A Computational Approach to Hedging Credit Valuation Adjustment in a Jump-Diffusion Setting
AA Computational Approach to Hedging Credit ValuationAdjustment in a Jump-Diffusion Setting
Thomas van der Zwaard a,b, ∗ , Lech A. Grzelak a,b , Cornelis W. Oosterlee a,c a Delft Institute of Applied Mathematics, Delft University of Technology, Delft, the Netherlands b Rabobank, Utrecht, the Netherlands c CWI - National Research Institute for Mathematics and Computer Science, Amsterdam, the Netherlands
Abstract
This study contributes to understanding Valuation Adjustments (xVA) by focussing on thedynamic hedging of Credit Valuation Adjustment (CVA), corresponding Profit & Loss (P&L)and the P&L explain. This is done in a Monte Carlo simulation setting, based on a theoreticalhedging framework discussed in existing literature. We look at CVA hedging for a portfoliowith European options on a stock, first in a Black-Scholes setting, then in a Merton jump-diffusion setting. Furthermore, we analyze the trading business at a bank after including xVAsin pricing. We provide insights into the hedging of derivatives and their xVAs by analyzingand visualizing the cash-flows of a portfolio from a desk structure perspective. The case studyshows that not charging CVA at trade inception results in a guaranteed loss. Furthermore,hedging CVA is crucial to end up with a stable trading strategy. In the Black-Scholes settingthis can be done using the underlying stock, whereas in the Merton jump-diffusion setting weneed to add extra options to the hedge portfolio to properly hedge the jump risk. In additionto the simulation, we derive analytical results that explain our observations from the numericalexperiments. Understanding the hedging of CVA helps to deal with xVAs in a practical setting.
Keywords: computational finance, dynamic hedging, Credit Valuation Adjustment (CVA),Merton jump-diffusion, counterparty credit risk (CCR), xVA hedging
1. Introduction
Since the 2007-2008 global financial crisis, financial institutions have been required to applyand report Valuation Adjustments (xVAs) for over-the-counter (OTC) derivatives and hedgethe associated risks. These adjustments to the risk-neutral price of a derivative account forpreviously neglected risks that were revealed during the crisis. Credit Valuation Adjustment(CVA), corresponding to Counterparty Credit Risk (CCR), was the first of many xVAs. CVAvolatility is one of the major drivers behind the large losses observed during the crisis. Linearcontracts are no longer linear when CCR is included in the valuation. Hence, not includingCCR in pricing results in an incorrect hedging policy. Next, xVA pricing evolved with variousother xVAs, see [1, 15] for more information. Calculating the increasing number of xVAs iscomputationally challenging, and has attracted significant academic and corporate interest.The first literature on xVA pricing appeared before and during the crisis [14, 19, 29]. Todate, several books have addressed the topic [1, 7, 15]. There are three streams in literature onxVA pricing, all describing the mathematical problem in different ways. First of all, there is theMonte Carlo approach where the xVA is expressed as an expectation. A set of risk factors issimulated in a Monte Carlo engine, after which the future exposures that contribute to the xVA ∗ Corresponding author at Delft Institute of Applied Mathematics, TU Delft, Delft, the Netherlands.
Email addresses:
[email protected] (Thomas van der Zwaard),
[email protected] (Lech A. Grzelak),
[email protected] (Cornelis W. Oosterlee)The views expressed in this paper are the personal views of the authors and do not necessarily reflect theviews or policies of their current or past employers. Linear contracts/instruments have contractual cash-flows that are a linear function of the underlying.
Version: May 22, 2020 a r X i v : . [ q -f i n . C P ] M a y etrics are evaluated along the simulated paths [2, 3, 8, 9, 10]. The Monte Carlo approachallows for scalable computations. Hence, this type of method is typically implemented bybanks. Second, there is the PDE approach that aims to solve the xVA PDE directly [5, 13].Dimensionality is one of the downsides of this approach, though low-dimensional non-linearproblems can be treated highly accurately. Last, there is the BSDE approach, where the mainpurpose is to get accurate Greeks [20, 21, 24, 25, 28]. This may be non-trivial in a regressionbased Monte Carlo approach. Literature on xVA has focussed on deriving the mathematicalpricing equations, as well as addressing the computational challenges that arise when solvingthese pricing equations. In our work, we closely examine xVA pricing and hedging. We focus onthe dynamic hedging of CVA, where we study the cash-flows of a portfolio from a desk structureperspective. This results in an improved understanding of the CVA hedging mechanics. Weshow that the CVA in the portfolio needs to be hedged to end up with a stable trading strategy.The trading strategy is a combination of all trading positions with a counterparty andthe corresponding hedging strategy [2, 3]. A wealth account [11] is connected to this tradingstrategy, it tracks the total wealth of the trading strategy over time. This is a cash account thataccrues interest. Bielecki and Rutkowski formalized this framework [28]. The generic tradingstrategy formulation provides a useful framework to analyze exchanges of assets and cash.Furthermore, the generic formulation is useful for the hedging of xVAs. We provide numericalexamples and insights in a Monte Carlo simulation setting, starting in the Black-Scholes world.The constant volatility is one of the known shortcomings of the Black-Scholes model. Theresulting flat implied volatility is in clear contrast with the market’s implied volatility smile.To properly manage this smile risk, several streams of modelling have emerged. Among theseapproaches are the local volatility models (e.g., Dupire [4], Derman and Kani [12]) and thestochastic (local) volatility models (e.g., Heston [27], Hagan et al. [22]). Although these modelsfit in the generic hedging framework, we choose to focus on jump-diffusion models, which exhibitheavy tails in the distribution of stock returns and fit well with the jump patterns observed forstocks. Jump-diffusion models extend the Black-Scholes model with independently distributedjumps driven by a Poisson process. Typical choices of jump size distributions are a doubleexponential distribution, Kou [26], or normal distributed jumps, Merton [23]. We chooseto work with the latter. Further research on the Merton jump-diffusion model has been oncalibrating the model and hedging the jump risk that partially drives option prices under thismodel [6, 16].Therefore, after studying the dynamic hedging of CVA in a Black-Scholes setting, we dothe same in a Merton jump-diffusion setting. The numerical analysis is performed by meansof a Monte Carlo simulation. In addition, we examine the impact of defaults on the portfolio.The Profit and Loss of the trading strategy is examined to assess the hedge’s performance.We show that, for a portfolio of European options, not including CVA in the pricing resultsin a guaranteed loss on average. Hence, we interpret CVA as a fair compensation for thecredit risk of the counterparty. Charging CVA to the client at trade inception overcomes theguaranteed loss at maturity. CVA can be treated as a cash amount, however, this ignoresthe dependencies of the CVA on the market variables. After including the CVA hedge in thestrategy, we assess the impact of jumps in the underlying stock on the portfolio. In particular,we find that in the context of CVA, the jump risk can be mitigated to a large degree by addingextra hedging instruments to the strategy. Rather than merely performing a simulation, wealso derive analytic results to understand and explain our numerical observations.This paper is organized as follows. In Section 2 we provide background information onProfit and Loss, which is used to examine the outcome of the hedging strategies. In Section 3we introduce the trading strategy and the corresponding wealth account. Next, the numericalsimulation of the future market states and results of the various hedging strategies will beaddressed in Sections 4 and 5 respectively. Finally, the work is concluded in Section 6.
2. Profit and Loss
Profit and Loss (P&L) is a financial institution’s income statement. This is an officiallyreported number that summarizes the change in
Mark-to-Market (MtM) value over a periodof time, normally one business day. During this period, the institution examines which partof the change in MtM can be attributed to market moves and other predefined effects, such2s the passing of time. This process is also called P&L attribution , or P&L explain , whichexplains the impact of the daily market movements on the value of the portfolio. The goal isto verify that the modelled risk factors satisfactorily explain the change in portfolio value. TheP&L explain process runs after the closing of every business day, so it is a backward lookingmeasure. Typically, the variance between two consecutive days is also examined.A financial institution wants to explain the P&L as well as possible. Hence the residualP&L, which from now on we address as P&L unexplained , should be as small as possible. Yetexplaining all P&L is insufficient, as extreme numbers that are completely explained are alsoundesirable. Therefore, institutions put thresholds/limits on both P&L and P&L unexplainedat a portfolio level, where the thresholds depend on the portfolio composition. In addition tothese P&L limits, market risk limits on particular risk types are in place for both traders andportfolios. The aim of these market risk limits is to make sure the trading activities remainwithin the institution’s predefined risk appetite and that exposure to certain market driversis bounded. This naturally limits the P&L as well. The portfolios are not always entirely flatin terms of risk, as in practice it is not feasible to rebalance the hedge daily, for example dueto transaction costs. Furthermore, having a flat portfolio is not always the goal as this maybe a way to express a view on the market. In practice, the residual risk of a portfolio is beingmonitored.For xVAs a separate explain process exists, which is analogous to the one for the MtMvalue. Limits on market risk drivers, P&L and P&L unexplained are present.In literature the following approaches to P&L explain can be found [1, 18]:1. A Taylor-based explain process where the partial derivatives of the instrument V withrespect to market data γ i are used to explain the difference in V ( t k − ) and V ( t k ), t k − 3. Hedging framework We start with a risk-neutral pricing framework, where we consider the various contributionsto the price of a derivative. Using the standard replicating portfolio argument [7], the price ofa derivative is equivalent to the price of a replicating portfolio containing other securities.We define a trading strategy as a combination of positions in a set of available tradinginstruments, accompanied by the hedging positions in hedging instruments, which mitigatethe market risk associated with the positions in the trading instruments. We assume that theshort-selling of all assets is possible. In practice, hedges do not necessarily mitigate all themarket risk, but certain limits are placed per market risk factor on a portfolio level. A traderwill make sure that the exposure of a portfolio to certain risk factors will remain below thesepredefined thresholds by taking the necessary positions in market instruments. Here we willassume that all the market risk is always reduced by the dynamic hedging strategy, i.e., wechoose the hedging positions that together replicate the risk profile of the trading positions asgood as possible.The trading strategy Π( t ) is generically defined for N trading instruments V i ( t ) in which trading positions ζ i ( t ) are taken, which are assumed to be exogenously provided. Analogously,we consider M hedging instruments H i ( t ) in which hedging positions η i ( t ) are taken. Thetrading strategy Π( t ) is then summarized as follows:Π( t ) = N (cid:88) i =1 ζ i ( t ) V i ( t ) + M (cid:88) i =1 η i ( t ) H i ( t ) . (3.1)We start in a Black-Scholes setting, where European options are considered for the sakeof clarity and ease of computation. However, the generic formulation allows for much moreflexibility, for example, for the case of a large portfolio of interest rate swaps. The underlyinginterest rate risks are then eliminated by taking hedging positions in the par instruments usedto build the underlying yield curve(s) that are in turn required to value the portfolio of interestrate swaps.Define the cash-flows for the trading and hedging instruments respectively by c Ti,j ( t ) and c Hi,j ( t ), denoting the time t value of the j -th cash-flow paid at time T j , corresponding torespectively instruments V i and H i , which in total generate respectively n i and m i cash-flows.We then define the time t value of the cumulative cash-flows corresponding to instrument V i and H i respectively as C Ti ( t ) = n i (cid:88) j =1 ζ i ( T j ) c Ti,j ( t ) { T j ≤ t } , C Hi ( t ) = m i (cid:88) j =1 η i ( T j ) c Hi,j ( t ) { T j ≤ t } , (3.2)which is to be interpreted as the quantity representing all cash-flows paid up and till time t ,taking into account the time value of money, B ( t ), being the bank account . This is the solutionof d B ( t ) = r ( t ) B ( t )d t , where B ( t ) = 1 and with risk-free rate r ( t ).4n parallel with the trading strategy Π( t ), we define a wealth process w ( t ) that representsthe total wealth realized over time, obtained by summing up all the profits and losses over time.There are two sources for changes in wealth: one is the rebalancing of positions in instruments,the other results from cash-flows associated with the various instruments. We assume that V i ( t ) and H i ( t ) denote the value after exchange of all cash-flows at time t . For example, ifa cash-flow takes place at date t , the values V i ( t ) and H i ( t ) do not contain the value of thiscash-flow. Rebalancing the trading positions, and consecutively the hedging positions, takesplace after the exchange of cash-flows. The wealth is a cash amount that accrues interest overtime at the risk-free rate. At this stage we do not distinguish between different levels of interestrate for paying and receiving interest.At time t , the wealth will be composed of the cost to enter the positions in all instruments,including any potential cash-flows taking place at t : w ( t ) = − Π( t ) + N (cid:88) i =1 C Ti ( t ) + M (cid:88) i =1 C Hi ( t ) . (3.3)The rebalancing of the trading and hedging positions is denoted by d ζ i ( t ) and d η i ( t ) re-spectively. We write the following (recursive) expression for the wealth at t ≤ t k − < t k : w ( t k ) = w ( t k − ) B ( t k ) B ( t k − ) − N (cid:88) i =1 B ( t k ) (cid:90) t k t k − V i ( u ) B ( u ) d ζ i ( u ) + N (cid:88) i =1 B ( t k ) (cid:90) t k t k − d C Ti ( u ) B ( u ) − M (cid:88) i =1 B ( t k ) (cid:90) t k t k − H i ( u ) B ( u ) d η i ( u ) + M (cid:88) i =1 B ( t k ) (cid:90) t k t k − d C Hi ( u ) B ( u ) . (3.4)The first term in Equation (3.4) is the wealth at the previous point in time t k − that has accruedinterest. This is followed by the re-balancing and cash-flows of respectively the trading andhedging instruments.The continuous time formulation from Equation (3.4) is discretized in time t < t < . . . 4. Market simulation The hedging framework from Section 3 will be used in a numerical simulation setting toassess Valuation Adjustments and the hedging thereof. The starting point of the numericalanalysis is choosing a portfolio of trading instruments and the corresponding hedging instru-ments. The economic value of a trading instrument is the combination of the risk-free valueand Valuation Adjustments, where we choose to look at CVA only. We consider only Europeanoptions for the trading instruments. DVA turns out to be trivial for this portfolio composition,so DVA is ignored in the analysis. CVA computations require the following two main compo-nents: market exposure and default probability. The former is introduced in Section 4.1, thelatter is discussed here directly.We model the jump-to-default by a Poisson process X P ( t ) with constant hazard rate λ ( t ) = ξ P , i.e. a homogenous Poisson process. The deterministic hazard rate (e.g., constant orpiece-wise constant) implies that credit events are independent of the interest rates anddeterministic recovery rates. In addition, assume that the recovery rate R is deterministic. To gain some intuition on how this Poisson processes works, look at its expected value E [ X P ( t )] = ξ P t ,which is the expected number of events in a time interval with length t . So say we have an interval [0 , T ] oflength T where we expect one event every 2 T , then we must set ξ P = T . A credit event is considered to be the first event of a Poisson counting process which occurs at somerandom time τ with probability P ( X P ( τ + d t ) − X P ( τ ) | X P ( τ ) = 0) , i.e., the probability of default in interval[ τ, τ + d t ) conditional on survival until time τ . urvival probability SP( t, T ) is the probability that the counterparty will survive until time T ,conditional on survival till time t :SP( t, T ) = E t (cid:34) exp (cid:40) − (cid:90) Tt λ ( s )d s (cid:41)(cid:35) = exp (cid:40) − (cid:90) Tt λ ( s )d s (cid:41) = e − ξ P ( T − t ) . A credit curve is the credit-analogue of the yield curve, where we do not extract discount factorsbut survival probabilities from the curve. In a market implied setting, survival probabilitiesSP( t , T ) are extracted from the market using quotes of highly standardized CDSs. For CVAcalculations we require knowledge about the probability of default PD( t, T ) of a counterpartyin a certain time interval [ t, T ]. The probability of default is related to the survival probability:PD( t, T ) = 1 − SP( t, T ).We choose to hedge all the market risk introduced by the risk-free value and CVA corre-sponding to the trading instruments. We do not consider the credit risk component introducedby the CVA. In Section 4.2 we look in more detail at the setup of the trading strategy. Given a call option V that runs from t till maturity t K , we create a grid of monitoringdates t < t < . . . < t k . . . < t K at which we compute exposures . The following convenientresult can easily be derived for the discounted expected positive exposure ( EPE ) of a call/putoption: EPE( t , t k ) = V ( t ) . (4.1)Using the result from Equation (4.1) in the formula for CVA, assuming no Wrong WayRisk (WWR) , yields: CVA( t ) = (1 − R ) K (cid:88) k =1 EPE( t , t k ) PD( t k − , t k )= (1 − R ) K (cid:88) k =1 V ( t ) PD( t k − , t k )= (1 − R ) V ( t ) PD( t , t K ) . (4.2)The results in Equations (4.1) and (4.2) are model-free up to the level how V ( t ) is computed. For our trading instruments we choose a constant long unit position (i.e., buy) in a Europeancall option V ( t ), i.e., N = 1 and ζ ( t ) = 1 ∀ t . The option is assumed to be an OTC dealwhere the underlying asset S ( t ) is a third-party asset, i.e., S ( t ) is not the asset of the optionseller. This implies the absence of WWR, meaning we assume that the creditworthiness of theoption seller and buyer and the underlying asset move independently. In addition, we assumea constant credit curve over time, i.e., no stochasticity for the credit is used. Furthermore,we assume the option has a cash-settled payoff. The market risk of the option is hedged bybuying/selling the underlying stock from/to an exchange, i.e., M = 1 and H ( t ) = S ( t ):Π( t ) = V ( t ) + η ( t ) S ( t ) , (4.3)where η ( t ) depends on a model chosen by the financial institution. Hedging positions η ( t ) arerebalanced on a daily basis. We assume no dividends are paid, though they can easily be addedto the framework in the form of cash-flows. The trading instrument will generate one cash-flow, CVA is in essence a compound option on the value of a portfolio. A compound option refers to an option onan option. Say that we consider a portfolio of a single option, then the max function in the expected exposurein the CVA formula makes the CVA a compound option. This reverse Black-Scholes hedge is an academic setup, as buying a call option and hedging the delta riskis not what one usually encounters. However, for illustrative purposes this particular setting is chosen. V i ( t ) can represent the risk-neutral value, which corresponds tothe case without Counterparty Credit Risk (CCR). On the other hand, it can also representthe economic value, which is the sum of the risk-neutral value and xVAs that are taken intoaccount, and corresponds to the case where CCR is taken into account, i.e., V ( t ) := V ( t ) , (the case without CCR) V ( t ) := V ( t ) − CVA( t ) (the case with CCR)= V ( t ) (1 − (1 − R ) PD( t, t K )) , (4.4)where for the case with CCR we used Equation (4.2). Recall that V ( t ) represents the risk-freevalue of the option, whereas V ( t ) represents the risky value of the option. This compositionof two terms also needs to be taken into account when determining hedging position η ( t ): dowe hedge only V ( t ) or also CVA( t )?First, the market ( S ( t ), V i ( t ) and H i ( t )) is simulated using a model (e.g., Black-Scholesor Merton jump-diffusion). While valuing the portfolio, these numbers are assumed to beexogenous, meaning there is no longer a model dependence. The model used to compute thehedging quantities and perform the P&L explain needs to be calibrated to the market. Whenthis model is the same as the model used to simulate the market, the calibration is trivial.We use two strategies, Π NoCCR and Π CCR , to assess the effect of CCR. The former cor-responds to the portfolio without simulation of defaults, while the latter includes CCR bysimulating default times. In all cases, the hedges are rebalanced daily and assumed to befree of CCR. The simulated default times τ = t d are drawn from the same distribution thatdrives the credit curve. So, the simulated default times are the first jumps of X P ( t ). In theexperiments, we assume a risk-free closeout , where we give back V to the defaulted counter-party, and in return receive R · V ( τ ) in case this value is positive. We assume stock S ( t ) tobe independent of any default events of the counterparty. At default, we re-enter the samedeal with another counterparty which is assumed to be credit-risk free, for example a clearinghouse. This approach is in line with considering the CVA as the cost of hedging counterpartycredit risk, regardless of a counterparty default. If the CVA is hedged, this hedge position isclosed at default. So far we have not assumed any model for S and V ( t ), only a choice of credit curve andsimulation of defaults was made. The next step is to assume a model for S and V ( t ). Our firstchoice is the Black-Scholes model that allows for analytical option prices and derivatives, in adeterministic interest rate setting. Recall the Black-Scholes SDE under real-world measure P :d S ( t ) = µS ( t )d t + σS ( t )d W P ( t ) . (4.5)For the simulation of the market scenarios, the SDE (4.5) is discretized using an Euler scheme.The risk introduced by the underlying stock is eliminated when choosing the Black-Scholesdelta hedging quantity: η ( t ) = − ∂V ( t ) ∂S . (4.6)In the risk-free case where V ( t ) = V ( t ), hedging quantity (4.6) holds directly. On the otherhand, in the risky case V ( t ) = V ( t ) − CVA( t ), where CVA is hedged, we can write the followingusing result (4.2): η ( t ) = − ∂V ( t ) ∂S (1 − (1 − R ) PD( t, t K )) . (4.7)The P dynamics given in Equation (4.5) are used for the simulation of a synthetic market.On the other hand, for pricing options and calculating risks as in Equations (4.6) and (4.7) weneed Q dynamics. To obtain these, in Equation (4.5) we choose µ = r and use a risk-neutralBrownian motion. 8he simulation is illustrated by a series of schematic drawings that indicate the flow ofcash and instruments, see Appendix C. There, we first consider the case without CCR to geta basic understanding of the mechanics of the trading strategy from the perspective of a bank.Then, CCR is introduced, and the bank’s trading desk and xVA desk are represented as asingle entity, referred to as the trading desk. Finally, we remove this assumption by examiningthe internal exchange of cash-flows and products between the desks. aim to overcome the Black-Scholes assumption of constant impliedvolatility, by introducing independently distributed jumps in the dynamics. They can generi-cally be defined as [7]:d X ( t ) = µ d t + σ d W P ( t ) + J ( t )d X P J ( t ) , S ( t ) = e X ( t ) . (4.8)The jumps J arrive according to Poisson process X P J ( t ) that is assumed to be independentof the Brownian process W P ( t ). Typical choices of jump size J distributions are a doubleexponential distribution as introduced by Kou [26] or normally distributed jumps as introducedby Merton [23]. We choose to work with the latter. Jump magnitudes in this model followdistribution J ∼ N (cid:0) µ J , σ J (cid:1) .For the simulation of a synthetic market we use an Euler discretization of the P dynamicsfrom Equation (4.8). On the other hand, for pricing options and calculating risks we need Q dynamics. To obtain these, in Equation (4.8) we should choose the drift as follows : µ = r − ξ J E (cid:2) e J − (cid:3) − σ = r − ξ J (cid:16) e µ J + σ J − (cid:17) − σ . Furthermore, we use a risk-neutral Brownian motion and jump process in the Q dynamics,which are again assumed to be independent. For European option prices under the Mertonjump-diffusion model an analytic expression exists, see Appendix A.1.After simulating the market S ( t ) = e X ( t ) with the Merton model, we compute the Mertonoption price using Equation (A.1). Here we do not make an assumption on V ( t ) being a risk-free or risky option value (without CVA versus with CVA). Distinguishing between the twocases can be done in a similar fashion as discussed in Section 4.3. For setting up the hedgingposition η ( t ), we consider the following three approaches.The first approach is applicable to the situation where the institution’s pricing model ismisaligned with the market. In our setup, this is represented by using a Black-Scholes deltahedge as in Section 4.3, even though the market does not follow these dynamics. This is doneby extracting the Black-Scholes implied volatility from the Merton option prices observed inthe market, and setting up a Black-Scholes delta hedge using the underlying stock. We confirmthe results by Naik and Lee [30] that this hedging strategy is not suitable for the case of anunderlying asset driven by both diffusion and jump risk.The second approach represents the case in which one’s pricing model is perfectly alignedwith the market. In this context this means that the Merton delta, as in Equation (A.5), canbe used to compute the hedging quantity. Equations (4.6) and (4.7) still hold in this case,given that the Black-Scholes deltas are replaced by Merton deltas.Up to this point, no attempt has been made to hedge the jump risk introduced by the Mer-ton model. The option pricing formula (A.1) contains an infinite sum of scaled Black-Scholesoption prices, which is a direct result of the jump size following a continuous distribution.Thus, in an attempt to hedge the jump risk introduced by the model, one would theoreticallyneed infinitely many options, which is practically infeasible. Hedging the jump risk has beenaddressed by adding a number of options to the hedging portfolio [6, 16]. This significantlyreduces the variance of the portfolio. In particular, a local minimal variance hedging strat-egy was examined, combined with a delta position in the underlying stock. In this paper, weuse the analytical jump parameter sensitivities from Appendix A.1 to determine the hedgingpositions that aim to eliminate the underlying jump risk. In this case we can use the known result that for A ∼ N (cid:0) µ, σ (cid:1) we know E (cid:104) e A (cid:105) = e µ + σ . H ( t ) = S ( t ) and H ( t ), which is a European optiondifferent from option V ( t ) we aim to hedge. Option H ( t ) is a risk-free option, such that noadditional xVAs are introduced. To summarize, we have:Π( t ) = V ( t ) + η ( t ) S ( t ) + η ( t ) H ( t ) . We choose stock position η ( t ) such that the portfolio is delta-neutral, i.e., such that ∂ Π( t ) ∂S = 0: η ( t ) = − ∂V ( t ) ∂S − η ( t ) ∂H ( t ) ∂S . This leaves the question of how to choose η ( t ). We use analytical jump parameter sensitivitiesto determine the remaining hedging position. As the strategy contains one hedging option, wemust choose one of the jump parameters that we consider most important. All jump parametershave a level effect, µ J also affects the skew, and σ J also affects the curvature, see AppendixA.2. The level effect from ξ J is more significant than that for µ J and σ J . Hence, we choose ξ J to set up the hedge. So, for η ( t ) we have: η ( t ) = − ∂V ( t ) ∂ξ J (cid:20) ∂H ( t ) ∂ξ J (cid:21) − . (4.9)The partial derivatives w.r.t. ξ J are computed analytically using the result in Equation (A.4).The hedging strategy as presented here is equivalent to first taking a position in the stock η ( t )to hedge the trading instrument (so Equation 4.6 but with the Merton delta), then takingposition (4.9) to hedge the jump risk, and then updating η ( t ) to account for the additionaldelta risk generated by this position in the hedging option. 5. Numerical results We implement the market simulation, as introduced in Section 4, in a Monte Carlo setting.In our experiments, we simulate the synthetic market and do all pricing under the Q dynamics.The algorithm used to obtain the numerical results is summarized below (Algorithm 1).All results are obtained with the following parameters: t = 0, T = 1, S ( t ) = 100, r = 0 . σ = 0 . K = 0 . ξ P = 0 . 2, and R = 0 . 5. For the Merton jump-diffusion parameters wechoose σ J = 0 . µ J = − . ξ J = 0 . 1. We use L = 10 Monte Carlo paths and 200time steps per year to create the set of monitoring dates. The results are displayed using anumber of 100 shares for the option, such that the controlled notional by the option is 10 .As a result, the vertical axes of the plots can be interpreted as errors in bps. Furthermore, inthe results the bank is represented as a single entity, meaning that the trading desk and xVAdesk do not have separate trading strategies and wealth accounts. This split in the results caneasily be made using the schematic drawings in Appendix C, but here we do not do this forsake of brevity. The first numerical results correspond to the Black-Scholes hedging setting as discussed inSection 4.3. In this section, the Black-Scholes model is used for market simulation, computinghedging quantities, and P&L explain. In Figure 1a we confirm a guaranteed loss at maturity if the counterparty could default,but no CVA is charged. This is done by examining the impact of simulated defaults on theportfolio, where a guaranteed loss is represented by E t [Π CCR ( t K ) + w CCR ( t K )] < 0. ForΠ NoCCR we have the desired result, namely E t [Π NoCCR ( t K ) + w NoCCR ( t K )] ≈ 0, with someresidual noise coming from the Monte Carlo simulation. The terminal wealth distribution inFigure 1b can be interpreted as a bimodal distribution. The large peak corresponds to thepaths without default, whereas the low and wide peak on the left corresponds to the lossesencountered as a result of defaults. 10 nput: Trading strategies Π NoCCR and Π CCR , risk factors γ , number of simulation paths L and dates K Output: Numerical results of the CVA hedging exercise Initialize two portfolios Π NoCCR and Π CCR Initialize simulation grid of risk factors γ for l ← to L do for k ← to K do Simulate all risk factors γ ( t k ) for path l Re-value Π NoCCR ( t k ) and Π CCR ( t k ) for path l for i ← to N do Re-value V i ( t k − , t k ) for path l to be used later in the P&L calculations end for for j ← to M do Re-value H j ( t k − , t k ) for path l to be used later in the P&L calculations end for end for Simulate default time τ l Perform closeout at default time τ l for Π CCR for k ← to K do Compute the relevant P&L quantities at time t k , prepare P&L E for path l Compute wealth w NoCCR and w CCR for path l end for end for Compute required output metrics and visualize results Algorithm 1: CVA hedging algorithm (a) Average Π( t ) and w ( t ). (b) Distribution of w CCR at maturity.Figure 1: CVA not included in the portfolio. Intuitively, E t [Π CCR ( t K ) + w CCR ( t K )] ≈ CVA( t ) B ( t K ) B ( t ) should hold. Hence, we add theCVA charge at inception as a cash amount to the wealth account, meaning that we perform alinear shift in initial wealth. This should result in E t [Π CCR ( t K ) + w CCR ( t K )] ≈ 0, which isindeed the case, see Figure 2a. So, the CVA charge proves to be a fair compensation of thecredit riskiness of the counterparty. Furthermore, we see in Figure 2b that the distribution of w CCR ( t K ) gets shifted to the right due to the CVA charge added at inception. The shift isprecisely the required amount such that the mean of the distribution is around zero.We see that charging the CVA at inception to the client and putting it on the wealth accountovercomes the issue of a guaranteed loss. However, treating the CVA charge as a cash numberwith no dependencies on underlying market variables is not useful in practice and naive. Inpractice, the CVA is charged to the client at trade inception. As time passes, changes in themarket result in a change in CVA. Next, we consider the CVA to be driven by market risk, but we decide not to hedge thisrisk. This means we use the hedging quantity from Equation (4.6) with the risk-free Black-Scholes delta, even though we have V ( t ) = V ( t ) − CVA( t ). From now on, we refer to the case11 a) Average Π( t ) and w ( t ). (b) Distribution of w CCR at maturity.Figure 2: CVA included in the portfolio as a cash amount. of Black-Scholes paths and delta as the pure Black-Scholes case. (a) Average Π( t ) and w ( t ). (b) Volatility of Π( t ) and w ( t ).Figure 3: Comparison of Π NoCCR and Π CCR using a Black-Scholes market and valuationmodel. CVA is not hedged. Figure 3a represents the case in which CVA was added to the portfolio but not hedged.We see that the expected terminal wealth condition E t [Π( t K ) + w ( t K )] = 0 is satisfied. InFigure 3b, we see that the variability of Π( t ) + w ( t ) for Π CCR is much higher compared to thatof Π NoCCR . This is expected as a result of the additional source of randomness introduced bythe simulated defaults in Π CCR . Π( t ) + w ( t ) must always be examined first when assessing theperformance of the portfolio, but it does not guarantee optimal performance.Therefore, consider Figures 4a and 4b, where we plot the mean and volatility of P&L P ( t ).One can clearly see that even though the mean P&L P for Π CCR is slightly lower compared toΠ NoCCR , the volatility is significantly higher over the majority of the lifetime of the option.Thus, this hedging strategy is incomplete. The two lines in Figure 4b overlap close to maturity,because the binarity of the payoff takes over. Furthermore, most of the simulated defaults haveoccurred by this time. Because after a default the same deal is entered with a credit risk freecounterparty, the two portfolios eventually exhibit similar behaviour. From Figures 4c and 4dwe see that a significant portion of the P&L P ( t ) can be explained using the underlying stock. We see that the potential defaults through the lifetime of the option contribute to a significantinitial difference in P&L U . This difference diminishes over time, as fewer defaults are expectedto occur before the maturity of the option. As the strong increase in variance is not promisingat a first glance, in Section 5.1.3 a thorough analysis of this behaviour can be found. All in all,the results indicate that CVA needs to be hedged to eliminate the majority of the underlyingrisk in the portfolio. In case a delta hedge was employed to hedge first order risks, this term was ignored in the P&L explainprocess to avoid taking the delta effect into account twice. a) Average P&L P ( t ). (b) Volatility of P&L P ( t ).(c) Average P&L U ( t ). (d) Volatility of P&L U ( t ).Figure 4: Comparison of Π NoCCR and Π CCR using a Black-Scholes market and valuationmodel. CVA is not hedged. Looking at Figure 4a, one might question why the average P&L P ( t ) for Π NoCCR is notequal to zero, as this would mean that a Black-Scholes delta-hedge is unable to hedge all therisk of the option. Figure 5a shows that this number converges to zero if d t → 0, so ourobservations are merely the result of the discretization. The volatility of P&L P ( t ) for Π NoCCR also converges to zero if the number of time-steps in the discretization is increased.We also confirm that the CVA as charged initially covers the otherwise experienced loss atdefault. We do this by confirming that the following error measure is approximately zero: ε ( t ) = E t (cid:20) { τ ≤ t K } (cid:20) B ( t ) B ( τ ) ( RV ( τ ) − V ( τ )) + CVA( t ) (cid:21) + { τ>t K } CVA( t ) (cid:21) ≈ L L (cid:88) l =1 { τ l ≤ t K } (cid:20) B ( t ) B ( τ l ) ( RV ( τ i ) − V ( τ l )) + CVA( t ) (cid:21) + { τ l >t K } CVA( t ) , (5.1)where L denotes the number of Monte Carlo paths used in the simulation. We analyze thebehaviour of this error measure by changing the number of Monte Carlo paths and number oftime-steps used in the simulation. From Figure 5b we see that for a given number of monitoringdates per year, increasing the number of paths results in | ε ( t ) | → 0. Here we clearly seethat using L = 10 provided unstable results, but we do observe converging behaviour whenincreasing L . As defaults occur infrequently, the number of paths must be large to properlyapproximate the numerical expectation from Equation (5.1). Furthermore, the choice of 200monitoring dates per year appears to be a very fine tradeoff between speed and accuracy. In Figure 4b we see an exploding behaviour in the volatility of P&L P as t → T , forboth Π NoCCR and Π CCR , which we want to understand. Therefore, we explain the distributionfeatures we observe by looking at the P&L P mean and variance for Π NoCCR . Thus, we examine13 a) Π NoCCR results of P&L P ( t ) average andvolatility for different number of monitoring dates. (b) | ε ( t ) | for different number of monitoringdates, per different number of Monte Carlo paths L .Figure 5 P&L P for the portfolio in Equation (4.3), with the Black-Scholes delta hedging quantity as inEquation (4.6). P&L P from Equation (3.10) can be rewritten as follows:P&L P ( t k ) = Π( t k − , S ( t k )) − Π( t k − , S ( t k − ))= [ V ( t k − , S ( t k )) − V ( t k − , S ( t k − ))] + η ( t k − ) [ S ( t k ) − S ( t k − )] . (5.2)Dividing both sides of Equation (5.2) by d S = S ( t k ) − S ( t k − ) , and using the definition of aforward finite difference approximation yields:P&L P ( t k ) S ( t k ) − S ( t k − ) = V ( t k − , S ( t k )) − V ( t k − , S ( t k − )) S ( t k ) − S ( t k − ) − ∂V ( t k − ) ∂S = d S ∂ V ( t k − , S ( t k − )) ∂S + O (cid:0) (d S ) (cid:1) , (5.3) ⇒ P&L P ( t k ) = (d S ) ∂ V ( t k − , S ( t k − )) ∂S + O (cid:0) (d S ) (cid:1) . (5.4)Equation (5.4) shows that two types of errors drive P&L P . First, there is the truncationerror of the forward finite differences which are used in Equation (5.3). Furthermore, there isa discretization error that disappears if d S → 0, which will happen if d t = t k − t k − → ∂ V ( t, S ) ∂S = K e − r ( T − t ) φ ( d ( t, S )) S σ √ T − t , (5.5) d ( t, S ) = ln SK + (cid:0) r − σ (cid:1) [ T − t ] σ √ T − t . Define X ∼ N ( µ X , σ X ) such that S ( t k ) d = S ( t k − )e X , see Appendix B for further details.Using this and the Black-Scholes gamma (5.5), we rewrite Equation (5.4) as follows:P&L P ( t k ) = (d S ) K e − rτ φ ( d ( t k − , S ( t k − ))) S ( t k − ) σ √ τ + O (cid:0) (d S ) (cid:1) = S ( t k − ) (cid:2) e X − (cid:3) K e − rτ φ ( d ( t k − , S ( t k − ))) S ( t k − ) σ √ τ + O (cid:0) (d S ) (cid:1) = K e − rτ σ √ τ (cid:2) e X − (cid:3) φ ( d ) + O (cid:0) (d S ) (cid:1) , (5.6)where τ := T − t k − and for ease of notation we write d = d ( t k − , S ( t k − )).14efine d ∼ N ( µ d , σ d ), see Appendix B. Then, using definitions (B.13) and (B.14) forrespectively f ( µ X , σ X ) and g ( µ X , σ X ), we obtain this expression of the variance of P&L P : V ar t (P&L P ( t k )) ≈ K e − rτ πσ τ f ( µ X , σ X ) · e − µ d σ d (cid:113) σ d − g ( µ X , σ X ) · e − µ d σ d σ d . (5.7)See Appendix B for a full derivation of this result. Figure 6: Numerical confirmation that the analytical variance for Π NoCCR from Equa-tion (5.7) is in line with the Monte Carlo simulation. From Figure 6 we see that the analytical result from Equation (5.7) and volatility from theMonte Carlo simulation are in line. The strong increase of P&L P volatility close to maturitycan be understood from the perspective of P&L P being driven by the option gamma, seeEquation (5.4). It is known for the option delta to be instable near maturity, causing anincreased gamma. This explains the increase in variance of P&L P ( t k ) as t k → T . Especiallywhen the option is close to the ATM point, the gamma is large, causing a large gamma volatility.Looking at Equation (5.7), the strong increase in volatility towards maturity can be ex-plained by looking at the variance as a scaled difference between the factors (cid:16)(cid:113) σ d (cid:17) − and (cid:0) σ d (cid:1) − . The first term is non-linear, while the second is linear. Approaching maturity,the non-linearity of (cid:16)(cid:113) σ d (cid:17) − increases, causing the increased variance. In Section 5.1.2 we concluded that CVA needs to be hedged. Hence, we now hedge themarket risk of the CVA using the underlying stock, summarized by the hedging quantityEquation (4.7). In terms of the zero average return, hedging the CVA yields the same resultsas not hedging. Comparing Figures 4a and 7a shows that the average P&L P for Π CCR is initiallylower for the case where we do not hedge the CVA than when we do. However, this difference isnot significant, especially not when taking the size of the volatility into consideration. In otherwords, the volatility appears to be a dominating factor in these results. Comparing Figures 4band 7b shows a benefit of hedging the CVA, as this yields a significantly lower volatility inP&L P . Hence, we again conclude that CVA must be hedged.Regarding the P&L U , the results from Figures 4c and 4d also hold for the situation withCVA hedge. This makes sense, as the CVA market risk is either hedged and then does notneed to be explained, or it is not hedged but then could be explained. So, hedging CVA doesnot affect the result of how much P&L can be explained. The numerical results in this section correspond to the Merton jump-diffusion hedgingsetting from Section 4.4. The market is simulated using the Merton jump-diffusion model. Thehedging quantities and P&L E are computed with various models, depending on the experiment.15 a) Average P&L P ( t ). (b) Volatility of P&L P ( t ).Figure 7: Comparison of Π NoCCR and Π CCR using a Black-Scholes market and valuationmodel. CVA is hedged using the underlying stock. Here, a Black-Scholes delta hedge takes place. Introducing jumps in the stock dynamicsresults in an extra source of randomness. In particular, Π( t ) + w ( t ) for Π NoCCR shows roughlytwice as much volatility through the option’s lifetime. For Π CCR , the simulated defaults appearto be dominating as there are no significant differences compared to the pure Black-Scholescase. (a) Average P&L P ( t ). (b) Volatility of P&L P ( t ).(c) Average P&L U ( t ). (d) Volatility of P&L U ( t ).Figure 8: Comparison of Π NoCCR and Π CCR using a Merton market and a Black-Scholesvaluation model. CVA is not hedged. There are two observations when comparing the P&L P mean in Figure 8a with the pureBlack-Scholes results from Figure 4a. First, the level of the average is higher during the lifetimeof the option. Second, a peak is observed close to maturity. This is the result of the Black-16choles delta’s inability to cope with jumps just before maturity. In particular, a jump in theunderlying stock for the paths where the option is close to the ATM point before maturitycan result in a change of the option being in or out of the money. The volatility of P&L P from Figure 8b is larger as well as significantly more variable over time when comparing withthe pure Black-Scholes case (see Figure 4b), though the shape stays roughly the same. Theseresults are in line with our expectations, as the Black-Scholes delta does not take into accountthe additional source of randomness from the stock jumps, so this risk is not hedged.The average P&L U in Figure 8c is slightly lower than the average P&L P . Yet the peakbefore maturity observed for P&L P remains present, indicating that the hedging strategy incombination with the chosen model does not result in the desired results. Recall that this peakis not present in the pure Black-Scholes case (see Figure 4c). For the volatility of P&L U inFigure 8d, the same reduction versus the P&L P volatility is observed as in the pure Black-Scholes case (see Figure 4d). However, close to maturity the P&L U volatility is even higherthan the P&L P volatility. This effect is the result of the gamma explain volatility just beforematurity, which has significant peaks. The option gamma is the rate of change in the optiondelta w.r.t. changes in the underlying price. For the ATM cases, the delta is extremely sensitiveto changes in the underlying asset. So, paths around the ATM level just before maturity causethis increase in P&L U volatility. Furthermore, we observe significant movement in the volatilityfor the Merton case just before the maturity of the option. All this together is a clear indicationthat the jump effects are missing in the explain process.The CVA hedge in this context has no effect on either Π( t ) + w ( t ) or P&L U , which is in linewith our observations for the pure Black-Scholes case. For the P&L P the same effect of theCVA hedge is observed as in the pure Black-Scholes case: the average P&L P of Π CCR overlapssignificantly with that of Π NoCCR after the introduction of the CVA hedge. Furthermore, theinitial volatility in the Π CCR is much lower in the case of a CVA hedge, and the volatilitiesoverlap much earlier in the case of a CVA hedge.Introducing jumps in the Merton dynamics indeed results in additional randomness whencomparing with the pure Black-Scholes case, already before considering the CCR. For the casewith CCR, the introduction of jumps in the stock seems to dominate the effect of the CCRin this case study. The CVA hedge has the same desired effects for both Black-Scholes andMerton paths, of course ignoring the additional randomness introduced by the Merton jumps.We can conclude that the Black-Scholes delta hedge in the case of a Merton market has itsshortcomings. Therefore, as a next step, we compare these results with a Merton delta hedge. We now use the Merton delta hedge in a Merton market, and do not hedge the CVA. Theonly difference compared to the Black-Scholes delta hedge is in the average P&L P and P&L U .For the P&L P , the peak close to maturity we observed in the case of a Black-Scholes deltahedge, see Figure 8a, has disappeared. For the P&L U , the big upward peak just before maturity,see Figure 8c, has disappeared too. A minor downward peak remains due to instability in thegamma explain. The Merton Greeks are not capable of mitigating these instabilities.The effect of the CVA hedge is clearly visible in Figures 9a and 9b. After filtering out theeffects of the Merton delta as described before, the CVA hedge yields the same conclusion asfor the Black-Scholes delta hedge.Overall, the impact of moving from the Black-Scholes delta hedge to the Merton deltahedge was significant, especially in the average P&L P . Yet some issues and undesired resultsremained, for example due to the instability of the gamma explain. Clearly we need anotherway to hedge away more of the jump risk associated with the Merton model. Therefore, we hedge the residual jump risk from the Merton model using an extra instru-ment in the hedging portfolio: one extra option. Typically, OTM options are chosen in thehedging portfolio as they are cheap. One of the difficulties resulting from Equation (4.9) is that ∂H ( t ) ∂ξ J tends to zero close to maturity, due to the option being OTM. ∂V ( t ) ∂ξ J may behave thesame, however, it is likely that ∂H ( t ) ∂ξ J will decay faster due to the option being deeper OTM.As a result, η ( t ) in Equation (4.9) will increase drastically close to maturity of the option. For17 a) Average P&L P ( t ). (b) Volatility of P&L P ( t ).Figure 9: Comparison of Π NoCCR and Π CCR using a Merton market and valuation model.CVA is hedged using the underlying stock. the hedging option we choose an OTM put with strike K = 90, maturity T = 1 and controllingthe same number of shares as the original option we hedge. In case we hedge the CVA, this isdone using both the underlying stock and same option we use to hedge the product itself. (a) Average P&L P ( t ). (b) Volatility of P&L P ( t ).(c) Average P&L U ( t ). (d) Volatility of P&L U ( t ).Figure 10: Comparison of Π NoCCR and Π CCR using a Merton market and valuationmodel. An additional option is added to the hedging portfolio. CVA is not hedged. The effect of the single option hedge is clearly visible in the volatility of Π( t ) + w ( t ) forΠ NoCCR . Where for the Black-Scholes hedge we observe roughly twice as much as volatilitythrough the lifetime of the option, we now observe results similar to the pure Black-Scholescase. So the single option hedge yields the desired effects. For Π CCR , the CCR still dominatesthe jumps from the stock. 18he mean P&L P in Figure 10a exhibits a non-linear behaviour due to the additional op-tionally in the portfolio. The P&L P volatility in Figure 10b is much lower compared to before,see Figure 8b. Again we observe the beneficial effects of the hedging option, as the results lookmore like the pure Black-Scholes case.The average P&L U in Figure 10c, compared to the Merton delta hedge, is comparable up tothe same non-linear effect we also observe for the P&L P . Overall, the volatility in Figure 10dis comparable to the previous results in Figure 8d, except for a significantly higher peak justbefore maturity. This is the result of an exploding gamma explain volatility close to maturity. (a) Average P&L P ( t ). (b) Volatility of P&L P ( t ).Figure 11: Comparison of Π NoCCR and Π CCR using a Merton market and valuationmodel. An additional option is added to the hedging portfolio. CVA is hedged usingboth the underlying stock and the additional option. As before, the CVA hedge has no effect on Π( t ) + w ( t ) or P&L U , which is in line with ourother observations so far. The effect of the CVA hedge on P&L P , see Figure 11, is similar to theeffect in the pure Black-Scholes case. Especially the volatility, which is the dominating factor,is comparable to the pure Black-Scholes case (see Figure 7b). For the average we observein Figure 11a that the Π CCR is slightly above Π NoCCR , which is also observed for the pureBlack-Scholes case (see Figure 7a).Adding the option to the hedging strategy has a clear effect of smaller P&L P volatility.Increasing the number of options in the hedging strategy will likely increase the hedging per-formance, however, the effect of the first option is already significant [6, 16]. The increase inP&L U volatility can be explained by the hedging option having a fixed strike. For some of thesimulated paths, this hedge option may become significantly OTM, meaning the sensitivity tothe jumps is close to zero. This results in a less effective hedge of the jump risk. 6. Conclusion In conclusion, dynamic CVA hedging is now better understood, both in a Black-Scholesand a Merton jump-diffusion setting. Starting from a theoretical hedging framework, we haveexamined the mechanics of a trading strategy which included CVA pricing and hedging. Wevisualized cash-flows and exchanges of traded instruments of the portfolio. For a case studyof a portfolio containing European options and the underlying stock, we used a Monte Carlosimulation to study hedging. Hedging performance was assessed by analyzing the tradingstrategy balance, including the corresponding wealth account. Furthermore, we studied theP&L behaviour of the strategy, as well as the performance of the P&L explain. In particular,analytic results helped us to explain and analyze P&L behaviour that was observed from thesimulation.First, in a Black-Scholes setting we showed that failing to charge CVA to a credit riskycounterparty will result in an average guaranteed loss. We can see CVA as a fair compensationfor this credit risk. Merely treating CVA as a cash amount turns out to be nave, as CVA isdriven by market risk. In fact, failing to hedge the market risk component of CVA resultsin an unstable trading strategy, and thus implies the hedge in the portfolio is incomplete.19e have shown that CVA in a Black-Scholes context can satisfactorily be hedged using theunderlying stock. For the P&L P , we saw a significant increase in volatility as the optionmaturity approached. We showed that P&L P is driven by the option gamma, which is knownto become unstable close to maturity, especially for options that are close to the ATM point.Thus, the strong volatility increase is a direct consequence of the parameters chosen in ourcase study. CVA hedging significantly lowers the volatility P&L P . Therefore, we conclude thathedging CVA is necessary for a stable trading strategy.Next, we examined our case study portfolio when the underlying stock is driven by a Mertonjump-diffusion process. A Black-Scholes delta hedge in this is incapable of dealing with thejump risk introduced by Merton’s model. Hedging the CVA in this situation is still a must,though clearly something is missing from the strategy. Though using the Merton option deltasfor the hedge is better than the Black-Scholes delta hedge, the effect is insufficient. Adding anoption as a hedging instrument mitigates a significant part of the jump risk. 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Woodward. Managing Smile Risk. Wilmott Magazine ,3:84–108, September 2002.[23] R.C. Merton. Option pricing when the underlying stock returns are discontinuous. Journal of FinancialEconomics , 3(1-2):125144, January 1976.[24] S. Crepey. Bilateral counterparty risk under funding constraints - Part I: Pricing. Mathematical Finance ,25(1):1–22, January 2015.[25] S. Crepey. Bilateral counterparty risk under funding constraints - Part II: CVA. Mathematical Finance ,25(1):23–50, January 2015.[26] S.G. Kou. A Jump-Diffusion Model for Option Pricing. Management Science , 48(8):10861101, August2002.[27] S.L. Heston. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bondsand Currency Options. Review of Financial Studies , 6(2):327–343, April 1993.[28] T.R. Bielecki, M. Rutkowski. Valuation and Hedging of Contracts with Funding Costs and Collateraliza-tion. SIAM Journal on Financial Mathematics , 6(1):594–655, July 2015.[29] U. Cherubini. Counterparty Risk in Derivatives and Collateral Policies The Replicating Portfolio Approach.July 2005.[30] V. Naik, M. Lee. General Equilibrium Pricing of Options on the Market Portfolio with DiscontinuousReturns. Review of Financial Studies , 3(4):493–521, October 1990. Appendix A. The Merton jump-diffusion model Appendix A.1. Analytical European option prices and sensitivities Here we re-iterate a result from [23, 7], namely the analytical expression for a European calloption price on stock S , maturity T , and strike K , under the Merton jump-diffusion model: V ( t ) = e − r ( T − t ) (cid:88) n ≥ [ ξ J ( T − t )] n e − ξ J ( T − t ) n ! V ( n ) , (A.1) V ( n ) = e ˆ µ X ( n )+ ˆ σ X ( n )( T − t ) Φ( d ) − K Φ( d ) , ˆ µ X ( n ) = log( S ( t )) + (cid:20) r − ξ J (cid:16) e µ J + σ J − (cid:17) − σ (cid:21) ( T − t ) + nµ J , ˆ σ X ( n ) = (cid:115) σ + nσ J T − t ,d = log (cid:16) S ( t ) K (cid:17) + (cid:104) r − ξ J (cid:16) e µ J + σ J − (cid:17) − σ + ˆ σ X ( n ) (cid:105) ( T − t ) + nµ J ˆ σ X ( n ) √ T − t ,d = d − ˆ σ X ( n ) √ T − t, where r is the risk-free interest rate, σ is the stock volatility. Furthermore, µ J , σ J , and ξ J are respectively the jump mean, volatility and intensity. In the case of a put option, one cansimply apply the put-call parity to the call option price from Equation (A.1).It can be shown that call price V ( t ) from Equation (A.1) has the following analyticalexpressions of derivatives w.r.t. jump parameters µ J , σ J and ξ J : ∂V ( t ) ∂µ J = e − r ( T − t ) (cid:88) n ≥ [ ξ J ( T − t )] n e − ξ J ( T − t ) n ! (cid:16) n − ξ J e µ J + σ J ( T − t ) (cid:17) e ˆ µ X ( n )+ ˆ σ X ( n )( T − t ) Φ( d ) , (A.2) ∂V ( t ) ∂σ J = e − r ( T − t ) (cid:88) n ≥ [ ξ J ( T − t )] n e − ξ J ( T − t ) n ! (cid:18) σ J (cid:16) n − ξ J e µ J + σ J ( T − t ) (cid:17) e ˆ µ X ( n )+ ˆ σ X ( n )( T − t ) Φ( d ) + Kφ ( d ) 1ˆ σ X ( n ) nσ J √ T − t (cid:19) , (A.3) ∂V ( t ) ∂ξ J = e − r ( T − t ) (cid:88) n ≥ [ ξ J ( T − t )] n e − ξ J ( T − t ) n ! (cid:18) V ( n ) (cid:20) nξ J − ( T − t ) (cid:21) − (cid:16) e µ J + σ J − (cid:17) ( T − t )e ˆ µ X ( n )+ ˆ σ X ( n )( T − t ) Φ( d ) (cid:19) . (A.4)21y the put-call parity it can easily be seen that the jump parameter sensitivities above holdfor both call and put options.In a similar fashion we can derive the following expression for the call option delta and gamma under the Merton jump-diffusion model: ∂V ( t ) ∂S = e − r ( T − t ) (cid:88) n ≥ [ ξ J ( T − t )] n e − ξ J ( T − t ) n ! · e ˆ µ X ( n ) − log( S ( t ))+ ˆ σ X ( n )( T − t ) Φ( d ) , (A.5) ∂ V ( t ) ∂S = e − r ( T − t ) (cid:88) n ≥ [ ξ J ( T − t )] n e − ξ J ( T − t ) n ! · e ˆ µ X ( n ) − log( S ( t ))+ ˆ σ X ( n )( T − t ) φ ( d ) S ( t )ˆ σ X ( n ) √ T − t , (A.6)The option pricing formula (A.1) contains an infinite sum of scaled Black-Scholes optionprices, being a direct result of the jump size following a continuous distribution. This clearlyshows that if one attempts to hedge the jump risk introduced by the model, one would theo-retically need infinitely many options to do so, which is infeasible from a practical perspective. Appendix A.2. Jump parameter impact on implied volatilities Here we assess the impact of the jump parameters on the Black-Scholes implied volatilities σ imp . This is done by varying the parameters µ J , σ J , and ξ J one by one while keeping all theother parameters constant. (a) Impact of jump size mean µ J . (b) Impact of jump size volatility σ J .(c) Impact of jump intensity ξ J .Figure A.12: Merton jump parameter impact on σ imp . Parameters: S ( t ) = 100, r =0 . σ = 0 . t = 0, T = 1. Jump parameters base values: µ J = − . σ J = 0 . ξ J = 0 . µ J ∈ {− . , − . , . , . , . } , σ J ∈ { . , . , . , . , . } , ξ J ∈{ . , . , . , . , . } . In Figure A.12a we observe that µ J has a clear level effect . In addition the slope , i.e., skew ,and direction of the slope are affected. For the simulations we will choose a negative value of µ J , which implies that the stock will be likely to go down. The reason of this choice is thattypically in the market we observe that OTM options are cheaper, which matches the higherlikelihood of a downward jump in stock. From Figure A.12b we conclude that σ J has an impacton both the level and the curvature of the implied volatility. This makes sense as a higher σ J means more uncertainty, hence higher option prices, so also a higher σ imp . The curvature effectcan be observed by the OTM strike region exhibiting a more pronounced hockey-stick effect22hich becomes larger as σ J increases. Finally Figure A.12c tells us that ξ J impacts the levelonly. For higher ξ J , jumps will occur more frequently, introducing more uncertainty, resultingin higher σ imp . Appendix A.3. Convergence of number of terms in expansion We have seen that the option price (A.1) as well as sensitivities (A.2-A.6) can be written asan infinite sum of scaled Black-Scholes option prices. The question arises how to approximatethese infinite sums in a practical situation by means of a numerical implementation. Each ofthe quantities we need to compute contain the following weights in the infinite summation:[ ξ J ( T − t )] n e − ξ J ( T − t ) n ! . These weights appear to be exponentially decaying, indicating that in a numerical implemen-tation the sum can simply be cut off at some point at which the desired accuracy is achieved.As an example we cut off the option price at a basis-point (bps) level, i.e., 10 − . (a) Impact of maturity T . (b) Impact of strike K .(c) Impact of jump size mean µ J . (d) Impact of jump size volatility σ J .(e) Impact of jump intensity ξ J .Figure A.13: Impact on number of terms in the Merton option price expansion. Tol-erance 10 − has been used to determine the cutoff point. Parameters: t = 0, T = 1, S ( t ) = 100, r = 0 . σ = 0 . K = 100, µ J = − . σ J = 0 . ξ J = 0 . From the experiment in Figure A.13 we can conclude that for a chosen impact ( T , K , µ J , σ J , ξ J ) all examined quantities (price, delta, jump parameter sensitivities) exhibit the same23attern regarding the number of terms required to achieve the desired bps accuracy. Indeed,using a higher tolerance for the cutoff point yields a higher number of terms required in theexpansion, see Figure A.14 where we cut off at 10 − .These results are in line with those presented by Boen on European rainbow option valuesunder the two-asset Merton jump-diffusion model [17]. Boen derives a semi-closed analyticalformula for European rainbow options, under the two-asset Merton jump-diffusion model.Depending on the size of ξ J ( T − t ) the number of terms used in the sum needs to be sufficientlylarge to guarantee accurate option prices. Convergence of this formula in the case of a Europeanput-on-the-min option is illustrated. (a) Impact of maturity T . (b) Impact of strike K .(c) Impact of jump size mean µ J . (d) Impact of jump size volatility σ J .(e) Impact of jump intensity ξ J .Figure A.14: Impact on number of terms in the Merton option price expansion. Toler-ance 10 − has been used to determine the cutoff point. Parameters: t = 0, T = 1, S ( t ) = 100, r = 0 . σ = 0 . K = 100, µ J = − . σ J = 0 . ξ J = 0 . ppendix B. Variance analysis derivations As is stated in Section 5.1.3, we derive an analytic expression for the mean and variance ofP&L P from Equation (5.6). But before that, we need the following two results. First, S ( t k ) d = S ( t k − )e X , (B.1) X = (cid:18) r − σ (cid:19) [ t k − t k − ] + σ (cid:112) t k − t k − Z ∼ N (cid:18)(cid:18) r − σ (cid:19) [ t k − t k − ] , σ [ t k − t k − ] (cid:19) =: N ( µ X , σ X ) , where Z ∼ N (0 , S ( t k − ) d = S ( t ) exp (cid:26)(cid:18) r − σ (cid:19) [ t k − − t ] + σ (cid:112) t k − − t ˜ Z (cid:27) , (B.2)where ˜ Z ∼ N (0 , Z . Using Equation (B.2) and τ := T − t k − we write d as: d = ln S ( t ) K + (cid:0) r − σ (cid:1) [ T − t ] σ √ τ + (cid:114) t k − − t τ ˜ Z ∼ N (cid:32) ln S ( t ) K + (cid:0) r − σ (cid:1) [ T − t ] σ √ τ , t k − − t τ (cid:33) =: N ( µ d , σ d ) . Appendix B.1. Mean We are interested in how P&L P from Equation (5.6) is distributed conditional on theinformation known at today ( t ). First, look at the mean: E t [P&L P ( t k )] ≈ E t (cid:20) K e − rτ σ √ τ (cid:2) e X − (cid:3) φ ( d ) (cid:21) = K e − rτ σ √ τ E t (cid:104)(cid:2) e X − (cid:3) (cid:105) E t [ φ ( d )] , (B.3)where we use the independency of (cid:2) e X − (cid:3) and φ ( d ) due to the independent increments.The first expectation from Equation (B.3) can be written as: E t (cid:104)(cid:2) e X − (cid:3) (cid:105) = V ar t (cid:0) e X − (cid:1) + (cid:0) E t (cid:2) e X − (cid:3)(cid:1) = (cid:16) e σ X − (cid:17) e µ X + σ X + (cid:16) e µ X + σ X − (cid:17) , (B.4)where we use the lognormality of e X . The second expectation can be written as: E t [ φ ( d )] = E t (cid:104) e − d (cid:105) √ π = e − µ d ( σ d ) √ π (cid:113) σ d , (B.5)where we use the fact that d follows σ d times a non-central chi-squared distribution withone degree of freedom ( k = 1), and with non-centrality parameter λ = µ d σ d , i.e., d ∼ σ d χ (cid:18) , µ d σ d (cid:19) . Using results (B.4) and (B.5) allows us to rewrite Equation (B.3) as: E t [P&L P ( t k )] ≈ K e − rτ σ √ τ e − µ d ( σ d ) √ π (cid:113) σ d (cid:18)(cid:16) e σ X − (cid:17) e µ X + σ X + (cid:16) e µ X + σ X − (cid:17) (cid:19) . (B.6)25 ppendix B.2. Variance For the variance of P&L P from Equation (5.6) we have: V ar t (P&L P ( t k )) = E t (cid:104) (P&L P ( t k )) (cid:105) − ( E t [P&L P ( t k )]) , (B.7)where the second term can be computed using Equation (B.6). By the independent incrementargument, the first term in Equation (B.7) can be written as: E t (cid:104) (P&L P ( t k )) (cid:105) ≈ E t (cid:34)(cid:18) K e − rτ σ √ τ (cid:2) e X − (cid:3) φ ( d ) (cid:19) (cid:35) = K e − rτ σ τ E t (cid:104)(cid:2) e X − (cid:3) (cid:105) E t (cid:104) ( φ ( d )) (cid:105) . (B.8)For the first expectation in Equation (B.8) we write: E t (cid:104)(cid:2) e X − (cid:3) (cid:105) = E t (cid:2) e X − X + 6e X − X + 1 (cid:3) = e µ X + σ X − µ X + σ X + 6e µ X + σ X − µ X + σ X + 1 , (B.9)using the properties of the moment generating function of X . The second expectation inEquation (B.8) can be written as: E t (cid:104) ( φ ( d )) (cid:105) = E t (cid:104) e − d (cid:105) π = e − µ d σ d π (cid:113) σ d , (B.10)where we once more use that d follows σ d times a non-central chi-squared distribution withparameters as stated before. Results (B.9) and (B.10) allow us to rewrite Equation (B.8) as: E t (cid:104) (P&L P ( t k )) (cid:105) ≈ K e − rτ σ τ e − µ d σ d π (cid:113) σ d (cid:16) e µ X + σ X − µ X + σ X + 6e µ X + σ X − µ X + σ X + 1 (cid:17) . (B.11)Using the results from Equations (B.6) and (B.11) in Equation (B.7) yields: V ar t (P&L P ( t k )) ≈ K e − rτ πσ τ f ( µ X , σ X ) · e − µ d σ d (cid:113) σ d − g ( µ X , σ X ) · e − µ d σ d σ d , (B.12)where f ( µ X , σ X ) := e µ X + σ X − µ X + σ X + 6e µ X + σ X − µ X + σ X + 1 , (B.13) g ( µ X , σ X ) := (cid:18)(cid:16) e σ X − (cid:17) e µ X + σ X + (cid:16) e µ X + σ X − (cid:17) (cid:19) . (B.14) Appendix C. Desk structure and flow of cash The simulation is illustrated by a series of schematic drawings that indicate the flow ofcash and instruments. First, in Appendix C.1, we consider the case without CCR to get abasic understanding. In Appendix C.2, CCR is introduced, and the trading desk and xVAdesk are represented as a single entity, referred to as the trading desk. Finally, in AppendixC.3, we remove this assumption by examining the internal exchange of cash-flows and productsbetween the desks. In all figures to follow, dotted lines indicate flows of cash, hence this relates26o wealth accounts w , w trading , and w xVA . Furthermore, solid lines indicate the exchange of aproduct/asset, hence this relates to portfolios Π, Π trading , and Π xVA . Values are seen from theperspective of the desks, i.e., an arrow away from a desk indicates the desk needs to pay and anarrow towards a desk indicates the desk receives. We assume that the wealth account at time t k − is positive, in the sense that this can be seen as a deposit with the Treasury departmentfor which interest is received. Appendix C.1. Case without CCR At this stage the xVA desk is not involved as we assume no xVAs are required. Here η ( t ) = − ∆( t ) = − ∂V ( t ) ∂S . (C.1)At t , see Figure C.15, the option is bought from the counterparty (blue lines) and theBlack-Scholes delta hedge (C.1) is constructed (red lines). BankWealthaccount w ( t )Treasury Trading deskΠ( t ) MarketExchangeClearinghouseCounterparty V ( t ) S ( t )∆( t ) S ( t )∆( t )∆( t ) shares of S Option VV ( t ) Figure C.15: Case without CCR, situation at t . The blue lines correspond to the trade;the red lines correspond to the hedge. Then for t < t k < t K , see Figure C.16, interest on the wealth account is received from theTreasury department (teal lines) and the hedge is rebalanced (red lines).Finally, the situation maturity t K is displayed in Figure C.17. We first again receive interest(teal lines), then close the hedge (red lines) and settle the payoff of the option in cash (bluelines). Appendix C.2. Case with CCR, single desk within the bank Assume that the trading desk and xVA desk are represented as a single entity, still referredto as the trading desk. Here η ( t ) = − ˆ∆( t ) = − ∂V ( t ) ∂S [1 − (1 − R ) PD( t, t K )] , (C.2)meaning that also the market-risk component of the CVA is hedged in the market.Initially at t , see Figure C.18, the risky option V is bought from the counterparty (bluelines). As in the previous case, we construct a hedge (red lines), but now with a position thattakes into account the hedging of CVA, see Equation (C.2). In the case that w ( t k − ) is negative we have to pay interest to the Treasury desk on the borrowed amountand the flows between the Treasury department and wealth account would be in the opposite direction. ankWealthaccount w ( t k )Treasury Trading deskΠ( t k ) MarketExchangeClearinghouse S ( t k )d∆( t k )Interest w ( t k − ) (cid:104) B ( t k ) B ( t k − ) − (cid:105) S ( t k )d∆( t k )d∆( t k ) shares of S Figure C.16: Case without CCR, situation at t < t k < t K . The red lines correspond tothe hedge; the teal lines correspond to the interest. BankWealthaccount w ( t K )Treasury Trading deskΠ( t K ) MarketExchangeClearinghouseCounterparty( S ( t K ) − K ) + K ( t K )∆( t K − )Interest w ( t K − ) (cid:104) B ( t K ) B ( t K − ) − (cid:105) S ( t K )∆( t K − )∆( t K − ) shares of S ( S ( t K ) − K ) + Figure C.17: Case without CCR, situation at t K . The blue lines correspond to the trade;the red lines correspond to the hedge; the teal lines correspond to the interest. For t < t k < t K if there is no default, we rebalance our hedge as before and receive intereston the wealth account. If a default does occur, see Figure C.19, the risk-free closeout takesplace (brown lines) and the same risk-neutral option is bought from the clearing house (violetlines). As always, we receive interest (teal lines). Regarding the hedge (red lines), therewe close the hedge that takes into account the CVA, see Equation (C.2), and construct a newhedge for the option bought from the clearing house. This new hedge is equivalent to theBlack-Scholes hedge for the risk-free case in Equation (C.1).The situation at maturity, see Figure C.20, is then the same as for the case without CCR,apart from the payoff of the option now being settled with the clearing house rather than thecounterparty. Here we assume the default takes place before maturity. The case of default at maturity is not discussedfor sake of brevity. The remaining case naturally extends from the demonstrated material. ankWealthaccount w ( t )Treasury Trading deskΠ( t ) MarketExchangeClearinghouseCounterparty V ( t )ˆ∆( t ) S ( t )CVA( t ) ˆ∆( t ) S ( t )ˆ∆( t ) shares of S Option V V ( t )CVA( t ) Figure C.18: Case with CCR, single desk within the bank, situation at t . The bluelines correspond to the trade; the red lines correspond to the hedge. BankWealthaccount w ( t k )Treasury Trading deskΠ( t k ) MarketExchangeClearinghouseCounterparty V ( t d )∆( t d ) S ( t d ) − ˆ∆( t d − ) S ( t d ) R · V ( t d )Interest w ( t d − ) (cid:104) B ( t d ) B ( t d − ) − (cid:105) ∆( t d ) S ( t d ) − ˆ∆( t d − ) S ( t d )∆( t d ) − ˆ∆( t d − )shares of S Option V R · S ( t d ) Option VV ( t d ) Figure C.19: Case with CCR, single desk within the bank, situation at t < τ = t d We now remove the assumption of a single desk, and examine the internal exchange ofcash-flows and products between the trading desk and xVA desk. Here η ( t ) = − ∂V ( t ) ∂S (cid:124) (cid:123)(cid:122) (cid:125) − ∆( t ) + ∂V ( t ) ∂S (1 − R ) PD( t, t K ) (cid:124) (cid:123)(cid:122) (cid:125) − ∆( t ) , meaning that also the market-risk component of the CVA is hedged in the market.Initially, the situation at t is displayed in Figure C.21, where the positions in trading andhedging instruments are assumed. Note the transfer of the CVA between the trading and xVA29 ankWealthaccount w ( t K )Treasury Trading deskΠ( t K ) MarketExchangeClearinghouse( S ( t K ) − K ) + ∆( t K − ) S ( t K )Interest w ( t K − ) (cid:104) B ( t K ) B ( t K − ) − (cid:105) ∆( t K − ) S ( t K )∆( t K − ) shares of S ( S ( t K ) − K ) + Figure C.20: Case with CCR, single desk within the bank, situation at t K . The redlines correspond to the hedge; the teal lines correspond to the interest; the violet linescorrespond to the new trade entered upon default. desk, indicating that the trading desk manages the risk-free component of the transaction,whereas the xVA desk takes responsibility for the CVA. BankWealthaccount w trading ( t )Wealthaccount w xVA ( t )Treasury Trading deskΠ trading ( t )xVA deskΠ xVA ( t ) MarketExchangeClearinghouseCounterpartyCVACVA( t ) V ( t )∆( t ) S ( t )∆( t ) S ( t )CVA( t ) ∆( t ) S ( t )∆( t ) shares of S ∆( t ) shares of S ∆( t ) S ( t )Option V V ( t )CVA( t ) Figure C.21: Case with CCR, separate trading desk and xVA desk within the bank,situation at t . The blue lines correspond to the trade; the red lines correspond to thehedge; the cyan lines correspond to the CVA hedge. At t < t k < t K , if there is no default, both desks rebalance their hedging positions andreceive interest on their wealth account. In case of a default at t < τ = t d < t K , see Fig-ure C.22, a risk-free closeout takes place between the bank and the counterparty. Furthermore,the xVA desk enters the same risk-free contract V with a clearing house and closes its hedgingposition on the CVA. The xVA desk then gives the new contract V to the trading desk, sothat they are immune to the default. The ‘damage’ of the default is thus visible at the xVAdesk level, which is precisely the place handling this risk.Finally, Figure C.23 represents the situation at t K , at which the option payoff is settled incash, and the trading desk closes its hedging positions. The output metrics as introduced inSection 3.1 can be carefully analyzed to draw conclusions about the hedging strategy used.30 ankWealthaccount w trading ( t d )Wealthaccount w xVA ( t d )Treasury Trading deskΠ trading ( t d )xVA deskΠ xVA ( t d ) MarketExchangeClearinghouseCounterpartyOption V CVA R · V ( t d ) S ( t d )d∆( t d )Interest w trading ( t d − ) (cid:104) B ( t d ) B ( t d − ) − (cid:105) R · V ( t d )∆( t d − ) S ( t d ) V ( t d )Interest w xVA ( t d − ) (cid:104) B ( t d ) B ( t d − ) − (cid:105) S ( t d )d∆( t d )d∆( t d ) shares of S ∆( t d − ) shares of S ∆( t d − ) S ( t d ) V ( t d )Option V Option V R · V ( t d ) Figure C.22: Case with CCR, separate trading desk and xVA desk within the bank,situation at t < τ = t d < t K . The red lines correspond to the hedge; the cyan linescorrespond to the CVA hedge; the teal lines correspond to the interest; the brown linescorrespond to the default; the violet lines correspond to the new trade entered upondefault. BankWealthaccount w trading ( t K )Wealthaccount w xVA ( t K )Treasury Trading deskΠ trading ( t K )xVA deskΠ xVA ( t K ) MarketExchangeClearinghouse( S ( t K ) − K ) + ∆( t K − ) S ( t K )( S ( t K ) − K ) + Interest w trading ( t K − ) (cid:104) B ( t K ) B ( t K − ) − (cid:105) Interest w xVA ( t K − ) (cid:104) B ( t K ) B ( t K − ) − (cid:105) ∆( t K − ) shares of S ∆( t K − ) S ( t K )( S ( t K ) − K ) + Figure C.23: Case with CCR, separate trading desk and xVA desk within the bank,situation at t K . The red lines correspond to the hedge; the teal lines correspond to theinterest; the violet lines correspond to the new trade entered upon default.. The red lines correspond to the hedge; the teal lines correspond to theinterest; the violet lines correspond to the new trade entered upon default.