Problem 25.3 Easy Free 3 variations

Call Price from Put-Call Parity

A non-dividend-paying stock trades at $S_0=8$ . Zero-coupon bonds paying $1 at time $T=1$ are available, with continuously compounded interest rate $r=0.02$ . European puts on the stock, all expiring at $T=1$ , trade at the following (strike, price) pairs: $(5,0.4)$ , $(10,3.2)$ , $(15,5.6)$ . Using put-call parity, find the time- $0$ price of a European call with strike $K=10$ and the same expiry. Give your answer to two decimal places.

Work it out, then compare with the answer.

SolutionShow the worked solution

Key idea. A long call and a short put at the same strike $K$ have terminal payoff

\max(S_T-K,0)-\max(K-S_T,0)=S_T-K .

This is the payoff of one share minus $K$ zero-coupon bonds. By no arbitrage, the time- $0$ values agree:

C-P=S_0-Ke^{-rT}.

Pricing the call. Only the put with the same strike, $K=10$ , is needed; the $5$ and $15$ puts play no role. With $S_0=8$ , $P=3.2$ , $r=0.02$ and $T=1$ :

C=P+S_0-Ke^{-rT}=3.2+8-10e^{-0.02}.

Since $10e^{-0.02}\approx 9.8020$ ,

C\approx 11.2-9.8020=1.3980 .

Check. The call is out of the money ( $S_0<K$ ), so it should be cheaper than the put, which it is. It also satisfies the bound $C\ge \max(0,\,S_0-Ke^{-rT})=0$ .

\boxed{C = 11.2-10e^{-0.02}\approx 1.40}

Variations

The same idea with a twist. Each solution gives only the changed step.

Variation 25.3a

A European call struck at $120$ on a futures contract is priced at $3, and the underlying futures price is currently $110$ . Assuming the discount factor between now and expiry is $1$ , use put-call parity for futures options ( $C-P = F-K$ ) to find the price of the European put struck at $120$ with the same expiry.

Work it out, then compare with the answer.

Solution

For options on a futures contract with discount factor $1$ , parity reads $C-P=F-K$ . Solving for the put:

P=C-(F-K)=3-(110-120)=13.

The put is in the money by $10$ . On top of that intrinsic value it carries the same $\$3$ of time value as the call.

\boxed{P=13}

Variation 25.3b

Stock ABC trades at $62, with interest rates and dividends both zero, and all options below expire in one month. Define a straddle as $+1$ call and $+1$ put at the same strike, and a call (put) spread as $+1$ call (put) at the higher strike minus $1$ call (put) at the lower strike... actually use standard: $+1$ in-the-money option $-1$ out-of-the-money option, same type, same expiry.

The $60$ -strike straddle costs $23. Using put-call parity to split the straddle into its call and put legs, find the prices of the $60$ call and the $60$ put. Then, given that the $60/65$ call spread (long the $60$ call, short the $65$ call) is worth $2.2, find the price of the $65$ call.

Work it out, then compare with the answer.

Solution

With zero rates and dividends, parity gives $C_{60}-P_{60}=S_0-K=62-60=2$ . The straddle gives $C_{60}+P_{60}=23$ . Adding and subtracting these two equations:

C_{60}=\tfrac{23+2}{2}=12.5,\qquad P_{60}=\tfrac{23-2}{2}=10.5.

The call spread is long the $60$ call and short the $65$ call, so $C_{60}-C_{65}=2.2$ . Hence

C_{65}=12.5-2.2=10.3.

This is consistent with call prices decreasing in strike, since $C_{65}<C_{60}$ .

\boxed{C_{60}=12.5,\quad P_{60}=10.5,\quad C_{65}=10.3}

Variation 25.3c

A non-dividend-paying stock trades at $S_0=15$ . European calls expiring at the same date trade at the following (strike, price) pairs: $(10,8.1)$ , $(15,4.2)$ , $(20,1.4)$ . Using put-call parity on each strike, find the time- $0$ price of the $10/15/20$ put butterfly (long one put at $10$ , short two puts at $15$ , long one put at $20$ ).

Work it out, then compare with the answer.

Solution

Let $B$ be the discount factor to expiry. For each strike, parity gives $P_K=C_K-S_0+KB$ . Applying this to the butterfly weights $(1,-2,1)$ :

P_{10}-2P_{15}+P_{20}=\bigl(C_{10}-2C_{15}+C_{20}\bigr)-(1-2+1)S_0+(10-30+20)B .

Both the stock term and the bond term vanish. This is why no interest rate is needed. The remaining call combination is

C_{10}-2C_{15}+C_{20}=8.1-8.4+1.4=1.1 .

So the put butterfly costs the same as the call butterfly, as it must, since their payoffs are identical.

\boxed{1.1}

Spotted an error? Email us