Question

Find the initial value of the portfolio replicating a call option if proportional transaction costs are incurred whenever the underlying stock is sold. (No transaction costs apply when the stock is bought.) Compare this value with the case free of such costs. Assume that \(S(0)=X=100\) dollars, \(u=0.1, d=-0.1\) and \(r=0.05\), admitting transaction costs at \(c=2 \%\) (the seller receiving \(98 \%\) of the stock value).

Short answer

Without transaction costs: 9.49. With transaction costs: 11.76.

Step-by-step solution

- Identify given values

Extract the given values from the problem: - Initial stock price: \(S(0) = 100\)- Exercise price: \(X = 100\) - Up factor: \(u = 0.1\) - Down factor: \(d = -0.1\) - Risk-free rate: \(r = 0.05\) - Transaction cost on selling: \(c = 0.02\) (2%)

- Calculate the up and down state prices

Calculate the stock prices in the up and down states: \(S_u = S(0) \times (1 + u) = 100 \times 1.1 = 110\) \(S_d = S(0) \times (1 + d) = 100 \times 0.9 = 90\)

- Determine the option payoffs

Calculate the payoffs of the call option in both up and down states (call option's intrinsic value): \(C_u = \text{max}(S_u - X, 0) = \text{max}(110 - 100, 0) = 10\) \(C_d = \text{max}(S_d - X, 0) = \text{max}(90 - 100, 0) = 0\)

- Calculate the risk-neutral probabilities

Calculate the risk-neutral probability: \(p = \frac{e^{r \times \text{time step}} - d}{u - d}\) Substituting the values, \(p = \frac{e^{0.05 \times 1} - (-0.1)}{0.1 - (-0.1)} = \frac{e^{0.05} + 0.1}{0.2} \ p \approx \frac{1.0513 + 0.1}{0.2} \approx 5.757\)

- Calculate call option price without transaction cost

Calculate the price of the call option without transaction costs using the formula, \(C = e^{-r} [p \times C_u + (1 - p) \times C_d]\). Substitute the values to solve for \(C\): \(C = e^{-0.05} [5.757 \times 10 + (1 - 5.757) \times 0] \approx 9.49\)

- Adjust for transaction costs

Adjust for the transaction costs: since there is a 2% transaction cost, multiply the down state stock price by 0.98. \(S_d^{'} = 90 \times 0.98 = 88.2\) Recalculate: \(p^{'} = \frac{e^{0.05} - 88.2/110}{1 - 88.2/110} \ p^{'} = \frac{1.0513 - 0.8018}{1 - 0.8018} \approx 1.25\). Finally, recalculate: \(C^{'} = e^{-0.05} [1.25 \times 10 + (1-1.25) \times 0] \approx 11.76\)

- Compare values

Compare the call option prices: Without transaction costs: \(C \approx 9.49\) With transaction costs: \(C^{'} \approx 11.76\)

Key concepts

portfolio replication

Portfolio replication is useful when pricing options. It involves creating a portfolio with stocks and risk-free assets that mimics the option's payoffs.
To replicate a call option, you'd ensure that for every possible price of the underlying stock, the portfolio has the same value as the call option.
Involves a few ingredients:

• If stock price goes up, so should the portfolio.
• If stock goes down, adjust portfolio similarly.

In the exercise, portfolio replication is crucial to determine the initial value needed to match the call option's payoffs in up or down states.

transaction costs

Transaction costs are fees incurred when buying or selling assets.
These are particularly important in option pricing because they affect the replicating portfolio. In the given exercise, only selling the stock incurs a transaction cost of 2%.
This changes the effective down state price as sellers receive only 98% of the stock value.
Thus, transaction costs increase the replicating portfolio's value, making it costlier compared to the case without such costs.

risk-neutral probability

Risk-neutral probability helps in pricing derivatives by eliminating the need to consider risk preferences.
It is calculated using the up and down factors along with the risk-free rate as shown in the solution ewline
ewline The formula is: ewline
ewline \( p = \frac{e^{r \cdot \text{time step}} - d}{u - d} \)ewline
ewline Risk-neutral probabilities ensure the expected return on the stock matches the risk-free rate. They are crucial in determining the fair price of the option.

option payoffs

Option payoffs represent the gains or losses from the option. For a call option, the payoff depends on whether the stock price exceeds the exercise price.
In our exercise, if the stock price goes up to 110, the payoff is \(\max(110 -100, 0) = 10\).
If it goes down to 90, the payoff is \(\max(90 - 100, 0) = 0\).
Payoffs help in determining the value of the option in various states and using risk-neutral probabilities, we can derive the present value of these expected payoffs.