Options derive their value from underlying assets. In the case of traded underlyings, the option value is affected by the asset liquidity. The impact of liquidity on equities and bonds is a well-known phenomenon: selling the asset pushes the price down, buying the asset moves the price up. Option hedging is nothing but selling and buying some quantity of the underlying asset. Liquidity can be viewed as part of a chain reaction in hedging: changes in the asset value result in the option owner re-hedging which in turn impacts the asset's liquidity and so on. Therefore, liquidity has to be taken into account when pricing traded options.
Asset liquidity can be determined as the total number of shares or bonds to be traded to move the asset price by one point. According to some traders' estimate, selling the average daily volume of an equity drives the share's price down by 3%. When there are no excessively large fluctuations in the average daily volume, liquidity can be approximated by a constant parameter. In general, liquidity has to be treated as a random variable. The intuition behind the liquidity-adjusted option pricing procedure is quite simple.
The starting point is to consider a modified equation of motion for the asset
where µS and are the asset price drift and the asset price volatility, respectively. Here W(t) is the Wiener process. The last term is due to the liquidity back reaction with L being the asset liquidity. At this point the functions µS , and L can be arbitrary. Correspondingly, dN is the change in the asset position in the time interval dt. It is clear that dN is equal to the change in the asset position due to delta-hedging. Therefore, we have to choose .
Following the optimal hedging strategy for a portfolio , where S is driven by the above equation of motion, one arrives at a liquidity-adjusted Black-Scholes equation for option value V:
Here r is the risk-free rate and q is the continuous dividend or coupon held.
Let us assume that L is constant. By numerically solving the given equation one can find liquidity-adjusted option values. These solutions have several remarkable properties. One is that the liquidity correction to the pricing equation naturally gives rise to the bid-offer spread in the option values. Indeed, the short value associated with the ask price is greater than the long value associated with the bid price. As the liquidity increases, that is L goes up, the spread gets narrower.
The Volatility Smile
The most important observation is that the solutions for option values produce implied volatilities which are consistent with the volatility smile of market option prices. This works as follows: We first calculate the liquidity-adjusted value of the option and then use the standard Black-Scholes equation with a constant volatility to compute the corresponding implied volatility which fits the given liquidity-corrected value. To compare the liquidity-generated implied volatility with the market implied volatility we use the fact that options are traded at prices close to the bid value. Therefore, we have to take the liquidity-adjusted theoretical option value corresponding to the long position. Further, we choose the diffusion part of the equation of motion in the standard log-normal form: *S= *S with * being constant historical volatility.
The results of our calculations are presented by two graphs. These are liquidity-generated implied volatilities for American puts and calls with different times to expiry.
The plots obtained for option values take into account transaction costs and discrete hedging. The liquidity coefficient is chosen as a large number because it is proportional to the total number of options in the market, which can be substantial--even greater than the daily average volume of the asset.
As one can see the smile, is more pronounced for bid options with shorter maturities, as is the case for options on equities.
Conclusion
To conclude the liquidity-adjusted Black-Scholes model offers a consistent theoretical description of a range of market effects. Liquidity appears to be responsible for such an important phenomenon as the volatility smile. Of course, the considered model is still an approximation because it does not take account of such events as price shocks which can be further integrated into the pricing equation via Poisson processes. Also, as we mentioned above, the liquidity parameter L can change randomly. So can the historic volatility. However, liquidity-adjusted solutions even with constant liquidity and historic volatility contain very important information about the market option value behaviour.
This week's Learning Curve was written by Oleg Soloviev and Oleg Rudenkoi, ceo and software developer, respectively, at Econophysicain London.
