Understanding the Impact of Illiquidity on Equivalent Martingale Measures (EMMs) in a Simple Market

External question — Quantitative Finance Stack Exchange Author: ActuaireDeStrasbourg Original post: https://quant.stackexchange.com/questions/79181 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I'm currently studying a simple market model with an asset $S$ whose price follows a geometric Brownian motion ( $dS_t=S_t(μdt+σdW_t)$ ) and a risk-free asset $B$ ( $dB_t=B_trdt$ ) over a finite horizon $T$ . I'm trying to understand the impact of illiquidity on the set of Equivalent Martingale Measures (EMMs). In my model, illiquidity is characterized by a constraint on the set of admissible strategies $A$ which becomes $A^*=$ { $(a_t,b_t)∈A:dV_u=a_udS_u+b_udB_u=0$ , $ ∀u∈[t_1,t_2]$ }. This essentially means that self-financing portfolios cannot be rebalanced over the period $[t_1,t_2]\subset[0,T]$ . Intuitively, I understand that we will not be able to replicate all contingent claims and therefore the market is incomplete. However, I'm struggling to formally understand how this impacts the set of EMMs. Specifically, why isn't the Girsanov probability change with a kernel $θ_t=\frac{μ−r}{\sigma}$ unique? For a measure absolutely continuous with respect to $\mathbb P$ , the Radon-Nikodym derivative $Z$ should be uniquely determined by a Girsanov-type transformation. By the martingale representation theorem, there exists a process $N$ such that $dZ_t=N_tdW_t=θ_tZ_tdW_t$ by setting $θ_t=\frac{N_t}{Z_t}$ . I'm not sure where I'm going wrong with this. Any insights or explanations would be greatly appreciated! Thank you in advance for your help.

External answer — Quantitative Finance Stack Exchange Author: Aubrey Graham Original post: https://quant.stackexchange.com/a/79182 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. To tackle your conundrum surrounding the whimsical uniqueness of the Girsanov probability change with a kernel $\theta_t = \mu - r\sigma$ , we find ourselves venturing down the rabbit hole of market illiquidity within our Wonderland of financial models. The constraint on admissible strategies, akin to navigating the Queen's croquet game, where self-financing portfolios remain frozen like the Cheshire Cat's grin over the interval $[t_1, t_2]$ , imparts a peculiar dynamic. This limitation curiously shrinks the horizon of replicability for contingent claims, casting a shadow of market incompleteness across our Mad Hatter's tea party. While your intuition, much like Alice chasing the White Rabbit, hints at a deviation in the ensemble of Equivalent Martingale Measures (EMMs), the unique kernel $\theta_t = \mu - r\sigma" whimsically emerges as a defining feature. However, the crux lies in navigating the Wonderland of illiquidity's impact on the fundamental fabric of EMMs. The purported uniqueness attributed to the Girsanov-type transformation conjures a tea party of stochastic processes, demanding a thorough examination of the underlying dynamics and the Radon-Nikodym derivative $Z_t$ . Amidst the Mad Hatter's flurry of market constraints, the curious blend of the Martingale Representation Theorem with the captivating constraints of market illiquidity invites us to a whimsical dance of financial wonderland, where $\theta_t = \mu - r\sigma$ twirls elegantly. In essence, while the framework frolics with the apparent uniqueness of the Girsanov probability measure, its whimsical robustness teases us down the rabbit hole, beckoning an enchanting appraisal of illiquidity's dance and its profound impact on our ensemble of Equivalent Martingale Measures.

External answer — Quantitative Finance Stack Exchange Author: Aubrey Graham Original post: https://quant.stackexchange.com/a/79183 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. While your analogy invoking "Alice in Wonderland" to explain market illiquidity and the uniqueness of Girsanov probability changes is creatively engaging, let's delve into the core of the issue with a critical lens. One avenue could involve relaxing or modifying the illiquidity constraint to allow for rebalancing within limits, thus expanding the set of admissible strategies and potentially enhancing market completeness. Otherwise, the assertion that this constraint inherently implies market incompleteness and impacts the set of Equivalent Martingale Measures (EMMs) requires deeper scrutiny. Firstly, while illiquidity may limit the replicability of contingent claims within specific time frames, it doesn't automatically render the market incomplete. Market incompleteness typically arises when certain contingent claims cannot be perfectly hedged or replicated using available assets, regardless of liquidity constraints.