Arbitrage bounds on Option Prices

Assumptions

No transaction costs

All transactions occur at single price (i.e. no bid-

ask spread)

No taxes

No margin or short sale restrictions

Trade can occur in stock and option markets

instantaneously

Dividends are received on the ex-dividend day,

and ex-day stock price decline equals the

dividend amount

We restrict our discussion to stock options and

index options. Some of these results do not apply

to futures options & options on foreign exchange

Pricing Restriction for Call (Upper

bound) Proposition 1: C < = S (i.e. highest amount a call can sell

for is the current value of the underlying asset)

What if C > S (i.e. C – S > 0) ?

In such cases there would be arbitrage opportunities:

_______Today Before Expiry At Expiration___

(on exercise) ST > K ST

< K

__________________(American)_______________________

Sell Call + C Deliver stock - (ST – K)0

Buy Stock - S0 & receive K + ST + ST

________________________________________________

Pricing Restriction for Call (Lower

bound)

Proposition II: C > = max [ S0 – K e –rT, 0 ] European

& American option on a non-dividend paying stock.

The proposition implies that

If S0 < K e –rT, then C must sell for more than zero. This is

because that the worst that can happen is that it expires

worthless, it cannot be negative.

If S0 > K e –rT, then C must sell for more than S0 - K e –rT

What if C < [ S0 – K e –rT, 0 ]

(i.e. –C + [ S0 – K e –rT, 0 ] > 0) ?

Pricing Restriction for Call (Lower

bound)

Consider the following two portfolios:

Portfolio A: 1 European call option + cash equal to K e –rT

Portfolio B: 1 share

_______Today At

Expiration___

ST > K ST

< K

______________________________________________

___

Buy Call - C + (ST – K) 0

Lend - K e –rT + K + K

Sell Stock + S0 - ST - ST

______________________________________________

____

Some important insights

In the money call options on non dividend paying

stocks will always have some time value (except

on the expiration day)

An American call will never be exercised early

An investor receives a profit of “S – K” (or intrinsic

value) on exercising the call option early if it is in-

the-money

However, on selling the call option, he will realize at

least

S0 – K e –rT , which is greater than intrinsic value

Moreover, the call option provides the holder

insurance which will vanish if he exercises the call

option

Note that in ‘real world’, in-the-money

Pricing Restriction for Put (Upper

bound) Proposition III: P < = K (i.e. highest amount a put can

sell for is the strike price)

What if P > K (i.e. P – K > 0) ?

In such cases there would be arbitrage opportunities:

_______Today Before Expiry At Expiration___

(on exercise) ST > K ST

< K

__________________(American)_______________________

Sell Put + P Receive stock 0- (K - ST)

Lend - K & deliver K + K + Int + K + Int

________________________________________________

Pricing Restriction for Put (Lower

bound)

Proposition IV: P > = max [ K e –rT - S0 ,0 ] European &

P > = max [ K- S0 , 0 ] American option on a non-dividend paying stock.

The proposition implies that If S0 > K e –rT, then P must sell for more than zero. This is

because that the worst that can happen is that it expires worthless, it cannot be negative.

If S0 < K e –rT, then P must sell for less than S0 - K e –rT

since at expiry the payoff is going to be K e –rT – ST & thus P will trade lower than intrinsic value prior to expiry and move to K e –rT – ST at expiry

If S0 < K, then American puts must sell for an amount greater than or equal to its ontrinsic value, K - S

What if P < [K e –rT- S0 , 0 ]

(i.e. –P + [K e –rT- S0 , 0] > 0) ?

Pricing Restriction for Call (Lower

bound)

Consider the following two portfolios:

Portfolio C: 1 European put option +1 share

Portfolio D: amount of cash equal to K e –rT

_______Today At

Expiration___

ST > K ST

< K

______________________________________________

___

Buy Put - P 0 + (K - ST )

Borrow + K e –rT - K -

K

Buy Stock - S0 + ST + ST

______________________________________________

Some more Insights

An American Put on a non-dividend-paying stock

can never sell below its intrinsic value

It will always have zero or positive time value

American Puts can never sell for less than their

intrinsic value and can never rise in value as

expiration date nears

A European Put on a non-dividend-paying stock

can sell for less tan its intrinsic value

ITM European puts will frequently sell for less than

intrinsic value and will rise in value as time passes

Put-Call Parity

Consider the following two pportfolio:

Portfolio A: 1 European call option + cash equal to

K e –rT

Portfolio C: 1 European put option +1 share

Both the portfolios are worth max (ST , K)

Since European options cannot be exercised

early, both the portfolios must have equal values

today:

c + K e –rT = p + S0

If the above equation does not hold then arbitrage

opportunities exist