Question

1. As the CEO of Gelta Airlines, your compensation is tied to the performance of firm. You know that fuel price has a significant impact on the performance of the firm. To avoid operating losses from high fuel prices, you are considering 100 call contracts (1 contract = 100 call options) with exercise price of $30/barrel. Its price is either $25 or $50 at the end of the year. The risk-free rate is 10% and the current market price of the stock is $28.
   1. What is the replicating portfolio strategy? Specify the investments.
   2. What is the no arbitrage condition?
   3. What is the value of these call contracts?
   4. If the exercise price is $20, then what are the answers for parts a, b, and c?
   5. Considering a frictionless market, how do shareholders react on the disclosure of such contracts?

Answer 1

K = 30; S0 = 28; Su = 50; Sd = 25; r = 10%; t = 1; N = 100 call contracts = 100 x 100 = 10,000 options

Part (a)

Let the replicating portfolio comprise of borrowing an amount B at risk free rate for 1 year and buy C barrel today. Hence, the value of the replicating portfolio should be same as that of the call contracts.

In the up state:

Replicating portfolio value = Value of the call contracts

Hence, C x Su - B x (1 + r)t = Cu

Or, C x 50 - B x (1 + 10%)1 = max (Su - K, 0) x N = max (50 - 30, 0) x 10,000 = 200,000

Hence, 50C - 1.1B = 200,000 -----------Eqn (1)

In the down state:

Replicating portfolio value = Value of the call contracts

Hence, C x Sd - B x (1 + r)t = Cd

Or, C x 25 - B x (1 + 10%)1 = max (Su - K, 0) x N = max (25 - 30, 0) x 10,000 = 0

Hence, 25C - 1.1B = 0

Hence, 1.1B = 25C

From eqn (1): 50C - 1.1B = 50C - 25 C = 25C = 200,000

Hence, C = 200,000 / 25 = 8,000

and B = 25C/1.1 = 200,000 / 1.1 =  181,818.18

Hence, the replicating portfolio is to borrow $  181,818.18 for a year and buy (long) 8,000 barrles of oil today.

Part (b)

The no arbitrage condition is:

the value of the call contracts today = Value of the replicating portfolio today

Part (c)

Value of the contracts = C x S0 - B = 8,000 x 28 - 181,818.18 = $  42,181.82

Part (d)

If K = 20 then, we will have the following two equations:

In the up state:

Replicating portfolio value = Value of the call contracts

Hence, C x Su - B x (1 + r)t = Cu

Or, C x 50 - B x (1 + 10%)1 = max (Su - K, 0) x N = max (50 - 20, 0) x 10,000 = 300,000

Hence, 50C - 1.1B = 300,000 -----------Eqn (2)

In the down state:

Replicating portfolio value = Value of the call contracts

Hence, C x Sd - B x (1 + r)t = Cd

Or, C x 25 - B x (1 + 10%)1 = max (Su - K, 0) x N = max (25 - 20, 0) x 10,000 = 50,000

Hence, 25C - 1.1B = 50,000 ---------------- Eqn (3)

Hence, Eqn (2) - Eqn (3): 25C = 250,000

Hence, C = 250,000 / 25 = 10,000

and from eqn (3); B = (25C - 50,000)/1.1 = (25 x 10,000 - 50,000)/1.1 = 181,818.18

Hence, the replicating portfolio is to borrow $ 181,818.18 for a year and buy (long) 10,000 barrles of oil today.

And the value of the contracts = C x S0 - B = 10,000 x 28 - 181,818.18 = 98,181.82

Part (e)

Call options are safe ways to hedge. Call options bring unlimited upside with limited downside and hence is a very popular hedging tool. Shareholders sholuld react positively on disclosure of these contracts.