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If I have a stock, with shares are currently trading at 200 dollars per unit. In 1 year from now, it is expected that the shares rise to 250 dollars with probability 0.5, and fall to 190 dollars with probability 0.5. The annual risk-free interest rate is 0.03. With exercise price 210.

Suppose this call option is valued at 10 dollars (per unit of asset traded).

For the portfolio Π0 = C0 − λS0 made at time 0. Which consists of purchasing the European call option (for the right to buy up to 1000 company shares), and short selling λ units of asset, where λ is parameter that you are free to choose.

What is the portfolio payoff at the expiry time when the shares rise to 250 dollars.

So far I have that the payoff for the call value is (250-210)*100 but I'm not sure how to proceed

Jacob Mitch
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1 Answers1

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The expected payoff is $0.5*(250-210)*100$.

In terms of the short sale, there is $0.5$ chance that the short will lose $50$ dollars for every share sold short. There is also a $0.5$ chance that there will be a $10$ payoff for every share sold short. Let $s_p$ be the expected payoff for one share sold short. $$ s_p = -0.5(50) + 0.5(10) = -25 + 10 $$ $$ s_p = -15 $$.

Bob
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  • the interest rate is does not continue compounding thanks. Would you happen to know what I should do about the short selling λ assets because I assume I would lose money if the price has gone up to to 250 – Jacob Mitch Feb 12 '20 at 19:57
  • @JacobMitch I did not read the question correctly. I will update my answer. – Bob Feb 12 '20 at 20:07
  • @JacobMitch I hope I have given you enough so you can take it from here. Feel free to make a follow up comment. – Bob Feb 12 '20 at 20:11
  • thank you how are the 50 and 10 values calculated is it 250-200 and 200-190? to calculate the portfolio payoff at expire time would I do the 0.5∗(250−210)∗1000 (the value of the call option for 1000 shares) minus Sp( λ) I'm just curious if the call option being worth $10 per share come into the problem. Thank you very much for your help – Jacob Mitch Feb 12 '20 at 20:17
  • @JacobMitch You are right about the $50$ and essential right about the $10$. It is $190-200$. If I understand the problem correctly, you are shorting both the stock and a call. Therefore, you should consider the pay off from the option. – Bob Feb 12 '20 at 21:48
  • Would I be right in thinking the total pay off for the call option on 1000 shares and shorted λ shares if the price reaches 250 from 200 to be (-The initial cost of the call option + value of call option at expiry-losses from the short) = -($101000)+(1000(250-210))- λ(250-200) I know i'm shorting the stock but im not sure if im shorting the call as well – Jacob Mitch Feb 12 '20 at 22:11
  • Oh sorry to add, i'm not sure how the interest rate 0.03 would come into it at expiry does not compound continuously – Jacob Mitch Feb 12 '20 at 22:20
  • @JacobMitch The expected payoff on the call option is $0.51000(250-210)$. The strike price is $210$ and there is only a $0.5$ chance that it will reach $250$. – Bob Feb 12 '20 at 22:57
  • Interest rates don't affect payoffs. It's extraneous information if all you are asked for is a payoff. – Barry Smith Feb 12 '20 at 23:20
  • thank you very much for your help, that bit of information was for a different question. If the stock was guaranteed to go to 250, ignoring probability, was I correct with my short selling calculations. Thank you – Jacob Mitch Feb 13 '20 at 01:02