^{1}

^{2}

^{3}

^{*}

^{1}

^{2}

^{3}

Edited by: Alexander Melnikov, University of Alberta, Canada

Reviewed by: Jianfeng Guo, Xian University of Posts and Telecommunications, China; Edward W. Sun, KEDGE Business School France, France

*Correspondence: Sergei Levendorskiĭ, Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

This article was submitted to Mathematical Finance, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We generalize the Piterbarg [

Counterparty credit risk and the cost of funding in the valuation of derivatives have become a paramount topic in the industry. Counterparty credit risk can be defined as the risk of a party to a financial contract defaulting prior to the contract's expiration and not fulfilling all of its obligations. Each party to a transaction runs the risk of a loss from the counterparty defaulting; there is also the possibility of a gain from an increase of the self-default risk which reduces the counterparty's expectation of the transaction value. The expected loss from the default of a counterparty is generally referred to as the credit valuation adjustment (CVA), whilst the expected benefit from self-default risk is referred to as debt valuation adjustment (DVA). The question of whether the gain from increasing self default risk can actually be monetized and therefore should be included in derivative valuation values remains debatable, and it is currently excluded from prudential capital calculations under CRD IV, but must be included under generally accepted accounting rule as specified under IFRS 13.

Counterparty credit risk can be mitigated by collateralization. Each party to a transaction agrees to post/receive collateral when the value of the transaction moves against / in favor of one of the party under a collateral agreement. Transactions are generally executed via central counterparties (CCP) under robust collateral agreements which require posting of both initial and variation margins, whilst regulations of over-the-counter (as proposed by the Basel Committee in September 2013) will soon also make it mandatory for both parties to post initial and variation margins. Collateral that are posted against derivative transactions generally receive risk-free returns but must be funded/borrowed at non-risk-free rates, therefore introduces an additional cost to each counterparty, and the expected value of this cost is generally referred to as funding valuation adjustment (FVA). Since collateral funding cost is a function of the collateral requirement, which in turn is a non-linear function of the value of the derivative, the pricing problem involving FVA is non-linear and hence non-trivial.

In numerous studies, these adjustments (CVA, DVA, and FVA) have been analyzed separately. Pykhtin and Zhu [

An initial analysis of collateralization and funding risk in is due to Piterbarg [

In the paper, we consider a generalization of Piterbarg's [

A rigorous study of such a complicated situation requires an explicit specification of objective functions of agents involved (the trader and the treasury/XVA desk), and the result of such a study will strongly depend on these objective functions and the bargaining power of the agents involved. We leave this study for the future. In the present paper, we make the following simplifying assumption which is close but not identical to the remark made by Piterbarg [

Under the above assumption, we therefore use an “exterior” equivalent martingale measure (EMM) to price all flows inside the tiny submarket that involves the trader and treasury/XVA desk. However, we explicitly take into account the rules of the asset pricing inside this tiny market that are specified by the regulatory requirements. We leave for the future an extremely important question whether these requirements are rational or not.

The paper organized as follows. The model is described in detail In Section 2, we formulate the model in detail, and formulate non-linear boundary problems for different types of options. Section 3 describes the Carr's randomization algorithm for the problem and its realization. The detailed numerical scheme are derived in Section 4, and explicit algorithms are formulated in Section 5. Numerical examples are given in Section 6, and Section 7 concludes. Technical details are relegated to Appendices.

Assume that, apart from the security that we model, the market is void of frictions and arbitrage free. Then, under additional technical regularity conditions, there exists an equivalent measure ℚ such that the price of any security traded in the market is the expected value of discounted instantaneous payoffs and payoff streams that the security gives the right to, under ℚ. Assume also that, under ℚ, the discounted prices of all securities are functions of a strong Markov process

In short rate models, it is natural to regard

The constructions below can be easily generalized to the case of the stochastic interest rate; the rate can be time-dependent. However, in the present paper, we assume that the riskless rate

If the external market is the Black-Scholes market, then Equation (2.3) is

In the model, which takes into account all the payments related to required collateral posting and funding, the stream of the total payoffs during the life-time of the contingent claim becomes rather involved. What is very important, and what makes the whole idea of the no-arbitrage pricing in the presence of FVA incorrect, strictly speaking, is the non-linearity of the pricing equation. Indeed, as it is shown below, the payoff stream

Before we go into the description of the stream

The stream that accounts for the cost of collateral posting/receiving. The stream is (_{C})_{C} is the collateral rate (for example the US effective FED rate).

The stream that accounts for the cost of unsecured funding. The stream for unsecured funding is (_{F})(_{F} is the unsecured funding rate.

The on-default cashflow. The stream is
^{+} = max{^{−} = min{_{i} is the recovery rate respectively for

The collateral depends on _{A}, _{B} are the collateral thresholds for _{i}, λ, _{i} are constants, hence, the stream of payoffs can be considered as a function of one argument

Adding up these streams, we obtain the following equality

The two extremes are uncollateralized and fully collateralized cases considered in Piterbarg [

In the latter case, _{C})_{C})

The underlying stock or index is modeled as

Recall that the characteristic exponent of a Lévy process _{t}}_{t ≥ 0} is a continuous function ψ:ℝ → ℂ satisfying ψ(0) = 0 and

From now on we will assume that there exist λ_{−} < −1 < 0 < λ_{+} such that the underlying Lévy process _{−}, λ_{+}). Roughly speaking, this means that the characteristic exponent ψ(ξ) admits analytic continuation into the open strip Im ξ ∈ (λ_{−}, λ_{+}). Moreover, ψ(ξ) grows at most polynomially as Re ξ → ±∞ within every closed strip Im ξ ∈ [ω_{−}, ω_{+}] ⊂ (λ_{−}, λ_{+}). A precise formulation (in terms of the Lévy density of

A Brownian motion (used in the classical Black-Scholes model [

In Merton's model [

A hyper-exponential jump-diffusion process introduced in Lipton [^{±} are positive integers and ^{+} = ^{−} = 1. A Lévy process with characteristic exponent Equation (2.9) is of exponential type

Lévy processes of the ^{1}_{−} < 0 < λ_{+} are called the _{−}, λ_{+}), so there is no conflict of notation.

Variance Gamma (V.G.) processes were first used in empirical studies of financial markets by Madan and collaborators [^{2}_{−} < 0 < λ_{+}, _{−}, λ_{+}).

Normal Inverse Gaussian (NIG) processes were introduced to finance by Barndorff-Nielsen [

Other examples of Lévy processes of exponential type can be found in Boyarchenko and Levendorskiĭ [

Let _{Eur.}(_{t}) be the price of an European option with maturity _{Eur.}(_{t}) is a unique bounded solution of the boundary problem

If

Let _{d.o.put}(_{t}) be the price of the down-and-out put option with maturity _{d.o.put}(_{t}) is a unique bounded solution of the boundary problem

Let _{f.t.d.}(_{t}) be the price of the down-and-in first touch digital option with maturity _{f.t.d.}(_{t}) is a unique bounded solution of the boundary problem

In the next section, we will use the Carr's randomization and backward induction to solve the boundary problems for European options and barrier options, with CVA and FVA.

Define the

Given any _{q} ~ Exp ^{−1}. The form of the Wiener-Hopf factorization (WHF) formula that is commonly used in probability theory is as follows:

Introducing the

Since the trajectories of the supremum process

Define operators

We also define the EPV operator

The

Carr's randomization [_{0} < _{1} < … < _{N} = _{s}, _{s + 1}] is replaced with an exponentially distributed random maturity period with mean Δ_{s} = _{s + 1} − _{s}. Moreover, these _{s} =

Denote by _{s}(_{N−s}, _{0}(

For all 0 ≤ _{s + 1}(^{h}, terminal payoff function _{s}(_{s} and a nonlinear stream _{s}(_{s}(

Then, we discretize the time derivative in equation Equation (2.15) by a finite difference and rewrite Equation (2.15) in the implicit-explicit form: for

We define

The function _{N}(_{N}(

In this case, Equation (3.8) holds on the whole line. Using the well-known equality

For all _{s + 1}(_{0}(

The formulas are as follows:
^{−1}, defined by ^{+}(^{−}(

We now suppose that there exist λ_{−} < 0 < λ_{+} such that the Lévy process _{−}, λ_{+}). (cf. Section 2.2). Under a certain regularity assumption on the characteristic exponent ψ(ξ) of

Here, ω_{±} are real numbers such that λ_{−} < ω_{−} < 0 < ω_{+} < λ_{+}, and such that there exists δ > 0 with Re(_{−}, ω_{+}].

Let us consider one of the formulas Equation (4.1), or the definition of the Fourier transform. With the standard approach to the numerical realization of these formulas, one truncates the improper integral on the right hand side replacing it with an integral over a bounded interval, and uses a suitable quadrature rule to approximate the latter integral with a finite sum. However, it is demonstrated in Boyarchenko and Levendorskiĭ [

Given functions

In order to discretize the problem, we assume given a uniformly spaced grid of points _{j} = _{1}+(_{j} = _{j}) for all _{j}, _{j+1}]:

Of course, Equation (4.5) is an exact equality for _{j} and for _{j + 1}; inside the interval (_{j}, _{j + 1}), the error of this approximation is controlled by the size of the second derivative

The next result is obtained by a straightforward computation.

_{1}, _{M}] using Equation (4.5), and let us approximate _{1}, _{M}]. This leads to the following approximation of the values of the function _{j} = _{j}), and the coefficients

In this subsection we recall the formulas for the enhanced convolution realization of the operators

We quote [

_{q}(

_{0}(ξ), _{1}(ξ) and _{Δ}(ξ) at ξ = 0 are, of course, removable, as one can easily verify using the power series expansion of the exponential function.

The enhanced convolution realization of _{j} = _{j}) for 1 ≤

The situation is similar to that of Section 4.4.2, except that if we view _{j} = _{j}) and

In previous section, we explain the backward induction scheme for CVA and FVA using the EPV operators and their realizations. As one may expect the non-linear stream would may lead to sizeable errors due to a large number of steps in the backward induction scheme. We introduce a correction term to overcome this problem. The suggested method is relatively straightforward.

We use piecewise linear function to approximate the function _{s}(_{s}_{s}(_{s}(_{s} = {−_{A}, 0, _{B}}. The piecewise linear approximation may give an inaccurate approximation to _{s} = _{B}.

Denote _{s,i} = _{s}(_{i}), and let _{s}. We the smallest _{i} such that _{i} ≥ _{B}, that is _{i} = min{_{B}}, and if is not large, then we use the linear interpolation to locate the kink

Since the price of the European put decays fast as _{s,i+1} − _{s,i} is not negligibly small.

The correction can be calculated using the standard Fourier transform technique. In Figure

We represent

Thus, the corrected

The Fourier transform of Ĝ(ξ) is easy to calculate

In this section, we formulate explicit algorithms for calculation of prices of European options, down-and-out barrier put options and down-and-in first-touch digital options. The algorithms are similar to the ones in [_{t} is a Lévy process of exponential type (λ_{−}, λ_{+}), with λ_{−} < −1 < 0 < λ_{+}. The detailed algorithms are as follows:

(1) One must describe the market by giving the riskless rate

(2) One must specify the collateral rate, unsecured funding rate, default intensity, recovery rate and collateral thresholds.

(3) One must specify the maturity date,

(4) We will use Carr's randomization in the situation where the maturity period, [0, _{t} =

These steps constitute the input of the initial data for the algorithms. The algorithms consist of blocks borrowed from Boyarchenko and Levendorskiĭ [_{N}(

(5) Choose a uniformly spaced grid _{j} = _{1}+(_{1} = −_{1} is

(6) Set ζ = 2π∕(_{2}, _{3} so that the dual grid
_{1} = _{2} · _{3} and ζ_{1} = ζ∕_{2}, is sufficiently long and sufficiently fine. Since one of the subsequent steps uses FFT for arrays of length 2_{1}, we recommend making the choices so that _{2} and _{3} are powers of 2.

(7) Calculate the values of

(8) Calculate the convolution coefficients

To complete the calculation of prices using Carr's randomization procedure, we must now consider the three types of options separately. At each step below, the inputs are function values on the chosen grid, and the output are the values of another function at the same grid. For simplicity, we write _{j} on the grid.

The remaining steps are as follows:

(8) Set _{0}(_{j}) = _{j}) for each

(9) For _{s}(_{j})) and its correction using the procedures in 4.5 then

(10) The vector _{N} is the approximation to the value function of the European option, at points of the chosen grid.

For a down-and-out barrier put options, the remaining steps are as follows:

(8) For 1 ≤ _{0}:

For 0 ≤ _{s}(_{s}_{s}(_{s}(_{1}(

(9) In the cycle for _{s}(

Calculate the correction for

(10) For

The vector _{N} is the desired approximation to the value function of the down-and-out barrier put option.

For down-and-in first touch digital options, it is convenient to work with the auxiliary function _{s}(_{s}(

(8) For _{0}(_{[h, +∞)}(_{s}(_{s}(

Calculate

(9) In the cycle for _{s}(_{s}(_{s}_{s}(_{s}(

(10) For

The vector _{N} is the desired approximation to the value function of the down-and-in first touch digital option.

All calculations and the results presented in this paper were performed in MATLAB R2014a, on a computer with processor Intel Core i5 (2.7 GHz), memory 8 GB 1600 MHz DDR3, under the Mac OS X 10.9.4 operating system. The tables are collected in Appendix C.1.

We calibrate some of the models to June 2015 put options on the Hang Seng index as of 9 January 2015. Namely, we calibrate the market data to the DEJD, KoBoL, VG and NIG models. We use the algorithms in Section 5 to calculate prices for European options, down-and-out barrier put options and down-and-in first touch digital options for these models. As one may expect the trade volume should be better when the option is close to maturity, say less than 1 month. Hence we then use the same procedures to re-calibrate the models again to June 2015 put options on the Hang Seng Index as of 28 May 2015 and price the options with adjustments. We observe that when the time to maturity decreases, the total adjustments become similar across different models but the linearization errors remain.

Prices and adjustments for European options are presented in Tables

_{C} _{F} _{B} _{B}

1354.24 | 1348.04 | 1349.70 | 1354.00 | 1353.70 | |

Total adjustments | −10.21 | −16.49 | −14.79 | −10.46 | −10.54 |

1364.45 | 1364.52 | 1364.49 | 1364.46 | 1364.24 | |

CVA (Separate) | −5.84 | −12.12 | −10.42 | −6.07 | −6.15 |

FVA (Separate) | −4.43 | −10.65 | −8.94 | −4.60 | −4.68 |

Combined adjustments | −10.26 | −22.77 | −19.36 | −10.67 | −10.83 |

Linearization error | 0.55% | 38.08% | 30.88% | 2.00% | 2.79% |

CPU time (in s) | 2.50 | 2.51 | 2.68 | 2.52 | 2.52 |

_{C} _{F} _{B} _{B}

798.69 | 798.72 | 797.92 | 798.09 | 798.12 | |

Total adjustments | −1.92 | −1.92 | −3.89 | −1.98 | −1.98 |

800.61 | 800.64 | 801.81 | 800.07 | 800.09 | |

CVA (Separate) | −1.16 | −1.16 | −3.12 | −1.21 | −1.21 |

FVA (Separate) | −0.89 | −0.89 | −2.84 | −0.93 | −0.93 |

Combined adjustments | −2.05 | −2.05 | −5.96 | −2.14 | −2.14 |

Linearization error | 6.71% | 6.72% | 53.35% | 8.21% | 8.13% |

CPU time (in s) | 2.41 | 2.42 | 2.41 | 2.42 | 2.38 |

_{C} _{F} _{B} _{B}

2045.67 | 1672.68 | 1349.69 | 1076.25 | 849.61 | |

Total adjustments | −18.58 | −16.58 | −14.79 | −13.16 | −11.65 |

2064.25 | 1689.26 | 1364.48 | 1089.41 | 861.26 |

_{C} _{F} _{B} _{B}

1348.34 | 1053.39 | 797.92 | 587.23 | 422.22 | |

Total adjustments | −4.48 | −4.20 | −3.89 | −3.52 | −3.09 |

1352.82 | 1057.59 | 801.81 | 590.75 | 425.31 |

_{C} _{F} _{B} _{B}

_{C} _{F} _{B} _{B}

0.5664 | 0.5484 | 0.5480 | 0.5469 | 0.5473 | |

Total adjustments | −0.0082 | −0.0084 | −0.0082 | −0.0080 | −0.0080 |

0.5746 | 0.5568 | 0.5562 | 0.5550 | 0.5553 | |

CVA (Separate) | −0.0056 | −0.0059 | −0.0057 | −0.0055 | −0.0055 |

FVA (Separate) | −0.0025 | −0.0029 | −0.0027 | −0.0025 | −0.0025 |

Combined adjustments | −0.0081 | −0.0088 | −0.0084 | −0.0080 | −0.0080 |

Linearization error | 0.75% | 4.44% | 1.93% | 0.63% | 0.57% |

CPU time (in s) | 2.50 | 2.48 | 2.68 | 2.52 | 2.51 |

_{C} _{F} _{B} _{B}

0.6009 | 0.6009 | 0.6125 | 0.6161 | 0.6140 | |

Total adjustments | −0.0016 | −0.0016 | −0.0016 | −0.0016 | −0.0016 |

0.6025 | 0.6025 | 0.6141 | 0.6177 | 0.6156 | |

CVA (Separate) | −0.0010 | −0.0010 | −0.0010 | −0.0009 | −0.0009 |

FVA (Separate) | −0.0005 | −0.0005 | −0.0005 | −0.0005 | −0.0005 |

Combined adjustments | −0.0015 | −0.0015 | −0.0015 | −0.0014 | −0.0014 |

Linearization error | 8.80% | 8.80% | 8.6% | 12.35% | 10.68% |

CPU time (in s) | 2.40 | 2.41 | 2.41 | 2.43 | 2.40 |

_{C} _{F} _{B} _{B}

0.7091 | 0.6301 | 0.5480 | 0.4668 | 0.3901 | |

Total adjustments | −0.0096 | −0.0090 | −0.0082 | −0.0073 | −0.0063 |

0.7187 | 0.6391 | 0.5562 | 0.4741 | 0.3964 |

_{C} _{F} _{B} _{B}

0.8057 | 0.7184 | 0.6125 | 0.4969 | 0.3834 | |

Total adjustments | −0.0019 | −0.0017 | −0.0016 | −0.0015 | −0.0013 |

0.8075 | 0.7201 | 0.6141 | 0.4984 | 0.3847 |

_{C} _{F} _{B} _{B}

Prices and adjustments for down-and-out put options are presented in Tables

_{C} _{F} _{B} _{B}

0.2643 | 0.4130 | 0.7063 | 1.6509 | 1.1543 | |

Total adjustments | −0.0009 | −0.0013 | −0.0021 | −0.0044 | −0.0032 |

0.2653 | 0.4143 | 0.7084 | 1.6553 | 1.1575 | |

CVA (Separate) | −0.0003 | −0.0003 | −0.0004 | −0.0006 | −0.0005 |

FVA (Separate) | −0.0007 | −0.0010 | −0.0017 | −0.0038 | −0.0027 |

Combined adjustments | −0.0009 | −0.0013 | −0.0021 | −0.0044 | −0.0032 |

Linearization error | 0.02% | 0.02% | 0.02% | 0.01% | 0.01% |

CPU time (in s) | 12.28 | 12.78 | 16.70 | 13.11 | 12.69 |

_{C} _{F} _{B} _{B}

0.84862 | 0.92727 | 2.95393 | 9.61763 | 23.39111 | |

Total adjustments | −0.00065 | −0.00071 | −0.00208 | −0.00651 | −0.01549 |

0.84927 | 0.92797 | 2.95602 | 9.62413 | 23.40660 | |

CVA (Separate) | −0.00008 | −0.00008 | −0.00011 | −0.00015 | −0.00009 |

FVA (Separate) | −0.00058 | −0.00063 | −0.00197 | −0.00636 | −0.01541 |

Combined adjustments | −0.00065 | −0.00071 | −0.00208 | −0.00651 | −0.01549 |

Linearization error | 0.0026% | 0.0024% | 0.0013% | 0.0006% | 0.0002% |

CPU time (in s) | 12.89 | 12.92 | 13.64 | 13.35 | 12.81 |

_{C} _{F} _{B} _{B}

0.2041 | 0.4750 | 0.7063 | 0.9138 | 1.1005 | |

Total adjustments | −0.0006 | −0.0014 | −0.0021 | −0.0027 | −0.0032 |

0.2047 | 0.4764 | 0.7084 | 0.9165 | 1.1038 |

_{C} _{F} _{B} _{B}

0.92658 | 1.64812 | 2.95393 | 4.07302 | 4.94760 | |

Total adjustments | −0.00066 | −0.00117 | −0.00208 | −0.00286 | −0.00346 |

0.92724 | 1.64929 | 2.95602 | 4.07587 | 4.95107 |

_{C} _{F} _{B} _{B}

0.8896 | 0.8681 | 0.8318 | 0.7622 | 0.7955 | |

Total adjustments | −0.0013 | −0.0015 | −0.0017 | −0.0022 | −0.0020 |

0.8909 | 0.8696 | 0.8335 | 0.7644 | 0.7975 | |

CVA (Separate) | −0.0008 | −0.0009 | −0.0011 | −0.0013 | −0.0012 |

FVA (Separate) | −0.0002 | −0.0002 | −0.0002 | −0.0002 | −0.0002 |

Combined adjustments | −0.0011 | −0.0011 | −0.0013 | −0.0015 | −0.0014 |

Linearization error | 20.53% | 23.88% | 28.08% | 33.30% | 31.12% |

CPU time (in s) | 11.60 | 11.76 | 12.27 | 12.29 | 14.42 |

_{C} _{F} _{B} _{B}

0.84372 | 0.84296 | 0.77286 | 0.68497 | 0.73357 | |

Total adjustments | −0.00047 | −0.00047 | −0.00057 | −0.00066 | −0.00046 |

0.84419 | 0.84343 | 0.77343 | 0.68564 | 0.73403 | |

CVA (Separate) | −0.00033 | −0.00033 | −0.00039 | −0.00044 | −0.00039 |

FVA (Separate) | −0.00017 | −0.00017 | −0.00017 | −0.00015 | −0.00002 |

Combined adjustments | −0.00050 | −0.00051 | −0.00056 | −0.00059 | −0.00041 |

Linearization error | 8.25% | 8.10% | 2.86% | 10.74% | 9.92% |

CPU time (in s) | 11.68 | 11.782 | 12.45 | 12.52 | 13.81 |

_{C} _{F} _{B} _{B}

0.9517 | 0.8877 | 0.8318 | 0.7797 | 0.7302 | |

CVA + FVA | −0.0007 | −0.0013 | −0.0017 | −0.0020 | −0.0023 |

0.9525 | 0.8890 | 0.8335 | 0.7817 | 0.7325 |

_{C} _{F} _{B} _{B}

0.92918 | 0.87463 | 0.77286 | 0.67616 | 0.58471 | |

CVA + FVA | −0.00034 | −0.00044 | −0.00057 | −0.00064 | −0.00066 |

0.92952 | 0.87507 | 0.77343 | 0.67680 | 0.58536 |

In this paper, we generalized the model of Piterbarg [

The limitations of the method developed in this paper is that portfolio netting effects cannot be modeled using the single model. While this deal-by-deal approach allows traders to access the impact of CVA/FVA for intra-day trading immediately. A possible extension of the current method is to introduce stochastic interest rate into consideration which remains for the future.

Since the financial crisis of 2008, there has been a regulatory drive towards better recognition and mitigation of derivative counterparty credit risk. Basel 3 (as implemented in law as CRD IV in Europe) introduced CVA and CVA risk capital which requires banks to include CVA in derivative valuation based on the market implied default probabilities (from CDS spreads) of the counterparty, and capitalize the additional volatility of derivative valuation due to fluctuation of CDS spreads. The Dodd-Frank Act and EMIR requires standardized derivative contracts to be traded with CCPs under robust collateral agreements. The Basel committee introduced mandatory exchange of initial and variation margins (making the collateral requirements similar to a CCP trade) for all OTC derivative transactions, whilst liquidity coverage ratios introduced in Basel 3 explicitly capture contingent liquidity requirements from derivative transactions. The combined effect of the above is that

some counterparty credit risk will be transformed into funding risk (the need to fund margin calls), and therefore derivative valuation must incorporate both CVA and FVA simultaneously;

the remaining counterparty credit risk will largely be driven by gap events, either through jumps in the underlying asset prices or the jump-to-default of a counterparty making it important to incorporate jumps in asset price dynamics in derivative valuation.

The present work is therefore extremely relevant in the context of current industry development and we see potential application of our work in:

investigating the combined effects of counterparty credit risk and collateral funding costs, particularly in the context of a centralized CVA/FVA desk quantifying the impact of jumps in asset prices when evaluating CVA and/or FVA;

calculating the upper bound (as portfolio netting effects cannot be modeled using this single model) of the CVA/FVA adjustment for intra-day trading or for valuing particularly large transactions where portfolio netting may be reasonably ignored.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank the participants of SIAM FM 2014, Chicago, November 2014, and especially Tomasz Bielecki and Harvey Sturm for valuable comments and suggestions. The suggestions made by two reviewers are especially appreciated.

In order to be able to implement the enhanced realization of the operators

As before, we consider a uniformly spaced grid of points _{k} = ξ_{1}+(_{−} < 0 < λ_{+} such that the Lévy process _{t}}_{t≥0} is of exponential type (λ_{−}, λ_{+}), and recall that ψ(ξ) denotes the characteristic exponent of

We obtain approximate formulas for ^{−2} as Re η → ±∞, it is sometimes necessary to use an η-grid that is longer than the ξ-grid for this discretization, in order to guarantee the desired precision of the calculation of

With these remarks in mind, and with the notation above, we present an algorithm for the approximate calculation of the values

Select a positive integer

Choose ω_{−} ∈ (λ_{−}, 0) and ω_{+} ∈ (0, λ_{+}) such that there exists δ > 0 with Re(_{−}, ω_{+}]. As a rule of thumb, we recommend taking ω_{±} = λ_{±}∕3 in the algorithms that are based on Carr's randomization method, since in these examples,

Define the η-grid as follows:

Using the simplified trapezoid rule to discretize Equation (4.3) leads to the following approximation:

The last formula can be rewritten as follows:

Noting that

Using the results of the previous step, calculate the right hand side of Equation (9.1).

The algorithm of calculating the convolution coefficients

We present the algorithm for calculating

We have

We use the dual grid

Calculate the integrand by

For each _{2}, and each _{3}, calculate inverse fast Fourier transform of

Finally take the sum

Given array _{j} = _{j}), we want to calculate the sums

Let

Let

Calculate the vector

The sum is

Let τ_{B}, τ_{C} be the default time of the bank and the counterparty respectively. Define the time of the first default event among the two parties as the stopping time

The close-out amount at default is the costs or losses that the surviving party incurs when replacing the terminated deal with an economic equivalent. The size of these costs will depend on which party survives and we define the close-out amount as
_{B, τ} is the close-out amount on the counterparty's default priced at time τ by the bank and _{A, τ} is the close-out amount if the bank defaults. We adopt the approach of Brigo [

If both parties agree on the exposure, that is _{B, τ} = _{C, τ} = _{τ}, when we take the risk-neutral expectation, we see that the price of the discounted on-default cashflow,

A margining procedure specifies the set of dates during the life of a deal when both parties post or withdraw collaterals, according to their current exposure, to or from an account held by the collateral taker. A realistic margining practice should allow for collateral posting only on a fixed time-grid {_{1}, …, _{n}}. Without loss of generality, we assume that the collateral account _{t} is held by the bank if _{t} > 0, and by the counterparty if _{t} < 0.

The CSA agreement holding between the counterparties ensures that the collateral taker remunerates the account at a particular accrual rate. We denote the collateral rate

Assume the interests accrued by the collateral are saved into the account itself. The cashflows originating from the bank and going to the counterparty if default events do not occur are

The bank opens the collateral account at the first margin date _{1} if _{t1} < 0 (the counterparty is the collateral taker);

The bank posts to or withdraws from the account at each _{k}, as long as _{tk} < 0, and the collateral account grow at the CSA rate

The bank closes the account at the last margining date _{m} if _{tm} < 0.

On the other hand, the counterparty considers the same cashflows for opposite values of the collateral account at each margining date with the CSA rate

Let τ be the time of the first default event. To introduce default event, we can stop collateral margining when they occur, so we have

If we use a first order expansion (for small

By taking the time limit, we have the expression for the stream of the collateral:

Denote _{t} the funding account and without loss of generality, we assume that the the trading desk is borrowing from the treasury if _{t} > 0, and otherwise lending to treasury if _{t} < 0. Similar to the case in collateral, we assume the trading desk enters a funding position on a discrete time-grid {_{1}, …, _{m}}.

Given two adjacent funding times _{j} and _{j + 1}, for 1 ≤ _{tj} at time _{j}. At time _{j + 1} the desk redeems the position again and either returns the cash to the treasury if it was a borrowing position and pays the funding costs on the borrowed cash, or it gets the cash back if it was a lending position and receives funding benefits as interest on the invested cash. We assume that these funding costs and benefits are determined at the start date of each funding period and charged at the end of the period.

Let

In other words, if the desk requires to borrow cash, this can be done at the funding/borrowing rate

Repeating the similar steps as in the part of collateral, the sum of discounted cashflows from funding is equal to

If we use a first order expansion (for small

By taking the time limit, we have the expression for the stream of the collateral:

Denote

Models parameters as of 9 January 2015: BM: σ = 0.18.

KoBoL: ν = 1.5, _{±} = 0.029, λ_{+} = 4.49, λ_{−} = −20.03, μ = 0.19, σ = 0.

VG: _{±} = 14.32, λ_{+} = 24.11, λ_{−} = −37.19, μ = 0.16, σ = 0.

DEJD: _{±} = 0.43, λ_{+} = 9.06, λ_{−} = −50, μ = −0.014, σ = 0.16.

NIG: α = 21.85, β = −6.73, δ = 0.66, μ = 0.16, σ = 0.

Models parameters as of 28 May 2015:

BM: σ = 0.189.

KoBoL: ν = 1.5, _{±} = 0.028, λ_{+} = 4.46, λ_{−} = −5.56, μ = −0.15,σ = 0.

VG: _{±} = 22.49, λ_{+} = 33.01, λ_{−} = −34.02, μ = −0.15, σ = 0.

DEJD: _{±} = 0.0011, λ_{+} = 25.93, λ_{−} = −50, μ = −0.17, σ = 0.189.

NIG: α = 19.33, β = −0.54, δ = 0.79, μ = −0.15, σ = 0.

Numerical parameters:

^{1}In the formulas below, and elsewhere in the text, we use the standard convention that ^{ν} = ^{ν·ln z} for any ν ∈ ℂ and any

^{2}What we present is not the most common way of writing the formula. Rather, we chose an expression that is equivalent to the standard one and makes the analogy with Equation (2.10) transparent.