13.7 Calibration methodology


13.7 Calibration methodology

13.7.1 Algorithm for constant mean reversion and volatility parameters

The following is an algorithm of the methodology to calibrate a reversion speed a and associated volatility parameter ƒ which give rise to the smallest mispricing.

 For valuation_date 1 to  n   a  = initial_value     While  a  is in the acceptable interval       For expiry 1 to  m  initialise    1  and    2  get Black_premium for this expiry and valuation date  P   1  = HW _option_premium (    1  ) - Black_premium  P   2  = HW_option_premium (    2  ) - Black_premium         While  P   1  > 1 * 10  -5  and  P   2  > 1 * 10  -5      mid   = (    1  +    2  )/2  P  mid   = HW_option_premium (     mid   ) - Black_premium           If  P   1  *  P  mid   < 0 Then    2  =     mid   ElseIf  P   2  *  P  mid   < 0 Then    1  =     mid   End If         Loop   =     mid   Next expiry     ave   = Average(   for expiry 1 to  m  )       ave_mispricing = Average((HW_option_premium (     ave   )                          - Black_premium)/Black_premium;                           for expiry 1 to  m  If ave mispricing < optimal mispricing then  a  optimal   =  a  update  a  Loop   Next valuation date 

For a given value of parameter a , determine the corresponding values of ƒ i for expiry date i = 1, , m . Since a and ƒ are parameters associated with the term structure, specifically the reversion speed and volatility of the short-term interest rate, they tell us what the price of the bond option implies about the characteristics of the short- term interest rate. Ideally, we would like the ƒ i s to be the same for all maturities i = 1, , m . This would imply all maturity bond options are priced consistently by our model of the short-term interest rate. However, in practice these ƒ i s may differ quite substantially. To determine an optimal value of a and corresponding ƒ , take the arithmetic average of ƒ i , i = 1, , m to be the proxy for ƒ . Pricing each bond option using the value of a and corresponding ƒ allows us to determine the degree of mispricing to market- observed option premia. The optimal value of a and corresponding ƒ (calculated using the averaging described above) are determined as those resulting in the smallest mispricing across the bond option maturities.

The methodology used in procedure HW_option_premium to calculate the HW option premium is as described in §13.5. To calculate the strike prices and hence premia of the sub-options comprising the coupon bond options we require A ( T,s i ) where T is the option expiry date and s i is a coupon payment date. From equation (13.17) we require P ( r, 0, s i ), P ( r, 0, T ) and in order to evaluate A ( T,s i ). Since T and s i will not necessarily fall on the available node points, we must interpolate the P ( r, 0, ·) and curves to obtain values for the correct maturity dates.

Given the initial interest rate term structure R ( r, 0, ·), make use of (13.3) to obtain the term structure of P ( r, 0, ·). We then apply the cubic spline interpolation to retrieve discount bond prices for any maturity date.

In order to evaluate , we first calculate ln P ( r, 0, ·) at each node point and then apply Lagrange interpolation to fit quadratic polynomials and hence evaluate derivatives at each node [5] . Applying cubic spline interpolation to these derivative values allows us to retrieve for any maturity date.

13.7.2 Algorithm for a flat volatility term structure

The following algorithm describes the methodology to calibrate a volatility parameter ƒ which gives rise to the smallest mispricing. This algorithm is essentially a subset of the previous algorithm since the methodology corresponds to a = 0 and hence only an optimal ƒ needs to be found.

 For valuation date 1 to  n  For expiry 1 to  m  initialise    1  and    2  get Black_premium for this expiry and valuation date  P   1  = HW_option_premium (    1  ) - Black_premium  P   2  =HW_option_premium (    2  ) - Black_premium       While  P   1  > 1 * 10  -5  and  P   2  > 1 * 10  -5      mid   = (    1  +    2  )/2  P  mid   = HW_option_premium (     mid   ) - Black_premium         If  P   1  *  P  mid   < 0 Then    2  =     mid   ElseIf  P   2  *  P  mid   < 0 Then    1  =     mid   End If       Loop   =     mid   Next expiry   Next valuation date 

To determine the strikes of the sub-options and hence their premia we require values of B ( T,s i ) and A ( T,s i ), where T is the option expiry date and s i the coupon payment date.

To determine B ( T,s i ) we must determine [6] B (0, ·) and . Given the initial interest rate term structure R ( r, 0, ·), make use of (13.21) to determine the values of B (0, ·) at all the node points. Applying cubic spline interpolation allows us to retrieve B (0, ·) for any maturity date. To determine , we fit a quadratic Lagrange polynomial at each node point and evaluate its first derivative. Then applying cubic spline interpolation to these derivatives allows us to retrieve for any maturity.

To determine A ( T,s i ) we also require . This is found by applying cubic spline interpolation to and then evaluating this integral. This involves the integral of the square of a cubic polynomial.

[5] This is the methodology described in the final paragraph of §13.4.1.

[6] See equation (13.9).




Interest Rate Modelling
Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132

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