Black Scholes Merton Model to Price DeFi Options (Part -2): Analyzing the Pricing ‘Systems’
A Systems Review of Adjustments Made to Black Scholes Merton Model by Various DeFi Protocols
In part -1 of this research series, we tried to analyze the various problems associated with the Black Scholes Merton Model’s (BSMM) assumptions as well as certain associated fractals. Now we will analyze how such problems are being solved in practical environments. It must be noted that it’s not appropriate to study a ‘mathematical model/equation operating in an engine’ as an ‘isolated entity’. This equation/model gets impacted, as well as impacts other parts of the system, and hence a ‘systems overview’ is required to get a comprehensive evaluation. Through this article, we 0xlol and Akhil Vajjhala try to investigate the adjustments made to standard BSMM by DeFi Protocols to increase its accuracy in pricing option positions on the blockchain. You can connect with us on twitter at 0xlol and Akhil.
Summarizing Part — 1
Read part -1 here
BSMM is based on the concept of dynamic replication of a portfolio such that the price of a derivative should be equivalent to the cost of replicating this portfolio. The model theoretically estimates the price of European-style options on a stock/asset that doesn’t pay any dividends (during the lifetime of that option). It takes into account certain assumptions which makes it ‘kind of’ unfit to price on-chain options.
The model assumes that this portfolio can finance itself i.e. no transaction fee or taxes are involved, contrary to which in DeFi — there exists significant gas-fee, costs to borrow collateral, etc. Further we see that the concept of risk free interest rates (RFIR) isn’t properly formed in DeFi. Since BSMM assumes that position’s strike price is discounted for a constant RFIR, this can introduce certain errors. Also dividends are pretty common in DeFi, which is in contrast to model’s assumption that there are no dividends throughout the life-time of the option.
BSMM assumes a constant level of volatility throughout the life-time of the option. This is certainly not true as probability distribution of rate of returns (RORs) on major crypto assets (ETH and BTC) are skewed. This can result in certain estimation errors, especially when estimating for implied volatility (IV).
Although we have majorly focused on assumptions taken by the model, it must be noted that in a trading environment — factors like trade size, demand-supply dynamics, IV skew, etc. impact the option pricing. And hence, it becomes necessary to factor them to have a healthy pricing model.
Now, let’s have a look at how such assumptions and problems are being adjusted for when pricing options on the blockchain infrastructure.
In this section, we will review how different protocols have been using BSMM with some adjustments. BSMM can also be considered a ‘base’ model, on top of which certain manipulations are carried out to make it appropriate for the immediate environment. As researchers, we have criticized the arrangements below to draw attention towards certain flaws, but it shall not be construed as ‘FUD’. No matter to what extent such protocols have been successful in their quest, pricing options accurately on-chain isn’t an easy task. Therefore, these experiments should be considered as stepping stones towards a better future for DeFi derivatives.
Siren is an options trading platform that uses the BSMM to price its options. It uses an on-chain oracle instead of an off-chain oracle to retrieve the IV. SIREN’s automated market maker (AMM) includes features that create a simulation of market depth, including price impact. This feature ensures that liquidity is always available for other traders, as the price impact will increase with an increase in trade size.
Additionally, the price impact creates an “urgency premium”. If a trader wants to complete a very large trade all at once, it will cost them more (benefitting SIREN’s liquidity providers (LP) in the form of increased premium profit). This prevents market manipulation and is a safeguard to option writers. These factors together form the basis for SIREN’s option pricing.
However, Siren’s variant of BSMM doesn’t deviate much from the standard version. We’ve already demonstrated this is part -1 that this doesn’t work well under the realms of DeFi. There’s a need to drastically improve this mechanism to bring stability to the system, as well as to safeguard the LPs from the adversarial impacts of using standard BSMM.
Lyra is an options trading protocol that utilizes an automated market maker (AMM) to trade options with their collateral in the pool. The protocol’s pricing mechanism uses BSMM as a base model, and adjusts IV by taking into account the skew (through skew ratio or SR) and the trade size (by the concept of standard size or SS). SS is linearly related to number of contracts traded.
One of the major problems with on-chain options is IV arbitrage, which can be attributed to skew. So, how does Lyra solve this problem?
Lyra uses a variable called ‘trade volatility’ which prevents arbitrage in the protocol by neutralizing the gains one can earn through arbitrage. Let’s suppose an on-chain arbitrageur or searcher comes across a front-running opportunity. The ‘trade volatility’ dynamically sets price impact in such a way that even if someone buys a few positions in hope of selling them higher, there’s a high probability that they have to sell them for a lower price. The graphs below illustrate this dynamic.
Trade volatility is calculated as a product of baseline IV and SR (ratio of the ‘IV at a particular strike price’ to the ‘baseline IV’ with the same expiry) to adjust the IV at the strike of a particular option in the set.
The protocol initializes a ‘baseline IV’ for each set of options with the same expiry. Baseline IV is the IV of an option whose strike price is equal to the spot price at a particular moment in time. This IV serves as a baseline or reference to adjust or fine-tune IVs of other option positions in the set. Also, it is adjusted according to the number of SS traded. It is increased by 1% when 1 SS is bought by AMM (or sold by pool) and vice versa.
At different points in time, different option positions will be at their strike price, and hence baseline IV is dynamic.
It must be noted that to accommodate the price impact created by the position size, for every SS bought, SR is increased by a constant value (initialized as 0.0075). Hence the SR is directly proportional to the IV of the option at the strike price and the position size.
In this way, Lyra adjusts for skew as well as the trade size. In the end, after these adjustments, a base premium (W) is generated by BSMM.
On the top of this base premium (W), to compensate for liabilities and fat-tailed events, Lyra incorporates two additional (dynamic) fees on the premium apart from the gas and exchange fee. So, how do these fees compensate for liabilities and fat-tailed events?
As discussed earlier, the ‘intensity’ of fluctuations in IV (and therefore option prices) changes according to the contract’s proximity to its strike price, resulting in a volatility smile. This sensitivity of option prices to fluctuations in the price of the underlying asset is called vega. A high amount of vega indicates the presence of increased liabilities for the system (as options with higher vega represent unstable entities in the system), especially to pools consisting of option’s writers collateral, and premium.
And hence, every other trade which contributes to ‘net liability of the system’ or ‘vega risk’ or ‘net vega exposure relative to the size of the pool’ is charged a dynamic fee to disincentivize such liabilities, and vice versa. This variable fee results in an ‘asymmetric spread’ around the base premium (W) obtained earlier to make option positions expensive for agents inducting extra risk in the system. Another dynamic fee (variance fee) is to hedge the system against the ill-effects of fat-tailed events such as AMM losing out to impermanent loss (IL)etc.
In its current form, Lyra is integrated with the Synthetix pools. This means that when a trader purchases a CALL option from the AMM, the AMM purchases the base asset (for example sETH) from the Synthetix debt pool. LPs are typically left with long exposure to the underlying asset, and are exposed to IL. These risks will be hedged in the next release of Lyra, called “Avalon.” Cross-chain IL protection may also be employed to hedge IL risk exposure.
In spite of all these adjustments, there are other harmful factors in the system like toxic order flow. The usual practice is to increase the fee which disincentivizes such flow. But at the same time, it results in illiquidity for the system. Lyra AMM allocates a certain capital for insurance or ‘delta-hedging’ to protect LPs’ capital (from certain negative exposures) along with keeping the system liquid (as LPs would prefer to provide capital in a system where it’s safe) while keeping the prices ‘competitive’.
Lyra does a good job by deploying corrective measures for all the potential hazards resulting from flawed assumptions of BSMM as well as for other harmful factors. However, certain ‘rigid adjustments’ in IV (a constant change of 0.0075 per SS trade, for example) can be made dynamic.
Also, even though the factors like baseline IV and SR are initialized from the liquid markets, moving further in the time, only SS determines their value majorly. And hence, there’s a need to factor in the supply-demand dynamics from the liquid markets to make the price-discovery more accurate.
Premia is a DeFi protocol that is concerned with American-style options. Its pricing mechanism consists of 3 parts: the modified BSMM, current pool level (adjusted for the impact of position size), and a coefficient to update the price level depending on the market’s liquidity.
The protocol uses a modified version of standard BSMM by using a full volatility surface (put on chain w/a Chainlink Oracle) rather than using single, constant volatility. The oracle takes the traded IV from the most liquid options markets for each cryptocurrency (BTC, ETH, etc.) and interpolates the entire surface for each asset based on all known points of IV.
Volatility surface is then parameterized and put on-chain through Chainlink’s decentralized network of nodes. This allows anyone to query the oracle on-chain to get the specific IV for any option (strike, maturity, call/put).
This IV is plugged into the BSMM in Premia’s contracts to get the base premium.
The price of an option on Premia is directly proportional to the C-value of the pool in which it is traded. The C-value, which represents a ‘dynamic multiple’ to standard BSMM (as a base model), is regulated by demand and supply dynamics as well as position size in the pool. When liquidity is added to the pool, the C-value decreases, and when liquidity is removed from the pool, the C-value increases. Initially, the pool has a very high C-value in order to incentivize LPs, and penalize liquidity removers (option buyers). If there are no option buyers, then C-value adjusts itself accordingly to create demand for the contracts.
Along with factoring in demand, the pricing mechanism makes sure that liabilities associated with a large position size are taken care of. A trader cannot buy out the entire pool of options at the same premium, hence discouraging the ‘fragmented liquidity’ in the pool. Also, the C-value acts as a ‘discrete liquidity adjustment coefficient’ as it is correlated with liquidity in the pool.
Even though the mechanism is sophisticatedly designed to mitigate risks provided by inappropriate pricing through standard BSMM, it introduces certain inefficiencies and illusions. The very-first LPs in the pool are lured to get the maximum yield on their deposits, but the probability of option buyers buying expensive positions is pretty low. And if someone buys these options, it is harmful for them.
This mechanism also creates a problem of illiquidity for option holders as they can only buy, and exercise positions from, and in the pool, there’s no way they can sell them back (as it introduces bi-directional risk to LPs). Along with this, there will always be excess liquidity in the pool as a result of the pool’s design (as initially, high prices for the contracts won’t create a demand for the liquidity provided at the same time). Instead of creating a zero-sum game, there’s a need to compensate the initial LPs with ‘real-yield’ to establish a sustainable incentive structure.
Updates: Premia has recently launched on Optimism where:
— they have enabled the sell-feature
— users can take advantage of cross-chain arbitrage.
Decentralized Options Exchange (Dopex) on the other hand takes a different approach. They price their options through BSMM, but with unique differences between highly liquid and less liquid assets. For highly liquid assets, Dopex uses volatility smiles from IV oracles. The IV taken from the weighted average (weighted proportionately to the liquidity at various centralized exchanges like FTX, Deribit etc.) of these smiles is plugged into BSMM and the option premium is calculated. For less liquid assets, the IV is calculated as a 30-day rolling average of the realized volatility (actual volatility) of the asset, but the same BSMM is used for the premium as well.
“Unlike protocols making use of flat IVs and black-scholes pricing which usually results in incorrect pricing across strikes and expiries especially farther out of the money, dopex uses a combination of RV (realized volatility), a volatility smile replicating formula and “delegates” who quote on multipliers that influence the steepness/dampness of curves formed by the pricing formula. This allows for the dopex option chain to be far more realistic and fair in terms of pricing.
Delegates are incentivized to quote fair pricing to gain more liquidity and usage of the platform to receive higher incentives such as pool rewards and platform fees. They are also penalized incase of inactivity and/or malicious/incorrect reporting. So who are these delegates? To begin with we handpicked 5 of the largest derivative trading firms in the crypto space to be dopex delegates since they have the expertise, knowledge, data and resources to provide accurate fair pricing for all dopex option chains.” — TzTok-Chad (Dopex’s Founder on Twitter)
But what happens when any of these major exchanges or delegates face a problem? Surely they get penalized, but what about the LPs and option buyers on the other end? Can’t that result in major arbitrage and frontrunning? Also, unlike other major protocols, Dopex doesn’t actively price in the effects of large trades on premiums. Wouldn’t that result in a fragmented liquidity problem?
The protocol also plans to use a loss-rebate token (rDPX) for option writers. It’s ‘little complicated’ value-accrual mechanism is inspired from SNX, LUNA, OHM, DPX, and LQTY. Unfortunately, the price-action doesn’t substantiate the claims to the said mechanism. There’s definitely a need to make it more sustainable so that the writers are fairly incentivized.
Though pricing less liquid assets using ‘realized volatility’ sounds like a good approach, it exposes them to arbitrage through front-running during periods of high volatility.
Rysk is a protocol that uses options in order to provide LPs with returns that are diversified, adjusted for risk, and uncorrelated (diversified portfolio). The protocol does this by continually buying and selling option positions in order to achieve a delta-neutral state. Rysk claims to protect their writers from volatility risk in the market, and allow for option buyers to find cheaper options than on other on-chain exchanges.
Options that would make the returns of a liquidity pool more uncorrelated — will be made cheaper. It incentivizes buyers to purchase these options, and vice versa. Option buyers can use arbitrage and benefit from cheaper options because of Rysk’s pricing adjustments. LPs provide collateral for options to be minted. They invest in Rysk for the uncorrelated returns so as to protect their investment from excess volatility and sharp changes in value.
Rysk prices option positions through BSMM with its IV skewed manually (based on market data and pushed to contracts, later to be done by an oracle), which is then adjusted to incentivize options in a way that will provide uncorrelated returns to LPs.
As we don’t have much information on how this IV skew will be dealt with, therefore we can’t comment anything here. But going through description on their mechanics, it’s clear that the protocol will be selling as well as buying option positions very frequently. First of all this creates a huge ‘gas-fee’ problem which impacts the sustainability of their portfolio management. Also, as protocol is still in its development phase, and we don’t have much information on how do they plan to tackle the problem of fat-tailed risks, vega-risk, IV arbitrage, toxic order flows, adjustments for RFIR and dividends, etc.
Further, Rysk has embraced the idea of arbitrage in DeFi, and fits it within the exchange. However, a big problem arises in the manual skew of IVs. There isn’t an automated way to skew the IV, and it could be adjusted later than needed during periods of high volatility, once again opening the doors to front-runners and trading bots.
The Black Scholes Merton model, which is very sophisticated, does not seem very suitable to price on-chain options. In fact, valuation models like BSMM are not designed to ensure accurate price discovery; they are better suited for ‘valuation’ purposes. Using BSMM ‘as it is’ to price on-chain options can give rise to severe problems such as arbitrage through front running, mispricing due to volatility smiles and skews, and overpricing due to other apparent or non-apparent costs.
Although some of the protocols have been successful in mitigating these problems to an extent, they either willfully or hopelessly ignore other problems. For example, all on-chain option protocols take RFIR as 0, which can lead to major consequences, as described here. Similar minor estimation errors in volatility can also drastically impact model outputs. Furthermore, we don’t see any adjustments made to factor-in the impact of dividends or additional costs (gas fee, borrowing interest, fee to get margin, fee for insurance, fee to adjust liabilities, etc.), which impacts the market spread. When you have to adjust for every other assumption and its side effects, it results in disincentivization of liquidity.
When it comes to TVL, we see only Premia (an American-style options protocol) has been able to retain significant liquidity out of four functional option protocols using BSMM. This is surprising because BSMM was only originally created to price European-style options. Is this a comment on protocol’s pricing mechanism? Well we don’t have any evidences! However, it would be worth investigating the reasons behind this phenomenon.
Now, let’s talk about an issue which doesn’t really exist at a protocol’s level, but is definitely due to these protocols: the problem of toxic order flows. Toxic flow originates when a market participant can be said to have an edge or an unfair advantage in the market which isn’t factored in by the system(s). Due to different pricing/mathematical models deployed by each protocol, there’s definitely a possibility of someone figuring out the best possible inter-protocol strategy/model to carve out the best returns. Our previous discussions show how on-chain option protocols are green fields for such toxic order flows. To fix this issue, there needs to be greater synchronization among these protocols despite their differences.
Crypto’s volatility has a centralization problem. Because on-chain exchanges are highly illiquid, every protocol has to source its IV from off-chain centralized sources. This means that too much of the volatility is dependent on exchanges like Deribit. If there is a major issue with one of these exchanges, it could have a significant impact on the market. In addition, blockchain scientists should focus on creating ‘pure’ on-chain volatility. Ribbon Finance’s RVOL can be taken as one of the examples.
In conclusion, it is unclear whether all of the changes made to BSMM are sufficient or if more need to be done. If further adjustments are needed, is it worth making all of those changes? Wouldn’t it be better to find a solution that is more compatible with decentralized ledger infrastructure?
Research and Writing: 0xlol and Akhil Vajjhala
Art Works: Darshan (Illustration), 0xlol (Conceptualization)
- Individuals: Thogard
- Teams: Rysk Finance, Premia Finance, Siren, Dopex, Lyra Finance