In this article, we will discuss the valuation of bitcoin futures. We also examine properties that classify bitcoin futures as a commodity derivative and compute bitcoin asset pricing formulae.
Futures on bitcoin
Bitcoin Futures were introduced by the Chicago Mercantile Exchange (CME) in December 2017. Bitcoin futures were listed with the following specifications; the contract unit is five bitcoins, as defined by the CME CF Bitcoin Reference Rate (BRR) which is an average of spot prices quoted by different exchanges., the price quotation is U.S. dollars and cents per bitcoin, with trading hours from Sunday — Friday 6:00 p.m. — 5:00 p.m. (5:00 p.m. — 4:00 p.m/ CT)
Below is a graph representing the open interest and daily volume of CME Bitcoin Futures.
Cryptocurrencies are comparable to commodities as they have ‘ storage’ properties. Cold storage means storing the private crypto keys in an offline environment to shield from hacking. Cold storage options are usually a paper wallet that contains a pair of private/public keys which are generated offline securely, and a hardware wallet which is an electronic device.
Theory of storage
The main similarity between bitcoin and commodities is a convenience yield. The convenience yield is the benefit or premium associated with holding an underlying product or physical good, rather than the associated derivative security or contract. Geman and Vasicek (1999) discuss the crucial issue of storability in the valuation of commodity futures. This theory is supported by the ‘Theory of storage’ by Working (1949).
The ‘Theory of storage’ describes a vital feature of commodity markets:
- When available inventory levels high, the buyers of that commodity keep their supply levels to the minimum. Therefore, Futures prices tend to be in contango. Contango describes the normal situation in which the spot or cash price of a commodity is lower than the forward price. Therefore, f(t,T)>S(t)
2. When available inventory levels of the commodity are low, buyers of the commodity tend to stock up on the goods. Thus, Futures prices tend to be in backwardation. Backwardation describes a situation in which the spot or cash price of a commodity is higher than the forward price. Therefore, f(t,T)<S(t)
Bitcoins as commodities: Convenience yield and ownership yield
Bitcoin has been regulated since 2015 in the US by the Commodity Futures Trading Commission (CFTC), and the CFTC chairman also declared Ethereum a commodity in September 2019. We argue that holding physical bitcoin presents a benefit because of frictions in the purchase of spot bitcoin; hence, the concept of a convenience yield can be extended to bitcoin. The recent insurance contracts offered at the end of 2019 to the institutions proposing warehousing to their customers is further evidence of the storability of bitcoin.
The best way to explain this concept is to show an equity example.
For the case of equities, there are two types of spot-forward relationships. This is the non- dividend-paying stock and the dividend-paying stock³.
- Non-dividend-paying stock
In the case of a non-dividend-paying stock and assuming no-arbitrage, the spot price S(t) is related to the T forward price f T(t) by the relationship:
Where r denotes the continuously compounded rate prevailing at date t for maturity T. The proof will come from the application of the no-arbitrage assumption to a well-chosen portfolio.
2. Dividend-paying stock
We assume that the stock pays a continuous dividend at rate g, and intermediate cash flows exist: the owner of the stock bought at date t receives the dividend gS(t)dt over the time interval (t, t+dt). We also assume that these dividends get immediately reinvested in the purchase of an extra quantity gdt of the stock, which leads to a total growth of e^g(T-t) over the period in the quantity of stock S detained. We start with a number e-g(T-t) of shares of stock S at date t that will grow to one share at date T.
Constructing a riskless position built at date t allows us to conclude that the sum of cash flows at date T is zero, hence,
In relation to bitcoin, the spot-forward relationship explains the ‘benefit’ component priced by the market for possessing physical bitcoins.
The spot-forward relationship for a storable commodity can be represented in many forms.
We denote this convenience yield y (defined as benefit minus storage cost, y = b-c) and write the evolution of the spot-forward relationship for cryptocurrencies as
Where f(t, T) denotes the price at date t of the futures contract maturing at date T, S(t) is the spot price of the crypto, r is the short term rate, and y is the convenience yield. Both r and y are assumed constant over the period (t, T).
A spot-forward example
For example⁴, the forward curve observed on 22 April 2019 (Figure 1) when the spot was US$5,310. It was increasing; this indicates that investors were optimistic, possibly because it was the time when Libra, the cryptocurrency proposed by Facebook.
From the spot-forward relationship written for the maturity T = 30 September 2019, we have that T-t = 5/12, and we derive from f(t, T) = $5,400 that r-y = 0.68%, hence y = 1.32% for a short term rate r = 2%.
From the above graph, we can see that the curve is in contango.
In this article, we discussed bitcoin asset pricing properties and a fundamental tool in all traditional asset pricing and valuation of futures. We also covered Working (1949) ‘Theory of Storage’ and the forward curve variations. We applied the spot-forward relation to bitcoin, which accounts for conditions of storability using the convenience yield.
In the next part, we’ll explore the valuation of Bitcoin Options. Since the CME introduced options in January 2020, we have enough data to ascertain whether the Bitcoin options market will validate assumptions of the Black-Scholes-Merton model.
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This article was motivated by my postgraduate financial engineering dissertation supervisor Helyette Geman (who also supervised Nassim Taleb’s PhD). I was delighted to know she was actively working on valuing cryptocurrency derivatives.
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: H Geman and O Vasicek (1999) ‘Forwards and Futures on Non-Storable Commodities: the Case of Electricity’, RISK
: H Working (1949) ‘The Theory of the Price of Storage’, American Economic Review 39, 1254–1262
: Geman, H., 2009. Commodities and commodity derivatives: modeling and pricing for agriculturals, metals and energy. John Wiley & Sons.