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If I have a stock is currently at $S_0$=$200, strike price 205 and risk free rate 0.02

The true value of this european call option I determined to be $11.8 per share (For the right to buy up to 1000 shares) expires in 1 year

If the call is misvalued and instead I buy the call option for $10 per share (for the right to buy up to 1000 shares) expires in 1 year

How would I calculate the abitrage profit from a combination of buying the $10 European call option and short selling X number of shares at t=0 and the coming out with a profit at expiry no matter what happens.

portfolio at time 0 =Call Option$_0$-X$S_0$

Any guidance would be greatly appreciated. Thank you

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Even if you are sure that the option is misvalued, you can't say that there is arbitrage gain. Why?? because the volatilty you used to price your option is almost sure different from the one used from the other side.

So if you are sure of your volatility, you can buy a variance or a volatility swap/option to make gain of your information.

Regarding your educational case, this three conditions should be satisfied ($P$ is the value of your portfolio):

$$P_0=10-X*200+B=0$$ (B is the amount invested in 1y bond) $$P_T(240)=35-X*240+B(1+r)>0$$ (r is the risk free rate and $P_T$ the portfolio value at maturity if the stock goes up to 240). $$P_T(180)=0-X*180+B(1+r)>0$$ the first equations gives: $$B=200X-10$$ you replace B by its value in the second and the third equations and you will get the condition for your X.

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  • $\begingroup$ in my calculation to calculate 11.8 the stock has an equal probability of going to 240 or 180 after a year So I used the one-step binomial model to calculate the 11.8. I'm just confused because the problem then says to value the call at 10 instead and short sell X units of assets to get arbitrage profit. I'm not sure on what a combination would look like $\endgroup$ – Jacob Mitch Feb 13 at 19:04
  • $\begingroup$ I edited my answer to solve your problem! $\endgroup$ – Valometrics.com Feb 13 at 21:14
  • $\begingroup$ hi thanks for your help but I'm just trying to calculate the arbitrage profit if from combination of short selling So far I have buy call option for 10dollars and short x number of stocks, invest proceeds at 2%. If Stock goes to 180 I have to payback the loan and buy shorted shares = (-10-200x)(e^0.02) -180x if it goes up to 240 I have to payback the loan exercise the option and purchase 240x shares = -(10-200x)(e^0.02) -240x+35 and I should come out $13.75-10=3.75 arbitrage profit, would it be ok to just double check this for me $\endgroup$ – Jacob Mitch Feb 15 at 18:48
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You said you've used the one-step binomial method to calculate 11.8 for one share. I'm not sure if that's right, since the call option $C_{0}$ $$ C_{0} = \left(\frac{S^+X^--S^-X^+}{S^+-S^-}\right)e^{-rT}+\left(\frac{X^+-X^-}{S^+-S^-}\right)S_{0}. $$

In your case $S^+=240, S^-=180 \implies X^+=S^+-E=240-205, X^-=0.$ using these I have $13.75 per a share.

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  • $\begingroup$ hi thanks yes that's correct I made a mistake in the numbers :) would you happen to know how to find the profit (13.75-10) buying the call instead at 10 dollars and shorting x stock. $\endgroup$ – Jacob Mitch Feb 15 at 18:52

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