Adapting the CVA model to Leland's framework
AAdapting the CVA model to Leland’s framework
P. Amster , and A.P. Mogni Departamento de Matem´atica,Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos Aires and IMAS - CONICETCiudad Universitaria, Pabell´on I, 1428 Buenos Aires, Argentina
E-mails : [email protected] — [email protected]
Abstract
We consider the framework proposed by Burgard and Kjaer in [1] that derives the PDE which governs theprice of an option including bilateral counterparty risk and funding. We extend this work by relaxing theassumption of absence of transaction costs in the hedging portfolio by proposing a cost proportional to theamount of assets traded and the traded price. After deriving the nonlinear PDE, we prove the existence ofa solution for the corresponding initial-boundary value problem. Moreover, we develop a numerical schemethat allows to find the solution of the PDE by setting different values for each parameter of the model.To understand the impact of each variable within the model, we analyze the Greeks of the option and thesensitivity of the price to changes in all the risk factors.
Keywords:
Nonlinear parabolic differential equations, Option pricing models, Transaction costs, CVA, Eulermethod.
Under the Black-Scholes pricing framework [2], the price of an option is derived by constructing a hedging port-folio consisting of a certain amount of the underlying asset and a money-market account. This methodologyrelies on a list of different assumptions that are set to simplify the model. For example, constant values ofvolatility and interest rates, the non-existence of dividend yields, the efficiency of the markets and the non-existence of transaction costs, among others. Nonetheless, the probability of default of both the issuer of theoption and the counterparty are not being considered when constructing the hedging portfolio. Therefore, it isexpected that the option price obtained by the standard modeling approach will be different when comparedwith the actual price of the contract.As explained in [3], the counterparty credit risk is the risk that a counterparty in a financial contract defaultsprior to the expiration of the contract and fails to make future payments. To include this probability in thepricing of a contract, the Credit Valuation Adjustment (CVA) is used. As described in [4], CVA is defined asthe difference between the price of the instrument including credit risk and the price where the counterparty ofthe transaction is considered free of risk. By definition, the CVA will be always positive if only the counterpartyrisk is considered. When the issuer credit riskiness is also taken into account, the Debit Valuation Adjustment(DVA) is included into the formula. The DVA acts oppositely as the CVA by adding value to the option whenthe issuer risk increases. If we set V BS as the Black-Scholes option price default free and V as the adjustedoption price we get V = V BS − CV A + DV A where
CV A is a cost and
DV A is a benefit.As mentioned in [4], before 2008 crisis, CVA was commonly calculated and charged only by tier one banks bychoosing either unilateral or bilateral models. After 2008 crisis, not only bilateral CVA started to be widelyused but also a different set of value adjustments began to be applied such as the Funding Valuation Adjustmentor FVA (including costs of of funding) and the Capital Valuation Adjustment or KVA (including cost of capital).1 a r X i v : . [ q -f i n . M F ] F e b everal works have been developed within the family of value adjustments for different OTC derivatives. In[5], the authors derive the CVA by decomposing the portfolio’s value into a set of binary states: positive ornegative cash flows directions, two possible default states and to receive or not the recovery rate. In [6], theauthors study a CDS pricing model which includes the probability of default of the counterparty. Credit spreadvolatilities are considered by assuming default intensities following a CIR dynamic with a correlation parameterwithin. The authors in [7] generalize these model in two ways. First, by allowing correlation not only betweendefault times but also with the underlying portfolio risk factors (interest rates). Second, by considering theprobability of default of the issuer.These previous works are then expanded by [8] in order to apply that methodology on IR Swaps and IR Ex-otics. DVA and FVA are included in the pricing framework by [9] where a recursive formula is obtained afterdiscounting all the cash flows occurring after the trading position is entered. These cash flows include not onlythe product cash flows (coupons, dividends, etc) but also cash flows required by collateral margining, fundingand investing procedures and default events.Finally, two works try to propose a general theory. In [1] the authors analyze the bilateral counterparty risk(CVA and DVA) with funding costs (FVA) by constructing a hedging portfolio composed by the underlyingasset, a risk-free zero-coupon bond and two default risky zero-coupon bond of both the issuer and the counter-party. A PDE is formalized by applying the standard self-financing assumption. In [10] (and further in [11, 12]),the author applies a risk-neutral pricing approach under funding constraints in order to obtain a reduced-formbackward stochastic differential equation (BSDE) approach to the problem of hedging and pricing the CVA.The second assumption covered in our model is the (non) existence of transaction costs. Following Leland’sapproach [13], transaction costs can be included in the pricing methodology by applying a discrete-time repli-cating strategy. A nonlinear partial differential equation is obtained for the option price , which is denoted by V ( S, t ); namely, ∂V∂t + 12 ˆ σ (cid:18) S ∂ V∂S (cid:19) S ∂ V∂S + rS ∂V∂S − rV = 0 , where ˆ σ is defined based upon the transaction costs function. This approach was then continued and improvedby [14] and [15].Different choices of transaction costs functions lead to variations on the nonlinear term of the partial differentialequation. In [16], the authors propose a non-increasing linear function and find solutions for the stationaryproblem. In [17], the concept of transaction costs function is generalized and the so-called mean value modi-fication of the transaction costs function is developed. This transformation allows the authors to formulate ageneral one-dimensional Black-Scholes equation by solving the equivalent quasilinear Gamma equation. Otherauthors [18, 19] also find solutions to the problem with constant transaction costs and relaxing the assumptionsof constant volatility and interest rate.The main distinctive aspect in the above-cited works is that they all consider only one asset within the partialdifferential equation. In [20] and [21], the author generalizes the Leland approach to cover different types ofmulti-asset options, developing the nonlinear partial differential equation and solving numerically a list of ex-amples. In [22] the authors combine a multidimensional approach with a general transaction cost function toderive a the dynamics of the option price under a fully nonlinear PDE.In this work we adapt the work of Burgard and Kjaer [1] in order to include the transaction costs generatedby trading the underlying assets and both the issuer and counterparty bonds. We propose an initial constanttransaction cost function proportional to the amount of assets traded. As a consequence, we derive a nonlin-ear PDE that extends the results found in [1] and prove the existence of a solution by applying the SchauderFixed-Point theorem. In the second part of the work, we develop a numerical approach to solve the PDE byconsidering a non-uniform grid on the spatial variable. The main greeks of the option (Delta, Gamma, Rho andVega) are calculated and analyzed to understand how both the value adjustments and transaction costs affectthe behavior of the option price. Nonetheless, a sensitivity analysis on the remaining parameters (hazard rate,recovery rates, etc) is performed to complete the study of the option price dynamics.The structure of the paper is as follows. In Section 2 we propose the market model that leads to the nonlineardynamic of the option price. In Section 3 we apply the Schauder Fixed-Point theorem to derive the existenceof solution for the original problem. Finally, in Section 4, the numerical framework is developed and different2esults are obtained to understand how the parameters affect the option price. The original paper of [1] derives the PDE for the value of a financial derivative considering bilateral counterpartyrisk and funding costs. For this purpose they propose an economy consisting of a risk-free zero-coupon bond,two default risky zero-coupon bond with zero recovery of parties B and C and a spot asset with no default risk.B will refer to the seller and C to the counterparty. Notation will be followed from the original work [1].The dynamics of the four tradable assets under the historical probability measure are defined as follows: dP R = P R r dt,dP B = P B r B dt − P B dJ B ,dP C = P C r C dt − P C dJ C ,dS = µ S dt + S dW t . The default risky zero-coupon bonds are modeled by considering both r B and r C interest rates and J B and J C the two independent point processes that jump from 0 to 1 on default of B and C respectively. The defaultrisk-free zero-coupon bond is a deterministic process with drift equal to r and the spot asset is modeled followinga geometric brownian motion with drift µ and volatility σ . Throughout this work the parameters r, r B , r C , µ and σ are positive and constant. Also we will use the following notation: x + = max ( x, x − = min ( x, . To derive the price of option ˆ V , we adapt the standard Black-Scholes framework [2] applied in [1] by consideringthe Leland’s approach [13]. Hence, we create a self-financing portfolio covering all the underlying risk factorsthat hedges the option. Let Π ( t ) be the seller’s portfolio which consists of δ ( t ) units of S ( t ), α B ( t ) units of P B ( t ), α C ( t ) units of P C ( t ) and β ( t ) units of cash. For hedging purposes we set Π ( t ) + ˆ V ( t ) = 0 and − ˆ V ( t ) = Π ( t ) = δ ( t ) S ( t ) + α B ( t ) P B ( t ) + α C ( t ) P C ( t ) + β ( t ) . (1)We define the transaction costs function for both default risky bonds P B and P C and the spot asset S as follows: T C B ( t, P B ) = C B | α B ( t ) | P B ( t ) ,T C C ( t, P C ) = C C | α C ( t ) | P C ( t ) ,T C S ( t, S ) = C S | δ ( t ) | S ( t )where C B , C C and C S are positive constants. This definition of transaction costs is the standard approachapplied initially in [13] and is the initial step to creating more complex dynamics. In this case, the costs aredefined to be proportional to the amount of assets traded multiplied by the price of each asset. For the purposeof enhancing clarity, we drop the dependencies on every function.By forcing the portfolio to be self-financing, we find that − d ˆ V = δ dS + α B dP B + α C dP C + dβ. (2)where dβ is decomposed as dβ = dβ S + dβ F + dβ C corresponding to the variations in the cash position due toeach of the three assets. In this step we consider the effect of the transaction cost in the hedging strategy. Oneach time step, there would be a decrease in the cash account because of the cost of buying or selling a differentamount of assets. Hence, the original calculations of [1] are modified as follows: • The share position provides a dividend income, a financing cost and a transaction cost. The variation inthe position is found to be dβ S = δ γ S S dt − δ q S S dt − dT C S (3)3 After the own bonds are purchased, if any surplus in cash is available, it must earn the free-risk-rate r . If borrowing money, the seller needs to pay the rate r F . In this case, transaction costs appear whencalculating the surplus after the own bonds purchasing. The variation in this position is determined by dβ F = r (cid:16) − ˆ V − α B P B − T C B (cid:17) + dt + r F (cid:16) − ˆ V − α B P B − T C B (cid:17) − dt = r (cid:16) − ˆ V − α B P B − T C B (cid:17) dt + s F (cid:16) − ˆ V − α B P B − T C B (cid:17) − dt (4)where s F = r F − r is the funding spread. • Finally, a financing cost due to short-selling the counterparty bond and its related transaction costs areconsidered for calculating the variation in the cash counterparty position as follows: dβ C = − α C r P C dt − dT C C . (5)By applying equations (3), (4) and (5) in (2), we obtain − d ˆ V = δ dS + α B P B ( r B dt − dJ B ) + α C P C ( r C dt − dJ C ) − dT C C − dT C S + (6) (cid:20) (cid:16) − ˆ V − α B P B − T C B (cid:17) + + r F (cid:16) − ˆ V − α B P B − T C B (cid:17) − + δ ( γ S − q S ) S − α C r P C (cid:21) dt = (cid:20) − r ˆ V + s F (cid:16) − ˆ V − α B P B − T C B (cid:17) − + ( r B − r ) α B P B + ( r C − r ) α C P C − r T C B + δ ( γ S − q S ) (cid:21) dt − dT C S − dT C C + δ dS − α B P B dJ B − α C P C dJ C . (7)Also, the Ito lemma adapted for jump processes provides another equivalent calculation of the variation of theprocess ˆ V : d ˆ V = ∂ ˆ V∂t dt + ∂ ˆ V∂S dS + 12 σ S ∂ ˆ V∂S dt + ˆ V B dJ B + ˆ V C dJ C (8)where ˆ V B and ˆ V C are calculated based on default conditions. In [1] it is showed that ˆ V B = ˆ V ( t, S, , − ˆ V ( t, S, ,
0) = − ˆ V + ˆ V + + R B ˆ V − ˆ V C = ˆ V ( t, S, , − ˆ V ( t, S, ,
0) = − ˆ V + R C ˆ V + + ˆ V − Hence, by adding (7) and (8), and in order to hedge the risks related to the corporate bonds and the spot assetwe find that δ = − ∂ ˆ V∂Sα B = − ˆ V + ˆ V + + R B ˆ V − P B α C = − ˆ V + R C ˆ V + + ˆ V − P C and0 = (cid:20) − r ˆ V + s F (cid:16) − ˆ V − α B P B − T C B (cid:17) − + ( r B − r ) α B P B + ( r C − r ) α C P C − r T C B + δ S ( γ S − q S ) + ∂ ˆ V∂t + 12 σ S ∂ ˆ V∂S dt − dT C S − dT C C (9)By recalling the definition of the transaction costs, we can compute dT C S and dT C C . Then, for the calculationof the transaction costs of the spot asset, we recall the value of δ and note that4 T C S = C S | dδ | S ∼ C S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π (10)where the approximation is made by taking the expected value of | dδ | and the lowest order O (cid:16) √ ∆ t (cid:17) as follows E ( | dδ | ) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dS ! = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt E [Φ]and setting Φ as a standard normal random variable.The variation of the transaction costs of the counterparty bond position is computed by applying the samerationale as before but over | dα C | in this case. Then, dT C C = C C | dα C | P C ∼ C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π (11)where in this occasion the approximation is obtained by taking the expected value of | dα C | as follows E ( | dα C | ) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dS ! = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt E [Φ]and setting again Φ as a standard normal random variable.By recalling (10) and (11) and applying those computations in (9), we obtain the following nonlinear parabolicpartial derivative equation0 = (cid:20) − r ˆ V + s F (cid:16) − ˆ V − α B P B − T C B (cid:17) − + ( r B − r ) α B P B + ( r C − r ) α C P C − r T C B + δ S ( γ S − q S ) + ∂ ˆ V∂t + 12 σ S ∂ ˆ V∂S dt − C S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π − C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π = (cid:20) − r ˆ V + s F (cid:16) − ˆ V − α B P B − C B (cid:12)(cid:12)(cid:12) − ˆ V + ˆ V + + R B ˆ V − (cid:12)(cid:12)(cid:12)(cid:17) − + ( r B − r ) α B P B + ( r C − r ) α C P C − r (cid:16) C B (cid:12)(cid:12)(cid:12) − ˆ V + ˆ V + + R B ˆ V − (cid:12)(cid:12)(cid:12)(cid:17) − ∂ ˆ V∂S S ( γ S − q S ) + ∂ ˆ V∂t + 12 σ S ∂ ˆ V∂S dt − C S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π − C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π (12)By setting λ B = r B − r , λ C = r C − r and applying the definitions of α B and α C , (12) becomes0 = (cid:20) − r ˆ V + s F (cid:16) − ˆ V + − R B ˆ V − − C B (cid:12)(cid:12)(cid:12) − ˆ V + ˆ V + + R B ˆ V − (cid:12)(cid:12)(cid:12)(cid:17) − + λ B (cid:16) − ˆ V + ˆ V + + R B ˆ V − (cid:17) + λ C (cid:16) − ˆ V + R C ˆ V + + ˆ V − (cid:17) − r (cid:16) C B (cid:12)(cid:12)(cid:12) − ˆ V + ˆ V + + R B ˆ V − (cid:12)(cid:12)(cid:12) (cid:17) − ∂ ˆ V∂S S ( γ S − q S ) + ∂ ˆ V∂t + 12 σ S ∂ ˆ V∂S dt − C S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π − C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ S √ dt r π (13)The absolute value that involves the transaction costs due to the own bonds purchase can be reduced by notingthat when ˆ V ≥
0, its value is 0 and when ˆ
V < R B −
1) ˆ V . Hence,5 (cid:12)(cid:12) − ˆ V + ˆ V + + R B ˆ V − (cid:12)(cid:12)(cid:12) = ( R B −
1) ˆ V − Using this reduction in (13), we get: (cid:16) − ˆ V + − R B ˆ V − − C B (cid:12)(cid:12)(cid:12) − ˆ V + ˆ V + + R B ˆ V − (cid:12)(cid:12)(cid:12)(cid:17) − = (cid:16) − ˆ V + − R B ˆ V − − C B ( R B −
1) ˆ V − (cid:17) − = (cid:16) − ˆ V + − ˆ V − [ R B − C B ( R B − (cid:17) − = ( − ˆ V if ˆ V ≥
00 if ˆ
V < − ˆ V + . (14)Thus, by implementing (14) in (13),0 = − r ˆ V − s F ˆ V + + λ B (cid:16) − ˆ V + ˆ V + + R B ˆ V − (cid:17) + λ C (cid:16) − ˆ V + R C ˆ V + + ˆ V − (cid:17) − r C B ( R B −
1) ˆ V − − ∂ ˆ V∂S S ( γ S − q S ) + ∂ ˆ V∂t + 12 σ S ∂ ˆ V∂S − σ S r π dt C S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + S − C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ! . (15)If we introduce the parabolic operator A t as A t ≡ σ S ∂ ˆ V∂S + ∂ ˆ V∂S S ( q S − γ S )then it follows that ˆ V is the solution of0 = ∂ ˆ V∂t + A t ˆ V − ( λ B + λ C + r ) ˆ V + ( λ B + λ C R C − s F ) ˆ V + + ( λ B R B + λ C − r ( R B − C B ) ˆ V − − σ S r π dt C S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + S − C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ! (16)Looking forward to compare (16) with equation 26 from [1], we rearrange the terms that involve ˆ V , ˆ V + andˆ V − and obtain the following nonlinear parabolic PDE ∂ ˆ V∂t + A t ˆ V − r ˆ V = s F ˆ V + + λ C (1 − R C ) ˆ V + + λ B (1 − R B ) ˆ V − − r (1 − R B ) C B ˆ V − + σ S r π dt C S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + σ S r π dt C C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (17)The first three terms on the right hand side of (17) are equal to the nonlinear terms of the original model. Theinclusion of the transaction costs in the hedging strategy brings to the model three new terms: • The fourth term in the right hand side of (17) corresponds to the amount of cash that is not invested at r rate when considering the surplus held by the seller after the purchase of its own bonds as it is shown in (4). • The fifth term is the effect of the transaction costs due to buying or selling δ assets of S . It shall be notedthat the term is equal to the corresponding one in Leland’s standard approach. • The sixth term is the effect of the transaction costs due to shorting the counterparty bond.6omparing (16) with Leland’s notation, we can define the modified volatility asˆ σ = σ − r π dt C S σ sgn ∂ ˆ V∂S !! (18)and noting that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R C ∂ ˆ V∂S + + ∂ ˆ V∂S − − ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ((cid:12)(cid:12)(cid:12) (1 − R C ) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12) if ˆ V ≥
00 if ˆ
V <
0= (1 − R C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (19)we obtain the following differential equation ∂ ˆ V∂t + 12 ˆ σ S ∂ ˆ V∂S + ∂ ˆ V∂S S ( q S − γ S ) − r ˆ V = s F ˆ V + + λ C (1 − R C ) ˆ V + + λ B (1 − R B ) ˆ V − − r (1 − R B ) C B ˆ V − + σ S r π dt C C (1 − R C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (20) Remark . The left-hand side of equation (20) is effectively a Black-Scholes operator with a volatility parameterˆ σ , a dividend yield γ S and a financing cost (different to the risk-free interest rate) q S . The right-hand side ofthe equation contains the nonlinear terms that arises from considering the existence of transaction costs anddefault risk. The inclusion of these ’extra’ costs can be thought as a perturbation to the original model. Byassessing the magnitude of the parameters of each term, it can be noted that they are indeed small. Hazardrates, recovery rates and interest rates are always below 1 and the transaction costs per unit of asset can bemodeled between 0 .
025 to 0 .
04 as it is done in [13].
For the purpose of finding the unique solution of the CVA problem with transaction costs, we set the mathe-matical framework in which we will work. Let Ω = ( S min , S max ) be an open subset of R + and Ω T = Ω × (0 , T ).Let 1 ≤ p ≤ ∞ and k ∈ N . We define the following Sobolev spaces W kp (Ω) = { u ∈ L p (Ω) | D α u ∈ L p (Ω) , ≤ | α | ≤ k } ,W k,kp (Ω T ) = n u ∈ L p (Ω T ) | D α ∂ βt u ∈ L p (Ω T ) , ≤ | α | + 2 β ≤ k o , where D α ∂ βt u is the weak partial derivative of u . These spaces are actually Banach spaces when assigning thefollowing norms k u k W kp (Ω) = X ≤| α |≤ k k D α u k L p (Ω) , k u k W k,kp (Ω T ) = X ≤| α | +2 β ≤ k (cid:13)(cid:13)(cid:13) D α ∂ βt u (cid:13)(cid:13)(cid:13) L p (Ω) , We are going to look for convex solutions of the problem (20). Problems of this kind refer to any derivativewhose payoff correspond to a convex function as it could be an European option call. Hence, the modifiedvolatility defined in (18) is changed to 7 σ = σ − r π dt C S σ ! . (21)Also, we apply the change of variables x = log ( S ) and τ = T − t . We define the parabolic operator L and thenonlinear operator N as L V = − ∂V∂τ + 12 ˆ σ ∂ V∂x + ∂V∂x (cid:18) q S − γ S −
12 ˆ σ (cid:19) − r V N V = V + [ s F + λ C (1 − R C )] + V − ( λ B − r C B ) (1 − R B ) + σ r π dt C C (1 − R C ) (cid:12)(cid:12)(cid:12)(cid:12) ∂V∂x + (cid:12)(cid:12)(cid:12)(cid:12) such as the problem reads as L ˆ V ( τ, x ) = N ˆ V ( τ, x ) in Ω × [0 , T ]ˆ V (0 , x ) = g ( x ) in Ω (22)ˆ V ( τ, x ) = f ( x ) in ∂ Ω × (0 , T ) . where g ( x ) is the initial condition (i.e. the payoff of the derivative) and f ( x ) is the boundary condition. Forexample, we define the conditions for an European call option as g ( x ) = (exp ( x ) − K ) + ,f ( x ) = (cid:26) x → x ) if x → ∞ . In order to find a solution of problem (22), we define an operator T : C , (cid:0) ¯Ω (cid:1) → C , (cid:0) ¯Ω (cid:1) such that T ( u ) = v ,where v ∈ W , p is the unique solution of the problem L v = N u . Our objective is to find a fixed point of theoperator T which at the same time will be the solution of problem (22). We will set three conditions that theparameters of the model must fulfill to assess the existence of a convex solution.The first condition is required to define a well-posed equation. As explained in [13], the modified volatilityshown in equation (21) must be positive. This can be addressed by setting a lower bound for the volatilityparameter as σ > r π dt C S (23)The second one is a sufficient condition which is required to find a fixed point of the operator T . Let c bepositive constant depending only on the domain, to be defined below and assume that the following inequalityholds: c [ s F + λ C (1 − R C )] + 2 ( λ B − r C B ) (1 − R B ) + σ r π dt C C (1 − R C ) ! < . (24)Given all the parameters of the model set, this assumption can be rewritten in terms of an upper bound for thevolatility parameter σ < − c ([ s F + λ C (1 − R C )] + 2 ( λ B − r C B ) (1 − R B )) c q π dt C C (1 − R C ) (25)The third and last condition shall be used to prove that the solution found is indeed convex. We shall assumethat the stock growth rate under the risk neutral measure has to be bounded, more specifically:8 S − γ S < M := max r + σ S max r π dt C C (1 − R C ) , r ! , (26)where r = r − [ s F + λ C (1 − R C )] and r = r − ( λ B − rC B ) (1 − R B ).The main theorem of the paper reads as follows. Theorem 3.1.
Suppose that assumptions (23) , (25) and (26) hold, that both the initial and boundary conditionsbelong to the W , p space and the initial condition is a convex function. Then, problem (22) admits at least onesolution. The main idea of the proof is to apply the Schauder fixed point theorem to the operator T previously defined.To this end, we shall define a nonempty convex, closed and bounded K ⊂ Ω such that T is a compact continuousmapping of K into itself.For a proof of Theorem 3.1, let us firstly recall Theorem 7.32 from [23] which shows that the operator T iswell defined and provides a lower estimate for L u . By adapting this result to our problem, we get the followinglemma: Lemma 3.2.
Let u ∈ C , (cid:0) ¯Ω (cid:1) , L be as in Theorem 7.32 from [23] and f := N u . Then there exists a uniquesolution of problem L v = f in Ω , v = g in { } × Ω and v = ϕ on ∂ Ω × (0 , T ) . Moreover, there exists C > independent of f such that v satisfies the estimate k v k W , p ≤ C (cid:16) kL v k p + k ϕ k W , p + k g k W , p (cid:17) . (27)The following lemma will be useful to address the continuity of the operator T . Lemma 3.3.
Let p > N and u n ∈ W , p a bounded sequence such that u n → u pointwise. Given G ( x ) =max ( x, it follows that G ( u n ) → G ( u ) in L p ∂G ( u n ) ∂x → ∂G ( u ) ∂x in L p (28) Proof:
The proof of the first statement follows from noting that | G ( x ) | ≤ k G ( u n ) − G ( u ) k p ≤k u n − u k p →
0. To address the second statement, we first note that ∂∂x ( G ◦ u ) = ( G ◦ u ) ∂u∂x . Then, we can rewrite( G ◦ u ) ∂u∂x − ( G ◦ u n ) ∂u n ∂x = ( G ◦ u − G ◦ u n ) ∂u∂x + ( G ◦ u n ) (cid:18) ∂u∂x − ∂u n ∂x (cid:19) For the first term, since | G ( u ) − G ( u n ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( G ( u ) − G ( u n )) ∂u∂x (cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x (cid:12)(cid:12)(cid:12)(cid:12) p Then, by dominated convergence theorem (cid:13)(cid:13)(cid:13)(cid:13) ( G ( u ) − G ( u n )) ∂u∂x (cid:13)(cid:13)(cid:13)(cid:13) pp = Z (cid:12)(cid:12)(cid:12)(cid:12) ( G ( u ) − G ( u n )) ∂u∂x (cid:12)(cid:12)(cid:12)(cid:12) p → . To assess the second term, we consider the inclusion W , p , → W , p to obtain a convergent subsequence in W , p . Nonetheless, given that | G ( u n ) | ≤
1, it follows that9 (cid:12)(cid:12)(cid:12) G ( u n ) (cid:18) ∂u∂x − ∂u n ∂x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x − ∂u n ∂x (cid:12)(cid:12)(cid:12)(cid:12) p → . and hence (cid:13)(cid:13)(cid:13)(cid:13) G ( u n ) (cid:18) ∂u∂x − ∂u n ∂x (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂x − ∂u n ∂x (cid:13)(cid:13)(cid:13)(cid:13) p → (cid:3) Given Lemma 3.2 and Lemma 3.3, we can now address the proof of Theorem 3.1. In order to apply SchauderFixed Point theorem to the operator T , we set K = B R ( u ) for some u ; then, we have to prove that T is acompact continuous mapping of K into itself. Within our proof we will set u = 0.Let us first verify the continuity of the operator T . Let u n , u ∈ W , p such that u n → u pointwise. By recallingLemma 3.2, there exists a constant C > k T u n − T u k W , p ≤ C kL T u n − L T u k p ≤ C kN u n − N u k p ≤ C [ s F + λ C (1 − R C )] (cid:13)(cid:13) u + n − u + (cid:13)(cid:13) p + C ( λ B − r C B ) (1 − R B ) (cid:13)(cid:13) u − n − u − (cid:13)(cid:13) p + C σ r π dt C C (1 − R C ) (cid:13)(cid:13)(cid:13)(cid:13) ∂u n ∂x + − ∂u∂x + (cid:13)(cid:13)(cid:13)(cid:13) p (29)(30)By applying Lemma 3.3, we know that k u + n − u + k p → (cid:13)(cid:13)(cid:13) ∂u n ∂x + − ∂u∂x + (cid:13)(cid:13)(cid:13) p →
0. The same lemma is validby changing the function G into G ( x ) = min ( x,
0) so that k u − n − u − k p → S ⊂ C , . By definition, thesubset S belongs to W , p and T ( S ) ⊂ W , p . Given that p >
1, the inclusion W , p , → C , guarantees that T ( S ) ⊂ C , is compact.Further, let R be a positive number such that R > k f k W , p + k g k W , p − c (cid:16) [ s F + λ C (1 − R C )] + 2 ( λ B − r C B ) (1 − R B ) + σ q π dt C C (1 − R C ) (cid:17) (31)and u such that k u k C , ≤ R . Then, there exists a constant c > W , p , → C , so k T u k C , ≤ c k T u k W , p . (32)Given the inequality presented in equation (32), we can use the result of Lemma 3.2. Hence, there exists aconstant c > k T u k C , ≤ c c (cid:16) kN u k p + k f k W , p + k g k W , p (cid:17) . (33)By recalling (22), the nonlinear term N is bounded by kN u k p ≤ [ s F + λ C (1 − R C )] (cid:13)(cid:13) u + (cid:13)(cid:13) p + ( λ B − r C B ) (1 − R B ) (cid:13)(cid:13) u − (cid:13)(cid:13) p + σ r π dt C C (1 − R C ) (cid:13)(cid:13)(cid:13)(cid:13) ∂u∂x + (cid:13)(cid:13)(cid:13)(cid:13) p < [ s F + λ C (1 − R C )] R + ( λ B − r C B ) (1 − R B ) 2 R + σ r π dt C C (1 − R C ) R< k R (34)10sing that k u k C , ≤ R where k = [ s F + λ C (1 − R C )] + ( λ B − r C B ) (1 − R B ) 2 + σ q π dt C C (1 − R C ).By applying (34) in (33), we get that k T u k C , ≤ c c k R + c c (cid:16) k f k W , p + k g k W , p (cid:17) .< R (35)which follows from the assumption (25) by setting c = c c and the lower bound of R .The last step of the proof is to show that the solution is indeed convex. To this end, we can analyze thesimilarity between equation (20) and a Black-Scholes equation with dividends and notice that given a convexinitial condition, the solution would remain convex. We analyze separately when ˆ V is positive or negative.When the solution is positive, equation (20) reduces to ∂ ˆ V∂t + 12 ˆ σ S ∂ ˆ V∂S + ∂ ˆ V∂S S ( q S − γ S ) − r ˆ V = [ s F + λ C (1 − R C )] ˆ V + σ S r π dt C C (1 − R C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ˆ V∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By rearranging terms and defining r = r − [ s F + λ C (1 − R C )] the equation above becomes ∂ ˆ V∂t + 12 ˆ σ S ∂ ˆ V∂S + ∂ ˆ V∂S S q S − γ S − sgn ∂ ˆ V∂S ! σ S r π dt C C (1 − R C ) ! − r ˆ V = 0 (36)When the solution is negative, equation (20) reduces to ∂ ˆ V∂t + 12 ˆ σ S ∂ ˆ V∂S + ∂ ˆ V∂S S ( q S − γ S ) − r ˆ V = ( λ B − rC B ) (1 − R B ) ˆ V .
By rearranging terms and defining r = r − ( λ B − rC B ) (1 − R B ) the equation above becomes ∂ ˆ V∂t + 12 ˆ σ S ∂ ˆ V∂S + ∂ ˆ V∂S S ( q S − γ S ) − r ˆ V = 0 (37)Equation (36) and (37) can be thought as a Black-Scholes equation with dividend yield γ S and free-risk interestrate r and r respectively. Moreover, the condition stated in (26) can be used to derive an upper bound for q S − γ S . If q S − γ S < M , we see that the growth rate of the stock under the risk-free measure is lower than thefree-risk interest rate. This dynamic is the one expected for a Black-Scholes model with dividends. Because theinitial condition of the problem is indeed convex the solution ˆ V is also convex. In this section we develop a numerical framework to solve the problem defined in (22) by applying a forwardEuler method. Hence, we recall the nonlinear problem L ˆ V ( τ, x ) = N ˆ V ( τ, x ) in Ω × [0 , T ]ˆ V (0 , x ) = g ( x ) in Ω (38)ˆ V ( τ, x ) = f ( x ) in ∂ Ω × (0 , T ) . with L and N defined in (22). For numerical convenience, we approximate the original smooth domain by adiscrete one ˆΩ T ⊂ [ a, b ] × [0 , T ], setting a and b in order to cover a set of feasible logarithmic stock prices.The step of the temporal variable is uniformly set as ∆ τ = T /T x being T x the number of grid points in the τ -direction. For the spatial variable, we decide to apply a non-uniform grid where the spacing is fine near thestrike and coarse away from the strike. In [24], the following grid is proposed11 i = x ∗ + α sinh (cid:18) c iN + c (cid:18) − iN (cid:19)(cid:19) (39)where c = sinh − (cid:18) x − − x ∗ α (cid:19) c = sinh − (cid:18) x + − x ∗ α (cid:19) . This is a transformation that maps the interval [0 ,
1] into [ x − , x + ] by concentrating the points near x ∗ . Thevalue of α sets how non-uniform the grid will be and N to be the amount of points within the grid. In ourproblem we set x ∗ = K and [ x − , x + ] accordingly to cover all the possible logarithmic prices. Hence, we definethe solution to the m -temporal step as ˆ V mi = ˆ V ( x i , m ∆ τ ) where 1 ≤ i ≤ N and 1 ≤ m ≤ T x . We also defineˆ U = max (cid:16) ˆ V , (cid:17) for numerical notation convenience.To derive the expression of the numerical framework we follow [25] and [26] in which this grid had been applied.By following the same steps, we obtain that the discretization of the first and second spatial derivatives aregiven by ∂ ˆ V∂x = ˆ V mi +1 − ˆ V mi x i +1 − x i ,∂ ˆ V∂x = h + i ˆ V mi +1 − ˆ V mi x i +1 − x i − h − i ˆ V mi − ˆ V mi − x i − x i − where h i = x i − x i − and h + i = 2 h i +1 ( h i +1 + h i ) ,h − i = 2 h i ( h i +1 + h i ) . Given that the temporal step is set uniformly, the finite difference framework is defined below L ˆ V = − ˆ V m +1 i − ˆ V mi ∆ τ ! + 12 ˆ σ h h + i (cid:16) ˆ V mi +1 − ˆ V mi (cid:17) − h − i (cid:16) ˆ V mi − ˆ V mi − (cid:17)i + ˆ V mi +1 − ˆ V mi x i +1 − x i (cid:18) q S − γ S −
12 ˆ σ (cid:19) − r ˆ V mi . (40) N ˆ V = max (cid:16) ˆ V mi , (cid:17) [ s F + λ C (1 − R C )] + min (cid:16) ˆ V mi , (cid:17) ( λ B − r C B ) (1 − R B )+ σ r π dt C C (1 − R C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ U mi +1 − ˆ U mi x i +1 − x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (41)By rearranging and combining terms we obtain the following iterative processˆ V m +1 i = ˆ V mi (cid:18) − ˆ σ ∆ τ (cid:0) h + i + h − i (cid:1) − ∆ τx i +1 − x i (cid:18) q S − γ S − ˆ σ (cid:19) − r ∆ τ (cid:19) + ˆ V mi − (cid:18) ˆ σ ∆ τ h − i (cid:19) + ˆ V mi +1 (cid:18) ˆ σ ∆ τ h + i + ∆ τx i +1 − x i (cid:18) q S − γ S − ˆ σ (cid:19)(cid:19) − ∆ τ max (cid:16) ˆ V mi , (cid:17) [ s F + λ C (1 − R C )] − ∆ τ min (cid:16) ˆ V mi , (cid:17) ( λ B − r C B ) (1 − R B ) − ∆ τ σ r π dt C C (1 − R C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ U mi +1 − ˆ U mi x i +1 − x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (42)If we let ˆ V i = g ( x i ), this framework can be used to find the solution of the problem (38) at each time step m . Further, results regarding the convergence, stability and consistency of the method adapted to non-uniformgrids on Black-Scholes problems can be found in [25] and [27].12 .2 Numerical Analysis In this section we analyze the behavior of the option price for an European call under different scenarios. Weperform a sensitivity analysis on the volatility, free-risk interest rate, transaction costs, recovery rates and haz-ard rates by stressing its values. Nonetheless, we compare our results with the ones obtained by the originalmodel proposed in [1] and calculate how the transaction costs impact the final CVA value. To further analyzethe behavior of the option price, we calculate its derivatives with respect to certain parameters. This deriva-tives are known as Greeks and consist of Delta (derivative with respect to the option price), Gamma (secondderivative with respect to the option price), Vega (derivative with respect to the volatility) and Rho (derivativewith respect to the interest rate).For notation purposes we recall BK to the original model and BK T C the model with transaction costs. Also,for each scenario, the parameters set for both models are defined in the caption of each figure and results areobtained at time τ = T . Within each figure, two types of vertical lines are included. The grey-shaded linescorrespond to the non-uniform x i grid defined in Section 4.1 and the black dashed-line represents the strike value. Figure (1) presents the two derivatives with respect to the option price, which are Delta and Gamma. Deltashows a similar behavior to an European call. It is known that for that vanilla option, Delta’s formula corre-spond to a normal cumulative function. When including CVA and transaction costs, it can be seen that whenthe option is deep out-of-the-money, Delta is near zero which implies that the portfolio defined in Equation (1)needs no shares of S to hedge the option. As the option gets at-the-money, Delta grows approximately up to0 .
5. The option is more sensitive to changes in the spot price so then almost 50% of the hedging portfolio hasto be covered with shares of S . This trend continues to converge to a Delta equal to 1 when the option getsdeeper in-the-money. At this point, the option price changes at the same rate with respect to the spot priceand hedging portfolio has to only be long shares of S to cover its hedging purpose.Since Gamma represents the second derivative of the option price with respect to the spot price, its maximumis actually reached when the option is at-the-money and diminishes when the option go either in-the-money orout-of-the-money. This behavior is again similar to the one seen on a vanilla European call and shows to ushow sensitive is Delta to movements in the spot price. O p t i on P r i c e D e l t a Spot Log price (a) Option Delta O p t i on P r i c e G a mm a Spot Log price (b) Option Gamma
Figure 1: C S = 0 . r = 0 . q S = 0 . γ S = 0 . σ = 0 . S f = 0, λ B = 0 . λ C = 0 . R B = 0 . R C = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8. Figure (2a) presents the sensitivity of the CVA to changes in the volatility parameter. The figure shows that thestrike price serves as threshold where the behavior of the CVA changes. When the option is out-of-the money(
S < K ), higher volatility produces higher CVA (more negative). However, when the option is in-the-money(
S > K ), the convexity changes leading to higher CVA as the volatility decreases. Figure (2b) expands theseresults over the entire set of possible volatilities. 13 C VA Spot Log price Vol=0.05Vol=0.2Vol=0.3Vol=0.4 (a) BK TC CVA for different volatilities C VA (b) BK TC CVA by Spot Price and Volatility
Figure 2: C S = 0 . q S = 0 . γ S = 0 . r = 0 . S f = 0, λ C = 0 . R C = 0 . λ B = 0 . R B = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8.In Figure (3a), the sensitivity of the option price with respect to different volatility parameters is presented.Under the usual Black-Scholes framework, it is expected to get higher option prices as volatility increases. Thispattern is confirmed up to a certain spot price. Under our framework, as the option gets deeper in-the-money,Delta (Figure (1a)), which represents the amount of shares to buy in the replicant strategy, tends to 1 and theimpact of the transaction costs increase by generating a decrease in the option price. This behavior can beconfirmed by assessing the first derivative of the option price with respect to the volatility (usually known asVega). In Figure (4), Vega is split with respect to the moneyness of the option. Figure (4a) shows that, whenthe option is out-of-the-money, Vega is positive as it is under the Black-Scholes model. Further, Figure (4b)demonstrate that not only Vega becomes negative as the option gets in-the-money but also that its sign changesin the same spot price as seen in Figure (3a). Hence, if we consider the impact of the volatility not only in theparabolic side of the PDE but also in the nonlinear term , it is expected to find these relationship between thevolatility parameter and the option price. O p t i on P r i c e Spot Log price Vol=0.05Vol=0.2Vol=0.3Vol=0.4 (a) BK TC Option Price for different volatilities D i ff e r en c e i n $ Spot Log price Vol=0.05Vol=0.2Vol=0.3Vol=0.4 (b) Difference between BK and BK TC Figure 3: C S = 0 . q S = 0 . γ S = 0 . r = 0 . S f = 0, λ C = 0 . R C = 0 . λ B = 0 . R B = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8. Figure (5a) presents the sensitivity of the CVA to changes in the interest rate. Figure (5b) expand these resultsto the entire interval of Spot Log prices. Both figures show that the CVA decreases as the interest rate increasesand the its size is larger when the option is deep in-the-money.Figure (6a) shows that the option price is a decreasing monotonic function with respect to the interest rate.This result is confirmed by analyzing the first derivative of the option price with respect to the interest ratepresented in figure (7), also known as Rho. When the option is out-of-the-money, Rho is approximately equal14 .1 0.2 0.3 0.4 0.5 1 1.5 202468 Spot Log PriceVolatility O p t i on V ega (a) Option Vega out-of-the-money O p t i on V ega (b) Option Vega in-the-money Figure 4: C S = 0 . r = 0 . q S = 0 . γ S = 0 . σ = 0 . S f = 0, λ B = 0 . λ C = 0 . R B = 0 . R C = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8. C VA Spot Log price r=0.02r=0.04r=0.06r=0.08 (a) BK TC CVA for different interest rates. C VA (b) BK TC CVA by Spot price and interest rate.
Figure 5: C S = 0 . q S = 0 . γ S = 0 . σ = 0 . S f = 0, λ C = 0 . R C = 0 . λ B = 0 . R B = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8.to zero. But as the option gets in-the-money, it is observed a negative slope. This result is counter-intuitive byconsidering that, under Black-Scholes model, the derivative is always positive. This discrepancy can be assessedby noting that, in equation (20), the coefficient of ∂ ˆ V∂S S is equal to ( q S − γ S ) instead of ( r − γ S ). Given thatunder the BK T C model, q S is being modeled as a constant function, the positive sensitivity of the option tothe interest rate is not observed. In order to match the expected behavior, an improvement of the modelingapproach of the financing cost and its relationship with the interest rate has to be done. Figure (8) presents the variation on the CVA due to changes in the transaction costs that arise of trading δ amount of shares S and α C amounts of bond P C . By recalling equation (20) it can be noted that an increase in C S leads to a decrease in the modified volatility. We actually can assume that the modified volatility behavessimilarly to the actual volatility, so that the analysis done in Section 4.2.2 can be applied. By considering thepattern showed in Figure (8) it can be seen that it is in line with the behavior of the CVA when varying thevolatility in Figure (8a). In both cases, the convexity changes near the strike value due to the same issuespresented in the aforementioned section.On the other side, the presence of C C in Equation (20) actually shows that larger costs generate a lower optionprice. Also, as transaction costs are multiplied by Delta, the gap widens as the option gets deeper in-the-money.This is the pattern that is observed in Figure (8b). 15 O p t i on P r i c e Spot Log price r=0.02r=0.04r=0.06r=0.08 (a) BK TC Option Price for different interest rates D i ff e r en c e i n $ Spot Log price r=0.02r=0.04r=0.06r=0.08 (b) Difference between BK and BK TC Figure 6: C S = 0 . q S = 0 . γ S = 0 . σ = 0 . S f = 0, λ C = 0 . R C = 0 . λ B = 0 . R B = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8. O p t i on R ho Figure 7: Option Rho
In Figure (9) the sensitivity of the CVA due to changes in the recovery rate of the counterparty bond is pre-sented. Given that the recovery rate determines the amount of instrument that can be recovered in case of adefault, the term (1 − R C ) estimates the loss that would arise in case of default. It is expected that higherrecovery rates imply lesser losses and then lesser CVA. Figure (9a) shows this monotonic relationship which isalso in line with the behavior seen in Equation (20).The hazard rate of the counterparty bond measures the likelihood that the bond will default at a certain pointof time. Hence, if the hazard rate increases, the probability of default of the bond also increases. Then, it isexpected to see a higher CVA value when deriving the option price. Figure (10a) presents the CVA value fordifferent hazard rates where the expected behavior is noticed. In this work we adapted the PDE formulation of CVA modeling originally proposed in [1] to consider the ex-istence of transaction costs when developing the hedging portfolio. We used the Leland approach to includeconstant transaction costs applied to trading the spot asset and both counterparty and seller bonds. We con-structed a nonlinear PDE model which explains the behavior of an option price considering both counterpartyrisk and transaction costs. Further, we proved the existence of a solution by applying a fixed-point approachunder a constraint in the volatility parameter. Numerical results showed that Delta and Gamma behave sim-ilarly as in a plain vanilla option but Vega and Rho presented differences in terms of the usual behavior. Itis observed that the presence of transaction costs impact on the way that volatility and risk-free interest rateaffects the option price. We also realized that the spot financing cost q S has to be linked with the risk-free16 C VA Spot Log price Cs=0.001Cs=0.008Cs=0.015Cs=0.022 (a) BK TC C VA Spot Log price Cc=0.001Cc=0.008Cc=0.015Cc=0.022 (b) BK TC Figure 8: r = 0 . q S = 0 . γ S = 0 . σ = 0 . S f = 0, λ C = 0 . R C = 0 . λ B = 0 . R B = 0 . C B = 0 . dt = 1 / τ = 1 / K = 8. C C = 0 .
001 for (8a) and C S = 0 .
002 for (8b). C VA Spot Log price Rc=0.02Rc=0.04Rc=0.06Rc=0.08 (a) BK TC D i ff e r en c e i n $ Spot Log price Rc=0.02Rc=0.04Rc=0.06Rc=0.08 (b) Difference between BK and BK TC Figure 9: C S = 0 . r = 0 . q S = 0 . γ S = 0 . σ = 0 . S f = 0, λ C = 0 . λ B = 0 . R B = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8interest rate to assure consistent results.Further work should include the extensions proposed in [1] such as modeling stochastic interest rates, stochas-tic hazard rates, proposing default time dependency or working with derivatives with more general payments.Nonetheless, the model can be extended by deriving not only a more complex transaction costs function suchas the ones used in [17] but also on studying the dynamics of the option price over a basket of assets as in [22]. This work was partially supported by projects CONICET PIP 11220130100006CO and UBACyT 20020160100002BA.17 C VA Spot Log price λ c=0.02 λ c=0.04 λ c=0.06 λ c=0.08 (a) BK TC D i ff e r en c e i n $ Spot Log price λ c=0.02 λ c=0.04 λ c=0.06 λ c=0.08 (b) Difference between BK and BK TC Figure 10: C S = 0 . r = 0 . q S = 0 . γ S = 0 . σ = 0 . S f = 0, λ B = 0 . R B = 0 . R C = 0 . C B = 0 . C C = 0 . dt = 1 / τ = 1 / K = 8.18 eferences [1] C. Burgard and M. Kjaer, “Partial differential equation representations of derivatives with bilateral coun-terparty risk and funding costs,” The Journal of Credit Risk , vol. 7, no. 3, p. 75, 2011.[2] F. Black and M. Scholes, “The pricing of options and corporate liabilities,”
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