Introduction to Derivative Securities :Introduction to Financial Engineering and Risk Management (Introduction to Financial Engineering and Risk Management) Answers 2025

Question 1 – Discount rate $d(0,4)$

Given spot rate for 4 years (annual compounding):

s4=8.1%=0.081s_4 = 8.1\% = 0.081s4​=8.1%=0.081

Discount factor:

d(0,4)=1(1+s4)4=1(1.081)4d(0,4) = \frac{1}{(1+s_4)^4} = \frac{1}{(1.081)^4}d(0,4)=(1+s4​)41​=(1.081)41​ (1.081)4≈1.364(1.081)^4 \approx 1.364(1.081)4≈1.364 d(0,4)≈11.364=0.733d(0,4) \approx \frac{1}{1.364} = 0.733d(0,4)≈1.3641​=0.733

✅ Answer: 0.733

Question 2 – 6-year Swap Fixed Rate

Swap fixed rate $c$ satisfies:

c∑i=16d(0,i)=1−d(0,6)c \sum_{i=1}^{6} d(0,i) = 1 – d(0,6)ci=1∑6​d(0,i)=1−d(0,6)

First compute discount factors:

Sum of discount factors:

$$\sum d(0,i)=4.606$$

Now compute swap rate:

c=1−0.6024.606=0.3984.606=0.0864c = \frac{1 – 0.602}{4.606} = \frac{0.398}{4.606} = 0.0864c=4.6061−0.602​=4.6060.398​=0.0864

✅ Answer: 8.64%

Question 3 – Hedging Using Futures

• Expected production = 150,000 lbs

• Contract size = 15,000 lbs

• Number of contracts:

N=15000015000=10N = \frac{150000}{15000} = 10N=15000150000​=10

• Futures price = 118.65 cents/lb

By shorting futures, the farmer locks in the futures price.

✅ Risk-free guaranteed price: 118.65 cents per pound

Question 4 – European Call Option (1-period Binomial)

Given:

• S0=100S_0 = 100S0​=100

• $u=1.05$, $d=1/1.05=0.95238$

• $R=1.02$

• $K=102$

Stock prices:

Su=105,Sd=95.238S_u = 105, \quad S_d = 95.238Su​=105,Sd​=95.238

Payoffs:

Cu=max⁡(105−102,0)=3C_u = \max(105 – 102, 0) = 3Cu​=max(105−102,0)=3 Cd=0C_d = 0Cd​=0

Risk-neutral probability:

p=R−du−d=1.02−0.952381.05−0.95238=0.692p = \frac{R – d}{u – d} = \frac{1.02 – 0.95238}{1.05 – 0.95238} = 0.692p=u−dR−d​=1.05−0.952381.02−0.95238​=0.692

Option value:

C0=11.02[0.692×3]=2.0761.02=2.04C_0 = \frac{1}{1.02} [0.692 \times 3] = \frac{2.076}{1.02} = 2.04C0​=1.021​[0.692×3]=1.022.076​=2.04

✅ Answer: 2.04

Question 5 – Cash Account in Replicating Portfolio

Delta:

Δ=Cu−CdSu−Sd=3105−95.238=0.307\Delta = \frac{C_u – C_d}{S_u – S_d} = \frac{3}{105 – 95.238} = 0.307Δ=Su​−Sd​Cu​−Cd​​=105−95.2383​=0.307

Cash investment $B$:

B=Cd−ΔSdR=0−(0.307)(95.238)1.02=−29.241.02=−28.667B = \frac{C_d – \Delta S_d}{R} = \frac{0 – (0.307)(95.238)}{1.02} = \frac{-29.24}{1.02} = -28.667B=RCd​−ΔSd​​=1.020−(0.307)(95.238)​=1.02−29.24​=−28.667

✅ Answer: −28.667

🧾 Summary Table (Final & Correct)