# Pricing of a call option in a one period binomial model

You are given a 5% call option worth \$2.66. The strike price, k is$41.00. S(0)=40, Sd= 35 (i.e the lower price of the stock at t=1) find Su (i.e the high price of the stock at t=1).

How would this be done? I know Cu= Su-\$41. C(0)=\$2.66 and I have something written in my notes that c(0)= B/(1+r) +ΔS(0), but I'm not sure what B is here. I assume to Find Su I need to first find Cu, but I'm not sure how to do this from the given information. Any help is appreciated.

## 1 Answer

I assume 5% is the interest rate. If we knew stock price in each state, formula for call option price in binomial model should be $$C_0 = \left(S_0 - \frac{S_d}{1+r}\right) \frac{S_u-K}{S_u-S_d}$$ Now just invert the formula to get $S_u$, given values of $C_0, S_0, S_d, K, r$.

Where did we get that formula for call option price in the first place? That's pretty important, so I'd recommend you to look into your notes more closely. In short, you want to form a portfolio of stocks and bonds that replicates the option payoff in the next period, i.e. find $a, b$ such that $$\begin{split} a S_u + (1+r)b &= S_u - K \\ a S_d + (1+r)b &= 0 \end{split}$$ (here, $a$ stands for how many stocks you buy, $b$ for how much you save into bonds)

If you solve this system, then the price of the option in current period must be just the value of this portfolio, $C_0 = a S_0 + b$ (otherwise one could make easy profits from the mispricing), and doing the algebra should yield the formula at the beginning.