Put Call Parity 4d641r

Put Call Parity RAVICHANDRAN

Notation c:

European call option price

C:

American call option price

p:

European put option price

P:

American put option price

S0 :

Stock price today

ST :

K:

Strike price

Stock price at option maturity

T:

Life of option

D:

PV of dividends paid during life of option

s:

Volatility of stock price

r

Risk-free rate for maturity T with cont. comp.

2

Put-call parity

• Put-call parity is a financial relationship between the price of a put option and a call option. • The put-call parity is a concept related to European call and put options. • The put-call parity is an option pricing concept that requires the values of call and put options to be in equilibrium to prevent arbitrage.

How does the put-call parity work?  Prices of put options, call options, and their underlying stock are very closely related.  A change in the price of the underlying stock affects the price of both call and put options that are written on the stock.  The put-call parity defines this relationship.

How does the put-call parity work? • The put-call parity relationship is specific in a way that a combination of any 2 components yields the same profit or loss profile as the third instrument. • The put-call parity says that if all these three instruments are in equilibrium, then there is no opportunity for arbitrage.

• The relationship is derived from the fact that combinations of options can make portfolios that are equivalent to holding the stock through time T, • and that they must return exactly the same gain or loss or an arbitrage would be available to traders.

What is the implication of put-call parity for synthetic positions? • The concept of put-call parity is especially important when trading synthetic positions. • When there is a mispricing between an instrument and its synthetic position, the put-call parity implies that an options arbitrage opportunity exists.

Put Call Parity Explanation • The put-call parity is a representation of two portfolios that yield the same outcome. • call option + bond = put option + stock • The left side represents a portfolio consisting of a call option and a bond. (zero coupon bond) • The right side represents a portfolio consisting of a put option and a stock.

Put Call Parity Explanation • call option + bond = put option + stock • If the price of the underlying stock raises, the put option expires worthless, the stock gains value, the call option ends in money, and the bond earns riskfree rate. Both portfolios have equal value at the end. • Regardless of whether the price of the underlying stock grows or falls, both sides of the equation balance each other. • If a portfolio on one side of the equation was cheaper, we could purchase it and sell the portfolio on the other side and profit from a risk-free arbitrage.

What is the put-call parity formula? • The put-call parity can be expressed as follows: • c + Ke -rT = p + S0 • c – European Call option price • P – European Put option price • S0 - Stock price at time t0 • K – Strike Price • Ke –rT - Present Value of Strike Price

Put Call Parity Relationship Derivation Assumptions : 1. Non dividend paying stocks 2. Option Style : European 3. call and put options - same strike price K and the same time to maturity T.

Put Call Parity Relationship Derivation Portfolio A • Long one call • Investment of PV(K) for maturity at T (Ke -rT ) (zero coupon bond equivalent strike price) Portfolio C Long one put Long one unit of stock

Put Call Parity contd. • The initial cost of Portfolio A is the cost of the call plus the amount of the investment, which is c + Ke -rT • The initial cost of Portfolio C is the sum of the prices of the put and the stock, which is p + S0

At time T • If ST < K: • The call (right to buy) in Portfolio A is worthless, while the investment is worth K. • Total value of Portfolio A: K • The put in Portfolio C is worth K − ST and the stock is worth ST . • Total value of Portfolio C: K • (ie. K- ST+ ST)

At time T • If ST ≥ K: • The call in Portfolio A is worth ST − K and the investment is worth K. • Total value of Portfolio A: ST • The put in Portfolio C is worthless, while the stock is worth ST . • Total value of Portfolio C: ST • In other words : Both the Portfolios are worth max(ST,K)

Put- Call Parity Expression • The portfolios have identical values in all circumstances at time T . • Both portfolios have no interim cash flows since there are no dividends on the stock and the options cannot be exercised early as they are European. • Therefore, the initial cost of the two portfolios must also be the same. • We must have c + Ke -rT = p + S0

Summary

Uses of Put-Call Parity • One of the most well-known results in option pricing, put-call parity is also one of the most useful. • The first and most obvious use of the result is in the valuation problem. • Once we can price European calls on nondividend-paying assets, we can derive the prices of the corresponding put options

Uses of Put-Call Parity • Put-call parity can be used to check for arbitrage opportunities resulting from relative mispricing of calls and puts. • For example, if we find c + Ke -rT > p + S0, then the call is overvalued relative to the put. • We can buy Portfolio C, sell Portfolio A, and make an arbitrage profit. • Conversely, if we find c + Ke -rT < p + S0, the put is overvalued relative to the call. • Arbitrage profits can be made by selling Portfolio C and buying Portfolio A.

Uses of Put-Call Parity • Third, rearranging the put-call parity expression tells us how to create synthetic instruments from traded ones. • For example, since put-call parity tells us that p = c + Ke -rT − S0, • we can create a synthetic long put by • buying a call, investing PV(K), and shorting one unit of the underlying. • Similarly we can create few other synthetic instruments.

Uses of Put-Call Parity • Put-call parity may be used to judge relative sensitivity to parameter changes, i.e., the difference in the reactions of calls and puts to changes in parameter values • put-call parity, we have c - p = S0 - Ke –rT • the difference in the changes in call and put values caused by a parameter change must be the same as the change in the right-hand side of the above equation.

• For example, suppose S0 changes by $1. Denote the change this causes in call and put values by dc and dp, respectively • Then we must have dc – dp = 1 • That is, the change in call value is a dollar more than the change in put value.

Put Call Parity On European Options on Dividend-Paying Assets • Modifying the put-call parity arguments to allow for dividends is easy • The only difference that dividends create is that in Portfolio A, there will be an interim cash flow when the underlying pays a dividend. • There is no corresponding interim cash flow in C.

Portfolio A Long one call Investment of PV(K) for maturity at T -Ke –rT Investment of PV(D) for maturity on the dividend date De –rT • Portfolio C • Long one put • Long one unit of stock • • • •

• This changes the initial cost of Portfolio A to • c + Ke -rT + De –rT the initial cost of Portfolio C (p + S0 ) remains the same. • The portfolios have the same value at T • c + Ke -rT + De –rT = (p + S0 )

Put Call Parity - American Options on Non-Dividend-Paying Assets • When the options concerned are American in style, it does not suffice to compare the portfolio values at maturity alone since one or both options may be exercised prior to maturity. • It becomes impossible to derive a “parity” (i.e., exact) relationship between the prices of calls and puts • However, an inequality-based relationship can still be derived

Arbitrage – Put Call Parity - Illustration Stock price is $31, Exercise price is $30, Risk-free interest rate is 10% per annum, Price of a three-month European call option is $3 • Price of a 3-month European put option is $2.25. • • • •

Arbitrage – Put Call Parity - Illustration • • • • • • •

c + Ke -rT = p + S0 Portfolio A = c + Ke -rT Portfolio C = p + S0 c + Ke-rT = 3 + 30 * e- 0.1* 3/12 = $32.26 p + S0 = 2.25 + 31= $33.25 Portfolio C is overpriced relative to portfolio A. An arbitrageur can buy the securities in portfolio A and short the securities in portfolio C.

Arbitrage – Put Call Parity - Illustration • An arbitrageur can buy the securities in portfolio A and short the securities in portfolio C. • The strategy involves buying the call and shorting both the put and the stock, generating a positive cash flow of • - 3 + 2.25 + 31 = $30.25 up front. • When invested at the risk-free interest rate, this amount grows to (Future value of $30.25) • 30.25 * e0.10*.25 = $31.02 in three months.

Arbitrage – Put Call Parity - Illustration • If the stock price at expiration of the option is greater than $30, the call will be exercised. • If it is less than $30, the put will be exercised. In either case, the arbitrageur ends up buying one share for $30. • This share can be used to close out the short position. • The net profit is therefore • $31.02 - $30.00 = $1.02

Summary – Arbitrage Example • Arbitrage opportunities when put–call parity does not hold. Stock price = $31; interest rate = 10%; call price = $3. Both put and call have strike price of $30 and three months to maturity.

Put-Call Parity relationship • Put-Call parity says that: • p + S0 = c + Ke-rT + D (in the case of dividend paying stock)

Put-Call Parity relationship  This means that one can replicate a put option through a combination of the underlying stock, the call option, and a position in the risk-free asset.  p = –S0 + c + Ke-rT + De –rT  Alternatively, one can replicate a call option through a combination of the underlying stock, the put option, and a position in the risk-free asset.  c = S0 + p – Ke-rT – De –rT

Put-Call Parity relationship  Alternatively, one can replicate a risk-free position through a combination of the underlying stock, the put option, and the call option.  Ke-rT = S0 + p – c – De –rT

 Finally, one can replicate the stock through a combination of the call option, the put option, and a position in the risk-free asset.

S0 = c – p + Ke-rT + De –rT

1. p = –S0 + c + Ke-rT + De –rT 2. c = S0 + p – Ke-rT – De –rT 3. Ke-rT = S0 + p – c – De –rT 4. S0 = c – p + Ke-rT + De –rT A “+” sign can be viewed as a long position in an asset, and a “–” sign as a short or written position in an asset.

• Also note that the relationship is valid right now, at time 0, and thus at anytime prior to expiration, up to and including expiration. • At expiration, put-call parity becomes: • p T + ST = c T + K • where pT , ST , and CT are the payoffs at expiration (time T) of the put option, the stock, and the call option respectively.