A PDE approach to option pricing in liquidity risk and large trader models
Matthias Fahrenwaldt (Leibniz Universitaet Hannover, Germany)

Tuesday June 3, 14:00-14:30 | session 2.9 | Liquidity | room H

We present a partial differential equation (PDE) approach for the pricing of contingent claims in a liquidity risk and price impacts model. Option prices under liquidity risk are shown to be solutions of a class of semilinear degenerate parabolic PDEs on bounded domains.
We prove the existence and uniqueness of weak solutions of this type of equation. The resulting derivative prices and their gradient are smooth on certain domains. We give a natural decomposition of derivative prices into a 'classical' part (without trade impact and liquidity costs) plus an error term reflecting trade impact and liquidity costs.
We build on various liquidity models and assume that prices are affected by trades through a change in the risk-return premium. We develop the framework in probabilistic terms then define the replication problem as the solution of a backward stochastic differential equation (BSDE), to which we associate a PDE.
We show the existence and uniqueness of this PDE and prove that its solution $u$ is also the solution of the associated BSDE. We choose to work with the concept of weak solutions as opposed to viscosity solutions for the simple reason that it allows us to obtain information about the growth of the gradient $D u$ in an $L^2$-space, which would be impossible to do in the viscosity framework. Indeed, an important feature of a PDE describing a liquidity setting is that the replication strategy, given by the derivative of the option price with respect to the underlying, converges to the replication strategy in a frictionless setting when the number of options replicated is small.
The detailed analysis leads to explicit and computable bounds on the $L^2$-norms for the derivative price and its delta. Due to the analysis being highly explicit, it serves as a basis for numerical approaches to the PDE. The high regularity of the option prices suggests that numerical approaches will have a fast convergence rate.

Optimal market making strategies under inventory constraints
Etienne Chevalier (University of Evry, France)
Joint work with M'hamed Gaigi, Vathana Ly Vath and Mohamed Mnif

Tuesday June 3, 14:30-15:00 | session 2.9 | Liquidity | room H

We study the problem of a market-maker acting as a liquidity provider by continuously setting bid and ask prices for an illiquid asset. We assume that the market maker has a contractual obligation to permanently quote bid and ask prices for the security and therefore to satisfy any sell or buy order from the asset's investors. On the opposite side of the trades, there are investors who act as liquidity takers by submitting either sell or buy market orders. The arrival of buy (and sell) market orders submitted by the investors is assumed to follow a Cox process with regime-shifting Markov intensity.
The role of the market maker is very important in the trading of illiquid assets as it acts as a facilitator of trades between different investors. The market maker may therefore benefit from the bid-ask spread but faces a number of constraints, in particular the liquidity and inventory constraints. The objective is to maximize the expected utility of the market maker's terminal wealth. We characterize our objective functions as unique viscosity solutions to the associated HJB system. We further enrich our study with some numerical results.

Optimal Debt Maturity Structure in a Model with Market and Funding Liquidity Risk
Eva Lütkebohmert (University of Freiburg, Germany)
Joint work with Daniel Oeltz and Yajun Xiao

Tuesday June 3, 15:00-15:30 | session 2.9 | Liquidity | room H

We present an integrated structural model for credit and liquidity risk which allows to determine an optimal debt maturity structure. We consider a firm which finances its risky assets by a mixture of short- and long-term debt as well as equity. The firm is exposed to its internal funding liquidity risk (rollover risk) through possible runs by short-term creditors. When creditors' have less faith in the firm's ability to either draw on its pre-committed credit lines from other financial institutions or to pledge its risky assets as collateral to raise new funding, this leads to increasing credit spreads to prevent creditors to withdraw their funding.
In our model credit spreads are derived endogenously at rollover dates while equity holders have to bear the gains and losses from rolling over short-term debt subject to their limited liability.
When funding liquidity dries up and the firm cannot raise further equity, it is forced to prematurely liquidate its assets on a secondary market for a firesale price. External market liquidity enters our model as firesale rates depend on the overall market situation. When the firm pledges its risky assets as collateral to raise new capital, investors impose margin requirements to ensure that the collateral is sufficient to cover the firm's value-at-risk or expected shortfall. Both quantities highly depend on the volatility of the firm's fundamental value which we model stochastically in a Heston framework.
When there is a liquidity shock to the overall financial market, volatilities increase. This induces creditors to become more risk averse and to require higher margins such that the assets sell at lower firesale rates. This again reinforces the firm's exposure to funding liquidity risk. Through this channel macro-economic effects, modelled as random shocks to the firm's volatility, influence market liquidity and feed back on the firm's internal funding liquidity through increasing margin requirements of investors.