1.1 Slide 52

Consider 1 unit of asset that pays a continuous dividend at rate \(\delta\) p.a. After \(T\) years, 1 unit of asset will grow to \(e^{\delta T}\) units of asset.

1. To have 1 unit asset at time \(T,\) how much unit of asset do we need to have at time \(t?\)
2. Assume an investor buys \(100,000\) stocks of XYZ company and holds them for \(3\) years. Each of the stocks held pays a continuous dividend yield of \(4\%\) p.a. and the investor reinvests all the dividends when they are paid. Calculate the additional number of shares the investor would have at the end of \(3\) years.

Solution.

1. If we have \(x\) units of asset at time \(t,\) the unit count of asset we have at time \(T\) is \[1=x\cdot e^{\delta(T-t)}\Rightarrow x=e^{\delta(t-T)}.\]
2. The share count owned by the investor in 3 years is \[100,000\cdot e^{4\%\cdot3}\approx112,749.685\approx112,749\] so the additional number of shares is \(112,749-100,000=12,749.\)

1.2 Slide 53

Consider a European call and a European put with the same underlying stock share \(S\) paying continuous dividend with rate \(\delta\) p.a., the same strike price \(K\) and expiration date \(T\) years. The risk-free interest rate is \(r\) p.a. compounded continuously. Prove the put-call parity \[C_t+K\cdot e^{-r(T-t)}=P_t+S_t\cdot e^{-\delta (T-t)}.\]

Solution. We first show that there exists an arbitrage opportunity if \[C_t+K\cdot e^{-r(T-t)}<P_t+S_t\cdot e^{-\delta(T-t)}\] as follows:

1. At time \(t:\)

1. We short sell \(e^{-\delta(T-t)}\) share with price \(S_t\cdot e^{-\delta(T-t)},\) write \(1\) put option and buy \(1\) call option.
2. We put the leftover money \[K_1=S_t\cdot e^{-\delta(T-t)}+P_t-C_t>K\cdot e^{-r(T-t)}\] into a saving account with interest rate \(r\) p.a.

2. At time \(T:\) the saving account balance is \(K_1\cdot e^{r(T-t)}>K.\)

1. If the put option is not exercised, we use the call option to buy back \(1\) share with price \(K\) for fulfilling the short sold share. The profit is \[K_1\cdot e^{r(T-t)}-K>0.\]
2. If the put option is exercised, we receive \(1\) share with price \(K\) from this put, and use it to fulfill the short sold share (without using the call option). The profit is \[K_1\cdot e^{r(T-t)}-K>0.\]

We next show that there exists an arbitrage opportunity if \[C_t+K\cdot e^{-r(T-t)}>P_t+S_t\cdot e^{-\delta(T-t)}\] as follows:

1. At time \(t:\)

1. We short sell \(1\) zero\(-\)coupon bond with face value \(K\) and earn \(K\cdot e^{-r(T-t)}.\)
2. We write \(1\) call option, buy \(1\) put option and \(e^{-\delta(T-t)}\) share.
3. We put the leftover money \[K_1=K\cdot e^{-r(T-t)}+C_t-P_t-S_t\cdot e^{-\delta(T-t)}>0\] into a saving account with interest rate \(r\) p.a.

2. At time \(T:\) the saving account balance is \(K_2=K_1\cdot e^{r(T-t)}>0.\)

1. If the call option is not exercised, we use the put option to sell the share and earn \(K\) for paying the bond’s face value. The profit is \(K_2>0.\)
2. If the call option is exercised, we hand over the share and earn \(K\) for paying the bond’s face value. The profit is \(K_2>0.\)

Therefore, to avoid arbitrage, the proposed equality must hold, as desired.

1.3 Slide 54

Suppose that a \(6-\)month European call option, with a strike price \(K=85,\) has a premium of $\(2.75.\) The underlying asset pays a continuous dividend with rate \(\delta=10\%\) p.a. The futures price for a \(6-\)month contract is worth $\(75\) and the risk-free rate \(r=5\%\) p.a. compounded continuously. Find the price of a \(6-\)month European put option written on the same underlying asset and with the same strike price.

Solution. The put-call parity implies \[2.75+85\cdot e^{-5\%\cdot6/12}=P_t+75\cdot e^{-10\%\cdot6/12}\Rightarrow P_t\approx14.31.\]

1.4 Slide 63

Given strictly positive interest rates, the best way to close out a long American call option position early (on a stock that pays no dividends) would be to

1. Exercise the call.
2. Sell the call.
3. Deliver the call.
4. Do none of the above.

Solution

b. When there is no dividend, there is never any reason to exercise an American call early. Instead, the option should be sold to another party.

2.1 Problem 2

Joe will receive $\(1,000,000\) in \(3\) months’ time in settlement of an account. He is concerned that the pound might strengthen in value against the dollar (more dollars to the pound) over the next \(3\) months. Today, \(£1=\)$\(1.8017.\) Joe could enter a forward contract or he could enter a call option (to buy pounds for dollars - a price agreed today). Describe carefully the advantages and disadvantages of entering a forward contract and a call option.

Solution.

The forward contract as well as the call option reduces uncertainty as the future exchange rate is fixed today in both cases.

A forward contract costs nothing at the inception to enter the contract. But a call option requires premium to be paid upfront.

A major disadvantage of a forward contract is that here is no limit to the losses from a forward contract. The potential unlimited loss which might arise from a forward contract, is restricted by having a call option.

The profit from a call option is reduced to the extent of the premium cost paid to buy the call option. In this respect, the profit would be higher from a forward contract.

2.2 Problem 5

Charlie buys a \(3-\)month put option (strike price \(£25\)) for \(£2.\)

1. Calculate the payoff and the profit if in \(3\) months’ time the value of the asset is (a) \(£28.2,\) (b) \(£24,\) (c) \(£22.99.\)
2. Draw a payoff & profit diagram for this put option.

o1 = Option(action = 'put', strike = 25, premium = 2)
(o1.payoff(28.2), o1.payoff(24), o1.payoff(22.99))

## (0, 1, 2.0100000000000016)

o1.pp_diagram()

2.3 Problem 11

Ben buys a call option (strike price $\(16.5\)) for $\(0.87\) and sells a call option (strike price $\(20\)) for $\(0.37.\) Both options expire in \(2\) months’ time. Draw a profit diagram for the bull spread created from these call options. Calculate the profit/loss from the bull spread if, in \(2\) months’ time, the underlying asset is valued at (i) $\(25,\) (ii) $\(18,\) (iii) $\(15.\)

Solution.

The profit/loss from the bull spread if, in two months’ time

\[\text{Profit/loss}=-0.87+0.37+max\{S_t-K,0\}+ max\{0,S_t-K\}=-0.5+max\{S_t-16.5,0\}+ max\{0,S_t-20\}\] \[\left\{\begin{matrix} S_t=25, \text{ Profit=}=3\\ S_t=18, \text{ Profit=}=1\\ S_t=15, \text{ Profit=}=-0.5 \end{matrix}\right.\]

2.4 Problem 14

Chien buys a call option (strike price = £60.00) for £5.00 and sells a call option (strike price = £50.00) for £8.50. Draw a pay-off diagram to illustrate the pay-off at maturity. Why is this portfolio called a bear spread? Calculate the profit/loss from the bear spread if at maturity the underlying asset is valued at

1. £85.00,
2. £54.50,
3. £45.00.

Solution.

The profit/loss from the bull spread if, in two months’ time

\[\text{Profit/loss}=-5+8.5+max\{S_t-60,0\}+ max\{S_t-50,0\}\\=3.5+max\{S_t-60,0\}+ max\{S_t-50,0\}\]

\[\left\{\begin{matrix} S_t=85, \text{ Profit=}=−6.5 \\ S_t=54.5, \text{ Profit=}=-1\\ S_t=45, \text{ Profit=}=3.5 \end{matrix}\right.\]

2.5 Problem 16

Petra buys a call option (strike price £35) for £1.5. She buys a call option with strike price £25 for £2.2 and sells two call options with a strike price of £30 for £1.8. All the options have the same expiry.

1. Draw a profit diagram for Petra’s portfolio.
2. Calculate the pay-off if the asset value at maturity is (a) £40, (b) £31, (c) £27.
3. Why might someone buy such a portfolio?
4. What is the name given to such a portfolio?

2.6 Problem 18

Henry bought a three-month call option (strike price $18) for $2 and a three-month put option (strike price $18) for $1.5. Calculate the profit from this portfolio if the value of the asset at maturity was (i) $22, (ii) $18, (iii) $14. Draw a diagram to illustrate the profit from this portfolio at maturity. What might cause an investor to set up such a portfolio? What is the name given to such a portfolio?

2.7 Problem 19

K9Inc publishes books about dogs and it is not doing well. I buy a six-month call option (strike price $15, cost $1) and two six-month put options (strike price $15, cost $2) on K9Inc shares. Draw a profit diagram for this portfolio. What is my profit when the value of a K9Inc share at maturity is (i) $20, (ii) $14, (iii) $10? Describe my thinking when I bought this portfolio. Describe a situation that might have caused me to modify my portfolio to meet a different expectation.

3.1 Problem 20

A trader creates a bear spread by selling a 6-month put option with a $25 strike price for $2.15 and buying a 6-month put option with a $29 strike price for $4.75. What is the initial investment? What is the total payoff (excluding the initial investment) when the stock price in 6 months is (a) $23, (b) $28, and (c) $33.

3.2 Problem 21

A trader sells a strangle by selling a 6-month European call option with a strike price of $50 for $3 and selling a 6-month European put option with a strike price of $40 for $4. For what range of prices of the underlying asset in 6 months does the trader make a profit?

3.3 Problem 22

Three put options on a stock have the same expiration date and strike prices of $55, $60, and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy. For what range of stock prices would the butterfly spread lead to a loss?

4.1 Slide 29

The stock price starts at $\(20\) and in each of \(4\) time steps may go up by \(10\%\) or down by \(10\%.\) Each time step is \(3\) months long and the risk-free interest rate is \(12\%\) p.a. compounded continuously. We consider a \(12-\)month option with a strike price of $\(21.\)

1. What is the option price?
2. Draw the binomial tree.
3. Write an R code to compute the stock price at each node of the binomial tree.
4. Compute the option price at each node.
5. Write an R code to compute the option price at each node.
6. Compute \(V_0\) directly from \(V_4.\)

4.2 Slide 30 - P1

A stock price is currently $100. Over each of the next two 6-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% p.a. with continuous compounding. What is the value of a 1-year European call option with a strike price of $100?

4.3 Slide 30 - P2

A stock price is currently $50. Over each of the next two 3-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% p.a. with continuous compounding. What is the value of a 6-month European call option with a strike price of $51?

4.4 Slide 31

The stock price starts at $100 and in each of three time steps may go up by a factor \(u=1.06\) or down by a factor \(d=1/u.\) Each time step is 3 months long. We consider a 9-month European call option with a strike price of $100.

1. What happens to the option price when the interest rate increases? Explain?
2. Compute the option price if the interest rate is 12% and 16% p.a. compounded continuously.

4.5 Slide 33

The stock price starts at $20 and in each of two time steps may go up by 10% or down by 10%. Each time step is 3 months long and the risk-free interest rate is 12% p.a. compounded continuously. We consider a 6-month put option with a strike price of $21.

1. Draw the binomial tree.
2. Compute the option price at each node.
3. Write a R code to compute the option price at each node.
4. Compute \(V_0\) directly from \(V_3.\)
5. Write a R code function to compute the option price at any number of time step \(N.\)

4.6 Slide 34

The stock price is \(S_0.\) A put option written on the stock has a maturity at time \(T\) and a strike price \(K.\) Consider a \(n-\)step binomial tree, with parameters \(u,d,r.\) Derive the formula to compute the option price.