Vasicek 1: binomial trees, the short rate processSuggested reading: chapter 9

HighlightsObjectives:Understand the shape of the term structureManage risk with non-parallel term structure shiftsValue options and bonds with embedded optionsReview: stock option pricing in a binomial modelA model of the short-term interest rate

Binomial TreesChapter 11Options Futures and Other Derivatives John Hull

A Simple Binomial Model

A stock price is currently $20In three months it will be either $22 or $18

Stock Price = $22Stock Price = $18Stock price = $20

A Call Option (Figure 11.1, page 242)A 3-month call option on the stock has a strike price of 21. Stock Price = $22Option Price = $1Stock Price = $18Option Price = $0Stock price = $20Option Price=?

Setting Up a Riskless PortfolioConsider the Portfolio:long D sharesshort 1 call option

Portfolio is riskless when 22D 1 = 18D or

D = 0.25

Valuing the Portfolio(Risk-Free Rate is 12%)The riskless portfolio is:

long 0.25 sharesshort 1 call optionThe value of the portfolio in 3 months is 22 x 0.25 1 = 4.50The value of the portfolio today is 4.5e 0.12x0.25 = 4.3670

Valuing the OptionThe portfolio that is

long 0.25 sharesshort 1 option is worth 4.367The value of the shares is 5.000 (= 0.25 x 20 )The value of the option is therefore 0.633 (= 5.000 4.367 )

Generalization (Figure 11.2, page 243)A derivative lasts for time T and is dependent on a stock

Generalization(continued)Consider the portfolio that is long D shares and short 1 derivative

The portfolio is riskless when S0uD u = S0dD d or

S0uD uS0dD d

Generalization(continued)Value of the portfolio at time T is S0uD uValue of the portfolio today is (S0uD u)erTAnother expression for the portfolio value today is S0D fHence = S0D (S0uD u )erT

Generalization(continued)Substituting for D we obtain

= [ pu + (1 p)d ]erT

where

p as a ProbabilityIt is natural to interpret p and 1-p as probabilities of up and down movementsThe value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate

p(1 p )

Risk-neutral ValuationWhen the probability of an up and down movements are p and 1-p the expected stock price at time T is S0erTThis shows that the stock price earns the risk-free rateBinomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rateThis is known as using risk-neutral valuation

Original Example Revisited

Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from

20e0.12x0.25 = 22p + 18(1 p )which gives p = 0.6523Alternatively, we can use the formula

S0u = 22 u = 1S0d = 18 d = 0S0 p(1 p )

Valuing the Option Using Risk-Neutral ValuationThe value of the option is e0.12x0.25 (0.6523x1 + 0.3477x0) = 0.633

Irrelevance of Stocks Expected Return

When we are valuing an option in terms of the the price of the underlying asset, the probability of up and down movements in the real world are irrelevantThis is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant

A Two-Step ExampleFigure 11.3, page 246

Each time step is 3 monthsK=21, r=12%

Valuing a Call OptionFigure 11.4, page 247

Value at node B = e0.12x0.25(0.6523x3.2 + 0.3477x0) = 2.0257Value at node A = e0.12x0.25(0.6523x2.0257 + 0.3477x0)

= 1.2823201.2823221824.23.219.80.016.20.02.02570.0ABCDEF

Binomial treesBinomial approach can be extended to many periods

Recombining trees (up followed by down down followed by up; tractable) vs. non-recombining trees

SSuSdSu2SudSd2Su3Su2dSud2Sd3etc

Small hLet h = time period between two nodes in the treeFor small h, many time periods and possible prices at the final date

=> binomial model can be quite realisticIn the limit, distribution of prices becomes continuous (e.g. log-normal)

Modeling bond pricesUnlike stocks, uncertainty on bond prices doesnt increase with time horizon (price = par at maturity); and prices cant go too high (which would imply

One-factor modelsAssume all bond prices are function of one random factor: the short-term interest rateWhy the short-term rate?Long-term bond yields should depend on expectations of future short ratesMore volatile than other (longer term) rates: seems to contain more information

The short-term rateSome facts on short-term interest rates:Mean-reversionTypically more volatile than long-term ratesTypically more volatile as they go upPositiveAgain, above binomial model doesnt work (cf. mean-reversion)

The Vasicek modelGoal: develop a model of the term structure that:is tractable (compute derivatives prices, manage risk)is reasonably accurate given historical data on interest ratesoffers no arbitrage opportunitiesSteps: (1) build a tree for the short rate (2) deduce (from no arbitrage) the evolution of the whole term structure (3) deduce: bond sensitivities (deltas) for risk management; no-arbitrage derivatives prices

The Vasicek short rateLet T (in years) = total amount of time modeled, m = nb of times this time line has been chopped up (in equal pieces) => h = T/m = time interval between two nodes = (annualized, continuously compounded) int. rate for a loan lasting hBetween 2 nodes (i.e. every h year(s)), can either jump up or down by the amount STEP =

The Vasicek short rateLet

The proba. of going up is qv, and the proba. of going down is 1 - qv, where:

qv = q* if 0 q* 1qv = 1 if q* > 1qv = 0 if q* < 0

The Vasicek short rate

The Vasicek short rateqv changes as a function of the interest rate. If < m, then qv > 50% and up is more likely. Conversely, if > m, then qv < 50%

=> mean-reversion toward long-term average int. rate m f measures speed of mean-reversion; 0 < f 1. The higher f, the less mean-reversion (f = 1 => qv = 50% = constant: no mean-reversion)

The Vasicek short rate s impacts the short rate volatilityTree recombiningProbabilities can hit 0 or 1. When proba. hits 0, the branch becomes irrelevant.Interest rates can become < 0

The short rate for small hGiven information available at time t, the short rate at time s has a normal distribution with:Expected value: m (1 - f(s-t))+ t f(s-t)Variance: s2 (1 f2(s-t))When s becomes large, the expected value of is m