Put-Call Parity and Delta Hedging for Bitcoin Options
Summary
The document explains two related ideas for European options settled in bitcoin: put-call parity for identifying a conversion or reversion mispricing, and delta hedging to manage exposure to the underlying. For matching calls and puts, parity relates their price difference to the discounted difference between the forward price and strike. A deviation can suggest an arbitrage opportunity, subject to trading costs and other market frictions.
For a position in a single option, the hedge size depends on the option’s sensitivity to the underlying, measured by delta. The answer suggests offsetting that exposure with an opposing position in the underlying or a forward, with forward delta being close to spot delta when rates and dividends are negligible. This can make the portfolio locally delta-neutral, but delta depends on the chosen pricing model and only addresses small price moves; model error and larger moves leave risk, while long options retain convexity exposure.
Key ideas
- Put-call parity links the prices of matching calls and puts to the forward price, strike, and discount factor.
- A parity deviation may indicate arbitrage, after accounting for trading costs and other frictions.
- An option’s delta estimates how its value changes with the underlying price.
- An opposing underlying or forward position can offset delta for small price moves.
- Delta is model-dependent, so delta hedging does not eliminate all risk.
Tags
Full text
# Options conversion/reversion arbitrage
# Options conversion/reversion arbitrage
I'm trading bitcoin option and i'm trying to find arbitrage opportunity with a synthetic short/long and a long/short future position.
The options are europeans style and settled in BTC. The contracts are future inverse contracts. My equity and PNL are in bitcoin and I want to be hedge at any time so I am protect to change of price agains't USD.
After spotting the ATM call and put to buy/sell, I'm having difficulties to figure out what amount of bitcoin I should long/short to earn money with this arbitrage and keep the value of my portfolio.
So far I understand that there is an arbitrage opportunity if:
> Price of the call - Price of the put - Underlying price = 0
What is the relationship to the amount of BTC to buy to be hedge WITH the call/put prices and quantities ?
Looking forward hearing from you
Thank you !
cryptonerd
## Answer by StackG (score 2, accepted)
https://quant.stackexchange.com/a/57138
Put-Call Parity says that \begin{align} C - P = D(F-K) \end{align} where $C$ and $P$ are the prices of two options at the same strike, $D$ is the discount factor to expiry (probably very close to $1$ right now...), $K$ is the strike and $F$ is the forward price, which you're trading via futures.
This is a model independent result, if it doesn't hold then you have an arbitrage opportunity (up to factors like trading costs).
If you just trade a single call option with price $C$ and want to hedge the underlying, you need to calculate the option delta, ${\frac {\partial C} {\partial S}}$, which shows the sensitivity of the price to the underlying, and trade $-1$ times that quantity of underlying (or in your case, the forward-delta ${\frac {\partial C} {\partial F}}$ which should be very close in a no-dividend, no-rates world). This will leave you with a delta-neutral portfolio, so for small moves in underlying your PnL will be 0 (if you're long the option, large moves will benefit you as you're long gamma/convexity).
Delta is a model-dependent quantity, however, so this will not remove all your risk as we don't have a perfect model for the market. You need to make a choice, the simplest is Black-Scholes delta but there are many other choices too...Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.