Calibration procedure

In this subsection, we resume the calibration procedure of PBS model using Variance Swap secu-rities and we propose some improvements and remarks.

The basic procedure is the following one:

Algorithm 6.5 (Calibration of Perturbative Black Scholes model)

1. for each maturity T, compute, using vanilla contingent claims prices founded on the market, the price of Variance Swap over the same period, and set the value on cumulated volatility σ₀(T)√

T to be equal to the square root of Variance Swap premium;

2. fix a bearable risk probability α <0.5(this parameter is the only exogenous one, an economic argument is needed);

3. evaluate the bid-ask spread on the market and, using theorem 6.2, fix the variance of cumu-lated volatility ǫ Γh

ςT

√Ti

σ0(T)√ T;

4. choose a really traded vanilla option with a strike around the money and compute the bias of cumulated volatility ǫ Ah

ςT

√Ti

σ0(T)√

T, thanks to equation (6.18).

Remark 6.4 Clearly, this procedure is a rough estimate and it is inconsistent, since PBS price for Variance Swap securities depends on the bias, see equation (6.14). Another drawback concerns the choice of At-The-Money option, calibration result depends deeply on this selection.

In order to avoid the inconsistence of previous calibration procedure, we propose to iterate it until the estimated parameters verify relation (6.5) with a set accuracy. At the end of a loop, we hold the previous value for bias and variance of cumulated volatility and we define a new estimation of this one, thanks to relation (6.5). In order to strengthen the calibration, we propose to fix a basket of at-the-money vanilla options and to fit the bias of volatility over this basket rather than to choose a single option.

6.5 Conclusion

In this chapter we have proposed a calibration procedure for Perturbative Black Scholes model, see chapter 4, based on prices of Variance Swap securities. Variance Swaps are a no-risk-model contingent claims, so their premium represents an implicit constraint to be verify by a calibrated model. In the last section, we have showed an iterative procedure that can be performed quickly, thanks to the closed-forms relations of each step. The only exogenous parameter remains the bearable risk probability accepted by the trader in exchange for the expected return.

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Part III

Partial Differential Equations and