Instructions

Please answer the questions and submit.

Exercises

1) Jafee Corp.

Jafee Corp. common stock is priced at $36.50 per share. The company just paid its $0.50 quarterly dividend. Interest rates are 6.0%. A $35.00 strike European call, maturing in 6 months, sells for $3.20. What is the price of a 6-month, $35.00 strike put option?

  1. $1.20

  2. $1.64 <– Answer

  3. $2.04

  4. $2.38

Solution

The Put-Call Parity Equation is as follows:

\(PV(F0,T - K) = Call(K, T ) - Put(K, T )\)

Where:

\(T\): at time 0 we enter into a long forward position expiring at time T.

\(K\): strike price at time T.

\(F0,T\): forward price.

\(PV(F0,T )\): is the present value of buying the index in the future is just the present value of the forward price.

\(PV(K)\): cost of buying a zero-coupon bond.

\(Call(K, T )\): Buying the call.

\(Put(K, T )\): Selling the put.

\(V_P(0)\): Value of the put at time 0.

\(V_C(0)\): Value of the call at time 0.

Then, we have as follows:

\(PV(F0,T ) = [Call(K, T ) - Put(K, T )] + PV(K)\)

Now, since there’s a dividend at play, we have as follows:

\(F^P_{0,T} = S_0\): the prepaid forward price for delivery at time T.

\(S_T\): stock price at time T.

\(E_0(ST)\): expected stock price at time T based on information we have at time 0.

\(F^P_{0,T} = E_0(S_T )e^{-\alpha T}\)

\(\alpha\): interest rate

A prepaid forward contract will entitle you to receive the stock at time T but without receiving the interim dividends. That means:

\[F^P_{0, T}(S) = S(0) - \sum^n_{i=1}{PV_{0,t_i}(D_{t_i})}\]

\(D\): stock dividend payments

And putting all toguether, we have the following equation:

\[V_P(0) = V_C(0) + K \cdot e^{-rT} - F^P_{0, T}(S)\]

Where

\[F^P_{0, T}(S) = S(0) - D \cdot e^{-rt_1} - D \cdot e^{-rt_2}\]

VC_0 <- 3.20   # call value at time zero
K <-35         # Strike price
r <- 6 / 100   # interest rate
S_0 <- 36.50   # prepaid forward price
D <- 0.50      # Dividends
T <-  6  / 12  # 1/2 - Expiration in years
t1 <- 3  / 12  # 1/4 - 3 months dividends in years
t2 <- 6  / 12  # 1/2 - 6 months dvidends in years

VP_0 <- VC_0 + K * exp(-r * T) - (S_0 - D * exp(-r * t1) - D * exp(-r * t2))

From above, the result is: 1.64

2) Dividends

A company is forecasted to pay dividends of $0.90, $1.20, and $1.45 in 3, 6, and 9 months, respectively. Given interest rates of 5.5%, how much dollar impact will dividends have on option prices? (Assume a 9-month option.)

  1. $3.45 <– Answer

  2. $3.90

  3. $4.22

  4. $4.50

Solution

A prepaid forward contract will entitle you to receive the stock at time T but without receiving the interim dividends. That means:

\[F^P_{0, T}(S) = S(0) - \sum^n_{i=1}{PV_{0,t_i}(D_{t_i})}\]

And just by calculating

\[D_{0,T} = \sum^n_{i=1}{PV_{0,t_i}(D_{t_i})}\]

We have as follows:

D1 <- 0.90      # dividend 1
D2 <- 1.20      # dividend 2
D3 <- 1.45      # dividend 3

t1 <- 3  / 12      # t1 for dividend 1
t2 <- 6  / 12      # t2 for dividend 2
t3 <- 9  / 12      # t3 for dividend 3

r <- 5.5 / 100  # interest rate

DT <- D1 * exp(-r * t1) + D2 * exp(-r * t2) + D3 * exp(-r * t3)

From above, the result is: 3.45

3) Prepaid swap

Given zero-coupon bond yields are 5.2%, 5.5%, and 5.8% in years 1, 2, and 3, respectively, calculate the prepaid swap price for corn. Assume corn forward prices for the proceeding 3 years are $2.10, $2.20, and $2.35, respectively.

  1. $5.96 <– Answer

  2. $6.04

  3. $6.12

  4. $6.20

Solution

In order to calculate the prepaid swap, we need to calculate as follows:

\[ Prepaid \:\: swap = \sum^n_{i=1}{Q_{t_i}F_{0,t_i}P(0,t_i)}\]

Where,

\(P (0, t_i)\): the price is the factor for discounting a fixed payment from date \(t_i\) to date 0.

\(F_{0,t_i}\): the forward prices on these dates.

\(Q_{t_i}\) units of the commodity.

And by solving, we obtain as follows:

\[Prepaid \: swap = \frac{2.10}{1 + 0.052} + \frac{2.20}{(1 + 0.055)^2} + \frac{2.35}{(1 + 0.058)^3}\]

P1 <- 2.10
P2 <- 2.20
P3 <- 2.35

r1 <- 5.2 / 100
r2 <- 5.5 / 100
r3 <- 5.8 / 100

R <- P1 / (1 + r1) + P2 / (1 + r2)^2 + P3 / (1 + r3)^3

From above, the result is: 5.96

4) Market value of the swap

Assume the net swap payment is $.50 on a reverse transaction involving a 3-year corn swap. What is the market value of the swap given interest rates on zero coupon treasury bonds are 5.2%, 5.6%, and 6.0% for 1, 2, and 3 years, respectively?

  1. $0.96

  2. $1.10

  3. $1.25

  4. $1.34 <– Answer

Solution

\[Prepaid \: swap = \frac{0.50}{1 + 0.052} + \frac{0.50}{(1 + 0.056)^2} + \frac{0.50}{(1 + 0.06)^3}\]

P1 <- 0.50
P2 <- 0.50
P3 <- 0.50

r1 <- 5.2 / 100
r2 <- 5.6 / 100
r3 <- 6.0 / 100

R <- P1 / (1 + r1) + P2 / (1 + r2)^2 + P3 / (1 + r3)^3

From above, the result is: 1.34

5) Yield to maturity

The price of a 3-year zero coupon government bond is 85.16. The price of a similar 4-year bond is 78.81. What is the yield to maturity (effective annual yield) on the 3-year bond?

  1. 4.6%

  2. 5.5% <– Answer

  3. 5.8%

  4. 6.7%

Solution

From above, we know that the 100% of face value is 78.81.

and by employing the formula for payments

\[B = {\frac{c}{(1 + y)^t}}\]

We have as follows:

c = 100

B = 85.16

t = 3

\[85.16 = {\frac{100}{(1 + y)^t}}\] And solving \(y\), we get:

B <- 85.16
c <- 100
t <- 3

y <- (c / B)^(1 / t) - 1

From above, the result is: 5.5 %.

6 Forward rate

The prices of 1, 2, 3, and 4-year zero coupon government bonds are 95.42, 90.36, 85.16, and 78.81, respectively. What is the implied 2-year forward rate between years 2 and 4?

  1. 4.8%

  2. 5.2%

  3. 5.5%

  4. 6.4%

NO CHOICE IS GIVEN: My answer is not listed; which is odd. Reference Page 199 & 200 from the text book.

Solution

The forward implied rate formula is as follows:

\[[1 + rt0(t1, t2)]^{t_2 - t_1} = Pt0(t0, t1)/Pt0(t0, t2)\]

That means:

\[rt0(2,4) = \sqrt{\frac{Pt0(0,2)}{Pt0(0,4)}} - 1\]

Pt02 <- 90.36
Pt04 <- 78.81

rt0_24 <- (Pt02 / Pt04)^(1/2) - 1

From above, the result is: 7.1 %.

7) Contango and backwardation

Which of the following terms most accurately describes the forward curve for soybeans over the next two years?

  1. Contango

  2. Backwardation

  3. Contango and backwardation <– Answer

  4. None of the above

Solution

If the forward curve is upward sloping, the forward prices more distant in time are higher; then we say the market is in contango.

If the forward curve is downward sloping, we say the market is in backwardation.

8) Farmer Jayne

Farmer Jayne decides to hedge 10,000 bushels of corn by purchasing put options with a strike price of $1.80. Six-month interest rates are 4.0% and the total premium on all puts is $1,200. If her total costs are $1.65 per bushel, what is her marginal change in profits if the spot price of corn drops from $1.80 to $1.75 by the time she sells her crop in 6 months?

  1. $248 loss

  2. $0 <– Answer

  3. $252 gain

  4. $1,500 loss

Solution

Preamble: A put option is a contract where the seller has the right to sell, but not the obligation. The buyer of the put controls exercise.

Put option payoff = max[0, strike price - spot price at expiration]

If we replace the values in the above equation, we have as follows:

Put option payoff = max[0, 1.75 - 1.80] = 0

The option profit is computed as

Purchased put profit = max[0, strike price - spot price at expiration] - future value of option premium

Now, let’s look at the differences

Purchased put profit = max[0, 1.80 - 1.80] - 1200 = -1200

Purchased put profit = max[0, 1.75 - 1.80] - 1200 = -1200

So, her marginal change in profits if the spot price of corn drops from $1.80 to $1.75 by the time she sells her crop in 6 months is 0.

9) The S&P 500 Index

The S&P 500 Index is priced at $950.46. The annualized dividend yield on the index is 1.40%. What is the price of a 6-month prepaid forward contract on the S & P 500 Index?

  1. $943.83 <– Answer

  2. $950.00

  3. $964.26

  4. $984.21

Solution

In order to find the annualized dividend yield, we must use:

\[F^P_{0,T} = S_0 e^{- \delta T}\]

\[F^P_{0,T} = 950.46 \cdot e^{-0.014·0.5}\]

S0 <- 950.46
T <- 6 / 12         # 6 months
r <- 1.40 / 100     # interest rate

F <- S0 * exp(-r * T)

From above, the result is: 943.83 %.

10) Profit or loss to a short

The spot price of the market index is $900. A 3-month forward contract on this index is priced at $930. What is the profit or loss to a short position if the spot price of the market index rises to $920 by the expiration date?

  1. $20 gain

  2. $20 loss

  3. $10 gain <– Answer

  4. $10 loss

Solution

We know as follows:

Payoff = max[0, 930 - 920] = 10

Profit = 10 - future value of option premium

Since there’s no future value option premium for the short position; the profit is $10 gain.

References

Chi Yau. 2013. R Tutorial with Bayesian Statistics Using Openbugs. USA: R-Tutor.com. http://www.r-tutor.com.

R Core Team. 2016. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

Robert L. McDonald. 2013. Derivatives Markets, Third Edition. United States: Northwestern University, Kellogg School of Management. https://www.pearson.com/us/higher-education/program/Mc-Donald-Derivatives-Markets-3rd-Edition/PGM44824.html.