(PPT) 08 - Vasicek 1 - DOKUMEN.TIPS (2024)

Vasicek 1: binomial trees, the short rate processSuggestedreading: chapter 9

HighlightsObjectives:Understand the shape of the termstructureManage risk with non-parallel term structure shiftsValueoptions and bonds with embedded optionsReview: stock option pricingin a binomial modelA model of the short-term interest rate

Binomial TreesChapter 11Options Futures and Other DerivativesJohn 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 onthe stock has a strike price of 21. Stock Price = $22Option Price =$1Stock Price = $18Option Price = $0Stock price = $20OptionPrice=?

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

Portfolio is riskless when 22D 1 = 18D or

D = 0.25

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

long 0.25 sharesshort 1 call optionThe value of the portfolio in3 months is 22 x 0.25 1 = 4.50The value of the portfolio today is4.5e 0.12x0.25 = 4.3670

Valuing the OptionThe portfolio that is

long 0.25 sharesshort 1 option is worth 4.367The value of theshares is 5.000 (= 0.25 x 20 )The value of the option is therefore0.633 (= 5.000 4.367 )

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

Generalization(continued)Consider the portfolio that is long Dshares 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 isS0uD uValue of the portfolio today is (S0uD u)erTAnother expressionfor 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 asprobabilities of up and down movementsThe value of a derivative isthen its expected payoff in a risk-neutral world discounted at therisk-free rate

p(1 p )

Risk-neutral ValuationWhen the probability of an up and downmovements are p and 1-p the expected stock price at time T isS0erTThis shows that the stock price earns the risk-freerateBinomial trees illustrate the general result that to value aderivative we can assume that the expected return on the underlyingasset is the risk-free rate and discount at the risk-free rateThisis known as using risk-neutral valuation

Original Example Revisited

Since p is the probability that gives a return on the stockequal 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 theoption 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 theunderlying asset, the probability of up and down movements in thereal world are irrelevantThis is an example of a more generalresult stating that the expected return on the underlying asset inthe 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 manyperiods

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 treeForsmall h, many time periods and possible prices at the finaldate

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

One-factor modelsAssume all bond prices are function of onerandom factor: the short-term interest rateWhy the short-termrate?Long-term bond yields should depend on expectations of futureshort ratesMore volatile than other (longer term) rates: seems tocontain more information

The short-term rateSome facts on short-term interestrates:Mean-reversionTypically more volatile than long-termratesTypically more volatile as they go upPositiveAgain, abovebinomial model doesnt work (cf. mean-reversion)

The Vasicek modelGoal: develop a model of the term structurethat:is tractable (compute derivatives prices, manage risk)isreasonably accurate given historical data on interest ratesoffersno arbitrage opportunitiesSteps: (1) build a tree for the shortrate (2) deduce (from no arbitrage) the evolution of the whole termstructure (3) deduce: bond sensitivities (deltas) for riskmanagement; no-arbitrage derivatives prices

The Vasicek short rateLet T (in years) = total amount of timemodeled, m = nb of times this time line has been chopped up (inequal pieces) => h = T/m = time interval between two nodes =(annualized, continuously compounded) int. rate for a loan lastinghBetween 2 nodes (i.e. every h year(s)), can either jump up or downby 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 interestrate. 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 fmeasures speed of mean-reversion; 0 < f 1. The higher f, theless mean-reversion (f = 1 => qv = 50% = constant: nomean-reversion)

The Vasicek short rate s impacts the short rate volatilityTreerecombiningProbabilities can hit 0 or 1. When proba. hits 0, thebranch 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:Expectedvalue: m (1 - f(s-t))+ t f(s-t)Variance: s2 (1 f2(s-t))When sbecomes large, the expected value of is m