# PUTCALLPARITY

PUT CALL PARITYThe Put Call Parity defines a price relationship between Call options, Put Options, and Underlying stock. The idea was first introduced by finance journalist Hans Stoll in 1969, in his paper ‘The Relationship Between Put and Call Prices’ in 1969. It is considered as one of the most important principle in Option Pricing (Hecht, 2019).There are two types of Options 1. XEO (European Options): the exercise of these options will only happen at the option’s expiry date.2. OEX (American Options): the exercise of these options can be at any time during their life.The Put Call Parity tends to work perfectly only with European options compared to the American Options (Hull, 2017).The formula applied to check the relationship between European put and call options with same strike prices and expiry isC + X  = P + SHere,C: Call PremiumP: Put Premiumr: risk free interest rate T: Time to maturity in terms of year.X: Strike price of Call and Put Option.S: Initial Price or Current Price of underlying. Put Call Parity Testing(XEO) European Put Call Parity                                 EUROPEAN PUT CALL PARITY OPTIONS ATM ITM OTM X 640 560 1190 PV(X) 617.2154797 540.0635448 1147.635033 C 627.7 706.5 140.35 LHS 1244.91548 1246.563545 1287.985033 S 1271.2 1271.2 1271.2 P 2.375 63.75 1.45 RHS 1273.575 1334.95 1272.65 DIFFERENCE -28.65952028 -88.38645524 15.33503261 OUTCOME DOES NOT HOLD TRUE DOES NOT HOLD TRUE HOLDS TRUE In the European Put Call Parity the options that hold true are the ones where the LHS is equal to the RHS, or they must be almost similar to each other with a small difference. In this scenario, from the above mentioned table we can conclude that OTM with X=1190 holds true. The other two figures were ITM with X=560 and ATM with X=640, since they have a slightly larger gap comparing to OTM. It does not hold true. This offers an arbitrage opportunity for the investors.(OEX) American Put Call Parity AMERICAN PUT CALL PARITY OPTIONS ATM ITM OTM C 636.8 716.1 143.45 D 2.16% 2.16% 2.16% PV(D) 0.020831022 0.020831022 0.020831022 X 640 1190 560 PV(X) 617.2154797 1147.635033 540.0635448 LHS 1254.036311 1863.755864 683.5343758 S 1271.2 1271.2 1271.2 P 2.375 64.85 1.475 RHS 1273.575 1336.05 1272.675 DIFFERENCE -19.53868925 527.7058636 -589.1406242 OUTCOME DOES NOT HOLD TRUE HOLDS TRUE DOES NOT HOLD TRUE In the American Options the Put Call Parity holds true when LHS is greater than or equal to RHS. As from the table above we can see that LHS < RHS when X=1190 when the option is in the money (ITM).   The Put Call Parity is denoted by the equation, P (So, T, X) = C (S0, T, X) – So + X (1+ r)-T Applying to the formula,1.45 = 140.35 – 1271.2 + 1147.6350332.375 < 16.785033As we can notice here that 16.785033 is greater, the call is over-priced. The investor is advised to sell his over-priced call in this situation and buy a put of the same value, as it becomes a perfect hedge. We could use an alternative equation i.e., S0 = C (So, T, X) – P (So, T, X) + X (1 + r) -T  Applying to the formula,1271.2 = 140.35 – 1.45 + 1147.6350331271.2 < 1286.535033When using the above formula, it indicates that the call, put and bonds are overpriced, so the investor should sell theses and buy the shares as they are underpriced. Therefore, to check arbitrage profits Put Call Parity variations can be used.Hecht, A., 2019.Options: The Concept Of Put-Call Parity. [online] The Balance. Available at: <https://www.thebalance.com/options-the-concept-of-put-call-parity-808888< [Accessed 12 April 2020].Hull, J., 2017.Fundamentals Of Futures And Options Markets. 10th ed. Harlow: Pearson Education Limited.