Question #13019

The current price of a stock is $22, and at the end of one year its price will be either $27 or $17. The annual risk-free rate is 6.0%, based on daily compounding. A 1-year call option on the stock, with an exercise price of $22, is available. Based on the binominal model, what is the option's value

Expert's answer

Current price& $22.00& Price at end of year:

Exercise price& $22.00& High& $27.00

RF& 6.00%& Low& $17.00

&

Step 1.& Payoff range, stock:& $27.00 − $17.00 = $10.00

Step 2.& Payoff range, option:

& If stock is high:

Price − Exercise = 27 − 22 = $ 5.00

& If stock is low:

(Price − Exercise) or $0 = & 0.00

& Option range:

$5 − $0 = $ 5.00

Step 3.& Equalize the ranges to find the number of shares of stock:

& Option range/Stock range = $5/$10 = shares of stock = 0.5

Step 4.& The payoff from 0.5 shares of stock will be either:& $13.50& or& $ 8.50

& The payoff from the option will be either: 5.00& or 0.00

& The portfolio's payoff will be either:& $ 8.50& or& $ 8.50

& So the portfolio's payoff is riskless, $8.50 regardless of which choice materializes.

Step 5.& The present value of $8.50 at the daily compounded risk-free rate is:

& PV = $8.50/(1 + (0.06/365))

& 365

& = $8.005.

Step 6.& The option price is the cost of the stock purchased for the portfolio minus the PV of

the payoff:

V = 0.5($22) − $8.01 = $2.99

& PTS:& 1& DIF:& Medium& OBJ:& 8.2& NAT:& AACSB: C; P

TOP:& Option price based on binomial model MSC:& Problem

Exercise price& $22.00& High& $27.00

RF& 6.00%& Low& $17.00

&

Step 1.& Payoff range, stock:& $27.00 − $17.00 = $10.00

Step 2.& Payoff range, option:

& If stock is high:

Price − Exercise = 27 − 22 = $ 5.00

& If stock is low:

(Price − Exercise) or $0 = & 0.00

& Option range:

$5 − $0 = $ 5.00

Step 3.& Equalize the ranges to find the number of shares of stock:

& Option range/Stock range = $5/$10 = shares of stock = 0.5

Step 4.& The payoff from 0.5 shares of stock will be either:& $13.50& or& $ 8.50

& The payoff from the option will be either: 5.00& or 0.00

& The portfolio's payoff will be either:& $ 8.50& or& $ 8.50

& So the portfolio's payoff is riskless, $8.50 regardless of which choice materializes.

Step 5.& The present value of $8.50 at the daily compounded risk-free rate is:

& PV = $8.50/(1 + (0.06/365))

& 365

& = $8.005.

Step 6.& The option price is the cost of the stock purchased for the portfolio minus the PV of

the payoff:

V = 0.5($22) − $8.01 = $2.99

& PTS:& 1& DIF:& Medium& OBJ:& 8.2& NAT:& AACSB: C; P

TOP:& Option price based on binomial model MSC:& Problem

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