1 Slide 34

Consider a European put option priced at $2.5 with strike price $22 and a European call option priced at $4.75 with strike price $30. What are the maximum loss to A, the writer of the put and B, the buyer of the call?

Solution. Let \(S_t\geq0\) be the stock price at maturity.

1. Assume that A sells the put to C. We consider two cases:

1. If \(S_t>22,\) then C does not exercise the option and A gains $2.5.
2. If \(S_t\leq22,\) then C exercises the option and A loses \[(22-S_t)-2.5=19.5-S_t\leq19.5.\] Hence the maximum loss for A is $19.5.

2. We consider two cases:

1. If \(S_t<30,\) then B does not exercise the option and loses $4.75.
2. If \(S_t\geq30,\) then B exercises the option and loses \[-[(S_t-30)-4.75]\leq4.75.\] Hence the maximum loss for B is $4.75.

2 Slide 40

At time \(t,\) consider a European call \(C_t\) and a European put \(P_t\) with the same underlying stock share \(S_t\) paying no dividend, the same strike price \(K\) and expiration date \(T.\) If the risk-free interest rate is \(r\) p.a. compounded annually, what is the corresponding put\(-\)call relationship?

Solution. We will show that the following equality holds: \[C_t+K\cdot(1+r)^{t-T}=P_t+S_t.\] We first show that there exists an arbitrage opportunity if \[C_t+K\cdot(1+r)^{t-T}<P_t+S_t,\] as follows:

1. At time \(t:\)

1. We short sell 1 share with price \(S_t,\) write 1 put option and buy 1 call option.
2. We put the leftover money \[K_1=S_t+P_t-C_t>K\cdot(1+r)^{t-T}\] into a saving account with interest rate is \(r\) p.a.

2. At time \(T:\) the saving account balance is \(K_1\cdot(1+r)^{T-t}>K.\)

1. If the put option is not exercised, we use the call option to buy back 1 share with price \(K\) for fulfilling the short sold share. The profit is \[K_1\cdot(1+r)^{T-t}-K>0.\]
2. If the put option is exercised, we receive 1 share with price \(K\) from this put, and use it to fulfill the short sold share (without using the call option). The profit is \[K_1\cdot(1+r)^{T-t}-K>0.\]

We next show that there exists an arbitrage opportunity if \[C_t+K\cdot(1+r)^{t-T}>P_t+S_t,\] as follows:

1. At time \(t:\)

1. We short sell a zero\(-\)coupon bond with face value \(K\) and earn \(K\cdot(1+r)^{t-T}.\)
2. We write 1 call option, buy 1 put option and 1 share.
3. We put the leftover money \[K_1=K\cdot(1+r)^{t-T}+C_t-P_t-S_t>0\] into a saving account with interest rate is \(r\) p.a.

2. At time \(T:\) the saving account balance is \(K_2=K_1\cdot(1+r)^{T-t}>0.\)

1. If the call option is not exercised, we use the put option to sell the share and earn \(K\) for paying the bond’s face value. The profit is \(K_2>0.\)
2. If the call option is exercised, we hand over the share and earn \(K\) for paying the bond’s face value. The profit is \(K_2>0.\)

Therefore, to avoid arbitrage, the proposed equality must hold, as desired.

3 Slide 47

At time \(t,\) consider a European call \(C_t\) and a European put \(P_t\) with the same underlying stock share \(S_t\) paying no dividend, the same strike price \(K\) and expiration date \(T.\) The risk-free interest rate is r compounded continuously. Show that:

1. \(C_t\geq S_t-K\cdot e^{-r(T-t)}.\)
2. \(P_t\geq K\cdot e^{-r(T-t)}-S_t.\)

Solution. We first prove the following lemma.

Lemma. The price of an option cannot be negative.

Proof. If the price of an option is \(\alpha<0,\) we buy 1 option and do not exercise it. We are ensured a guaranteed profit of \(-\alpha>0,\) so this is an arbitrage opportunity.

Back to the main problem. Note that the put\(-\)call parity is \[C_t+K\cdot e^{-r(T-t)}=P_t+S_t.\]

1. By the Lemma, \(P_t\geq0\) so \[C_t=P_t+S_t-K\cdot e^{-r(T-t)}\geq S_t-K\cdot e^{-r(T-t)}.\]
2. By the Lemma, \(C_t\geq0\) so \[P_t=C_t+K\cdot e^{-r(T-t)}-S_t\geq K\cdot e^{-r(T-t)}-S_t.\]

4 Slide 48

Given the current spot price $55 of an asset paying no dividend, we consider a European call option and a European put option with premiums $1.98 and $0.79, respectively on a common strike price K = $58 and having the same expiry time T. By setting the risk-free interest rate r = 3% p.a. compounded continuously, find T.

Solution. The put\(-\)call parity implies \[1.98+58\cdot e^{-3\%\cdot T}=0.79+55\Rightarrow T\approx2.499.\]

5 Slide 49

Suppose we have a quote for a 3-month European put option, with a strike price $60 of $1.25. The current stock price is $62 and the risk-free interest rate is 5% p.a. compounded annually. Owing to small trading in call options, there is no listing for the 3-month $60 call (a call option price with strike $60 expiring in 3 months). Suppose the stock does not pay any dividend. Find the price of the 3-month European call option.

Solution. The put\(-\)call parity implies \[C_t+60\cdot(1+5\%)^{-3/12}=1.25+62\Rightarrow C_t\approx3.977.\]