Basket options are valuable tools in structuring financial products. A typical basket consists of several weighted underlyings, and the basket spot is given by:
Ignoring volatility smile/skew, a European option on a basket can be priced using the moment matching or geometric conditioning method. In the presence of volatility smile/skew, however, the pricing is more complicated and it is important to understand the basket smile/skew.
It is possible to construct a historic basket volatility surface from the historic time series of the individual basket components. One such approach is to minimise the relative entropy (J. Zou and E. Derman, "Strike-Adjusted Spread: A New Metric For Estimating The Value Of Equity Options", Goldman Sachs Quantitative Strategies Research Notes, July 1999). Subject to constraints of the forward and at-the-money implied volatility, a risk neutral probability density function (PDF) for the basket can be built. The risk neutral PDF can then be used to construct a historic volatility surface for the basket.
While the historic basket volatility surface provides valuable smile/skew information of the basket, it is relatively static. When the market moves, the historic information changes little, given that recent market information only constitutes a small portion of the time series. For day-to-day trading and hedging purposes, one requires different approaches to cater for rapidly changing market conditions. This article illustrates a technique to construct a basket implied volatility surface from implied volatility surfaces of the individual basket components. The implied volatility surfaces of individual components are typically easier to obtain from the market than that of the basket.
Constructing Basket Implied Volatility Surfaces
Given the implied volatility surfaces of each basket component, the Cumulative Distribution Functions (CDF) at chosen time slices for the individual component can be built. If we use simple correlation structures at chosen time slices, the basket spot path can be simulated by sampling the correlated component CDFs. Assuming there are n individual components in the basket, at a given time slice T, the following steps can be taken to construct a basket implied volatility surface:
* Build CDFs for all basket components at the chosen time
slice T. This can be done by calculating digital Calls or
Puts at the appropriate strikes. The CDFs are obviously
distributed between 0 and 1, and they can be used to
sample the basket component spot path given a uniformly
distributed random number;
* Generate n independent random Gaussian numbers (g i)
(Wiener process);
* Correlate n Gaussian numbers by Cholesky decomposition.
Given the decomposed correlation matrix Cij , the
correlated Gaussian number is given by Gi = Cij * gi ;
* Convert the correlated Gussian numbers back to uniformly
distributed numbers by reversing the Wiener process
Ui = W-1 (G i);
* Use the correlated uniform number (U i) to sample the
basket component spots (S i) from the CDFs, and calculate
the basket spot
* Calculate either a Call or Put on the basket for a given
strike K:
* Calculate the basket implied volatility from either the Call
or Put by reversing the Black-Scholes formula
The above steps can be repeated for different strikes (K) and maturities (T) to build an entire basket implied volatility surface. The constructed basket implied volatility surface can be used to price and manage a portfolio of basket products of certain types consistently. It is important to note that in order to obtain a high quality basket implied volatility surface using this technique, some numerical smoothing techniques are needed in the process.
Results
To simplify the presentation, the results in this section are for a basket of three underlyings with an identical volatility surface. The implied volatility surface of one underlying is shown in Fig. 1. The basket implied volatility surface constructed using a correlation of 30% is shown in Fig. 2. As can be seen in Fig. 2, the basket smile/skew is quite pronounced. The implied volatility points in the near right corner of the basket volatility surface for the short maturities and high strikes are not fully recovered. In practice, this is not a problem given that the missing points are all in the very low probability region and extrapolation into this region is relatively straightforward.
The correlation effects on the basket implied volatility is shown in Fig. 3. The basket implied volatility curve at the time slice year two for different correlations is plotted in the figure. It is apparent that the higher the correlation, the higher the basket volatility level. This is a well understood fact as the higher correlations indicate that the basket components move more coherently, resulting in the higher basket volatility.
Fig. 3 also shows that the slope of the volatility curve changes with the correlation. This indicates that the skew is also a function of the correlation. Fig. 4 plots the basket skew versus correlation at the time slice year two. The basket skew is defined as the volatility slope at the strike of 100%. As can be seen in Fig. 4, the basket skew increases as the correlation increases. This is because the larger correlation makes the constituents of the basket move together more often and increases the probability of the basket spot reaching large extreme values. The resulted probability density function can only be matched by that from an implied volatility curve with a larger skew.
Conclusions
Given the implied volatility surfaces of individual basket components and a relatively simple correlation structure, one can construct a basket implied volatility surface. The basket implied volatility surface can be used to extract basket smile/skew information, and to manage certain types of basket options in a consistent manner. Both volatility and skew are the positive functions of the correlation. It is possible to extend the basket volatility surface construction method to more complex correlation structures. In practice, given how the basket correlation products are hedged, a simpler correlation structure is generally more desirable.
This week's Learning Curve was written by Dong Qu, head of equity derivatives quantitative research and applications at Abbey National Financial Products in London.
