Static Replication

What does it all mean?

No-Arbitrage Pricing

No-arbitrage pricing is the concept that a financial instrument should be priced as the sum of components that replicate its economic payoff. Put-call parity is one example. Put differently, two instruments with identical cash flows in all future states of the world must be priced equally today.

Dynamic Replication

The classic approach to hedging a derivative involves maintaining an ever-changing position in the underlying asset, also called dynamic delta hedging. According to Black-Scholes theory, a stock option behaves like a weighted portfolio of risky stock and riskless zero-coupon bonds. The Black-Scholes options formula tells you how to calculate the portfolio weights. They depend on the stock price level, the dividend yield, the stock volatility, the riskless interest rate and the time to expiration.

Put differently, this means you can in principle own a portfolio of stock and riskless bonds, and achieve exactly the same returns as the option. To do so, you must continuously adjust the weights in your portfolio according to the formula as time passes and/or the stock price moves. This portfolio is called the dynamic replicating portfolio. Options traders ordinarily hedge options by shorting the dynamic replicating portfolio against a long position in the option to eliminate all the risk related to stock price movement.

There are difficulties with this hedging method. First, continuous weight adjustment is impossible, and so traders adjust at discrete intervals (e.g. every few minutes/hours). This causes small errors that compound over the life of the option, and result in replication whose accuracy increases with the frequency of hedging. Second, there are transaction costs associated with adjusting the portfolio weights which grow with the frequency of adjustment and can overwhelm the profit margin of the option. Traders have to compromise between accuracy and cost.

Static Replication

Static replication, on the other hand, refers to hedging a position without needing to alter or adjust components with the passage of time. A static hedge may involve setting up a portfolio of simple European options that is guaranteed to match the payout of the instrument to be hedged. By leveraging Frequent Batch Auctions, Elektro introduces a way for derivatives to be replicated statically within the protocol, by decomposing them into atomic instruments. This has huge implications for liquidity.

What does this enable?

• Elektro allows for internally consistent ‘per auction forward’ to settle ‘within auction’ options and to trade options against delta.

• Ability to trade collateral swaps, spot/forward swaps, forward forward swaps as standard conditional orders.

• Organic ability to accommodate exponential growth inherent in order book combinatorics, by enforcing linear ‘ring wise’ relationships by pair-wise replication at the building block level.

• Embedded put-call parity relationships providing for liquidity bootstrapping by allowing for a variety of potential counterparties for each option trade (in its most simple form, a buyer of a call, is not just matched with a seller of a call, but also can be potentially matched against a seller of a funded put).

• Binary options being among the atomic instruments onto which all payoffs can be decomposed into allows for direct solution to the hedging approximation required for digital risk in markets (all of them) without that standard building block. This, in turn, implies the ability to include all statically replicable structured products as simple order entries into the auctions as they subsequently get perfectly replicated within the protocol into atomic instruments (path dependence as a direction of future research has already been explored with intriguing potential consequences).

• Direct option funding equivalence relationships directly incorporated into matching engine; box spreads, calls + puts vs forwards; variance swaps as multi legged conditional orders, etc

• Variance swaps as multi legged conditional orders

• Market fungible VAR based trigger margin loans that allow for market discovery of risky funding curves; a new huge market that has traditionally been outside the purview of exchanges

Statically replicated structured products

• Discount certificates/reverse convertibles

• Risk/reward ratio certificates

• Variance swaps

• Outperformance certificates

• Bonus certificates

• Twin-win certificates

Example 1: Outperformance Certificates

Outperformance certificates enable investors to participate disproportionately in price advances in the underlying coin/pair if it rises higher than a specified threshold value.

Example 2: Bonus Certificates

Bonus certificates are participation that feature full upside participation and a conditional capital protection as long as the underlying coin/pair doesn’t cross a predefined threshold (barrier). Bonus certificates tend to perform well in both sideways and rising markets.

Example 3: Twin-win Certificates

Twin-win certificates generate a profit for the investor not only when the price of the underlying coin/pair goes up, but also if it declines to a certain extent.