Page 85 - DCOM510_FINANCIAL_DERIVATIVES

Financial Derivatives

Notes A major advance in option pricing was accomplished by Hans Stoll in 1969, when he used the no
arbitrage argument to link up the price of a call option and a put option. This principle is called
put-call parity. Let us understand the put-call parity in details.

6.1.2 Put-Call Parity

Put-call is nothing but a relationship that must exist between the prices of European put and call
options having same underlying assets, strike price and expiration date. That relationship is
derived by the help of arbitrage arguments. Put-call parity is a classic application of arbitrage-
based pricing – it does not instruct us on how to price either put or call options, but it gives us an
iron law linking the two prices. Hence, if call options can be somehow priced, then the price of
the put option is immediately known.
Since put-call parity is a canonical arbitrage argument, we will spell it out in detail here.

Suppose a person has one share of Reliance and buys a put option at ` 300 which can be exercised
T years in the future. In this case, the person faces no future downside risk below ` 300, since the
put option gives him the right to sell Reliance at ` 300. Suppose, in addition, the person sells a
call option on Reliance at ` 300. In this case, if the price goes above ` 300, the call holder will
exercise the call option and take away the share at ` 300. The sale of the call eliminates upside
risk above ` 300.
Hence, the following portfolio {one share, plus a put option at ` 300, minus a call option at ` 300}
risklessly obtains ` 300 on date T. This payoff is identical to a simple bond which yields ` 300 on
date T.
Suppose the interest rate in the economy is r, then this bond has the present price 300/ (1 + r) .
T
This is a situation to which the law of one price applies: we have two portfolios which yield the
identical payoff:
T
1. 300 / (1 + r) invested in a simple bond, which turns into ` 300 on date T for sure, and
2. A portfolio formed of S + P – C, which turns into ` 300 on date T for sure.

By the law of one price, if two portfolios yield the identical payoffs then they must cost the same.
T
Hence we get the formula: S + P – C = X/ (1 + r) , where S is the spot price, P is the put price, C
is the call price, X is the exercise price and T is the time to expiration.
If prices in any economy ever violate this formula, then risk less profits can be obtained by a
suitable combination of puts, calls and shares. In summary, put-call parity links up the price of
a call and the price of a put. If one is known, then we can infer the other.
The put-call parity states that the difference in price between a call-option and a put-option with
the same terms should equal the price of the underlying asset less the present discounted value
of the exercise price. A put and call option written on the same stock with the same exercise price
and maturity date must sustain, if there are to be no risk less arbitrage opportunities. This
condition is known as the ‘put-call parity’ (Kester and Backstrand, 1995). This relationship can be
expressed as follows:
V – V = P – X
c p a
where,
V = the price of a call option,
c
V = the price of a put option,
p
P = the price of the underlying asset,
a
X = present discounted value of the exercise price.

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