¡Descarga Valuing Options with Non-Parallel Term Structures: Binomial Trees & Short Rate y más Monografías, Ensayos en PDF de Finanzas solo en Docsity! 1 Vasicek 1: binomial trees, the short rate process Suggested reading: chapter 9 2 Highlights • Objectives: – Understand the shape of the term structure – Manage risk with non-parallel term structure shifts – Value options and bonds with embedded options • Review: stock option pricing in a binomial model • A model of the short-term interest rate 5 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option (Figure 11.1, page 242) A 3-month call option on the stock has a strike price of 21. 6 • Consider the Portfolio: long shares short 1 call option • Portfolio is riskless when 22– 1 = 18 or = 0.25 22– 1 18 Setting Up a Riskless Portfolio 7 Valuing the Portfolio (Risk-Free Rate is 12%) • The riskless portfolio is: long 0.25 shares short 1 call option • The value of the portfolio in 3 months is 22 x0.25 – 1 = 4.50 • The value of the portfolio today is 4.5e – 0.12x0.25 = 4.3670 10 Generalization (continued) • Consider the portfolio that is long shares and short 1 derivative • The portfolio is riskless when S0u– ƒu = S0d– ƒd or dSuS fdu 00 ƒ S0u– ƒu S0d– ƒd 11 Generalization (continued) • Value of the portfolio at time T is S0u– ƒu • Value of the portfolio today is (S0u – ƒu)e–rT • Another expression for the portfolio value today is S0– f • Hence ƒ = S0– (S0u– ƒu )e–rT 12 Generalization (continued) • Substituting for we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT where p e d u d rT 15 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 = 0.6523 • Alternatively, we can use the formula 6523.0 9.01.1 9.00.250.12 e du de p rT S0u = 22 ƒu = 1 S0d = 18 ƒd = 0 S0 ƒ p (1– p ) 16 Valuing the Option Using Risk- Neutral Valuation The value of the option is e–0.12x0.25 (0.6523x1 + 0.3477x0) = 0.633 S0u = 22 ƒu = 1 S0d = 18 ƒd = 0 S0 ƒ 0.65 23 0.3477 17 Irrelevance of Stock’s 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 irrelevant • This is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant 20 Binomial trees • Binomial approach can be extended to many periods • Recombining trees (up followed by down <=> down followed by up; tractable) vs. non- recombining trees S Su Sd Su2 Sud Sd2 Su3 Su2d Sud2 Sd3 etc… 21 “Small h” • Let h = time period between two nodes in the tree • For small h, many time periods and possible prices at the final date => binomial model can be quite realistic • In the limit, distribution of prices becomes continuous (e.g. log-normal) 22 Modeling bond prices • Unlike stocks, uncertainty on bond prices doesn’t increase with time horizon (price = par at maturity); and prices can’t go too high (which would imply <0 int. rates) • Doesn’t make sense to assume constant interest rate • Above binomial model won’t work. Need a new approach 25 The Vasicek model • Goal: develop a model of the term structure that: – is tractable (compute derivatives prices, manage risk) – is reasonably accurate given historical data on interest rates – offers no arbitrage opportunities • Steps: (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… 26 The Vasicek short rate • Let 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 h • Between 2 nodes (i.e. every h year(s)), can either jump up or down by the amount STEP = r h ln2 r 27 The Vasicek short rate • Let • The proba. of going up is qv, and the proba. of going down is 1 - qv, where: qv = q* if 0 ≤ q* ≤ 1 qv = 1 if q* > 1 qv = 0 if q* < 0 8 ln 2 1* hr q tt r r 30 The Vasicek short rate • impacts the short rate volatility • Tree recombining • Probabilities can hit 0 or 1. When proba. hits 0, the branch becomes irrelevant. • Interest rates can become < 0 … 31 The short rate for small h • Given information available at time t, the short rate at time s has a normal distribution with: – Expected value: (1 - (s-t))+ t (s-t) – Variance: 2 (1 – 2(s-t)) • When s becomes large, the expected value of is r r r
