Black-Scholes Prices and Portfolio Greeks

Suppose that on December 1, 2017, the spot price of Amazon, Inc. (AMZN) stock is 1176.75 with an annualized
volatility of 21.38% and an annualized dividend yield of 0.00%. Suppose the risk free rate is 1.62%. Furthermore,
suppose that AMZN European options expiring on June 15, 2018 are available at strikes 1140.00, 1160.00,
1180.00, 1200.00 and 1220.00.
a) Use the Black-Scholes Formula to fill in the European-style option premiums (prices) in the chart below.
(Round your prices to 2 decimal places.)
Strike Call Price Put Price
1140.00
1160.00
1180.00
1200.00
1220.00
b) Suppose the following market prices are observed. Find the implied volatility for each option. (Express your
answers in percentage terms rounded to 2 decimal places.)
Strike Call Price Implied Vol Put Price Implied Vol
1140.00 115.03 73.20
1160.00 108.96 82.40
1180.00 96.09 91.76
1200.00 89.18 101.12
1220.00 77.80 112.45
Math 4355/7355, Fall 2017 – Final Exam 2
1. (20 points) Black-Scholes Prices and Portfolio Greeks (continued)
c) List the option transactions from part a) (purchase or write, call or put, strikes) that would constitute a
straddle. Assume that the minimum quantity for purchase or sale is 100 options. Use K = 1180.00 as the ATM
strike.
d) Calculate the portfolio Greeks for this position. (Round your values to 4 decimal places.)
Sensitivity Value Unit
∆ per 1 $ increase in share price
Γ 0.5 per 1 $ change in share price squared
ν per 100 bps increase in volatility
Θ per 1 day decrease in time to expiration
ρ per 1 bp increase in risk-free rate
Ψ per 1 bp increase in dividend yield
e) Explain the signs for ν and Θ. Are they consistent with the market view associated with this trading strategy?
Math 4355/7355, Fall 2017 – Final Exam 3
1. (20 points) Black-Scholes Prices and Portfolio Greeks (continued)
f) List the option transactions from part a) (purchase or write, call or put, strikes) that would constitute an bull
spread. Assume that the minimum quantity for purchase or sale is 100 options. Use K = 1180.00 as the ATM
strike and K = 1200.00 as the OTM strike.
g) Calculate the portfolio Greeks for this position. (Round your values to 4 decimal places.)
Sensitivity Value Unit
∆ per 1 $ increase in share price
Γ 0.5 per 1 $ change in share price squared
ν per 100 bps increase in volatility
Θ per 1 day decrease in time to expiration
ρ per 1 bp increase in risk-free rate
Ψ per 1 bp increase in dividend yield
h) Explain the signs for ∆ and ν. Are they consistent with the market view associated with this trading strategy?
Math 4355/7355, Fall 2017 – Final Exam 4
2. (20 points) Currency Options
Suppose that on December 1, 2017, U.S. risk-free rate (yield) is 1.62% while the comparable yield on European
Central Bank (ECB) bonds is −0.54%. On that same day, the dollar to euro conversion rate is 1.1904 dollars per
euro and the annualized volatility in the dollar-euro exchange rate is 11.27%.
a) Calculate C$
(x, K, T) and the related Greek sensitivities for the following dollar denominated call on 1 euro
with strike K = 1.1900 expiring 06.08.2018. (Round your values to 4 decimal places.)
Sensitivity Value Unit
C$
(1.1904, 1.1900, 189 days)
∆ per 1 $ increase in dollars per euro
Γ 0.5 per 1 $ change in dollars per euro squared
ν per 100 bps increase in volatility
Θ per 1 day decrease in time to expiration
ρ per 1 bp increase in U.S. yield
Ψ per 1 bp increase in ECB yield
b) Calculate Pe

1
x
,
1
K
, T
and the related Greek sensitivities for the following euro denominated put on 1 dollar
with strike 1/K = 0.8403 expiring 06.08.2018. (Round your values to 4 decimal places.)
Sensitivity Value Unit
Pe(0.8401, 0.8403, 189 days)
∆ per 1 e increase in euros per dollar
Γ 0.5 per 1 e change in euros per dollar squared
ν per 100 bps increase in volatility
Θ per 1 day decrease in time to expiration
ρ per 1 bp increase in ECB yield
Ψ per 1 bp change in U.S. yield
c) Demonstrate that the currency option price scaling formula C$
(x, K, T) = xKPe

1
x
,
1
K
, T
holds.
Math 4355/7355, Fall 2017 – Final Exam 5
3. (20 points) Bond Rates
Suppose the following par coupon rates for Treasury securities were observed. Suppose also that these coupons
are paid once per year at the end of each year.
Maturity Par Coupon
1 1.6200%
2 1.7800%
3 1.9000%
5 2.1400%
7 2.3100%
10 2.4200%
a) Use the bootstrapping method to calculate the zero coupon price and zero coupon yield for each maturity.
(For zero coupon yields, express your answers in percentage terms rounded to 4 decimal places. For zero coupon
prices, express your answer in unit terms, par = 1.0000, rounded to 6 decimal places.)
Maturity Par Coupon Zero Coupon Price Zero Coupon Yield
1 1.6200%
2 1.7800%
3 1.9000%
4 2.0332%
5 2.1400%
6 2.2428%
7 2.3100%
8 2.3916%
9 2.4402%
10 2.4200%
Math 4355/7355, Fall 2017 – Final Exam 6
3. (20 points) Bond Rates (continued)
b) Use your results in a) to calculate implied 1-year rates for year 1 to year 2, year 2 to year 3, …, year 9 to year
10. (Express your answers in percentage terms rounded to 4 decimal places.)
r0(t1, t2) Implied Forward 1-Year Rate
r0(0, 1)
r0(1, 2)
r0(2, 3)
r0(3, 4)
r0(4, 5)
r0(5, 6)
r0(6, 7)
r0(7, 8)
r0(8, 9)
r0(9, 10)
Math 4355/7355, Fall 2017 – Final Exam 7
4. (20 points) Duration and Convexity
Consider a $10,000 5-year Treasury note that pays semi-annual interest and returns principal at maturity.
Suppose the bond’s annual coupon rate is 2.14% and that the market yield is y = 2.14% as well.
a) Calculate the bond values B(y), B(y − ∆y) and B(y + ∆y) where ∆y = 25 bps. (Round your answers to the
nearest cent.)
Bond Value
B(y)
B(y − ∆y)
B(y + ∆y)
b) Calculate Macaulay duration DMac, modified duration DMod and convexity Conv for this bond when
y = 2.14%. (Round your answers to 4 decimal places.)
Value
DMac
DMod
Conv
c) Demonstrate how DMod and Conv can be used to estimate B(y − ∆y) and B(y + ∆y).
Math 4355/7355, Fall 2017 – Final Exam 8
5. (20 points) Eurodollar Futures and Swap Rates
On the Question 5 tab of the final exam template, you will find Eurodollar futures prices for 12.01.2017. Find the
swap rate for each maturity on the template. Provide these rates for the maturities listed below. (Express your
answers in percentage terms rounded to 4 decimal places.)
Maturity (years) Swap Rate
1
2
3
5
7
10
Math 4355/7355, Fall 2017 – Final Exam 9
6. (20 points) Exotic Options
a) A standard European-style call option has exercise value ST − K if ST > K. Exercise is not required even if
ST > K.
Consider a non-standard European-style call option with exercise value ST − K1 if ST > K2 where K1 6= K2.
Furthermore, assume this option must be exercised if ST > K2. Adapt the Black-Scholes formula to price this
option as C(S, K1, K2, σ, r, T, δ). Provide the main formula and formulas for d1 and d2.
b) Suppose K1 > K2. What option pricing constraint must be relaxed to accommodate this option?
Math 4355/7355, Fall 2017 – Final Exam 10
6. (20 points) Exotic Options (continued)
c) A standard European-style call option gives the option purchaser the right to exchange K dollars in cash for 1
share of stock valued at ST if ST > K.
Consider a non-standard European-style call option which gives the option purchaser the right to exchange 1
share of some other stock valued at RT for 1 share of stock valued at ST if ST > RT . (In this simplified example,
we assume that S0 = R0 at t = 0.) Adapt the Black-Scholes formula to price this option as C(S, R, σS,R, δR, T, δS).
Provide the main formula and formulas for d1 and d2.
d) Suppose σS is the volatility of the first stock, σR is the volatility of the second stock and ρ is the correlation
between the two stock prices over time. How should σS,R be calculated?