Funding, repo and credit inclusive valuation as modified option pricing

D Brigo,C Buescu,M Rutkowski

doi:10.1016/j.orl.2017.10.009

Abstract

We take the holistic approach of computing an OTC claim value that incorporates credit and funding liquidity risks and their interplays, instead of forcing individual price adjustments: CVA, DVA, FVA, KVA. The resulting nonlinear mathematical problem features semilinear PDEs and FBSDEs. We show that for the benchmark vulnerable claim there is an analytical solution, and we express it in terms of the Black–Scholes formula with dividends. This allows for a detailed valuation analysis, stress testing and risk analysis via sensitivities.

Highlights

Prior to the financial crisis of 2007-2008, institutions tended to ignore the credit risk of highly-rated counterparties in valuing and hedging contingent claims traded over-the-counter (OTC), claims which are bilateral contracts negotiated between two default-risky entities
One of the explosive manifestations of this crisis was the sudden divergence between the rate of overnight indexed swaps (OISs) and the LIBOR rate, pointing to the credit and liquidity risk existing in the interbank market
This forced dealers and financial institutions to reassess the valuation of OTC claims, leading to various adjustments to their book value

Summary

Introduction

Prior to the financial crisis of 2007-2008, institutions tended to ignore the credit risk of highly-rated counterparties in valuing and hedging contingent claims traded over-the-counter (OTC), claims which are bilateral contracts negotiated between two default-risky entities. Since the market model under study is linear, it is equivalent to postulate that φ replicates the payoff X and to set Pt = Vtφ Note, that this step would not be possible if trading in primary assets was nonlinear (for instance, in models with differing lending and borrowing treasury rates). The process P is governed under Qβ by dPt − rtC Pt dt = ∆tσSt dWtβ with terminal condition PT = (ST − K)+ where W β is the Brownian motion under Qβ This leads to the following probabilistic representation for Pt. Pt = e−rC(T −t) Eβ [(ST − K)+ | Ft] = e−(rC −fβ)(T −t) Eβ [e−fβ(T −t)(ST − K)+ | Ft], which in turn yields (10) through either standard computations of conditional expectation or by noting that it is given by the Black-Scholes formula with the interest rate f β and no dividends. The effective funding rate f β should be replaced by f β − δ

Vulnerable Call Option Pricing by Adjusted Cash Flows Approach

Sensitivity Analysis t