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Testing a Call-and-Short-Stock Arbitrage with a Bond Position

Article Quant Q&A · Author: Jacob Mitch
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Summary

The discussion examines whether buying a European call and shorting a chosen number of shares can guarantee a profit at expiry. It sets up a portfolio containing the call, short stock, and a bond. The initial portfolio value is constrained to zero, then the portfolio’s terminal value is checked in both the up and down stock scenarios. Substituting the bond investment implied by the initial condition gives inequalities that can be used to find whether any share quantity satisfies both outcomes.

The replies also caution that a perceived option mispricing does not itself establish arbitrage: the volatility assumption used to value the call may differ from the market’s pricing assumptions. One answer separately questions the stated one-step binomial call value and calculates a different value using the given up and down stock prices and strike. The example is limited to a simple two-state setup; it does not provide enough information to determine a real-world trade’s profitability, including the risk-free rate and full market frictions.

Key ideas

  • A zero-cost arbitrage test must include the option, short shares, and a bond position.
  • The portfolio must have a nonnegative terminal value in every modeled stock outcome and a positive value in at least one.
  • The zero initial-value condition determines the bond investment as a function of the short-share quantity.
  • A disagreement about volatility assumptions can explain apparent option mispricing without creating arbitrage.
  • A one-step binomial valuation depends on the specified up and down prices, strike, and discounting assumptions.

Tags

OptionsArbitrageDerivatives pricing
Full text
# European Call option combined with Short selling


# European Call option combined with Short selling












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 Any guidance would be greatly appreciated. Thank you

## Answer by Valometrics.com (score 0, accepted)

https://quant.stackexchange.com/a/51144

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.

## Answer by MNic (score 0)

https://quant.stackexchange.com/a/51158

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.