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Consider a one period binomial stock model with $S_0=4$, $S_1(H)=8$ and $S_1(T)=2$. The interest rate is $25$%. Let's say I buy a call option for $1.20$ with strike price $K=5$ which expires at time $1$.

I'm trying to figure out how to invest in the stock and money markets such that at time $1$ I have $1.50$ no matter how the stock behaves, that is to say my portfolio mimics what would happen if I had just taken my $1.20$ and instead of buying the call option put it straight into the money market. And I want to do this without spending any more of my own money than the $1.20$ I spent on the call option.

$$V_1=\Delta_0S_1+(S_1-5)^+-\frac{5}{4}4\Delta_0$$ $$V_1(H)=3\Delta_0+3$$ $$V_1(T)=-3\Delta_0.\;\;\;$$

Now I set these equations both equal to $1.50$ and obtain the solution $\Delta_0=-.5$. And if I'm interpreting this correctly that means I should short sell a half-a-share of stock and invest that money in the money market account. Is that correct?

Also $\Delta_0$ is over-determined by these equations, I only need one of them to find it, and if I wanted any other return than $1.50$ there'd be no solution, what is that meaning of that? Does that mean if I wanted any other return I'd need to borrow some money from the money market or invest more of my own money; thereby introducing a second variable?

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1 Answers1

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You already have a call option worth \$1.20. You are trying to convert it into a riskless portfolio by taking additional positions in money market and stock market without making any new investment. Suppose this requires position of $\Delta$ shares of stock. Then since stock price is \$4, you need to borrow $4\Delta$. This is the position in the money market. At $t=1$, your call option yields $(S_1-5)^+$. The stock position can be sold for $\Delta S_1$. The amount borrowed is returned with 25% interest so you repay $5\Delta$. Your portfolio value equals $V_1=(S_1-5)^+ + \Delta S_1 - 5\Delta$. Since you want your position to be riskless, you specify $V_1(H)=V_1(T)$ to get $(8-5)^+ + 8\Delta - 5\Delta =(2-5)^+ + 2\Delta - 5\Delta$ or $\Delta = -0.5$. This is the only equation to determine $\Delta$ for a riskless portfolio.

You however specified two equations: $V_1(H)=1.5$ and $V_1(L)=1.5$. They both work because (i) you assume option is correctly priced (at \$1.2) and (ii) a payoff of $1.5$ equals riskless rate. If your portfolio is riskless, you will get same payoff in both states.

If your option is correctly priced and you demanded return of 30% (or anything other than 25%) in high state, then this is more than riskfree rate so your portfolio cannot be risk-free. That means the return in the low state cannot also be 30% with the same portfolio.

Similarly, suppose you wanted return of 25% but your option was incorrectly priced at $1 then 25% return is partly the return because you got the option cheap and partly the economic return of less than 25%. So again your portfolio will not be riskless and you will get different numbers from the two equations.

In practice, the replicating portfolios you are setting up can be used to price option, which you already seem to know. If you didn't know the option price $C$, you will have: $V_1=(S_1-5)^+ + \Delta S_1 - 5\Delta$. For the portfolio to be riskless, you specify $V_1(H)=V_1(T)$ to get $\Delta = -0.5$. Substitute $\Delta=-0.5$ to determine $V_1(H)=V_1(L)=\$1.5$. Then you argue that this is a return of 25% on your investment of $C$ in buying a call option initially so $C=\$1.5/1.25=\$1.2$.

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