## Empirical results

The 2by10 and 5by10 swaptions are priced for time series 6 April 2004-28 January 2005 using the pricing formula from Section 12.3. When computing the forward interest rates' correlator M(x, x'; t) the daily swaption prices are stable when more than 270 days of historical data for ZCYC were used; but a 270-day average does not give the best fit of the predictions of model swaption price with the swaption's market value. This may be due to too much old information creating large errors in the predictions for the present-day swaption prices. However, averaging on less historical data causes the swaption price curve to fluctuate strongly since it is likely that new information dominates swaption pricing and makes the price too sensitive to small changes.

The empirical study showed that a moving averaging of 180 days of historical data gives the best result for this period. One can most likely improve the accuracy by higher frequency sampling of 180 days of historical data.

The results obtained from the quantum finance model are compared with daily market data and are shown in Figures 12.3(a) and 12.3??; the normalized root mean square of errors are 3.31% and 6.31% respectively. The perturbative model result given in Eq. (12.5) had to be rescaled by an overall factor of 1/.Jn to match it with the market swaption values [18]; the explanation of this single overall factor needs further analysis.

The results for the swaption volatility and correlation discussed in Section 12.4 are derived for the change on the same instruments. From Eq. (12.9)

SCi = Ci(t0 + e) — Ci(t0) = Ci(t0 + e, Rs) — Ci(t0, Rs) (12.22)

where CI (t0 + e) and CI (t0) are the same contract being traded on successive days. Par fixed interest rate Rp is determined when the contract is initiated at time t0, and the swaption CI(t0) is at the money. However, in general, CI(t0 + e) is away from the money; the reason being that the swaption depends on the forward bond prices Fl, and these change every day and hence there is a daily change in the par fixed rate Rp.

110000-, 10500010000095000 -90000: 85000 -80000 -75000 -

110000-, 10500010000095000 -90000: 85000 -80000 -75000 -

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co 3

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100 150 Time series (b)

Figure 12.3 (a) 2by10 swaption price versus time t0 (6 April 2004-28 January 2005), for both market (unbroken line) and the quantum finance model (dashed line). Normalized root mean square error = 3.3l%. (b) 5by10 swaption price versus time t0 (6 April 2004-28 January 2005), both market (unbroken line) and model (dashed line). The normalized root mean square error = 6.31%.

Figure 12.3 (a) 2by10 swaption price versus time t0 (6 April 2004-28 January 2005), for both market (unbroken line) and the quantum finance model (dashed line). Normalized root mean square error = 3.3l%. (b) 5by10 swaption price versus time t0 (6 April 2004-28 January 2005), both market (unbroken line) and model (dashed line). The normalized root mean square error = 6.31%.

Figure 12.4 (a) Swaption variance {C\)c, (C2f)c and covariance [C 1C2)c versus time io (15 June 2004-27 January 2005) computed from the quantum finance model, with the value of the forward bond prices taken from market data. The unbroken line is the variance of a 2by10 swaption, the dashed line is the variance of a 5by10 swaption, and the dotted line is the covariance of the two swaptions. (b) 2by10 swaption price, at the money, from the market (unbroken line), from the quantum finance model (large dashes), and from the HJM model (dotted line). Time t0 is in the interval (6 April 2004-28 January 2005). The normalized root mean square error for HJM = 18.87% compared with the far more accurate quantum finance swaption formula with error = 3.31%.

Bloomberg provides historical daily data only for the prices of the swaptions at the money; swaption prices 'in the money' and 'out of the money' are not quoted. Hence, only the swaption volatility and correlation computed from the model are shown in Figure 12.4(a), without any comparison made with the market value for these quantities.

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