Question

Consider a stock which is currently selling at $4.5. The stock price will either go up to $5 + x with probability 0.5 or go down to $5 − x with probability 0.5 one period later. The one-period riskless rate of interest is 5%.
(a) What are the market prices of at-the-money call options that expire at the end of the period when x is set equal to $0.5, $1, $1.5, $2, and $2.5, respectively?

Answer 1

1. When x is set equal to $0.5 :

Current price of the stock : S is $4.5.

Stock price will either go up to $5 + x = $5 + 0.5 = $5.5 with probability p = 0.5.

Price of the stock at the upside, uS = $5.5

It will go down to $5 - x = $5 - 0.5 = $4.5 with probability, 1-p = 0.5.

Price of the stock at the downside, dS = $4.5

The one-period riskless rate of interest : r = 5%.

The call options are at the money, so the option strike price is equal to the current stock price, hence X = $4.5

Call option payoff at the upside : C+ = Max of (0, uS - X) = Max of (0, 5.5 - 4.5) = Max of (0, 1) = $1

Call option payoff at the downside : C- = Max of (0, dS - X) =  Max of (0, 4.5 - 4.5) = $0

As per the Binomial Option Pricing Model,

Market Price of a Call Option = (p x C+ ) + (1-p x C-) / 1 + r

= 0.5 x $1 + 0.5 x $0 / 1 + 0.05

= $0.5 + $0 / 1.05

= $0.48 (rounded off)

2. When x is set equal to $1 :

Stock price will either go up to $5 + x = $5 + 1 = $6, uS = $6

It will go down to $5 - x = $5 - 1 = $4, dS = $4

p = 1 - p = 0.5

r = 5%

strike price X = $4.5 (same as the current stock price for an at the money call option)

Call option payoff at the upside : C+ = Max of (0, uS - X) = Max of (0, 6 - 4.5) = Max of (0, 1.5) = $1.50

Call option payoff at the downside : C- = Max of (0, dS - X) = Max of (0, 4 - 4.5) = Max of (0, -0.5) = $0

Market Price of a Call Option = (p x C+ ) + (1-p x C-) / 1 + r

= (0.5 x $1.5) + (0.5 x $0) / 1 + 0.05

= $0.75 / 1.05

= $0.71 (rounded off)

3. When x is set equal to $1.5 :

Stock price will either go up to $5 + x = $5 + 1.5 = $6.5, uS = $6.5

It will go down to $5 - x = $5 - 1.5 = $3.5, dS = $3.5

p = 1 - p = 0.5

r = 5%

strike price X = $4.5

Call option payoff at the upside : C+ = Max of (0, uS - X) = Max of (0, 6.5 - 4.5) = $2

Call option payoff at the downside : C- = Max of (0, dS - X) = Max of (0, 3.5 - 4.5) = Max of (0, -1.0) = $0

Market Price of a Call Option = (p x C+ ) + (1-p x C-) / 1 + r

= (0.5 x $2) + (0.5 x $0) / 1 + 0.05

= $1 / 1.05

= $0.95 (rounded off)

4. When x is set equal to $2 :

Stock price will either go up to $5 + x = $5 + 2 = $7, uS = $7

It will go down to $5 - x = $5 - 2 = $3, dS = $3

p = 1 - p = 0.5

r = 5%

strike price X = $4.5

Call option payoff at the upside : C+ = Max of (0, uS - X) = Max of (0, 7 - 4.5) = $2.5

Call option payoff at the downside : C- = Max of (0, dS - X) = Max of (0, 3 - 4.5) = $0

Market Price of a Call Option = (p x C+ ) + (1-p x C-) / 1 + r

= (0.5 x $2.5) + (0.5 x $0) / 1 + 0.05

= $1.25 / 1.05

= $1.19 (rounded off)

5. When x is set equal to $2.5 :

Stock price will either go up to $5 + x = $5 + 2.5 = $7.5, uS = $7.5

It will go down to $5 - x = $5 - 2.5 = $2.5, dS = $2.5

p = 1 - p = 0.5

r = 5%

strike price X = $4.5

Call option payoff at the upside : C+ = Max of (0, uS - X) = Max of (0, 7.5 - 4.5) = $3

Call option payoff at the downside : C- = Max of (0, dS - X) = Max of (0, 2.5 - 4.5) = $0

Market Price of a Call Option = (p x C+ ) + (1-p x C-) / 1 + r

= (0.5 x $3) + (0.5 x $0) / 1 + 0.05

= $1.5 / 1.05

= $1.43 (rounded off)