Article Contents
Article Contents

# An alternative tree method for calibration of the local volatility

• * Corresponding author: Zuoliang Xu

This work is supported by National Natural Science Foundation of China(11571365) and the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China(18XNH107)

• In this paper, we combine the traditional binomial tree and trinomial tree to construct a new alternative tree pricing model, where the local volatility is a deterministic function of time. We then prove the convergence rates of the alternative tree method. The proposed model can price a wide range of derivatives efficiently and accurately. In addition, we research the optimization approach for the calibration of local volatility. The calibration problem can be transformed into a nonlinear unconstrained optimization problem by exterior penalty method. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S & P 500 index. Numerical results confirm the excellent performance of the alternative tree pricing model.

Mathematics Subject Classification: Primary: 91G20; Secondary: 91G60.

 Citation:

• Figure 1.  The left figure presents CRR method and steps while the right figure presents TTM and steps. The blue line denotes the BS price. The red line denotes the CRR and TTM price. The green line denotes CRR price with odd steps while the black line denotes CRR price with even steps

Figure 2.  The alternative tree

Figure 3.  CRR and TTM price with different time steps

Figure 4.  Volatility function $\sigma_{ex}(t)$ and volatility estimation for $n = 7$

Figure 5.  Stability analysis of the algorithm

Figure 6.  Comparison of the exact value and the optimal with alternative tree, TTM and CRR tree

Figure 7.  Volatility calibrated by linear and quadratic penalty method

Figure 8.  Local volatility and calibrated volatility with $\frac{K}{S_0} = 100\%, 110\%$

Table 1.  Some tree methods for calibration of the local volatility

 Auther Tree method volatility function Derman(1996), Barle(1999) Recombining TTM $\sigma=\sigma(S, t)$ Li(2001) Recombining BTM $\sigma=\sigma(S, t)$ Crépey (2003) TTM with regularization $\sigma=\sigma(S, t)$ Charalambous et al. (2007) Nonrecombining BTM $\sigma=\sigma(t)$ Lok and Lyuu (2017) Recombining waterline tree $\sigma=\sigma(S)\sigma(t)$ Gong and Xu (2019) Nonrecombining TTM $\sigma=\sigma(t)$
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