Financial Derivatives: Options and Futures Pricing Problems
Interest Rates, Forwards, and Futures Contracts
Alba, having worked all summer, would like to deposit her money to earn extra income. A bank offers her 5% compounded semiannually. A friend starting a new business offers her 4.92% compounded quarterly. Which option is better and why?
Solution: We compare the effective annual rates.
- Bank Offer: \((1 + \frac{0.05}{2})^2 – 1 = 5.0625\%\)
- Friend’s Offer: \((1 + \frac{0.0492}{4})^4 – 1 \approx 5.012\%\)
The first option is better because it yields a higher effective annual interest rate, even though the second option compounds more frequently.
Calculate the future value of €1,000 after 30 years with an interest rate of 10% using:
- Compound interest rates.
- Continuously compounded interest rates.
Which of the two options yields a larger amount, and why?
Solution:
- (a) Compound Interest: €1,000 * \((1 + 0.10)^{30}\) = €17,449.40
- (b) Continuously Compounded Interest: €1,000 * \(e^{(0.10 \cdot 30)}\) = €20,085.54
The continuously compounded option is larger because it represents the theoretical limit of compounding as the number of periods per year approaches infinity, resulting in constant reinvestment of interest.
You are taking a short position in a one-year futures contract for shares in XY Corp. The spot price of XY shares is €100 when you enter the contract. The one-year risk-free interest rate is 10% (continuously compounded). What is the corresponding futures price?
Solution: The futures price is €100 * \(e^{0.10 \cdot 1}\) = €110.52.
Suppose you enter into a 6-month forward contract on a stock when the stock price is €50 and the risk-free interest rate (with continuous compounding) is 5%. What is the forward price?
Solution: The forward price is €50 * \(e^{0.05 \cdot 0.5}\) = €51.27.
Consider a 10-month forward contract on a stock with a current price of €100. The risk-free rate is 7% per annum for all maturities. Dividends of €0.50 per share are expected after 3, 6, and 9 months. What is the forward price?
Solution: First, calculate the present value (PV) of the dividends:
PV = €0.50 * (\(e^{-0.07 \cdot 3/12} + e^{-0.07 \cdot 6/12} + e^{-0.07 \cdot 9/12}\)) ≈ €1.45.
The forward price is (€100 – €1.45) * \(e^{0.07 \cdot 10/12}\) = €104.47.Consider a 10-month forward contract on a stock with a current price of €100. The risk-free rate is 7% per annum, and the dividend yield is 3% per annum for all maturities. What is the forward price?
Solution: The forward price is €100 * \(e^{(0.07 – 0.03) \cdot 10/12}\) = €103.39.
For a 1-year futures contract, it costs €3 per unit to store the asset, with payment made at the end of the year. The spot price is €200 per unit, and the risk-free rate is 5% per annum. What is the forward price?
Solution: The present value of the storage cost is €3 * \(e^{-0.05}\). The forward price is (€200 + €3 * \(e^{-0.05}\)) * \(e^{0.05}\) = €213.25.
Consider a 1-year futures contract. The spot price is €200 per unit, the risk-free rate is 5% per annum, the convenience yield is 1% per annum, and the storage cost is 2% per annum. What is the forward price?
Solution: The forward price is €200 * \(e^{(0.05 + 0.02 – 0.01) \cdot 1}\) = €212.37.
A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is €40 and the risk-free rate is 10% per annum with continuous compounding.
- What are the forward price and the initial value of the forward contract?
- Six months later, the stock price is €45, and the risk-free rate is still 10%. What are the new forward price and the value of the original forward contract?
Solution:
- Initially, the forward price is €40 * \(e^{0.10 \cdot 1}\) = €44.21. The initial value of the contract is zero.
- After six months, the new forward price for a new 6-month contract is €45 * \(e^{0.10 \cdot 0.5}\) = €47.31. The value of the original contract is €45 – €44.21 * \(e^{-0.10 \cdot 0.5}\) = €2.95.
Suppose the 1-year interest rates in the UK and the EU are 1% and 2% per annum, respectively, and the spot exchange rate is 1.10 EUR per GBP. What is the 1-year forward exchange rate? (Assume EUR is the local currency).
Solution: The forward rate is 1.10 * \(e^{(0.02 – 0.01) \cdot 1}\) = 1.111 EUR per GBP.
Options Pricing: Black-Scholes & Binomial Models
Assume a stock price \(S_0\) = $100, a risk-free rate \(r\) = 1% (continuously compounded), and volatility \(\sigma\) = 20%. What is the Black-Scholes price for a European call option with a strike price \(K\) = $100 and time to maturity \(T\) = 1 year? What is the price of the corresponding put option?
Solution: Call price \(c\) ≈ $8.43, Put price \(p\) ≈ $7.44.
Under the Black-Scholes model, what is the price of a European call option with \(S_0\) = $100, \(K\) = $110, \(r\) = 5% (continuously compounded), \(\sigma\) = 30%, and \(T\) = 2 years? What is the price of the corresponding put option?
Solution: Call price \(c\) ≈ $16.99, Put price \(p\) ≈ $16.52.
Assume a non-dividend-paying stock with \(S_0\) = $100, \(r\) = 1% (continuously compounded), \(T\) = 1 year, and \(K\) = $100. If the corresponding call option price is $12.368, what is the implied volatility?
- 0.1 (10%)
- 0.2 (20%)
- 0.3 (30%)
Using the same data as in Exercise 1 (\(S_0\) = $100, \(K\) = $100, \(r\) = 1%, \(\sigma\) = 20%, \(T\) = 1), compute the call option price using a binomial model. Compare the results with the Black-Scholes price for one-step, two-step, and three-step models. What do you observe?
Solution:
- With one step, \(c\) ≈ $10.41
- With two steps, \(c\) ≈ $7.53
- With three steps, \(c\) ≈ $7.58
Observation: As the number of steps in the binomial model increases, the calculated option price converges toward the Black-Scholes price of $8.43.