Put-Call Parity for Options on Spot and Futures
Summary
The note explains why put-call parity can appear to differ between formulas for options on spot and options on futures. For spot options, the call-minus-put relationship uses a discount factor applied to the forward-minus-strike amount. For options on futures, the parity relation can instead be written as call minus put equals futures price minus strike, without an additional discount factor in that expression.
The answer gives two intuitions. A futures position is marked to market, while cash held as margin can continue earning interest, so the futures price does not require the same discounting treatment as spot. It also relates the prices by noting that options on futures convert into futures at option expiry, and derives the undiscounted relation under assumptions about expiry timing and ignoring delivery-period mismatches. The note cautions that conventions differ by underlying: options on an equity index itself use the spot-option form.
Key ideas
- Put-call parity for options on spot includes discounting of the forward-minus-strike amount.
- Options on futures can use a parity relation without an additional discount factor on futures price or strike.
- Daily futures mark-to-market and interest earned on posted cash help explain the distinction.
- The derivation assumes compatible expiry timing and ignores delivery-period mismatches.
- Index options written on the index itself use the spot-option relation.
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Full text
# Why is put-call parity defined differently by CME and Wikipedia? # Why is put-call parity defined differently by CME and Wikipedia? In general, Wikipedia defines Put-Call parity as: ``` C - P = D(F - K) ---------------- C = call price P = put price F = *FORWARD* price K = strike ``` which can be re-written as: ``` C - P = S - D(K) ---------------- C = call price P = put price S = spot price K = strike ``` Why does CME define Put-Call parity differently as: ``` F - C + P - K = 0 which can be re-written as: C - P = F - K ---------------- C = call price P = put price F = *FUTURE* price K = strike ``` Why is there no discount factor (D) on F or K in the CME formula? ## Answer by uday (score 3, accepted) https://quant.stackexchange.com/a/47109 There are two ways to look at it, a mathematical way or an alternative, intuitive way. The alternative way can be to look at F as an alternative S with 0 interest rate discounting because we still have the cash (minus a small posted margin, and ignoring this) which earns the interest rate. So for the F’s value itself every day’s time value of money effect is zero and the daily mark-to-market makes a PNL transfer from the cash posted. More specifically , for S we need to use discounting to arrive at F price, but when F itself is the new spot, we don’t need further discount, as we don’t pay the value of F. In the traditional way, $(C_s - P_s) = D (F-K)$ is correct when both $C_s$ and $P_s$ are options on Spot. But in the case of CME options, the options are all options on futures. Let $C_f, P_f$ be options on futures. At option expiry, the option gets converted to a future not a spot , which has a discounting factor vs spot. Making some assumptions on the options expiry date (which in practice is on or before futures expiry date, and also ignoring a delivery period which causes a further mismatch between futures expiry and spot conversion): Similar to $S = DF$, one can write $C_s - P_s = D(C_f-P_f)$. So it becomes: $D(C_f-P_f) = D(F-K)$, or $C_f -P_f = F-K$. However, in the case Equity indices , usually options on the Index (e.g. on CBOE, or on Eurex) are more popular than options on futures. For these options the original wiki formula would apply .
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.