4. Model Analysis
4.2 Fixes for the Characteristic Function Instabilities
4.2.3 An SDE Based Approximation With a More Sophisticated Pa-
Instead of directly using the SDE approach to inform the parameter values, we can also use this as an initial ”guess” at a functional form for the estimate of E[√
t], and combine this with the technique used byGrzelak and Oosterlee(2012) to estimate their parameter values. Doing this with the following approximation:
E[
√t] = ae−bt. (4.3)
we get:
a=√
0,
b= − logΛ(1) a
,
where Λ(t)is defined in (A.1). This new approximation does not suffer the short-comings of approximation (2.19), as it simply requires that √
0 > 0, which is as-sumed to be the case. This approximation is therefore consistent with the general model structure. Interestingly, the a term above is the same as that of the SDE ap-proach taken to determine parameter approximations. This suggests that the two
4.2 Fixes for the Characteristic Function Instabilities 30
methods are reasonably consistent as well.
The fundamental difference between the functional forms of these approxima-tions is that the original form converges to a over time, while this new form con-verges to 0. Converging to 0 is desirable on account of the fact that A2(t)is in-creasing in τ . This means that, if it isn’t dampened by some other function, it will eventually result in A(u, τ ) becoming greater than 0 and consistently increasing thereafter.
Repeating this approximation for the stochastic volatility term in the LMM used byGrzelak and Oosterlee(2012), we have the following approximation:
E[
pV (t)] ≈ ave−bvt,
where
av =p V (0), bv = − log
Λv(t) av
.
Chapter 5
Results from the Adjusted Models
In this section we present the results using the suggested model adjustments in Section4. The results from these models will initially be compared, using the same parameter values as in Section3, to the market values in AppendixD. Thereafter, they will be compared using the parameter values which are known to cause issues in the current model’s characteristic function.
5.1 Comparison of the Adjusted Models with Market Prices
Here, we use the parameter values and tenor structure from (3.1) and (3.2) to com-pare the adjusted models in Sections4.2.1, 4.2.2and4.2.3. These results are pre-sented in Table5.1.
The application of the adjustment in Section 4.2.1 results in a mean absolute error of 4.107% when compared to the market values with the same strike and ma-turity values as those provided byGrzelak and Oosterlee(2012). When applying the adjustment in Section4.2.2, there is a mean absolute error of 4.02%. Interest-ingly, this is a lower error than is the case when using the old approximation for E[√
t]. This would imply that, even using this simple function, for the current model this new functional form of the approximation term is a better description than the old functional form. Finally, when applying the adjustment from Section 4.2.3, the mean absolute error is 3.58%, which is the lowest error achieved when im-plementing the constant volatility LMM. This is reasonable evidence that this new approximation function is more appropriate than the original. This is compounded by the fact that the implementation of this new approximation function is more ef-ficient than that of the original.
5.1ComparisonoftheAdjustedModelswithMarketPrices32
Tab. 5.1:Table comparing results from the adjustments in Sections4.2.1,4.2.2and4.2.3 Strike
European Call Price
T2 T5 T10
Section4.2.1 Section4.2.2 Section4.2.3 Section4.2.1 Section4.2.2 Section4.2.3 Section4.2.1 Section4.2.2 Section4.2.3
40% 0.6418 0.6418 0.6418 0.7018 0.7011 0.7015 0.7823 0.7784 0.7803
80% 0.3298 0.3303 0.3300 0.4632 0.4613 0.4632 0.6194 0.6091 0.6150
100% 0.2146 0.2156 0.2151 0.3718 0.3694 0.3721 0.5545 0.5414 0.5493
120% 0.1326 0.1339 0.1333 0.2975 0.2949 0.2982 0.4984 0.4829 0.4925
160% 0.0479 0.0489 0.0484 0.1908 0.1881 0.1921 0.4071 0.3882 0.4004
200% 0.0183 0.0187 0.0185 0.1243 0.1217 0.1255 0.3371 0.3162 0.3300
240% 0.0078 0.0080 0.0078 0.0828 0.0805 0.0839 0.2825 0.2606 0.2752
5.2 Comparison of the Adjusted Models with Updated Parameter Values 33
5.2 Comparison of the Adjusted Models with Updated Parameter Values
Now, we compare the results from the constant volatility model and the adjusted constant volatility model for the same parameter values, except for ρx,i = −0.94.
This value of ρx,i allows for the illustration of the numerical instability of the H1-LMM characteristic function, while ensuring that the instantaneous covariance ma-trix is still definite. The instantaneous covariance mama-trix is not positive-definite for ρx,i values close to −1. The result of using ρx,i = −0.94is shown in Table5.2. The shorter maturity prices are very similar, but there is a large devia-tion when applied to the longer dated opdevia-tions. This implies that, even though the ODE corrected method does reduce the instability, the additional approximation likely increases the error. The reason for this appears to stem from this method forcing the affect of the A2(t)term into the non-real part of the characteristic func-tion when ρx,i is negative. This is not observed with the adjusted approximation function method. However, because this function tends to 0 over time, it reduces the affect of the A2(t)term to to zero over time. However, it does not diminish the affect of this term entirely, and from the outset. Thus, it would likely give a more accurate approximation of the H-LMM characteristic function.
There is also a large deviation when applying the adjustment method in Section 4.2.2and Section4.2.3. This suggests that approximation (4.2) does not diminish the affect of A2(t)as well as approximation (4.3).
Tab. 5.2:Table comparing results from the adjustments in Section4.2.1, and using approximations (4.2) and (4.3), using ρx,i = −0.94
Strike
European Call Price
T5 T10
Adj (4.2.1) Adj (4.2) Adj (4.3) Adj (4.2.1) Adj (4.2) Adj (4.3)
40% 0.6982 0.6994 0.6988 0.7694 0.7764 0.7731
80% 0.4488 0.4528 0.4493 0.5815 0.6028 0.5914
100% 0.3521 0.3571 0.3517 0.5049 0.5329 0.5172
120% 0.2739 0.2793 0.2724 0.4386 0.4725 0.4528
160% 0.164 0.1695 0.161 0.3326 0.3748 0.3489
200% 0.0989 0.1039 0.096 0.2543 0.3009 0.2712
240% 0.0609 0.0653 0.0589 0.1964 0.2444 0.213
Chapter 6
Application of the Characteristic Function Adjustments to the
H1-LMM Model
In this section we review the affects of the characteristic function adjustment meth-ods on the H1-LMM model. The same parameter values are used as in their article and in3.2and3.1. The additional parameter values that were used byGrzelak and Oosterlee(2012) are as follows:
λ = 1, V (0) = 1, η = 0.1.
Note that the parameter values used for the additional approximations used in Grzelak and Oosterlee(2012) are found analogously to those used in Section4.2.2 and4.2.3.
Tab. 6.1:Table showing a sample of results from the H1-LMM model with the ad-justment from Section4.2.1and market data
Strike
European Call Price
T2 T5 T10
ChF Market ChF Market ChF Market
40% 0.6418 0.642 0.7018 0.702 0.7823 0.779 80% 0.3298 0.329 0.4632 0.461 0.6194 0.612 100% 0.2146 0.212 0.3718 0.368 0.5545 0.546 120% 0.1326 0.131 0.2975 0.293 0.4983 0.489 160% 0.0479 0.051 0.1908 0.188 0.4071 0.397 200% 0.0183 0.023 0.1243 0.125 0.3371 0.328 240% 0.0078 0.012 0.0828 0.086 0.2825 0.275
The results from the application of the adjustment in Section4.2.1are presented
Chapter 6. Application of the Characteristic Function Adjustments to the H1-LMM Model 35
in Table6.1. These results give a mean absolute error of 4.11%. This is still lower than the vanilla implementation which suggests that, for ρx,i > 0, this method is adequate.
Tab. 6.2:Table showing a sample of results from the H1-LMM model with the ap-proximation adjustment (4.2) and market data
Strike
European Call Price
T2 T5 T10
ChF Market ChF Market ChF Market
40% 0.6418 0.642 0.7007 0.702 0.7778 0.779 80% 0.3296 0.329 0.4591 0.461 0.6073 0.612 100% 0.2143 0.212 0.3663 0.368 0.5391 0.546 120% 0.1324 0.131 0.2910 0.293 0.4801 0.489 160% 0.0477 0.051 0.1835 0.188 0.3846 0.397 200% 0.0181 0.023 0.1172 0.125 0.3121 0.328 240% 0.0076 0.012 0.0766 0.086 0.2563 0.275
The results from the application of the adjustment in Section4.2.2are presented in Table6.2, which give a mean absolute error of 5.09%. This is the highest mean absolute error so far, which is evidence that the double approximation may be in-appropriate. Thus, we apply the alternatively derived approximation function.
Tab. 6.3:Table showing a sample of results from the H1-LMM model with approx-imation adjustment (4.3) and market data
Strike
European Call Price
T2 T5 T10
ChF Market ChF Market ChF Market
40% 0.6418 0.642 0.7014 0.702 0.7800 0.779 80% 0.3299 0.329 0.4625 0.461 0.6140 0.612 100% 0.2148 0.212 0.3712 0.368 0.5479 0.546 120% 0.1331 0.131 0.2971 0.293 0.4908 0.489 160% 0.0482 0.051 0.1908 0.188 0.3983 0.397 200% 0.0184 0.023 0.1243 0.125 0.3276 0.328 240% 0.0078 0.012 0.0828 0.086 0.2727 0.275
Finally, the results from applying the adjustment in Section4.2.3are presented in Table6.3. These results give a mean absolute error of 3.55%, which is the low-est across all the implementations relative to the sample of the market data. This
Chapter 6. Application of the Characteristic Function Adjustments to the H1-LMM Model 36
should be compared to the adjustment in Section4.2.2, which resulted in the high-est mean absolute error. Thus, it seems that the new functional form works for both of the volatility approximations. However, in the case of the H1-LMM model, one needs to be careful how one approximates the parameters in the final approxima-tion funcapproxima-tion.
Chapter 7
Conclusion
The utility of the H-LMM model is without question; long-dated options cannot be sufficiently evaluated without some form of stochastic interest rate assumption nor can interest-equity products. However, the sheer number of parameters that need to be calibrated in this model results in the requirement of an efficient method of calibration. As such, the use of Fourier pricing techniques is paramount.
These Fourier techniques all require a characteristic function to be applied.
Thus, having a characteristic function that is numerically unstable for a range of parameter values is a large drawback. This would require one to restrict the pa-rameter values before use, severely limiting the potential of the final model.
Thus, the methods suggested in this dissertation are important in that they al-low for efficient calibration techniques to be used, whilst not necessarily restricting the parameter values. Moreover, both cases of the adjusted approximation method are more efficient than the technique proposed byGrzelak and Oosterlee (2012), while approximation (4.3) has a lower error for the given data set.
While the ODE adjusted method ensures that the characteristic function does not explode, the additional approximation that is required to derive a closed-form solution results in the error term increasing relative to simply using the adjusted approximation method. Thus it is recommended to keep ODE method in mind should alternative characteristic functions need to be derived. However, there is sufficient evidence to conclude that the adjusted approximation method is more appropriate.
In sum, the adjusted approximation method is effective for the H1-LMM model characteristic function to be applied for calibration of the H-LMM model, as well as for the constant volatility LMM variant of the H-LMM model.
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Appendix A
Derivation of the Parameter Values in Equation (2.19)
We derive the parameter values for the expectation of the asset volatility process, as well as the parameters for the analogous approximation of the expectation of the interest rate volatility process used inGrzelak and Oosterlee(2012).
The dynamics for the relevant processes are:
dt= κ(¯ − t)dt + γ√
tdW(t), dV (t) = λ(V (0) − V (t))dt + ηp
V (t)dWVN(t).
Note that tis the variance process of the underlying asset, while V (t) is the vari-ance process of the LMM. Note further that, with no stochastic volatility in the LMM, this simplifies considerably.
We use the following approximations:
√t≈ E[√
t] = a1+ b1e−c1t, pV (t) ≈ E[p
V (t)] = a2+ b2e−c2t.
Define ˜Λ(t) := a + be−ct. We seek to optimise: mina,b,c||Λ(t) − ˜Λ(t)||n, where || · ||n
is any nthnorm, and:
Λ(t) = s
c(t)(ω(t) − 1) + c(t)d + c(t)d
2(d + ω(t)). (A.1)
Now, for the case of t, we also have:
c(t) = γ2(1 − e−κt)
4κ ,
w(t) = 4κ0e−κt γ2(1 − e−κt), d= 4κ¯
γ2 .
Appendix A. Derivation of the Parameter Values in Equation (2.19) 41 Re-arranging the above yields:
a1=
Appendix B
Simplification of (2.20) and (2.21)
Equation (2.22) can be re-written as:
Thus we must have:
a0 =
Appendix B. Simplification of (2.20) and (2.21) 43
Next we re-write (2.23) as:
Σi,j = (c0)ij+ (c1)ij value corresponds to the variable being dealt with. As a result, k = 1 corresponds to Xt, k = 2 corresponds to t, and k = j corresponds to Lj−2(t)for j > 2.
This allows us to write (2.23) more generally as:
Σi,j= (c0)ij+
This can be written in matrix form as:
c1 =
where 0 is a (n × 1)-dimensional matrix of 0’s.
Appendix B. Simplification of (2.20) and (2.21) 44
First, observe that B(u, τ ) can be written in terms of the components specific to each element in the state vector:
B(u, τ ) =
Appendix B. Simplification of (2.20) and (2.21) 45
Then, we can write (2.20) as:
∂
Appendix B. Simplification of (2.20) and (2.21) 46
wherePN
i=N +1:= 0. The final matrix multiplication required is:
B(u, τ )Tc0B(u, τ ) =
Thus, we are able to write (2.21) as:
∂
Appendix C
Integral of Equation (2.29)
Integrating (2.29) with respect to τ gives:
A(u, τ ) = A(u, 0) +
Appendix C. Integral of Equation (2.29) 48
where the proof follows by noting that A1(t)and A2(t)are constant over the given interval.
Appendix D
Market data
D.1 Zero Coupon Bond prices
Tab. D.1: ZCB prices Bond Price 0.9512 B(0, 1) 0.9048 B(0, 2) 0.8607 B(0, 3) 0.8187 B(0, 4) 0.7788 B(0, 5) 0.7408 B(0, 6) 0.7047 B(0, 7) 0.6703 B(0, 8) 0.6376 B(0, 9) 0.6065 B(0, 10)
D.2 European Call Option Market Prices
Tab. D.2:European call market prices Strike
European Call Price
T1 T2 T3 T4 T5 T10
40% 0.62 0.642 0.663 0.683 0.702 0.779 80% 0.271 0.329 0.378 0.421 0.461 0.612 100% 0.143 0.212 0.271 0.322 0.368 0.546 120% 0.067 0.131 0.19 0.244 0.293 0.489 160% 0.015 0.051 0.095 0.141 0.188 0.397 200% 0.004 0.023 0.051 0.086 0.125 0.328 240% 0.001 0.012 0.03 0.055 0.086 0.275 260% 0.001 0.009 0.024 0.045 0.073 0.253 300% 0 0.005 0.016 0.031 0.053 0.216