Chapter 3, Problem 15

[dstefan]

Chapter 3, Problem 15

Assume that interest rates are constant and equal to r. Show that, unless the price of a call option C with strike K and maturity T on a non-dividend paying asset with spot price S satisfies the inequality
S e^{-q T} - K e^{-r T} ~\leq~ C ~\leq~ S e^{-q T},
arbitrage opportunities arise.

Show that the value P of the corresponding put option must satisfy the following no--arbitrage condition:
K e^{-r T} - S e^{-q T} ~\leq~ P ~\leq~ K e^{-r T}


Solution:

One way to prove these bounds on the prices of European options is by using the Put--Call parity P + S e^{-q T} - C = K e^{-r T}.
For example, for the bounds on the price of the call, note that C ~=~ S e^{-q T} - K e^{-r T} ~+~ P. The payoff of the put at time T is max(K-S(T),0) which is less than the strike K. The value P of the put at time 0 cannot be more than K e^{-r T}, the present value at time 0 of K at time T. Also, the value P of the put option must be greater than 0. Thus,
0 ~\leq~ P ~\leq~ K e^{-r T} and therefore
S e^{-q T} - K e^{-r T} ~\leq~ S e^{-q T} - K e^{-r T} ~+~ P ~=~ C ~\leq~ S e^{-q T}

A more insightful way to prove these bounds would be by using the no-arbitrage Law of One Price.

Consider a portfolio made of a short position in one call option with strike K and maturity T and a long position in e^{-q T} units of the underlying asset. The value of at time 0 of this portfolio is
V(0) ~=~ S e^{-q T} - C.
If the dividends received on the long asset position are invested continuously in buying more shares of the underlying asset, the size of the asset position at time T will be 1 unit of the asset. Thus,
V(T) ~=~ S(T) - C(T) ~=~ S(T) - \max(S(T)-K,0) ~\leq~ K,
since, if S(T) > K, then V(T) = S(T) - (S(T)-K) = K, and, if S(T) \leq K, then V(T) = S(T) \leq K.

From the Generalized Law of One Price we conclude that
V(0) ~=~ S e^{-q T} - C ~\leq~ K e^{-r T},
and therefore S e^{-q T} - K e^{-r T} \leq C
All the other inequalities can be proved similarly.

[doug reich]

Is there an extra (implied) step to show how carry out the arbitrage?

I suppose if the Call price is too high, you would want to short it, and then sell short the dividend-discounted number of the underlying, sell one put option, and purchase a discount risk-free bond worth K at maturity.

Conversely, if the call price is too low, you would go long, and purchase one put, the dividend-discount number of the underlying, and short a discount bond worth K at maturity.

Then the initial investment is (plus or minus): C - Se^{-qT} - P + Ke^{-rT}... I can see that without knowing the value of the puts, you can't really compute the payoff.

[chrisd]

When making an arbitrage argument to prove the inequality about the calls, there's no need to use a put. For example, to prove the lower bound, buy a call, short e^-{qt} units of stock, and buy e^{-rt} risk free bonds. This will have initial value less than zero.

Then your payout is: max (S(T) - K, 0) - (S(T) - K)

If S(T) > K, your payout is 0, and if S(T) < K, your payout is K - S(T) . So you've got negative initial value, and all payouts are nonneggative, and some are positive, hence an arbitrage.

[doug reich]

I was stuck in the mindset of put-call parity, which is where my solution came from.

[chrisd]

Also, assuming your initial portfolio has negative value, you should be able to earn interest on that, guaranteeing a positive payout in all scenarios.