Reading 29: The Arbitrage-Free Valuation Framework

Question 1

Arbitrage-free valuation

Answer 1

-value securities such that no market participant can earn an arbitrage profit in a trade involving that security

Question 2

Arbitrage Opportunities

Answer 2

• Value additivity: whole differs from the sum

- Dominance: one asset trades at lower price that another asset with identical characteristics.

Question 3

Stripping/Reconstitution

Answer 3

• use when value additivity does not hold
• Ex:
  
  • stripped bond pieces vs whole bond

Question 4

Backward Induction

Answer 4

See model Kaplan page 35

Question 5

Three Binomial tree process rules

Answer 5

1) interest rate tree should generate arbitrage-free values
2) adjacent forward rates are two standard deviations apart (e^2*variance)
3) The middle forward rate is approx equal to the implies on-period forward rate for that period

Question 6

Pathwise Valuation

Answer 6

-n period equates to 2^(n-1) paths

Ex:
0 = 3%
1 = 5.7883% and 3.88%
2 = 10.7383% and 7.1981% and 4.8250%

Value of bond in path 1 (SUU)=
3/(1.03) + (3/(1.031.057883)) + (103/(1.031.057883*1.1107383))

Question 7

Path Dependency

Answer 7

• An important assumption of binomial valuation model is that the value of cash flows at a given point are independent of the path that interest rates followed until that point….aka cash flows are not path dependent.
• This is why binomial valuation doesnt work for MBS securities….ex where rates drop to 4% from 6%, go back up to 6 and then drop to 4% ago. Not alot of refinancing left. Monte Carlo simulation better because it allows path dependency (drift adjusted? adjust rates if out of whack)

Question 8

Term structure models

Answer 8

-attempt to capture the statistical properties of interest rate movements and provide us with quantitatively precise descriptions of how interest rates will change

Question 9

Equilibrium term structure models

Answer 9

-attempt to describe changes in the term structure through the use of fundamental economic variable that drive interest rate

Question 10

Cox-Ingersoll-Ross model

Answer 10

-based on the idea that interest rate movements are driven by individuals choosing between consumption today versus investing and consuming at a later time

dr = a(b-r)dt + variance(rdz^.5)

a=speed of mean reversion
b=long-run value of short-term interest rate
r=short term interest rate
t=time
dt=small increase in time
variance=volatility
dz=small random walk movement

**volatility rises with interest rates

Question 11

Vasicek model

Answer 11

• interest rate are mean reverting in long run
• dr = a(b-r)dt + variance(dz)
• *interest rates do not affect volatility
• **disadvantage = model does not force interest rates to be non-negative

Question 12

Arbitrage-free models

Answer 12

-assume bonds trading in market are correctly priced

Question 13

Ho-Lee model

Answer 13

dr = theta(dt) + variance(dz)

theta = time-dependent drift term

• uses market prices to fund the time-dependent drift term that generate term strucutre
• use to price zero coupon bonds and determine spot curve
• assumes constant volatility and produces normal distribution of future rates

Question 14

Kalotay-william-fabozzi (KWF) Model

Answer 14

• does not assume mean revesion, assumes constant volatility and constant drift.
• assumes short-term rate is lognormally dsitributed

dln(r) = theta(dt) +variance(dz)