CAIA L2 - 6.1 - Valuation and Hedging Using Binomial Trees Flashcards
List
Four Key Components of
Risk-Neutral Modeling
6.1 - Valuation and Hedging Using Binomial Trees
- The number of sets of P-measures (e.g., default probability, recovery rates, risk premiums) that equate to a derivative’s current price is usually unlimited.
- The risk premium is zero in a risk-neutral world, which results in a single set of Q-measures that can be derived from the derivative’s current price.
- A derivative valuation obtained from Q-measures is often identical to the zero-arbitrage price derived from P-measures.
- Q-measures are easy to deal with (tractable), making them suitable for risk-neutral modeling
Reasoning:
* Bond price depends on δ (default discount) and π (risk premium)
V0 = $100e–(δ+π)
* Current bond price is a function of investors’ numerous perceptions of default probability and required risk premiums => unlimited P-measures
*There is only one combination that π=0 (risk premium = zero) “The default probability consistent with zero-risk premium (i.e., risk neutrality) is known as the risk-neutral Q-measure”
* Q-measure is referred to as a risk-neutral probability
* Assuming risk neutrality => assuming investors are risk neutral = premium for risk is zero => discount rate is the risk-free rate
* risk-neutral probability of default > actual default probability (se π >=0)
6.1 - Valuation and Hedging Using Binomial Trees
Formula
Risk-neutral
probability of default
6.1 - Valuation and Hedging Using Binomial Trees
Current price = [Par × (1–p) + (Par× RR × p)] / (1+ rf)
p = prob of default
RR = recovery rate
rf = risk free rate
6.1 - Valuation and Hedging Using Binomial Trees
Identify
Three fallacies
generated by
averaging compounded rates of return
6.1 - Valuation and Hedging Using Binomial Trees
“leveraged ETFs destroy investor wealth when volatility increases, even in an efficient market”
although the geometric average expected return will decrease, the NPV remains at zero, which means wealth is unaffected.
‘–
“inverse ETFs destroy investor wealth”
The reality is that, in an efficient market, the illusion of underperformance occurs because of the same factors discussed earlier with leveraged ETFs (e.g., focusing on average compound growth rather than expected dollar investment value)
‘–
“rebalancing of a portfolio’s assets creates wealth through better diversification”
In fact, rebalancing cannot create a positive NPV for the portfolio if the underlying assets are efficiently priced (i.e., each asset has a zero NPV). Again, the illusion comes from studies focusing on expected compound rates of return.
6.1 - Valuation and Hedging Using Binomial Trees
How to calculate
CRR model - Binomial Tree of Stock Prices
(Cox, Ross, Rubinstein Model)
List:
How to value options using CRR model
6.1 - Valuation and Hedging Using Binomial Trees
u=e^(σ√Δt)
d=1/u
p’u’ = (r–d)/(u–d)
r = 1+ risk free rate
p’d’ = 1- p’u’
‘—
u = upward move multiplier
d = downward move multiplier
σ = assumed volatility or standard deviation of stock returns
∆t = time between periods (nodes)
p’u’ = risk-neutral probability of an up move
p’d’ = risk-neutral probability of an down move
‘—
Order:
1. Construct a binomial tree to model stock price evolution until the expiry of the option. (u = e^(σ√Δt) ; d= 1/u)
2. Compute the option payoffs (last node)
3. calculate probability of rising and falling (p’u’ = (r-d)/(u-d) ; p’d’ = 1- p’u’)
4. Perform backward induction of probability-weighted payoffs using the risk-free rate - value in prior node=(puVu + pdVd)/(1+rf)
‘–
Obs:
- Black-Scholes model cannot be used for options that can be exercised before maturity
- Binomial tree converges with Black-Scholes => as shorter period interval is used
6.1 - Valuation and Hedging Using Binomial Trees
How to calculate
Convertible bond
using binomial tree
6.1 - Valuation and Hedging Using Binomial Trees
- Construct a binomial tree to model stock price evolution until the expiry of the bond (see valuation of equity options) (u = e^(σ√Δt) ; d= 1/u)
- Take the binomial tree of stock prices and multiply each stock price by the conversion ratio (number of shares the bond can be converted into) to get a tree of conversion prices.
- calculate probability of rising and falling (p’u’ = (1+r-d)/(u-d) ; p’d’ = 1- p’u’)
- To value a previous period, V=(puVu + pdVd)/(1+rf) + coupon!!
- What to exchange? If the value is less than face+coupon => INCREASE to face+coupon (that is, you do not exercise, which means staying with face+coupon)
Note: convertible bond: great for investors to hold until maturity; suboptimal for issuer to maintain until maturity => issuer seeks to place call option to force early conversion
6.1 - Valuation and Hedging Using Binomial Trees
How to calculate
Callable bond
using binomial tree
6.1 - Valuation and Hedging Using Binomial Trees
- calculate movement rate (i’1,U’ = i’1,L’e^(2σ) ; i’1,L’ = i’1,U’e^(-2σ)
- p’u’=p’d’ = 0.5
- bond value at the last node = face + coupon
- To value a previous period, V=(puVu + pdVd)/(1+spot or forward) (do not add coupon yet)
- What to change? If the value is greater than face value => REDUCE to face value and then add with coupon
Obs:
Call price = V’straight’ - V’callable’
6.1 - Valuation and Hedging Using Binomial Trees
List
Two advantages of
using both programming languages
and a spreadsheet approach
6.1 - Valuation and Hedging Using Binomial Trees
- Permits the programmer to verify the output of the computer program and identify programming errors.
- Facilitate communication of the model with managers who do not understand the complexities of computer programming.
6.1 - Valuation and Hedging Using Binomial Trees
formula for a two-period geometric mean
average compound return = ((1 + period 1 return)(1 + period 2 return))½ – 1
Geometric mean of three periods would be ^1/3 etc
Black Derman Toy model (BDT)
The BDT model produces an arbitrage-free tree of interest rate evolution over time under a given interest rate volatility assumption.
The probabilities of up and down moves in the binomial interest rate tree will always be 0.5. A binomial interest rate tree assumes interest rates can take two possible values in the next period. The rates in a binomial interest rate tree are one-period forward rates. The interest rate at T0 is a spot rate.
Formula for calculating the the upper and lower rate in a binomail interest rate tree
For down movements use negative 2volatility
Calculating a straight bond price using a two-period binomial tree
The methodology of calculating the price of a callable bond using a two-period binomial tree is the same as the backward induction of the straight bond with one exception. If a backward induction value (at T1) is greater than the call price at T1, the backward induction value is eliminated and replaced with the call price
Why would the current price of an investment be least likely equal to the present value of its expected returns?
Markets have been slightly inefficient.
Inefficient markets lead to differences between an investment’s current price and the present value of its expected returns.
LO 6.1.2
