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

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.

• Intuitively if you told me I couldn’t delta hedge an option for some period (t1,t2) it would make the option more risky to sell (or buy). However it may not move the theoretical mid market value. Not sure if that helps
– dm63
Commented May 30 at 7:08

## 2 Answers

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.

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.

• You may want to address the question directly. Your references to Alice in Wonderland are fun to read but are not helpful to answering the question. Commented Apr 30 at 4:44
• As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center.
– Community Bot
Commented Apr 30 at 4:45