Funding Adjustments in Equity Linear Products
aa r X i v : . [ q -f i n . M F ] J un Funding Adjustments in Equity Linear Products
Stefania Gabrielli ∗ Andrea Pallavicini † Stefano Scoleri ‡ First Version: January 11, 2019. This Version: June 7, 2019
Abstract
Valuation adjustments are nowadays a common practice to include credit andliquidity effects in option pricing. Funding costs arising from collateral procedures,hedging strategies and taxes are added to option prices to take into account theproduction cost of financial contracts so that a profitability analysis can be reliablyassessed. In particular, when dealing with linear products, we need a precise evalua-tion of such contributions since bid-ask spreads may be very tight. In this paper westart from a general pricing framework inclusive of valuation adjustments to derivesimple evaluation formulae for the relevant case of total return equity swaps whenstock lending and borrowing is adopted as hedging strategy.
JEL classification codes:
G12, G13, G32.
AMS classification codes:
Keywords:
Funding, Valuation Adjustments, FVA, Collateral, Equity Swap, Total ReturnSwap, Stock Lending, Dividend Tax, Tobin Tax. ∗ Be Consulting, Piazza Affari 2, 20123 Milano, Italy. Email: [email protected] . † Department of Mathematics, Imperial College, London SW7 2AZ, UK and Banca IMI, Largo Mattioli3, 20121 Milano, Italy. Email: [email protected] . ‡ Be Consulting, Piazza Affari 2, 20123 Milano, Italy. Email: [email protected] , correspondingauthor. ontents The opinions here expressed are solely those of the authors and do not represent in any way those of theiremployers. . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products After the Financial crisis started in 2008, it was realized that the collateralization andfunding mechanisms of OTC derivatives typically have a sizeable effect on their valuation.It is now customary to adjust the risk-free value of a derivative contract by adding somequantities, collectively known as XVAs, in order to deal with such effects and to charge thecounterparty with the corresponding costs. Recent literature on the subject can be founde.g. in Brigo et al. (2019), Bielecki et al. (2018), Cr´epey (2015), Bichuch et al. (2018),Burgard and Kjaer (2013).In the present work, we consider the problem of pricing linear equity products includingall funding sources, both on the derivative side and on its hedge. In particular, we focuson Total Return Swaps (TRS), where the total return on an underlying asset is exchangedagainst floating cash-flows (LIBOR plus spread). A bank usually hedges a TRS withan opposite position in the underlying asset, so as to replicate the derivative cash-flows.In particular, stock lending and borrowing are commonly used as hedging strategies, eventhough direct purchase of the stock (buy-and-hold) is possible in principle. All the financingcosts generated by the chosen strategy should be carefully taken into account and includedinto the TRS spread. These costs typically come from funding the collateral accounts,from taxes paid on dividends and other forms of taxation such as the so-called Tobin tax .Notice that, even when the TRS is fully collateralized, the hedge is not, since a haircutis applied to repo transactions. As a result, funding unsecured is always needed in a certainamount, giving rise to an FVA contribution. These effects are reflected in the disountingcurves used for pricing, as we will detail in the following sections.Another peculiarity of funding equity TRS contracts is the role played by taxes. We canidentify four different tax contributions, arising from the derivative and hedging mechanics:first of all, the reduction of dividend flows due to taxes comes with different amountsdepending on the party receiving the dividends (the stock borrower in the repo market,the investor in the stock market, the equity receiver in the TRS market). We show how theTRS price is impacted by these tax asymmetries. Secondly, the EU FTT implemetationof the Tobin tax requires that a percentage of the stock market value, ranging from 10 to20 basis points in most markets, is paid each time a certain amount of shares is bought,e.g. for hedging purposes. In some cases, the Tobin tax contribution cannot be neglectedsince it is of the same order of the TRS spread.The paper is organized as follows: in section 2 we set up the theoretical pricing frame-work, extending Duffie (2001) approach to the presence of collateral agreements in differentcurrencies, and apply it to the computation of equity forward prices in general situations.In section 3 we apply our pricing framework to the case of Total Return Swaps and com-pute the TRS spread inclusive of all funding costs when different hedging strategies areadopted. In section 4 we support the results obtained in the previous section with somenumerical examples and in section 5 we draw our conclusions. Similar topics are covered In the European Union, a Tobin-style tax on financial transactions (FTT) has been first proposed in2011. Currently, it is adopted by 10 Member States. See https://ec.europa.eu/taxation_customs . . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products We consider an asset traded on the market. We term respectively S t and D t the price andcumulative dividend processes. If the asset requires a margining procedure, we term C t the price process of the margin account. Trading on the market requires a funding bankaccount. We term B t its price process. Here we consider symmetric rates, i.e. lending andborrowing are assumed to be made at the same rate. Furthermore, CVA and DVA are notconsidered in the present analysis. These assumptions can be relaxed, see e.g. Brigo et al.(2018).We can normalize asset prices w.r.t. the bank account by defining the deflated priceand dividend processes. In general, we assume that all deflated processes are expressed inthe same currency of the bank account, on the other hand dividends and periodic marginpayments may be expressed in a different currency. We can define (see Duffie (2001) andMoreni and Pallavicini (2017)) the deflated price and cumulative dividend processes asgiven by ¯ S t := χ f t B t S f t (1)and ¯ D t := Z t χ f u B u dπ f u + Z t χ g u B u ( dC g u − c g u C g u du + d h log χ g , C g i u ) (2)where f and g refers respectively to the dividend currency and to the margin accountcurrency with χ f t and χ g t the corresponding FX spot prices which convert the cash flowin the bank account currency. The terms in the integral have the following meaning: (i)contractual coupons, (ii) collateral posts due to the margining procedure, (iii) additionalfees to match the margin account accrual rate, (iv) wrong-way risk of funding a marginaccount in a currency different from the bank account one. We can define also a deflatedmargin account price process as given by¯ C t := χ g t B t C g t (3)The profit-and-loss generated by a buy-and-hold strategy is defined as the gain process G t . The deflated gain process can be written as¯ G t := ¯ S t + ¯ D t − ¯ C t (4)In order to avoid classical arbitrages among all the admissible trading strategies werequire the existence of a risk-neutral measure Q equivalent to the physical one such that¯ G t is a martingale under such measure. A direct consequence of the martingale condition is . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products B t d ¯ G t = B t (cid:0) d ¯ S t + d ¯ D t − d ¯ C t (cid:1) (5)= − χ f t r t S f t dt + χ f t dS f t + S f t dχ f t + d h χ f , S f i t + χ f t dπ f t + χ g t ( r t − c g t ) C g t dt − C g t dχ g t Then, we assume under the risk-neutral measure in the bank-account currency the followinggeneral dynamics for the asset and the FX spot prices dS f t = S f t µ f t dt + dM f t , dχ f t = χ f t ν f t dt + dN f t , dχ g t = χ g t ν g t dt + dN g t (6)where M t and N t are two risk-neutral martingales. Thus, we can write B t d ¯ G t = χ f t dπ f t + χ f t S f t (cid:0) µ f t + ν f t − r t (cid:1) dt + d h χ f , S f i t − χ g t C g t ( c g t + ν g t − r t ) dt + . . . (7)where the dots on the right-hand side represent the martingale part. Hence, by takingrisk-neutral expectations of both sides, and using the martingale property, we get µ f , Q t := µ f t = r t − ν f t − θ t − ∂ t π f t S f t + χ g t C g t χ f t S f t ( c g t + ν g t − r t ) (8)where we define the Itˆo correction θ f t dt := d h χ f , S f i t χ f t S f t (9)which represents the drift correction of an asset in currency f when observed under the risk-neutral measure in the bank-account currency. If we change measure to the risk-neutralmeasure in currency f we finally obtain µ f , Q f t = r t − ν f t − ∂ t π f t S f t + χ g t C g t χ f t S f t ( c g t + ν g t − r t ) (10)We can now calculate forward prices F f t ( T ) by substituting the expression for µ f , Q f t intothe asset price dynamics under the risk-neutral measure in currency f. ∂ T F f t ( T ) = E Q f t (cid:20) (cid:0) r T − ν f T (cid:1) S f T − ∂ T π f T + χ g T χ f T C g T ( c g T + ν g T − r T ) (cid:21) (11)In case of deterministic interest-rates we get ∂ T F f t ( T ) = (cid:0) r T − ν f T (cid:1) F f t ( T ) − ∂ T E Q f t (cid:2) π f T (cid:3) + E Q f t (cid:20) χ g T χ f T C g T (cid:21) ( c g T + ν g T − r T ) (12)We assume that the contractual dividends are constituted by absolute dividends q h k (possibly expressed in currency h) plus a proportional repo fee ℓ f t , while margins are alwaysproportional. π f t = Z t ℓ f u S f u du + X k χ h t k χ f t k q h k { t>t k } , C g t = (1 + α t ) χ f t χ g t S f t (13) . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products Remark 2.1.
The assumption of proportional collateralization can be questionable sinceit depends on the funding costs of the investor. Alternatively we can introduce a securityprice process subject to perfect collateralization (i.e. haircut α t equal to zero), and use itto evaluate the collateral in the original problem. See Fries and Lichtner (2014). We can proceed by defining the net q and gross Q forward dividend price in foreigncurrency f as given by q f t ( t k ) := E Q f t (cid:20) χ h t k χ f t k q h k (cid:21) , Q f t ( t k ) := 11 − ρ q f t ( t k ) (14)where ρ is the dividend tax (or other contractual reductions). Then, we substitute theabove formula in the forward expression to get ∂ T F f t ( T ) = (cid:2) − α T ( r T − ν f T ) + (1 + α T ) (cid:0) c g T + ν g T − ν f T (cid:1) − ℓ f T (cid:3) F f t ( T ) − X k q f t ( t k ) δ ( T − t k )(15)We can define the effective funding and collateral curves in currency f as given by r f t := r t − ν f t , c f t := c g t + ν g t − ν f t (16)as long as the repo-adjusted blended discounting curve in currency f z f t := − α t r f t + (1 + α t ) c f t − ℓ f t (17)to write the forward price ODE ∂ T F f t ( T ) = z f T F f t ( T ) − X k q f t ( t k ) δ ( T − t k ) (18)which can be solved to obtain F f t ( T ) = S f t P t ( T ; z f ) − X k P t ( t k ; z f ) P t ( T ; z f ) q f t ( t k )1 { t k We can approximate the spots inside the expected values in (49) with theforwards (neglecting any convexity adjustment) and, therefore, the expected dividends with Q f t ( t k ) . Otherwise, we can choose to rely on Black’s dynamics and explicitly compute theexpected values, which introduces a dependence on the volatility of the stock. In the sameway, regarding the prices of the calls for the Tobin tax valuation, we can approximate themwith their intrinsic value, retaining the dependence on the forward prices, otherwise we canuse Black formula. The following numerical examples will make use of Black dynamics. In the previous derivation the TRS is equity receiver. In the case of an equity-payer TRSwe can adopt the buy-and-hold or stock-lending hedging strategy. In this case, the Tobintax contribution changes as π Tobin , f t = − τ " n − X i =1 S f T i − S f T i − ! + S f T i { t>T i } (39)giving rise to a sequence of forward starting puts (instead of calls) on the performance.Moreover, if we adopt a buy-and-hold strategy, we get the following effective forward rate: Z BH , f t ( T i ) := R f t ( T i ) := 1 x i (cid:18) P t ( T i − ; r f ) P t ( T i ; r f ) − (cid:19) (40)otherwise, if we adopt a stock-lending strategy, we get Z SL , f t ( T i ) := 1 x i (cid:18) P t ( T i − ; z f ) P t ( T i ; z f ) − (cid:19) (41)It is also possible to make a blending of the two strategies, where the blended rates anddividend taxes are given by convex combinations of the BH and SL ones with a weight w : z f t = w (cid:2) − α t r f t + (1 + α t ) c f t − ℓ f t (cid:3) + (1 − w ) r f t (42) ρ = wρ B + (1 − w ) ρ I (43) . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products If we consider a resetting notional equity-receiver TRS (28), the time t price is given by V f t = X i P t ( T i ; y f ) F f t ( T i ) − F f t ( T i − ) S f0 + (1 − ρ T ) X k P t ( t k ; y f ) Q f t ( t k ) S f0 { t k 140 5 10 15 − − . . ρ B (%) K ( % ) w = 1 w = 0 . w = 0 . w = 0 . w = 0 . w = 0Figure 1: TRS spread K vs Repo dividend tax ρ B for different values of the blendingparameter w . 0 0 . . . . − w K ( % ) ρ B = 0% ρ B = 5% ρ B = 10% ρ B = 15% ρ B = 20% ρ B = 25% ρ B = 30%Figure 2: TRS spread K vs blending parameter w for different values of the Repo dividendtax ρ B . . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products 150 5 10 1500 . ρ B (%) K ( % ) w = 1 w = 0Figure 3: Accessibility region (in gray) for the TRS spread K when the Repo dividend tax ρ B and the blending parameter w are let vary.In Figure 4 we show how the par spread K decreases with increasing repo fees. Weshow the results for the case ρ B = 5%.In Figures 5 and 6 we show how the par spread K changes with different levels of thefunding and collateral rates: of course, the entity of this impact depends on the degreeof collateralization and on the value of the blending between SL and BH strategies: inthe present case, even though the TRS contract is perfectly collateralized, there is still animpact of the funding rate r t due to the funding mechanism of the buy-and-hold and of the(over-collateralized) stock lending strategy. Increasing funding and collateral rates implya proportional increase in the value of the TRS par spread. We again show the results forthe case ρ B = 5%.All the results are coherent with the behaviour derived in (55) and (56).The equity sensitivities of the par spread are much smaller than those of the rate curves(few basis points for 10% changes in the spot and dividend levels).Finally, we notice that, even though we described here the case of a resetting notionalTRS, similar results hold for the constant-notional version. In this work we have discussed the impact of funding costs on linear equity derivativessuch as Total Return Swaps. We adopted a martingale pricing approach in the presence ofdividend-paying assets and collateralized contracts, including partial collateralization of thederivative and its hedge (the latter assumed to be performed through repo transactions).We have shown that funding costs are reflected in the choice of the discounting curves:in the case of partially collateralized TRS, standard arguments imply that a correction . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products 160 10 20 30 40 − − ℓ (bps) K ( % ) w = 0 w = 0 . w = 0 . w = 0 . w = 0 . w = 1Figure 4: TRS spread K vs Repo fees ℓ for different values of the blending parameter w .The Repo dividend tax is set to 5%.0 0 . . . . . w K ( % ) r t + 10 bpsr t − bps Figure 5: TRS spread K in terms of blending parameter w for different levels of the fundingrates. The TRS is fully collateralized. The Repo dividend tax is set to 5% . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products 170 0 . . . . . w K ( % ) c t + 10 bpsc t − bps Figure 6: TRS spread K in terms of blending parameter w for different levels of thecollateral rate. The TRS is fully collateralized. The Repo dividend tax is set to 5%proportional to the funding spread of the bank is added to the OIS-discounted TRS spread.Moreover, the funding costs of the hedge also affect the TRS spread, not only through thefunding spreads coming from the repo-adjusted blended rate (17), but also through thedifferential on dividend taxes and through the Tobin tax, as it is clear from equations(55) and (56). The choice of the hedging strategy, together with the differences betweenthe involved tax regimes, allow for different values of the TRS spread, hence of its price.These arguments are important when a profitability analysis is carried out before a tradeis closed.In the present analysis, we have neglected any contribution coming from counterpartyrisk: while it is justified on the derivative side, since TRS are usually traded under CSA,a residual CVA is still present on the hedge side in the case of stock lending/borrowing,since the latter transactions are usually over-collateralized. We plan to address the problemof pricing the default of the counterparty in the stock lending transaction in an updatedversion of this paper. References Bielecki, T.R., Cialenco, I., Rutkowski, M. (2018)Arbitrage-free pricing of derivatives in nonlinear market models. Probability, Uncertaintyand Quantitative Risk https://ssrn.com/abstract=3131352 . . Gabrielli, A. Pallavicini, S. Scoleri, Funding Adjustments in Equity Linear Products Mathematical Finance 28 (2) 582–620.Brigo, D., Francischello, M., Pallavicini, A. (2019).Nonlinear Valuation under Credit, Funding, and Margins: Existence, Uniqueness, In-variance, and Disentanglement EJOR Risk Magazine , 12.Cr´epey, S. (2015)Bilateral Counterparty Risk under Funding Constraints. Mathematical Finance 25 (1),1–50 (2015)Duffie, D. (2001).Dynamic Asset Pricing Theory. Princeton University Press , 3rd edition.Fries, C.P. and Lichtner, M. 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