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
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. $$ s_p = -0.5(50) + 0.5(10) = -25 + 10 $$ $$ s_p = -15 $$.