Pricing Interest Rate Derivatives: The Vasicek and Hull

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Pricing Interest Rate Derivatives The Vasicek
and Hull White Models
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Term Structure Models versus Blacks Model

• Blacks model is concerned with describing the
  probability distribution of a single variable at
  a single point in time
• Example
• When valuing a swap option, we assume that the
  swap rate at the maturity date of the option is
  lognormal
• We dont assume anything about swap rate at any
  other date
• A term structure model describes the evolution of
  the whole yield curve

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The Zero Curve

• In a world in which the spot rate account is the
  numeraire, the value f(t) at time t of a
  derivative with payoff fT at time T is
• where is the average interest rate over the
  period from t to T
• For the price P(t,T) at t of a zero-coupon bond
  with maturity date T we get
• If P(t,T) would not be given by the above
  expression there would be an arbitrage
  opportunity.
• R(t,T) is the continuously compounded interest
  rate at time t for a maturity of T-t

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• So we have
• Which leads to
• This equation shows that the complete term
  structure of interest rates at some time t is
  determined by the risk-neutral dynamics of the
  spot rate
• US zero rates at 20/02/2008

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One-factor Models

• Consider the following physical dynamics for the
  spot rate r
• The spot rate is driven by a single Brownian
  motion z.
• Such models are called one-factor models
• The risk-neutral dynamics are then given by
• Example Rendleman Bartter Model
• The physical dynamics are
• The risk-neutral dynamics are
• This is geometrical Brownian motion, as for a
  stock

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Mean Reversion

• The assumption that the spot rate follows a
  geometrical Brownian motion is not realistic.
• Over time interest rates are pulled back to some
  long-run average level
• The Rendleman Bartter model does not include
  this feature

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• Mean reversion of interest rates is linked to the
  business cycle.
• When the economy is at its peak, demand for funds
  is high, leading to high interest rates.
• As the economy cools down, demand for funds
  decreases, and interest rates go down.
• Central banks try to prevent the economy from
  overheating by increasing short term rates, and
  they try to stimulate the economy during a
  slowdown by reducing short-term rates.

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The Vasicek Model

• The physical dynamics of the spot rate are given
  by
• We assume that the market price of risk is a
  constant ?. The risk-neutral dynamics are then
  given by
• where
• Calibrating the model to data, one always finds
  that ?0, and thus that .
• Both with the physical and the risk-neutral
  dynamics, the future spot rate has a normal
  distribution.

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• On can show that in the Vasicek model one has
• where A(t,T) and B(t,T) are given by
• From the expression for P(t,T) we get
• We see that all zero rates are linear functions
  of the spot rate r.
• Weak point of the Vasicek model interest rates
  can become negative. Although for realistic
  values for the parameters this is very unlikely.

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The Cox-Ingersoll-Ross Model

• The physical dynamics of the spot rate are given
  by
• We assume that the market price of risk is of the
  form
• The risk-neutral dynamics are then given by
• where
• Calibrating the model to data, one always finds
  that ?0, and thus that .
• Both with the physical and the risk-neutral
  dynamics, the future spot rates do not have a
  normal distribution.

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• Both for the physical ad the risk-neutral
  dynamics, we have an equation f the following
  type
• If r becomes very small (close to zero), the
  diffusion term goes to zero as well, and the
  drift term is strictly positive.
• As such, r is pushed away from zero, and can not
  become negative (or zero).
• Also in the CIR model one has
• But now, A(t,T) and B(t,T) are given by
• where

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Possible Term Structuresin Vasicek CIR Models

• In both models the zero rates that make up the
  term structure are linear functions of the spot
  rate.
• Therefore, the shape of he term structure at some
  future date t, is not driven by the value of
  r(t), but is mainly determined by the shape of
  the functions A(t,T) and B(t,T)
• r(t) mainly drives the level of the term structure

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Two-factor Models

• The assumption that a single factor (dz) drives
  the entre term structure is quite restrictive.
• As an immediate consequence we have that all zero
  rates are completely determined by the spot rate.
• Two possible generalizations
• Assume that the mean-reversion level b is
  stochastic and follows its on stochastic process
• Assume that the volatility s is stochastic
• In both cases, the term structure is driven by
  two factors
• The spot rate and the mean-reversion level of it
• The spot rate and its volatility

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No-Arbitrage Models

• In an equilibrium model todays term structure
  is an output
• The term structure observed in the market can not
  be fitted by the model.
• The Vasicek and CIR models only have thee
  parameters
• As a result, the zero-coupon bond prices implied
  by the model will not be the same as those
  obtained from market prices.
• This could potentially lead to arbitrate
  opportunities, as prices of options would not be
  consistent with prices of traded bonds.
• In a no-arbitrage model todays term structure
  is an input
• This is achieved by making one of the parameters
  time-depend

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Developing No-Arbitrage Model for r

• A model for r can be made to fit the initial term
  structure by including a function of time in the
  drift
• The Ho-Lee Model
• The risk-neutral dynamics are
• dr ?(t)dt sdz
• The variable ?(t) determines the average
  direction of the movement of the spot rate r at
  time t.
• Just as a implied volatilities allow the BS
  model to replicate observed market prices for
  European options on stocks, the parameter ?(t)
  allows the Ho-Lee model to be calibrated to the
  observed term structure.
• Interest rates normally distributed
• One volatility parameter, s

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• A time t0 the variable ?(t) can be calculated
  as
• The average direction that the spot rate will
  move to at a future date t is given by the
  current slope of the forward curve.
• Many analytic results for bond prices and option
  prices
• For zero-coupon bonds we have
• where
• In these equations time zero is today, and t and
  T and dates in the future with tT.

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The Hull and White Model

• The risk-neutral dynamics are
• dr q(t) ar dt sdz
• or
• dr aq(t )/a r dt sdz
• The Hull-White model is the no-arbitrage version
  of the Vasicek model.
• Many analytic results for bond prices and option
  prices
• Two volatility related parameters, a and s
• Interest rates normally distributed
• The function q(t) is given by

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• Bond prices in the Hull and White model are again
  given by
• where now
• We still have that all zero rates are determined
  by the spot rate
• The slope of he yield curve at some future date t
  is not driven by the level of r(t) but by the
  functions A(t,T) and B(t,T).

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Options on Zero-Coupon Bonds

• In Vasicek and Hull-White model, price of call
  maturing at T on a bond lasting to s is given by
  the following expression
• LP(0,s)N(h)-KP(0,T)N(h-sP)
• The price of put is
• KP(0,T)N(-h s P)-LP(0,s)N(-h)
• The variable s P is the standard deviation of
  the log of the bond price at time T.

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• This shows that we obtain an expression of the
  same structure as the Black Scholes model for
  equity options.

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• For the CIR model the expressions are more
  complicated and involve complicated integrals

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Options on Coupon Bearing Bonds

• In any one-factor model a European option on a
  coupon-bearing bond can be expressed as a
  portfolio of options on zero-coupon bonds.
• Consider a call with strike K and maturity date
  T.
• We first calculate the critical value for the
  spot rate r(T) at the option maturity for which
  the coupon-bearing bond price equals the strike
  price K at maturity
• The strike price for each zero-coupon bond is set
  equal to its value when the interest rate equals
  this critical value
• The value of the option on the coupon bond is
  then given by the sum of the options on the
  corresponding zero-coupon bonds, with strike
  prices as calculated above.

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Calibrating the Vasicek Model

• The physical dynamics
• The physical dynamics of the spot rate are given
  by
• or
• for some fixed time-step ?t
• From this we get
• We can obtain estimates for the three parameters
  by running a regression.

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• We need to estimate the intercept and the slope
  of the regression, as well as the standard
  deviation of the errors.
• At this point we have the parameters for the
  physical dynamics.
• The risk-neutral dynamics
• The risk-neutral dynamics, used for pricing, are
  given by
• The values for a and s we know already.
• We need to estimate the value of bQ

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• The parameter bQ is given that value which
  minimizes the pricing errors for selection of
  bonds.
• We minimize the mean squared error
• The risk-neutral dynamics An alternative
• In order to allow the model to better fit the
  current term structure, one can re-estimate the
  parameter a.
• In this case, only the estimate from the
  regression analysis that is used for pricing is
  the estimate for s.
• Typically, the two estimates for a are different