Fair value of European Call

Question

Assume that the ABC stock pays no dividend and is currently priced at $S0 = \$10$.

Assume that, at the expiry time $T > 0$, the stock price goes up to $u*S0$ with probability $0 < p < 1$ and down to $d*S0$ with probability $1 − p$. We know that $d < 1 < u$ but do not know $d$ or $u$.

Assume that there is no arbitrage and the interest rate is zero.

Consider the following three options with the same expiry $T$ on the ABC stock. Assume that a European put option with strike price $\$9$ is priced at $\$ 14/9$ while a European put option with strike price $\$8$ is priced $\$ 8/9$.

What is the fair value of a European call option with a strike price of $\$7$? Explain your answer.

I'm having trouble solving this question. I have never dealt with such questions before and have no background in finance. I'd appreciate any help any one can give, thanks!

Answer 1

The payoff of a put option at expiry is $\max(0, K - S_T)$ where $K$ is the strike price and $S_T$ is the stock price at $T$. The present value is the expected payoff discounted by the risk-free rate. Since the risk free rate is assumed to be $0$ we can ignore the discounting and the put price for a given strike $K$ is

$$P(K) = p \max(0,K - 10u) + (1-p)\max(0,K - 10d)$$

The price $S_T = S_0 u = 10u > 10$ with probability $p$ and $S_T = 10d <10$ with probability $1-p$. Since the strikes are all below $10$ the term multiplied by $p$ is zero and the given option prices are

$$P(9) = (1-p)\max(0,9-10d) = 14/9 \\ P(8) = (1-p)\max(0,8 - 10d) = 8/9$$

Since the options have non-zero value we know that the payoffs are not zero and we can write

$$(1-p)(9-10d) = 14/9 \\ (1-p)(8 - 10d) = 8/9$$

Solve these linear equations to get $1-p = 2/3$ and $d = 2/3$.

The value of the put option with strike $K = 7$ is

$$P(7) = (1-p)(7 - 10d) = 2/9$$

The value of the call option with strike $K = 7$ by Put-Call Parity:

(Put-Call parity basically states that if you are long a Call and short a Put, you own the stock at the strike price) $$C(7) =P(7) + S_0 -K*e^{-rt}$$

Since interest rates are zero in this case:

$$C(7)=P(7)+S_0−K$$

$$C(7) =2/9 + 10 -7$$ $$C(7) =29/9$$

Answer 2

We know both of the put options must have the chance of finishing in the money if they have any value. Thus, it must be the case that $10d < 8.$ So we know the payoff of the put options are $0$ in the up state and $K-10d$ in the down states.

If we have a portfolio $aS+b$ of underlying and cash, then the value is $10au + b$ in the up state and $10ad+b$ in the down state. If we choose $a$ and $b$ to replicate the put with strike $9$ we have $$ 10au + b = 0\\10ad+b = 9-10d.$$ The value of the put must be the value of the replicating portfolio, so we have $$ 10a + b = 14/9.$$ Eliminating $a$ and $b$ from these equations will give a relationship between $u$ and $d.$

Then we can do the same for the put with strike 8 and write down $$ 10au + b = 0\\10ad+b = 8-10d \\ 10a+b = 8/9$$ and again eliminate $a$ and $b$ to get a relationship between $u$ and $d.$ Then we can combine with the previous relationship to find $u$ and $d.$ Doing this, I get $u=5/3$ and $d=2/3.$

Then we can replicate the call with strike $7.$ We know its value in the up state is $\frac{50}{3}-7 = \frac{29}{3}$ and its value in the down state is zero. So $a$ and $b$ for the replicating formula must obey $$ \frac{50}{3}a + b = \frac{29}{3}\\\frac{20}{3}a+b = 0$$ so you can solve for $a$ and $b.$ Then the price of the call is just the value of the replicating portfolio $10a +b.$