Project 8
MGMTMFE 405
Instructor: L. Goukasian
You will need to write codes for all the parts of the project. Make sure the codes work properly and
understand the ideas behind each problem below. You may be asked to demonstrate how the codes
work, by running them, and interpret the results. Code clarity and accuracy will determine the
grades.
Submit your codes and a PDF file of your answers to questions (including graphs, histograms,
but no codes, in this PDF file) by 11PM PDT on Next Wednesday.
1. Assume the dynamics of the short-term interest rate, under the risk-neutral measure, are given by
the following SDE (Vasicek model):
𝑑𝑟𝑡 = 𝜅(𝑟̅− 𝑟𝑡
)𝑑𝑡 + 𝜎𝑑𝑊𝑡
with 𝑟0 = 5%, 𝜎 = 10%, 𝜅 = 0.82, 𝑟̅= 5%.
(a) Use Monte Carlo Simulation (assume each time step is a day) to find the price of a pure discount
bond, with Face Value of \$1,000, maturing in 𝑇 = 0.5 years (at time 𝑡 = 0):
𝑃(𝑡, 𝑇) = 𝔼𝑡

[\$1,000 ∗ 𝑒𝑥𝑝 (− ∫ 𝑟(𝑠)𝑑𝑠
𝑇
𝑡
)]
(b) Use Monte Carlo Simulation to find the price of a coupon paying bond, with Face Value of
\$1,000, paying semiannual coupons of \$30, maturing in 𝑇 = 4 years:
𝑃(0, 𝐶, 𝑇) = 𝔼0

[∑𝐶𝑖
8
𝑖=1
∗ 𝑒𝑥𝑝 (− ∫ 𝑟(𝑠)𝑑𝑠
𝑇𝑖
0
)]
where 𝐶 = {𝐶𝑖 = \$30 for 𝑖 = 1,2, … ,7; and 𝐶8 = \$1,030},
𝑇 = {𝑇1, 𝑇2, 𝑇3, 𝑇4, 𝑇5, 𝑇6, 𝑇7, 𝑇8
} = {0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4}.
(c) Use Monte Carlo Simulation to find the price of a European Call option on the pure discount
bond in part (a). The option matures in 3 months and has a strike price of 𝐾 = \$980. Use the
explicit formula for the underlying bond price (only for the bond price).
(d) Use Monte Carlo Simulation to find the price of a European Call option on the coupon paying
bond in part (b). The option matures in 3 months and has a strike price of 𝐾 = \$980. Use Monte
Carlo simulation for pricing the underlying bond.
(e) Find the price of a European Call option of part (d) by using the explicit formula for the
underlying bond price, and reconcile the findings with the ones of part (d).
2. Assume the dynamics of the short-term interest rate, under the risk-neutral measure, are given by
the following SDE (CIR model):
𝑑𝑟𝑡 = 𝜅(𝑟̅− 𝑟𝑡
)𝑑𝑡 + 𝜎√𝑟𝑡𝑑𝑊𝑡
with 𝑟0 = 5%, 𝜎 = 12%, 𝜅 = 0.92, 𝑟̅= 5.5%.
(a) Use Monte Carlo Simulation to find at time 𝑡 = 0 the price 𝑐(𝑡, 𝑇, 𝑆) of a European Call option,
with strike price of 𝐾 = \$980, maturity of 𝑇 = 0.5 years on a Pure Discount Bond with Face
Value of \$1,000, that matures in 𝑆 = 1 year:
𝑐(𝑡, 𝑇, 𝑆) = 𝔼𝑡

[𝑒𝑥𝑝 (− ∫ 𝑟(𝑢)𝑑𝑢
𝑇
𝑡
) ∗ max(𝑃(𝑇, 𝑆) − 𝐾, 0)]
(b) Use the Implicit Finite-Difference Method to find at time 𝑡 = 0 the price 𝑐(𝑡, 𝑇, 𝑆) of a European
Call option, with strike price of 𝐾 = \$980, maturity of 𝑇 = 0.5 years on a Pure Discount Bond
with Face Value of \$1,000, that matures in 𝑆 = 1 year. The PDE is given as
𝜕𝑐
𝜕𝑡 +
1
2
𝜎
2
𝑟
𝜕
2
𝑐
𝜕𝑟
2 + 𝜅(𝑟̅− 𝑟)
𝜕𝑐
𝜕𝑟 − 𝑟𝑐 = 0
with 𝑐(𝑇, 𝑇, 𝑆) = max(𝑃(𝑇, 𝑆) − 𝐾, 0), and 𝑃(𝑇, 𝑆) is computed explicitly.
(c) Compute the price 𝑐(𝑡, 𝑇, 𝑆) of the European Call option above using the explicit formula, and
compare it to your findings in parts (a) and (b) and comment on your findings.
3. Assume the dynamics of the short-term interest rate, under the risk-neutral measure, are given by
the following system of SDE (G2++ model):
{
𝑑𝑥𝑡 = −𝑎𝑥𝑡𝑑𝑡 + 𝜎𝑑𝑊𝑡
1
𝑑𝑦𝑡 = −𝑏𝑦𝑡𝑑𝑡 + 𝜂𝑑𝑊𝑡
2
𝑟𝑡 = 𝑥𝑡 + 𝑦𝑡 + 𝜙𝑡
𝑥0 = 𝑦0 = 0, 𝜙0 = 𝑟0 = 3%, 𝑑𝑊𝑡
1𝑑𝑊𝑡
2 = 𝜌𝑑𝑡, 𝜌 = 0.7, 𝑎 = 0.1, 𝑏 = 0.3, 𝜎 = 3%, 𝜂 = 8%.
Assume 𝜙𝑡 = 𝑐𝑜𝑛𝑠𝑡 = 3% for any 𝑡 ≥ 0.
Use Monte Carlo Simulation to find at time 𝑡 = 0 the price 𝑝(𝑡, 𝑇, 𝑆) of a European Put option,
with strike price of 𝐾 = \$950, maturity of 𝑇 = 0.5 years on a Pure Discount Bond with Face
value of \$1,000, that matures in 𝑆 = 1 year. Compare it with the price found by the explicit
formula and comment on it.
4. [Optional Not for grading]
Consider a European Put option, with strike price of 𝐾 = \$970, maturity of 𝑇 = 0.5 years on a
Pure Discount Bond with Face Value of \$1,000, that matures in 𝑆 = 1.5 years.
Which of the two models below would result in a more expensive price for the option?
(a) The Vasicek model 𝑑𝑟𝑡 = 𝜅(𝑟̅− 𝑟𝑡
)𝑑𝑡 + 𝜎𝑑𝑊𝑡 with 𝑟0 = 5%, 𝜎 = 12%, 𝜅 = 0.82, 𝑟̅= 5%.
(b) The CIR model 𝑑𝑟𝑡 = 𝜅(𝑟̅− 𝑟𝑡
)𝑑𝑡 + 𝜎√𝑟𝑡𝑑𝑊𝑡 with 𝑟0 = 5%, 𝜎 = 54%, 𝜅 = 0.82, 𝑟̅= 5%.
Answer by using explicit formulas or by Monte Carlo simulations.
Is the answer consistent with your intuition?

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