Interest Rate Models, FIS

Question 1

Describe the term structure of interest rates under the Vasicek model

Answer 1

When r_t is low, the Vasicek model implies an increasing term structure
When r_t is high, the Vasicek model implies a decreasing term structure
All point on the term structure move with perfect correlation
- drawback of all one factor models

Question 2

Describe a method to estimate the real world parameters if the Vasicek model

Answer 2

Sigma - calculate historical volatility
R bar - average short term rate over a sample period
Gamma - slope from regressing changes in the interest rate in r_t*delta

Question 3

Describe a method for estimating the risk neutral parameters of the Vasicek model

Answer 3

Minimize the pricing errors between the model and the market ZCB prices

Question 4

Define a relative value trade

Answer 4

A trade that is constructed to exploit a potential arbitrage opportunity when the market prices for interest rate securities are not consistent with the prices implied from an interest rate model

Question 5

Describe the steps of Monte Carlo simulation for valuing a security

Answer 5

1. Simulate M interest rate scenarios
2. For each interest rate path, compute the PV of the instrument payoff
3. Average the payoffs to obtain the Monte Carlo estimate

Question 6

Compare Risk Neutral and Risk Natural Monte Carlo simulation

Answer 6

Risk Neutral
- pricing interest rate securities
Risk Natural
- distribution of P&L
- value at risk
- expected shortfall

Question 7

Describe two common trends on the risk premium of zero coupon bonds

Answer 7

1. Return premium from holding long term ZCBs is typically higher than from holding short term bonds
2. For a given maturity, a lower spot rate is associated with a higher risk premium

Question 8

Describe the intuition behind the real interest rate in the utility function model

Answer 8

Rho - higher value means more utility by consuming today
hg - high growth in economy, borrow more to consume today, higher real rate
1/2h^2sigma_y^2 - when volatility GDP growth is high, higher risk of lower consumption in the future, prompts consumers to save more, decreases real rate

Question 9

Describe the intuition behind the inflation risk premium in the utility model

Answer 9

When covariance between GDP growth and inflation is negative, a short term nominal investment will be more risky to the investor

Question 10

Describe the intuition behind the risk neutral interest rate in the utility model

Answer 10

Since covariance between expected inflation and GDP growth is typically negative, the adjustment for risk is generally positive, so the risk neutral rate is higher than the real world rate

The risk neutral rate will increase as risk aversion increases, the covariance becomes more negative, or the mean reversion speed decreases

Question 11

List steps to derive the term structure parameter for the Ho-Lee model

Answer 11

1. Derive the corresponding spot rates from STRIPs
2. Interpolate spot rates to obtain a twice differentiable spot rate curve
3. Compute the instantaneous forward curve
4. Approximate theta using the forward curve

Question 12

State pros and cons of the Ho-Lee model

Answer 12

Pros
- simple and has closed form solutions for many derivatives
- no arbitrage model to replicate market ZCB prices
Cons
- short rate is not stationary
- implies flat term structure of volatility
- one factor model assumes all points on the term structure are correlated

Question 13

Describe pros and cons of the Hull-White model

Answer 13

Pros
- no arbitrage model
- has mean reversion, keeps the variation of interest rates bounded
- implies the yields of long term bonds have lower volatility, which is consistent with market data
- has closed form solutions
Cons
- interest rates can be negative
- one factor model assumes all points on the term structure are correlated

Question 14

Describe the impact of the mean reversion speed on the Hull-White model

Answer 14

Higher mean reversion means lower S(), which means lower option price
Higher mean reversion reduces B() and dispersion of interest rates

Question 15

State a drawback of lognormal models

Answer 15

Hard to find closed forms for bond prices

Question 16

Compare and contrast flat and forward volatilities

Answer 16

Flat volatility - quoted volatility for every caplet to obtain the market cap price
- same caplet can have a different quoted volatility depending on which cap it is a part of
- simply a quoting convention
Forward volatility - no arbitrage volatility that characterizes a particular caplet
- not dependent on which cap the caplet belongs to

Question 17

Describe a payer and receiver swap

Answer 17

Payer swap - pay the fixed rate

Receiver swap - receive the fixed rate

Question 18

State the key property under the forward risk neutral measure

Answer 18

The forward rate is lognormally distributed

Question 19

State the key assumptions of the LIBOR Market Model

Answer 19

The forward rate is lognormally distributed and follows a lognormal diffusion process under the T-forward risk-neutral dynamics

Question 20

State the formula for an arbitrary T-forward measure (that may be based on multiple forward rates)

Answer 20

[placeholder]

Question 21

State simplifying assumptions of the volatility function in the LIBOR Market Model

Answer 21

Volatility of forward rates only depends on the time to maturity
Function S(.) is constant on each expiry period of the cap
- Var(r_n) = S_i^2(T_i - t) + S_i-1^2Delta + … + S_1^2*Delta

Question 22

State the drawback of volatility rates only depending on time to maturity

Answer 22

Model may not be able to fit the term structure of volatilities if they are decreasing very rapidly as a function of time
- will cause S_i^2 < 0, but forward volatilities cannot be negative

Question 23

State two reasons the LIBOR Market Model is more commonly used than the Heath Jarrow Morton framework

Answer 23

HJM focuses on continuously compounded forward rates, which require additional calculations
- LMM focuses on simply compounded LIBOR-based forward rates, which are traded instruments
The LMM framework is better for pricing complex securities

Question 24

Define an in-arrears payment

Answer 24

A payment where the payoff is on the same date as the fixing of the reference rate

Question 25

Outline a methodology for calibrating a two factor Vasicek model to ZCB prices

Answer 25

1. Choose proxies for the ST and LT yields
2. Fix the volatility of the ST and LT yields to their empirical values
3. For convenience, assume phi*_2 = 0
4. Search for the set of parameters that minimize the error function

Question 26

Compare the one factor and two factor Vasicek model ability pies to fit ZCB prices

Answer 26

Two factor does a much better job of matching the term structure
- additional parameters enhance the ability to add curvature
- calibrating second factor mean reversion speed is negative (mean averting, or explosive)
One factor model has difficulty fitting term structures with curvature
- assumes all points on the curve are perfectly correlated

Question 27

Compare how well interest rate models fit the term structure of spot rate volatilities

Answer 27

One factor Vasicek - not great, one parameter to target volatility of yields
Two factor Vasicek (independent) - good job, more parameter to match ST and LT end of term structure
Two factor Vasicek (correlated) - worse job than 1fV because the optimization also includes the correlation factor
Two factor Hull White - good job

Question 28

Compare how well interest rate models fit the term structure of spot rate correlations

Answer 28

One factor Vasicek - not good, correlation is a flat line at 100%
Two factor Vasicek (independent) - slightly better, but model correlations still high
Two factor Vasicek (correlated) - even better, but overestimates correlation between short tenor spot rates and the short rate
Two factor Hull White - good job

Question 29

Describe a yield curve steepener

Answer 29

A derivative security whose payoff is calculated based on two points on the yield curve