Towards FVA Pricing: A Martingale Approach
Tal Morgenstern (University of Sydney, Australia)

Thursday June 5, 14:00-14:30 | session 8.6 | Liquidity | room L

We have recently observed the traditional derivative pricing framework being extended to include several adjustments. These adjustments involving for instance the recognition of counterparty credit risk or funding costs were intended to fix the assumptions of the prevailing models that failed to hold in the last global financial crisis.
In the last two years, the Funding Value Adjustment (FVA) acquired greater significance, being the subject of many papers and academic discussions. However, the insertion of this adjustment in particular and of the funding costs more generally, adds another layer of complexity to the valuation of contingent claims given the recursive nature of funding decisions and therefore requires a delicate treatment using robust definitions.
In the present work, we construct a formal mathematical framework to be utilized in the valuation of derivative instruments in a market under funding costs, consequently enabling an explicit formulation for FVA. To achieve this goal, we make use of the well known martingale approach customizing it to fit our purpose.
We show that even under different funding accounts the arguments behind the market price of risk still hold in nature but at the same time old formulations are not longer valid. Although the martingale setting is still useful for the problem at hand many aspects of the classical risk-neutral valuation framework will suffer modifications. These changes affect for example the properties of the numeraires, the specification of the state price densities and of course the representation of the equivalent martingale measures.
Making use of our framework we prove that many formulations for the FVA present in the academic literature, while comparable at first glance, are truthfully inconsistent when strictly analysed. In order to examine our setting in a practical context, we apply our theoretical approach to interest rate derivatives obtaining expressions for the FVA of those instruments.

Optimal investment and contingent claim valuation in illiquid markets
Teemu Pennanen (King's College London, UK)

Thursday June 5, 14:30-15:00 | session 8.6 | Liquidity | room L

In incomplete financial markets, the classical hedging argument for valuation of contingent claims has two natural generalizations. The first one has important applications in financial supervision and accounting while the second one is more relevant in trading of financial products. In the presence of illiquidity effects, these values become nonlinear functions of the underlying cash-flows. This paper extends basic results on arbitrage bounds and attainable claims to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. Explicit consideration of swap contracts is essential in illiquid markets where the valuation of swaps cannot be reduced to the valuation of cumulative claims at maturity. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of optimal investment under conditions that extend the no-arbitrage condition in the classical linear market model. All results are derived with the ``direct method'' without resorting to duality arguments.

A pricing theory on finite number of issued securities
Yoshihiko Uchida (Bank of Japan, Japan)
Joint work with Daisuke Yoshikawa

Thursday June 5, 15:00-15:30 | session 8.6 | Liquidity | room L

We consider a new style of pricing theory which is affected by the market clearing condition with finite number of issued securities. This feature isn’t explicitly considered in the traditional finance theory which usually assumes both homogeneous agents and market completeness. Once we introduce the assumption of heterogeneous agents and finiteness of issued securities, the security price shifts from the level dominated only by market uncertainty.
We utilize a simple and convincing setting, that there are only two types of market participants with different risk aversions. We also consider the effect of the market clearing condition with finite number of issued securities. The security price is described as a result of transactions between different types of market participants. While the market clearing condition is defined such that all participants have to retain all the issued securities as a whole, we derive the first order condition of utility maximization problem for each type of market participants. This gives us the form of the security price; more precisely, the optimal payment formula for each market participant. We also show the uniqueness and existence of the security price.
By the procedure shown above, we deduce the premium due to the constraint of the finite number of issued securities; we call it `finite number premium’. We can define the rational range of finite number premium as well, where the security price meets no-arbitrage condition.
As an empirical analysis, we apply our model to the JGB and JGB futures markets. Utilizing these market data, we derive the finite number premium and did likelihood ratio test. As a result, we show the significance of the finite number premium in the market of JGB futures. It implies the validation of our model.