Put-Call Parity and Bid-Ask Spreads in Futures Arbitrage
Summary
The document considers whether a Euro Stoxx futures contract and matching European call and put options offer an arbitrage when only one of each contract may be traded. It introduces put-call parity and rearranges it to compare the call-minus-put price with the discounted futures-minus-strike value. A synthetic forward can be formed from a long call and short put, then compared with the futures contract to identify which side is relatively expensive.
The example suggests assuming a zero interest rate to simplify the comparison, but the original question does not provide the rate or time to maturity needed for a full parity calculation. Bid and ask prices also matter: executable trades must buy at the ask and sell at the bid, so midpoint comparisons can overstate an opportunity. The response explains the arbitrage direction in broad terms but does not calculate a definitive maximum profit from all quoted prices or discuss transaction costs and contract specifications.
Key ideas
- A long call combined with a short put can replicate a forward exposure at the strike.
- Put-call parity relates the synthetic forward price to the futures price and strike, adjusted for discounting.
- A trader should buy the relatively cheap side and sell the relatively rich side.
- Bid and ask quotes determine executable arbitrage prices and can eliminate apparent profits.
- The example lacks rate and maturity details for a definitive parity calculation.
Tags
Full text
# Determine the maximum arbitrage profit from the given contracts
# Determine the maximum arbitrage profit from the given contracts
I really have tough time trying to figure this out.
> An investor observes the following prices in the market: Euro-Stoxx-Future DEC 148.02-148.03; Euro-Stoxx-Future Call-Option DEC 148.00 1.13-1.15; Euro-Stoxx-Future Put-Option DEC 148.00 1.19-1.21. What is the maximum profit the investor can achieve with the right arbitrage strategy, when only trading one contract each?
I am assuming that all the options mentioned in the question are european and I think this has something to do with the put-call-parity: $$c+Ke^{-rT}=p+F_0e^{-rT}$$ relation. But from the task at hand we don't know the rate $r$ nor the Time to maturity $T$.
Another thing that came to mind was the reverse conversion; long call, short put and short future. Which would pay: $$-148-1.15+1.19+148.02=0.06$$ But I feel like walking on a thin ice here because I don't really/fully understand where am I making the profit in the reverse conversion trade.
So please, if anyone has time to give me any feedback with the problem, comment on the current solution and explain the reverse conversion trade (or provide some futher reading link etc.) I would very much appreciate it.
Thanks!
## Answer by 0xFEE1DEAD (score 1)
https://quant.stackexchange.com/a/44915
You're on the right track.
The idea is to buy (sell) a synthetic forward (short put + long call) and sell (buy) the future.
You can re-arrange the formula as c - p = (F0 - K) * e^(-rT)
For simplicity's sake, lets assume r = 0:
c - p = F0 - K
If everything were priced correctly, both sides of the equation would be equal. To figure out the max profit, you need to consider the bid-ask spreads. Then you sell the one that's rich and buy the one that's cheap for a risk-less profit.
Here are a few links:
https://www.investopedia.com/articles/optioninvestor/05/011905.asp https://www.investopedia.com/university/conversion-arbitrage/conversion-arbitrage3.aspShown 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.