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You are given:

(i) The price of a stock is 43.00$

(ii) The continuously compounded risk-free interest rate is 5%

(iii) The stock pays a dividend of 1 three months from now

(iv) A 3-month European call option on the stock with strike 44 costs 1.90

You wish to create this stock synthetically, using a combination of 44-Strike Options expiring in 3 months and lending: Determine the amount of money you should lend.

So far I understand that using the Put-Call parity:

$C-P=S_0-PV(Dividend)-Ke^{-rt}$

and for a synthetic stock: $S_0=C-P+PV(Dividend)+Ke^{-rt}$

Hence:

$S=1.90-P+44e^{-0.0125}+e^{-0.0125}$

When I check the solution, it says that the amount to lend is

$45e^{-0.0125}=44.4410$

Where did this answer come from? How can I approach this type of problem?

Seraphim
  • 709

1 Answers1

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By Put-Call parity $$ \begin{align*} C(S, K, T)−P(S, K, T) &=S−PV(Divs)−K \,\mathrm{e}^{−rT}\\ \Longrightarrow\quad S &= C(S, K, T)−P(S, K, T) +\overbrace{K\, \mathrm{e}^{−rT}+PV(Divs)}^{\text{amount to lend}} \end{align*} $$ and then $$ \begin{align*} C(S, 44, 0.25)−P(S, 44, 0.25) &=S−PV(Divs)−K \,\mathrm{e}^{−0.05(0.25)}\\ \Longrightarrow\quad S &= C(S, 44, 0.25)−P(S, 44, 0.25)+\overbrace{K\, \mathrm{e}^{−0.0125}+PV(Divs)}^{\text{amount to lend}}\\ &=C(S, 44, 0.25)−P(S, 44, 0.25)+44\, \mathrm{e}^{−0.0125}+\mathrm{e}^{−0.0125}\\ &=C(S, 44, 0.25)−P(S, 44, 0.25)+45\, \mathrm{e}^{−0.0125} \end{align*} $$ So the amount to lend is $45\, \mathrm{e}^{−0.0125}=44.4410$.

alexjo
  • 14,976