Options Pricing: Put/Call Parity
Put/call parity is an options pricing concept first identified by economist Hans Stoll in his 1969 paper "The Relation Between Put and Call Prices." It defines the relationship that must exist between European put and call options with the same expiration and strike price (it does not apply to American style options because they can be exercised any time up to expiration). The principal states that the value of a call option, at one strike price, implies a fair value for the corresponding put and vice versa. The relationship arises from the fact that combinations of options can create positions that are identical to holding the underlying itself (a stock, for example). The option and stock positions must have the same return or an arbitrage opportunity would arise. Arbitrageurs would be able to make profitable trades, free of risk, until put/call parity returned.
Arbitrage is the opportunity to profit from price variances on one security in different markets. For example, arbitrage would exist if an investor could buy stock ABC in one market for $45 while simultaneously selling stock ABC in a different market for $50. The synchronized trades would offer the opportunity to profit with little to no risk.
Put options, call options and the underlying stock are related in that the combination of any two yields the same profit/loss profile as the remaining component. For example, to replicate the gain/loss features of a long stock position, an investor could simultaneously hold a long call and a short put (the call and put would have the same strike price and same expiration). Similarly, a short stock position could be replicated with a short call plus a long put and so on. If put/call parity did not exist, investors would be able to take advantage of arbitrage opportunities.
Options traders use put/call parity as a simple test for their European style options pricing models. If a pricing model results in put and call prices that do not satisfy put/call parity, it implies that an arbitrage opportunity exists and, in general, should be rejected as an unsound strategy.
There are several formulas to express put/call parity for European options. The following formula provides an example of a formula that can be used for nondividend paying securities:
c= S + p – Xe – r(Tt) p = c – S + Xe – r(Tt) 
Where c = call value S = current stock price p = put price X = exercise price e = Euler\'s constant (exponential function on a financial calculator equal to approximately 2.71828 r = continuously compounded risk free rate of interest T = Expiration date t = Current value date 
Many trading platforms that offer options analysis provide visual representations of put/call parity. Figure 7 shows an example of the relationship between a long stock/long put position (shown in red) and a long call (in blue) with the same expiration and strike price. The difference in the lines is the result of the assumed dividend that would be paid during the option's life. If no dividend was assumed, the lines would overlap.
Figure 7 An example of a put/call parity chart created with an analysis platform. 
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