The Arbitrage-Free Valuation Framework Flashcards
How do we calculate the number of possible paths in Binomial trees?
2^(n-1)
Interest rate tree i2LL referes to?
The one year forward rate at time 2, assuming the lower rates at time 1 and 2
In determining the appropriate level of volatility to use in modeling paths interest rates, we would most likely NOT use
Implied volatility based on observed prices of option-free Government bonds.
What does the log-normal random walk volatility capture?
The volatility of the one-year rate
When are Callable Bonds more valueable?
During a downward sloping yield curve
Call Option is valuable when yield curve flattens
When are Putable Bonds more valueable?
When the yield curve is upward sloping
Put Option is valuable when yield curve steepens
Formula for Value of issuer call opion
Value of stright bond - Value of callable bond
Formula for Value investors Putable bond
Value of putable bond - Value of stright bond
If volatility increases, what will happen to the value of callable bond
The new value will be lower than the previous price.
Value of Call = V Stright - V Callable
If volatility increases, what will happen to the value of Putable bond
The new value will be greater than the previous value
Explain the relationship of what will happen to the value of callable and putable if volatility increases
Callable bond value decreases
Putable bond value increases
If the OAS (Option Adjusted Spread) for a bond is higher than its peers, it is considered to be…
Undervalued
OAS for a bond is higher than the OAS of its peers, it is considered to be undervalued i.e. attractive
investment meaning it offers a higher compensation for a given level of risk (cheap).
If the OAS (Option Adjusted Spread) for a bond is Lower than its peers, it is considered to be…
Overvalued!!
bonds with
low OAS relative to peers are considered to be overvalued (rich) and should be avoided.
It offers lower compenstation for a given level of risk
What is the formula for Option Cost
OAS - Z Spread
What is the formula for Z-Spread
OAS - Option cost
Z-Spread ≥ OAS
Callable bond
Z-Spread ≤ OAS
Putable bond
Option cost for Callable bonds when Volatility increases
Positive Option cost
Callable Bond = Z-Spread ≥ OAS =
Option cost for Callable bonds when Volatility Decreases
Negative option cost
Callable Bond = Z-Spread ≥ OAS
Option cost for Putable bonds when Volatility increases
Negative option cost
Putable Bond = Z-Spread ≤ OAS
Option cost for Putable bonds when Volatility Decreases
Positive option cost
Putable Bond = Z-Spread ≤ OAS
What is the most appropriate duration to use for bonds with embedded options?
Effective duration
How do we interpret 1.97 effective duration?
for a 100 bsp change in the interest rates, the bond price will change by 1.97% on average.
What is effective duration?
A parallel shift in the yield curve. (benchmark yield curve)
- assuming no change in the bond’s credit spread, but it is not an accurate.
- measure of interest rate sensitivity to non-parallel shifts in the yield curve like those described by ‘Shaping Risk’.
- Shaping Risk refers to changes in portfolio value due to changes in the shape of the benchmark yield curve. However, parallel shifts explain more than 75% of the variation in bond portfolio returns.
Please explain deep in-the-money embedded option bonds
When the embedded option (call or put) is deep in the money, the effective duration of the bond with an embedded option resembles that of the straight bond maturing on the first exercise date,
reflecting the fact that the bond is highly likely to be called or put on that date.
Please explain the relationshop of out-of -the-money embedded option bonds
Effective Duration Callable ≤ Effective Duration Straight
Effective Duration Putable ≤ Effective Duration Straight
Effective Duration ZCB ≈ Maturity of the Bond
Effective Duration Fixed Rate Coupon < Maturity of the Bond
Effective Duration Floater ≈ Time in Years to Next Reset
Please explain the relationshop of At -the-money embedded option bonds
The effective duration of the callable bond shortens when interest rate falls, which is when the call
option moves into the money, limiting the price appreciation of the callable bond.
The effective duration of the putable bond shortens when interest rates rise, which is when the put option moves into the money, limiting the price depreciation of the putable bond.
While effective duration of straight bonds is relatively unaffected by changes in interest rates.
What kind of relationship does call option value have with interest rates?
Inverse relationship
Effective convexity of the callable bond turns negative when the call option is near the money
which indicates that the upside for a callable bond is much smaller than the downside. When rates are high, callable bonds are unlikely to be called and will exhibit positive convexity.
What kind of relationship does Put option value have with interest rates?
Direct relationship
Putable bonds always have positive convexity.
When the option is near the money, the upside for a putable bond is much larger than the downside
because the price of a putable bond is floored by the price of the put option, if it is near the exercise
date.
Which type of bonds can experience negative convexity?
Callable bonds.
Convertible Bonds: Conversion Value
Share price x Conversion Ratio
Convertible Bonds: Conversion Ratio
Bond Price / Conversion Price
Convertible Bonds: Conversion Price
Par or issue price / Conversion ratio
Convertible Bonds: Market Conversion premium per share
( Market value of convertible bond / Conversion Ratio ) - share price
Convertible Bonds: Market Conversion premium per share RATIO
[ ( PV convertible bond / Conversion Ratio ) / Share price ] -1
one-sided durations
Effective durations when interest rates go up or down, which are better at capturing the interest rate sensitivity of bonds with embedded options that do not react symmetrically to positive and negative changes in interest rates of the same magnitude
Effective Duration indicated the sensitivty of the bonds full price to a 100 bsp shift in the government
Par Curve
Downward sloping yeild curve -> Upward sloping yield curve
Put option: Increases
Call option: Decreases
Upward sloping yeild curve -> Downward sloping yield curve
Put option: Decreases
Call option: Increases
What will happen to OAS when volatility increases
OAS decreases
What will happen to OAS when Volatility Decreases
OAS increases
On-sided duration for Callable bonds
On-sided up duration > on-sided down duration
On-sided duration for Putable bonds
On-sided Down duration > on-sided up duration
Which combination will lead to lower Call option value
- Lower Volatility
- Higher call price (strike)
- Low cupon price
Which combination will lead to lower put option value
- Lower Volatility
- Lower put price (strike)
- Higher coupon
What is OAS (Option Adjusted Spread)
Option Adjusted Spread (OAS) is the constant spread, when added to all one-period forward rates on the interest rate tree which makes the arbitrage-free value of the bond (calculated value) equal to its market price.
The option-adjusted spread (OAS) is the measurement of the spread of a fixed-income security rate and the risk-free rate of return, which is then adjusted to take into account an embedded option.
Typically, an analyst uses Treasury yields for the risk-free rate. The spread is added to the fixed-income security price to make the risk-free bond price the same as the bond.
Value of floored floater bond
Value of stright bond + Value of embedded floating floor
Value of convertible bond
Value of stright bond + value of call option on issuer stock
Explain why putable bond will be trading higher than callable bonds when rates decrease?
When rates drops, callable bonds will increase in value only to a certain limit, 100$.
Stright bonds and putable bonds will always have higher value than callable bonds because of more oppertunites to exercise the option.
The minimal value of a convertible bond is equal to the greater of …
Conversion value and the value of the option free underlying bond
Convertible bond
Premium over stright
[Convertible bond price / Option free bond price] - 1
Formula for conversion price
Issue Price / Conversion Ratio
If we only know the 2 year par rate, and we know the 1 year, par, spot, and forward rates, how to we find the 2 year spot rate and forward rate?
First we need to find the 2 year spot rate.
Use the 1 year spot rate as the discount factor, and use the 2 year par rate as the payment.
Solve for spot rate at year 2 as the discount factor.
That will provide us the 2 year spot rate which we can use to solve the forward rate
Value of a floor floater
Value of stright bond + Value of embedded floor
Value of callable bond
Value of stright bond - value of issuer call option
