# seminars

**Seminar 6**

*c = 8.4, p = 7.4.*

2. What is the price, under the Black-Scholes model, of a European call
option with S0 = 100, K = 110, r = 5% (continuously compounded) ,
σ = 30% and T = 2? What is the price of the corresponding put option?

*c=16.99, p=16.52* 3. Assume a non-paying dividend stock such that S0 = 100, r = 1% (continuously
compounded), T = 1 and K = 100. If the corresponding call
option price is 12.368, what is the corresponding implied volatility?

*• 0.3 (30%)*• 0.2 (20%) • 0.1 (10%)

4. Consider the same data as in Exercise 1. Compute the corresponding call
option price assuming a 1-step binomial model. Compare the result with
the corresponding Black-Scholes price. What do you observe? Do the
exercise again assuming a 2-steps binomial model and a 3-steps binomial
model.

*With one step, c = 10.41. With two steps c = 7.53, with 3 steps c = 7.58.*

**Seminar 2**1. Alba has been working all the summer. She would like to put her money
into a deposit to get some extra money. A bank pays her 5 % compounded
semiannually. One of her friends is starting a start up and needs
some money. Her friend offer her 4,92% compounded quarterly. Which
one is the best? Why? We have that
1 + 5%
2
2
− 1 = 5.0625% and1 + 4.92%
4
4
− 1 = 5.012%. The first is bigger, it pays more interest even
the second one pays more often.

2. Calculate the value of 1.000 euros at 30 years with an interest rate of a
10% using:
(a) Compound interest rates. 1.000 (1 + 10%)30 = 17.449, 40
(b) Continuously compound interest rates. 1.000 exp (10% · 30) = 20.085, 54
Which one of the two options is the biggest. Why? The continuous
compounded is the biggest. It is the limit of the compounded interest when
the number of payments goes to infinity. It has a continuous reinvestment.

3. You are taking a short position in a one-year futures contracts of shares
in XY Corp. The spot price of XY shares in the market is 100 euros when
you enter into the future contract. The one-year risk free interest rate
is 10%. What is the corresponding future price? (Assume interest are
continuously compounded) 100e
0.1 = 110.51

4. Suppose that you enter into a 6-month forward contract on a stock when
the stock price is 50 euros and the risk-free interest rate (with continuous
compounding) is 5%. What is the forward price? (Assume interest are
continuously compounded) 50e
0.025 = 51.27

5. Consider a 10-month forward contract on a stock when the stock price is
100 euros. We assume that the risk-free rate of interest is 7% per annum
for all maturities. We also assume that dividends of 0.5 euros per share are
expected after 3 months, 6 months and 9 months. What is the corresponding
forward price? (Assume interest are continuously compounded)The
PV of the dividend is 0.5(e
−0.07/4 + e
−0.07/2 + e
−0.07∗9/12) = 1.449. Then
the forward price is (100 − 1.449)e
0.07∗10/12 = 104.47.

6. Consider a 10-month forward contract on a stock when the stock price
is 100 euros. We assume that the risk-free rate of interest is 7% per
annum and the dividend yield is 3% per annum for all maturities. What
is the corresponding forward price? (Assume interest are continuously
compounded) The The forward price is (100)e
(0.07−0.03)∗10/12 = 103.39

7. Consider a 1-year futures contract, it costs 3 euros per unit to store the
asset, with the payment being made at the end of the year. Assume that
the spot price is 200 euros per unit and the risk-free rate is 5% per annum
for all maturities. What is the corresponding forward price? (Assume
interest are continuously compounded) (200 + 3e
−0.05)e
0.05 = 213.25

8. Consider a 1-year futures contract of Bren. Assume that the spot price is
200 euros per unit and the risk-free rate is 5% per annum, the convenience
yield is 1% per annum and the storage cost is a 2% per annum for all
maturities. What is the corresponding forward price? (Assume interest
are continuously compounded) 200e
(5%+2%−1%) = 208.16

9. A one-year long forward contract on a non-dividend-paying stock is entered
into when the stock price is 40 euros and the risk-free rate of interest is
10% per annum with continuous compounding.

• What are the forward price and the initial value of the forward contract?
The forward price is 40e
0.1 = 44.21 euros, the forward contract
is worth zero.

• Six months later, the price of the stock is 45 euros and the risk-free
interest rate is still 10%. What are the forward price and the value
of the forward contract? The delivery price is set at 44.21, the value
of the contract is 45 − 44.21e
−0.1∗0.5 = 2.95. The forward price is
45e
0.1∗0.5 = 47.31

10. Suppose that the 1-year interest rates in UK and the EU are 1% and
2% per annum, respectively, and that the spot change rate is 1,1 EUR per
GBP. What is the the 1-year forward price? ((Assume interest are continuously
compounded and consider the EUR as the local currency).1.1e
0.02−0.01 =
1.11