FULL SYNTHETIC STRADDLE (ONE LOT)
The synthetic was priced at shares so there is no issue there. A short straddle is short one call, short one put, one strike. You can replicate a straddle two ways, both with shares:
Long 100 shares; short two calls at n-strike.
Short 100 shares; short two puts at n-strike.
The above shows parity as the shares cancel out.
Why trade the synthetic? (hypothetical) Say it's Sep and you're in a long term hold in shares and vol is high and/or you're concerned that the shares have gotten frothy. You are already long shares ($205 last) and you're intent on shorting the Jan 240C at 40-vol to effect the synthetic straddle.
The Jan 240C are $18 bid so you short two at $18 which results in shares from $205, short two of the $240 calls from 18. We calc the synthetic straddle premium by pricing what would be the intrinsic put value from 240.
Long from $205
Short 35 points from the strike
240 - 205 = 35
$36 prem rec'd from the calls
$35 + $36 = $71
A straddle is short a call, short a put. The two lot call prems represents the extrinsic premium in the combo... 18 in the call (all extrinsic) and 18 in the put. The full value of the ITM leg (put) is simply the value of the call + the diff between the shares and the stock price--this represents the intrinsic put prem. Remember that we have already priced (in) the extrinsic put prem (=call prem). The only difference between the put and the call is intrinsic value.
Straddle value (equivalence):
OTM prem = 18
ITM prem = OTM prem (18) + intrinsic value (35) = 53
18 + 53 = 71
You're short the synthetic straddle from $71 prem. You can be aggressive with your fills as you maintain a tax deferral on the trade. You're far less inclined to cover the shares now that your Jan terminal BE is $169 on shares. You're being paid to ride out any near-term vola and your terminal BE is 169 while you're neutral delta to the strike that you're short.
I'll get back tonight to discuss the actual half-straddle.