- CFA Exams
- CFA Level I Exam
- Study Session 14. Derivatives
- Reading 38. Valuation of Contingent Claims
- Subject 5. Black Option Valuation Model

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**CFA Practice Question**

An interest rate put option based on a 90-day underlying rate has an exercise rate of 5.5% and expires in 150 days. The forward rate is 5.25% and the volatility is 0.08. The continuously compounded risk-free rate is 4%. Calculate the price of the interest rate put option using the Black model. The notional principal is $10 million.

Correct Answer: $6,625

f

X = 0.055

d

d

N(d

N(d

1. Use the forward rate to discount the result back from day 240 to day 150: 0.002708 x e

2. Convert the result to a periodic rate based on a 90-day rate, using the customary 360-day year: 0.00265 x (90/360) = 0.0006625.

The time to maturity is T = 150/365 = 0.4110.

f

_{0}(T) = 0.0525X = 0.055

d

_{1}= [ln(f_{0}(T)/X) + (σ^{2}/2)/T] / (σ T^{1/2}) = [ln(0.0525/0.055) + 0.08^{2}/2 x 0.4110] / (0.08 x 0.4110^{1/2}= -0.8815d

_{2}= d_{1}- σ T^{1/2}= -0.8815 - 0.08 x 0.4110^{1/2}= -0.9327N(d

_{1}) = N(-0.8815) = 1 - N(0.8815) = 0.1894N(d

_{2}) = N(-0.9327) = 1 - N(0.9327) = 0.1762p = e

^{-0.04 x 0.4110}[0.055 x (1 - 0.1762) - 0.0525 x (1 - 0.0.1894)] = 0.002708Two adjustments:

1. Use the forward rate to discount the result back from day 240 to day 150: 0.002708 x e

^{-0.0525 x (90/365)}= 0.00265.2. Convert the result to a periodic rate based on a 90-day rate, using the customary 360-day year: 0.00265 x (90/360) = 0.0006625.

Therefore, the price is 10,000,000 x 0.0006625 = $6,625.

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**User Contributed Comments**
4

User |
Comment |
---|---|

danlan2 |
Do not understand the two adjustments. |

Rotigga |
Note the adjustment for step #2 is 90/360, not 90/365. |

ptyson |
the last part of the notes in this section explains it best. |

Nando1 |
Don't worry about it. We're not asked to calculate the model for the exam. |