Modeling Crypto Carry as Currency Interest or Commodity Convenience Yield

# Crypto currency or crypto asset


# Crypto currency or crypto asset












Under the risk-neutral measure the S&P500 price process can be written as $$ \frac{dS_t}{S_t} = (r_t-q_t) dt + \sigma_t dW_t $$ We can regard this as an exchange rate, where $S$ is the number of USD required to buy 1 S&P500, $r$ is the domestic (USD) interest rate and $q$ the foreign interest rate.

Indeed, one can imagine living in an economy/state where the currency is S&P500, where in this currency, assuming no default, the 'bank' pays interest according to $$ d\tilde S_t = q_t \tilde S_t dt $$

So here is my question: since cryptos like BTC and ETH pay no dividends, if they are regarded as currencies then the BTC/ETH interest rate is zero, right? And if the carry cost of crypto is substantial then surely it's due to it's commodity like nature and not its currency like nature?

And at the end of the day, does it even matter if we model the BTC price in USD as an USD asset or exchange rate?

I'm sure I'm releasing multiple brain farts above, but the currency/exchange rate vs asset distinction (or lack thereof) is starting to confuse me - thoughts?

## Answer by river_rat (score 2, accepted)

https://quant.stackexchange.com/a/84066

Ok lets consider the simplest model for this where the distinction matters. We have a stochastic convenience yield short rate process $$dq = \sigma_q dW^q_t$$ But we have to state under which measure we are working here. If we are treating our risky asset as a currency then we would typically be using the foreign risk neutral measure for that process (as we have a foreign bank account, bonds, swap, caps/floors and swaptions etc we need to be consistent with) and would need to measure change it to the domestic risk neutral measure. In this simple model we know what the foreign bond looks like in the foreign risk neutral measure $$dP^q(t,T) = qP^q(t,T)dt - \sigma_q(T-t)dW^{q,f}_t$$ Where we have added the $f$ superscript to keep track of which measure we are working with. We also know that under the domestic risk neutral measure $X_t\times P^q(t,T)$ must be a martingale where $X_t$ is the foreign exchange process between the two economies and satisfies the SDE $$dX_t = (r-q)X_tdt+\sigma_xX_tdW^{x,d}_t$$ Some ito formula chasing here and the martingale representation theorem gives us that $$dq = -\rho\sigma_x\sigma_qdt+\sigma_qdW^{q,d}_t$$ and $$d\left[ W^{x,d}_t,W^{q,d}_t\right] = \rho dt$$ i.e. We get a convexity adjustment term when we go from the foreign risk neutral measure to the domestic risk neutral measure in the FX case. Now if we are in the commodity case then we don't have to worry about any of this bond induced convexity, instead we just do the normal change of measure from the $\mathbb{P}$ real world measure to the $\mathbb{Q}$ risk neutral measure and see that $$dq = \mu dt + \sigma_qdW_t^q$$ for some drift $\mu$ that depends on the market price of convenience yield risk. So we have an extra free parameter in this case we need to estimate and calibrate to.