Black-Scholes-Merton (BSM) is an option pricing model for valuing European options. It was developed in the 1970s by Fisher Black, Myron Scholes, and Robert Merton, of whom two were awarded the Nobel Prize in Economic Sciences in 1997 for their work. The BSM model has become one of the most widely accepted pricing models for options and is used by both quantitative analysts and traders to determine the fair value of options.
The BSM model relies on several assumptions about the underlying stock price and its volatility, as well as the option holder’s risk preferences. Over the years, several advanced models have been developed to address the issue of constant volatility by using stochastic volatility models. Very few models, however, take into account the liquidity of the underlying assets, which can have a significant impact on option prices.
Reference [1] tackled the issue of liquidity in option pricing. It does so by using a mean-reverting stochastic process to describe market-wide liquidity,
This article addresses the problem of pricing European options when the underlying asset is not perfectly liquid. A liquidity discounting factor as a function of market-wide liquidity governed by a mean-reverting stochastic process and the sensitivity of the underlying price to market-wide liquidity is firstly introduced, so that the impact of liquidity on the underlying asset can be captured by the option pricing model. The characteristic function is analytically worked out using the Feynman–Kac theorem and a closed-form pricing formula for European options is successfully derived thereafter. Through numerical experiments, the accuracy of the newly derived formula is verified, and the significance of incorporating liquidity risk into option pricing is demonstrated.
In short, a closed-form formula was developed for pricing European options in which liquidity plays a role as a discounting factor.
This article provides a formal proof that liquidity has an effect on options price. We note that practitioners have been using a “volatility haircut” for a long time in order to incorporate liquidity into the pricing of complex financial instruments. It’d be interesting to see a formal link between the volatility haircut and liquidity.
References
[1] Puneet Pasricha, Song‑Ping Zhu, and Xin‑Jiang He, A closed‑form pricing formula for European options in an illiquid asset market, Financial Innovation (2022) 8:30
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