4 Suppose the current stock price is 1100 and the continuous

4. Suppose the current stock price is 1100 and the continuously compounded risk-free rate is 5%. The stock pays continuous dividends. You observe a 9-month forward price of 129.257 i) What dividend yield is implied by this forward price? ii) Suppose you believe the dividend yield over the next 9 months will be only 0.5%. What arbitrage would you undertake? iii) Suppose you believe the dividend yield will be 3% over the next 9 months. What arti- trage would you undertake?

Solution

i. What dividend yield is implied by the forward price?

F₀ = S₀e^(r-)T

= r - (1/T)ln(F₀/S₀)

=0.05 - (1/0.75)*LN(1129.257/1100) (in excel)

= 0.015000381

= 0.0150 = 1.50%

ii. Suppose that you believe the dividend yield over the next 9 months will be only 0.5%. What arbitrage would you undertake?

With a dividend yield of 0.005, the fair forward price would be:

=1100*EXP((0.05 - 0.005)*0.75) (in excel)

= 1137.758592 = 1137.76

The market forward rate is too low relative to our forecasted dividend yield. We would therefore buy the forward and create a synthetic short forward using a reverse cash and carry arbitrage:

Today

In 9 Months

Long forward

0

S_T - F₀

Short index (tailed)

+S₀e^-T

-S_T

Lend S₀ (tailed)

-S₀e^-T

S₀e^(r-)T

Total

0

S₀e^(r-)T - F₀

We will have,

Today

In 9 Months

Long forward

0

S_T - 1129.257

Short index (tailed)

1095.88

-S_T

Lend S₀ (tailed)

-1095.88

1137.759

Total

0

8.502

iii. Suppose that you believe the dividend yield over be 3% over the next 9 months. What arbitrage would you undertake?

With a dividend yield of 0.03, the fair forward price would be:

=1100*EXP((0.05 - 0.03)*0.75) (in excel)

= 1116.624371 = 1116.62

The market forward rate is too high relative to our forecasted dividend yield. We would therefore short the forward and create a synthetic long forward using a cash and carry arbitrage:

Now we engage in cash-and-carry arbitrage:

Today

In 9 Months

Short forward

0

F₀ - S_T

Buy index (tailed)

-S₀e^-T

+S_T

Borrow S₀ (tailed)

+S₀e^-T

-S₀e^(r-)T

Total

0

F₀ - S₀e^r(r-)T

We will have,

Today

In 9 Months

Short forward

0

1129.257 - S_T

Buy index (tailed)

-1075.53

+S_T

Borrow S₀ (tailed)

1075.53

-1116.624

Total

0

12.633

F0 = S0e(r-)T

= r - (1/T)ln(F0/S0)