Pay-later options Pay-later options are options for which the buyer is not requiredto pay the premium up front ie at the time that the contract is entered into Atexpirationthe holder of a pay-later op

We cannot use Black-Scholes formula to price a pay-later option, as it assumes that the premium is paid upfront. Instead, we can use a binomial tree to price the option.

We divide the 10-month period into 10 one-month intervals. At each interval, the stock price can either go up or down. We assume that the up and down factors are given by:

u = e^(σ√(1/12)) = e^(0.2√(1/12)) = 1.0577 d = 1/u = 0.9443

The expected return at each interval is given by:

E(R) = e^(rΔt) = e^(0.1/12) = 1.0083

We can now construct a binomial tree as follows:

         S₀ = $12
          / \
     uS₀ = $12.69
        /   \
 u²S₀ = $13.47  dS₀ = $11.34
       /     \
u³S₀ = $14.31  d²S₀ = $10.68
     /         \
u⁴S₀ = $15.20  d³S₀ = $10.07
   /             \
u⁵S₀ = $16.14  d⁴S₀ = $9.50
  /               \
...                ...

At each node, we calculate the value of the option as the maximum of two values: the difference between the stock price and the strike price, or the expected value of the option at the next time step discounted at the risk-free rate. For example, at the node uS₀, the option value is:

max(0, $14 - $12.69) = $1.31

At the node S₀, the option value is:

max(0, $14 - $12) = $2

We can then work backwards from the final nodes to the initial node, calculating the expected value of the option at each node as the weighted average of the values at the two possible next nodes. For example, at the node u²S₀, the expected value of the option is:

E(V) = (0.5 × $0) + (0.5 × $1.31) = $0.655

Finally, the price of the option is the expected value of the option at the initial node, discounted at the risk-free rate. At the node S₀, the expected value of the option is:

E(V) = (0.5 × $0) + (0.5 × $2) = $1

Discounting this back to the present using the risk-free rate, we get:

C₀ = $1 / e^(0.1/12) = $0.994

Therefore, the price of a call option on CCC stock maturing in 10 months’ time with a strike price of $14 is approximately $0.994