An Interest Rate Model

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1 An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal distribution, proportional volatility, basis point volatility, independent increments, limiting distribution An Interest Rate Model 1

2 Review of No Arbitrage Pricing Approac to contingent claims pricing I. starting wit te possible future payoffs of sort- and long-term zeroes II. replicate te payoffs of a derivative wit a portfolio or trading strategy using two zeroes III. use te law of one price to set te claim price equal to te price of te replicating portfolio Review of Risk-Neutral Probabilities Equivalent approac I. determine state-contingent claims prices from te original prices and payoffs of te zeroes II. derive "risk-neutral" probabilities from te state-contingent claims prices III. represent te no arbitrage price of a derivative as te "risk-neutral" expected value of its future payoff, discounted at te riskless rate. An Interest Rate Model 2

3 Starting wit Risk-Neutral Probabilities Conceptually, we start wit current prices and a set of future possible payoffs, and ten derive te risk-neutral probabilities. Once we ave a teory tat says tese riskneutral probabilities exist, owever, it is often more practical to start wit tem immediately. From a financial engineering standpoint, it is easier to set risk-neutral probabilities of te up and down states to eac, and ten work out wat te future payoffs must be to fit current prices. Interest Rate Modeling GOAL: build interest rate models tat capture basic properties of interest rates wile also fitting te current term structure Some basic properties are nonnegative interest rates non-normal distribution mean-reversion stocastic volatility and te level effect. Tis lecture will develop a specific interest rate model and explore some of its properties. Te next lecture will sow ow to calibrate te model to fit te current term structure. An Interest Rate Model 3

4 Building Price Tree from Rate Tree and Risk-Neutral Probabilities As motivation, note tat once we ave a tree of oneperiod rates ("sort" rates) and risk-neutral probabilities, we can price any term structure asset. For example, suppose we assume tat six-mont rates and risk-neutral probabilities are as follows: 5.54% Time 0 Time Time % 4.721% 6.915% 5.437% 4.275% Building Price Trees... Tis information will determine te price trees for te -year zero, te 1-year zero, and te 1.5 year zero. Examples: Te time 1, up-up price of te zero maturing at 1.5: 1/( /2) = Te time 1, up-down price of te zero maturing at 1.5: 1/( /2) = Te time up price of te zero maturing at time 1: 1/( /2) = Te time up price of te zero maturing at 1.5: (x x0.9735) x = An Interest Rate Model 4

5 Building Price Trees... Eventually, we can fill out te wole tree of prices for eac zero. Eac six-mont zero price in te tree comes directly from te six-mont rate. Te price of eac long zero is te discounted, riskneutral expected value of its future price. Time 0 Time Time Building Price Trees... Once we ave te tree or "model" of zero prices, we can price any interest rate derivative product. We price derivatives at teir replication cost. We compute te replication cost by discounting riskneutral expected payoffs. Pricing boils down to building te interest rate model. Time Time Time An Interest Rate Model 5

6 Lognormal Interest Rate Model Definition: A random variable Y as a lognormal distribution if ln(y) as a normal distribution (i.e., if Y=Exp(X) were X as a normal distribution). A lognormal model of interest rates gives bot non-negative interest rates iger volatility at iger interest rates. We will work wit a discrete-time binomial approximation of tis lognormal model. Log Model of Interest Rates Time 0 Te sort rate (te rate on -year bonds): re re Time m +σ 1 m σ 1 Time 2 m + m 2σ Notice tat eac date te sort rate canges by a multiplicative term: m±σ e Te exponential is always positive, wic guarantees tat interest rates are always positive in tis model. re re re m m m + m 2σ 1 2 An Interest Rate Model 6

7 Description of te Model is te amount of time between dates in te tree measured in years. For example, in a semi-annual tree, =. In a montly tree, = 1/12 = Eac value in te tree represents te sort rate or interest rate for a zero wit maturity. Eac date te (risk-neutral) probability of moving up or down is. Te parameters of te model are te drift terms m 1, m 2,...wic are known (nonstocastic) but can cange eac period and te proportional volatility σ wic is constant. Example: Semi-Annual Tree Suppose (details later) te time steps are 6 monts (=) te current 6-mont rate is 5.54% te drift over te first 6 monts is m1= te drift over te second 6 monts is m2 = te proportional volatility σ=0.17 An Interest Rate Model 7

8 Example: Semi-Annual Tree Time 0 Time Time 1 Te sort rate 5.54% 6.004% 4.721% 6.915% 5.437% 4.275% For example, at time, up, te 6-mont zero rate is e = e x 1+ x X exp(x) X+1 Diff exp(x) X X An Interest Rate Model 8

9 Volatility and te Level Effect Te volatility of te sort rate itself is not constant, but is instead approximately proportional to te level of te sort rate. To see tis, note tat for small x: Terefore, e x 1+ x m+ σ re r( 1+ m + σ ) re m σ r( 1+ m σ ) vol(new r) old r σ ) Example of te Level Effect Time Time % 6.915% 5.437% Suppose we arrive at te up state at time so te current spot rate is 6.004%. Te future spot rate is eiter 6.915% or 5.437%. Te (risk-neutral) expected future spot rate is (6.915%+5.437%) =6.176%. Te volatility of te future spot rate is ( ) 2 + ( ) 2 = 74 bp An Interest Rate Model 9

10 Example of te Level Effect... Time Time % 5.437% 4.275% In te down state at time te current spot rate is 4.721%. Te future spot rate is eiter 5.437% or 4.275%. Te (riskneutral) expected future spot rate is (5.437%+4.275%)=4.856%. Te volatility of te future spot rate is ( ) 2 + ( ) 2 = 58 bp Basis Point Volatility In tis model, Time Time 1 volatility is 6.915% proportional to te 6.004% level of te interest vol = 74 bp rate % Te parameter σ is 4.721% called te vol = 58 bp 4.275% proportional volatility. Te unannualized basis point volatility is approximately rσ : up state : = 72 bp down state : = 57 bp Te annualized basis point volatility is approximately rσ. An Interest Rate Model 10

11 Te Log of te Sort Rate ln( r) + m + m σ ln(r) ln( r) + m 1 + σ ln( r) + m 1 σ ln( ) + m + m r 1 2 ln( r) + m + m2 1 2σ Canges in te Log of te Sort Rate Te log of te rate always canges by an additive term, m + σ m σ Te mean cange is m. Te standard deviation of te cange is a constant, σ. Te standard deviation of te annual cange is σ. Wy? Te annual cange is te sum of te canges over eac period. Tere are 1/ canges eac year. Te canges or increments are independent (tere is no mean reversion in tis model), so te variance of te sum is te sum of te variances: 2 σ = σ 2 1 An Interest Rate Model 11

12 Te Limiting Distribution Suppose we old fixed te total calendar time spanned by te tree, but divide te time into smaller intervals ( goes to zero), so tat te number of intervals goes to infinity. Ten te distribution of te log of te terminal sort rate approaces a normal distribution te distribution of te terminal sort rate approaces a lognormal distribution. Review: Using te Interest Rate Tree to Build a Bond Price Tree Time 0 Time 6.004% 1 Sort rate 5.54% =1/( /2) Zero maturing at time Zero maturing at time 1? 4.721% =1/( /2) Te tree implies tat te price of te zero maturing at time 1 is x( )x = An Interest Rate Model 12

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