Final Exam FIN552 2022
Please submit electronically to the related Module in Canvas. Note that this is an INDIVIDUAL assignment. Unlike the group project, which was a GROUP assignment, this take-home testis designed to measure your individual knowledge of the material. Full credit will be awarded for all questions only to detailed answers. Any required derivations must be reasonably complete and detailed as well. Also, note that all variable names used below were defined in the lecture notes
1. (30 points)
Consider a Swaption at time 0, with maturity Tα , strike price K, and the underlying swap will mature at Tβ
a. Write down the discounted payoff of a receiver swaption using Forward rate. Make sure to explicitly specify the payoff
function using an underlying Interest Rate Swap contract.
b. Express the above payoff using Forward Swap rate and prove that a)àb)
c. Use the form of payoff in part b) to motivate the definition of moneyness for swaptions.
2. (20 points)
Essay Questions about Vasicek Model
a. How does each parameter of Vasicek model drt = K(θ 一 rt ) +σdWt Q influence the probability of generating negative short rates by forecasting under Q-measure over the next 30 years? Discuss. How about r0 – what impact does the starting point of your short rate forecast have on the above probability?
b. BRIEFLY (one or two sentences each) discuss two valid
criticisms of the Vasicek model. For each criticism you write, BRIEFLY (one sentence) argue whether the Hull-White model is capable or incapable of resolving your criticism.
3. (10 points)
Please list the standard Market Model for Cap Market and Swaption Market respectively. Explain the difference between these two models.
4. (10 points)
Why Proposition 6.3.1 (Page 213) is useful? Give an example when
one would need to know the dynamic of Fk under Ti-forward measure?
5. (30 points)
Consider a forward swap rate
a. Define anumeraire under which the above forward swap rate is a martingale.
b. Write down the Stochastic Differential Equation (SDE) for Sα,β(t)
under the measure defined by thenumeraire in Question 3(a).
c. Assume that β = α+2, please derive SDE of the forward swap rate
Sα, α+2(t) under thenumeraire P (t, Tα).
i. Write down the diffusion component.
ii. Derive the drift component by change-of-measure toolkit.