A simple efficient approximation to price basket stock options with volatility smile

Abstract

This paper develops a new approach to obtain the price and risk sensitivities of basket options which have a volatility smile. Using this approach, the Black–Scholes model and the Stochastic Volatility Inspired model have been used to obtain an approximate analytical pricing formula for basket options with a volatility smile. It is found that our approximate formula is quite accurate by comparing it with Monte Carlo simulations. It is also proved the option value of our approach is consistent with the option value generated by Levy’s and Gentle’s approaches for typical ranges of volatility. Further, we give a theoretical proof that the option values from Levy’s and Gentle’s works are the upper bound and the lower bound, respectively, for our option value. The calibration procedure and a practical example are provided. The main advantage of our approach is that it provides accurate and easily implemented basket option prices with volatility smile and hedge parameters and avoids the need to use time-consuming numerical procedures such as Monte Carlo simulation.

Appendices

Appendix 1: Lemma 1

Let $\mathfrak {R}=\{\hbox { semi-martingales on }[0,T]\}$. The quadratic covariance process of $X,Y\in \mathfrak {R}$ is denoted by $\langle X,Y\rangle$. Let $\mathfrak {R}_+ =\{X\in \mathfrak {R}:X_t >0, \forall t\}$ . If $X,Y\in \mathfrak {R}_+$, Ito’s lemma applied to $\hbox {In}(X)$ gives:

$$\begin{aligned} \left\langle \hbox {In} X, \hbox {In }Y\right\rangle =\left\langle \int {\frac{dX}{X}} \int {\frac{dY}{Y}} \right\rangle =\int d \frac{\langle X,Y \rangle }{XY}. \end{aligned}$$

Appendix 2: Proof of Theorem 2

The proof that the option value in Eq. (5) is the upper bound of the option value in Eq. (3) is given below.

Since the natural logarithm is strictly concave, the finite form of Jensen’s inequality gives

$$\begin{aligned} \ln \left( {w_1 x_1 +w_2 x_2 +\cdots +w_n x_n } \right) \ge w_1 \ln (x_1 )+w_2 \ln (x_2 )+\cdots +w_n \ln (x_n ). \end{aligned}$$

The functional equations of the natural logarithm imply:

$$\begin{aligned} \ln \left( {\sum \limits _{i,j=1}^n {a_i a_j \exp \left( \int \limits _0^T {\sigma _i (k,s)} \sigma _j (k,s)\rho _{ij} ds\right) } } \right) \ge \sum \limits _{i,j=1}^n {a_i a_j } \int \limits _0^T {\sigma _i (k,s)} \sigma _j (k,s)\rho _{ij} ds \end{aligned}$$

From (5), we have:

$$\begin{aligned} \hat{{c}}=e^{-rT}(MN(\hat{{d}}_1 )-KN(\hat{{d}}_2 )), \end{aligned}$$

where,

$$\begin{aligned} \hat{{d}}_1 =\frac{\hat{{m}}-\ln (K)+\hat{{v}}^{2}}{\hat{{v}}}=\frac{\ln (M)-\ln (K)+\frac{1}{2}\hat{{v}}^{2}}{\hat{{v}}}. \end{aligned}$$

Therefore, comparing Eq. (5) with Eq. (3) the only difference is the volatility. Since the Black–Scholes price is monotone in volatility, the larger volatility has the larger option value.

Appendix 3: Proof of Theorem 3

When $\tilde{K}=0$, we have $K=M-Me^{x}$. It has been proved that $c\ge \tilde{c}$ in the case (b).

For the case:

$$\begin{aligned} M-Me^{x}\le K\le M, \end{aligned}$$

we can set

$$\begin{aligned} K=M-aMe^{x}, \end{aligned}$$

where,

$$\begin{aligned} 0\le a\le 1. \end{aligned}$$

Therefore,

$$\begin{aligned} \tilde{K}=Me^{x}(1-a), \end{aligned}$$

and

$$\begin{aligned} Ke^{x}=Me^{x}(1-ae^{x}). \end{aligned}$$

Since

$$\begin{aligned} e^{x}\le 1, \end{aligned}$$

we have:

$$\begin{aligned} Ke^{x}\ge \tilde{K}. \end{aligned}$$

Consider the put option value:

$$\begin{aligned} \tilde{p}=E[\max ((\tilde{K}-G),0)]. \end{aligned}$$

and

$$\begin{aligned} p=E\left[ \max \left( \frac{1}{e^{x}}(Ke^{x}-G),0\right) \right] . \end{aligned}$$

Then

$$\begin{aligned} p\ge \tilde{p}. \end{aligned}$$

From call-put parity, we have:

$$\begin{aligned} c-p=M-K, \end{aligned}$$

and

$$\begin{aligned} \tilde{c}-\tilde{p}=M-K. \end{aligned}$$

Therefore, we have:

$$\begin{aligned} c\ge \tilde{c}. \end{aligned}$$

For the case: $K\ge M,$

we can set

$$\begin{aligned} K=M+aMe^{x}, \end{aligned}$$

where

$$\begin{aligned} a\ge 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \tilde{K}=(a+1)Me^{x}. \end{aligned}$$

and

$$\begin{aligned} Ke^{x}= & {} (M+aMe^{x})e^{x}=Me^{x}+aMe^{2x} \\\le & {} (a+1)Me^{x}. \end{aligned}$$

Now,

$$\begin{aligned} e^{x}\le 1, \end{aligned}$$

so,

$$\begin{aligned} c=E\left[ \max \left( \frac{1}{e^{x}}(G-Ke^{x}),0\right) \right] . \end{aligned}$$

As

$$\begin{aligned} \tilde{c}=E[\max ((G-\tilde{K}),0)]; \end{aligned}$$

we have:

$$\begin{aligned} c\ge \tilde{c}. \end{aligned}$$

Appendix 4: Dataset

Tables 7, 8, 9, 10, 11, and 12 are used for the numerical examples in Sect. 4.2. The raw data are drawn from Bloomberg and are omitted here to conserve the space.

The forward prices for the assets BNS, CNR, CM and ECA in the basket are listed in Table 7.

Table 7 The forward price for assets BNS, CNR, CM and ECA

The correlation between the assets BNS, CNR, CM and ECA is calculated by the historical data from 2015/06 to 2010/06 in the Table 8.

Table 8 The correlations between BNS, CNR, CM and ECA

The strikes of the basket portfolio used in European call option are listed in the first row, the log-moneynesses $(\ln (K/F))$ which will be used in the SVI model are in the second row in the Table 9.

Table 9 The strike and log-moneyness for the basket portfolio

Table 10 The SVI parameters for the assets BNS, CNR, CM and ECA

Table 11 The SVI parameters for the assets BNS, CNR, CM and ECA

Table 12 The SVI parameters for the assets BNS, CNR, CM and ECA

The SVI parameters for each asset BNS, CNR, CM and ECA are calibrated from market implied volatilities at the date 2015/06 for each slice. The calibration of SVI parameters are discussed in Sect. 4.1. The details for the interpolation and extrapolation of calibrated slices can be found in Gatheral and Jacquier (2014).