CAIA L2 - 6.1 - Valuation and Hedging Using Binomial Trees

Question 1

List

Four Key Components of
Risk-Neutral Modeling

6.1 - Valuation and Hedging Using Binomial Trees

Answer 1

1. 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.
2. 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.
3. A derivative valuation obtained from Q-measures is often identical to the zero-arbitrage price derived from P-measures.
4. Q-measures are easy to deal with (tractable), making them suitable for risk-neutral modeling

Raciocínio:
* Bond price depende de δ (default discount) e π (risk premium)
V0 = $100e–(δ+π)
* Preço atual do bond é função de inúmeras percepções de probabilidade de default e required risk premiums dos investidores => unlimited P-measures
* Há apenas uma combinação que 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

Question 2

Formula

Risk-neutral
probability of default

6.1 - Valuation and Hedging Using Binomial Trees

Answer 2

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

Question 3

Identify

Three fallacies
generated by
averaging compounded rates of return

6.1 - Valuation and Hedging Using Binomial Trees

Answer 3

“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

Question 4

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

Answer 4

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
‘—
Ordem:
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. calcular probabilidade de subir e cair (p’u’ = (1+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

Question 5

How to calculate

Convertible bond
using binomial tree

6.1 - Valuation and Hedging Using Binomial Trees

Answer 5

1. 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)
2. 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.
3. calcular probabilidade de subir e cair (p’u’ = (1+r-d)/(u-d) ; p’d’ = 1- p’u’)
4. Para valorar um período anterior, V=(puVu + pdVd)/(1+rf) + cupom!!
5. O que trocar? Se o valor for menor que face+cupom => AUMENTAR para face+cupom (isto é, voce não exerce, o que significa ficar com face+coupon)

Obs: convertible bond: ótimo para investidor segurar até maturity; subótimo para emissor manter até vcto => emissor busca colocar call option to force early convertion

6.1 - Valuation and Hedging Using Binomial Trees

Question 6

How to calculate

Callable bond
using binomial tree

6.1 - Valuation and Hedging Using Binomial Trees

Answer 6

1. calcular movimento da taxa (i’1,U’ = i’1,L’e^(2σ) ; i’1,L’ = i’1,U’e^(-2σ)
2. p’u’=p’d’ = 0.5
3. valor do bond no último nó = face + cupom
4. Para valorar um período anterior, V=(puVu + pdVd)/(1+spot or forward) (não colocar cupom ainda)
5. O que trocar? Se o valor for maior que face = > REDUZIR para valor de face e depois somar com cupom

Obs:
Call price = V’straight’ - V’callable’

6.1 - Valuation and Hedging Using Binomial Trees

Question 7

List

Two advantages of
using both programming languages
and a spreadsheet approach

6.1 - Valuation and Hedging Using Binomial Trees

Answer 7

1. Permits the programmer to verify the output of the computer program and identify programming errors.
2. Facilitate communication of the model with managers who do not understand the complexities of computer programming.

6.1 - Valuation and Hedging Using Binomial Trees