Financial Derivatives 600 words + tables+ excels sheet

PUT CALL PARITY

The 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 is

C + X = P + S

Here,

C: Call Premium

P: Put Premium

r: 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 + 1147.635033

2.375 < 16.785033

As 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 + 1147.635033

1271.2 < 1286.535033

When 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.