aa r X i v : . [ q -f i n . P R ] M a y SABR smiles for RFR caplets
Sander WillemsNatWest Markets ∗ First version: April 3, 2020This version: May 05, 2020
Abstract
We present a natural extension of the SABR model to price both backward and forward-looking RFR caplets in a post-Libor world. Forward-looking RFR caplets can be priced usingthe market standard approximations of Hagan et al. (2002). We provide closed-form effectiveSABR parameters for pricing backward-looking RFR caplets. These results are useful forsmile interpolation and for analyzing backward and forward-looking smiles in normalizedunits.
Following a speech by the Financial Conduct Authority in July 2017, it became apparent thatLibor is expected to cease after end 2021. Regulators have actively pushed for the developmentof new interest rate benchmarks that are firmly anchored to actual transactions, rather thanexpert judgement of a handful of panel banks. These new benchmarks are generally referredto as RFRs, short for risk-free rates. We refer to them in this paper simply as overnight rates,since that is what they are based on. Libor is currently by far the most important interest ratebenchmark. The transition away from Libor therefore has widespread consequences for financialproducts, see for example Henrard (2019a) for a quantitative finance perspective. We focus inthis paper in particular on the consequences for interest rate caps and floors.A standard Libor caplet pays the difference between a Libor rate and a strike rate with a floorat zero. The payment occurs at the end of the accrual period to which the Libor rate refers,while the Libor rate itself, and therefore the caplet’s payoff, is already known at the start ofthe accrual period. In a post-Libor world, caplets will possibly exist in two different formats.The first one, which we refer to as the backward-looking caplet, replaces the Libor rate by thecompounded overnight rate over the accrual period. This product is conceptually different froma Libor caplet, since its payoff is only known at the end of the accrual period. The secondone, which we refer to as the forward-looking caplet, replaces the Libor rate by the par rate ofa single-period Overnight Index Swap (OIS) over the accrual period. This one is conceptuallysimilar to a Libor caplet, but it requires a reliable OIS rate benchmark to be developed. ∗ The statements and opinions expressed in this article are my own and do not represent the views of NatWestMarkets Plc, NatWest Markets N.V. (and/or any branches) and/or their affiliates. In January 2020, the ICE Benchmark Administration launched a market consultation on the introduction ofan ICE swap rate based on SONIA, see ICE Benchmark Administration (2020).
Caplets in a post-Libor world
Consider two times τ < τ and define the continuously compounded overnight rate as R = 1 τ − τ (cid:16) e R τ τ r ( s ) d s − (cid:17) , where r ( t ) denotes an instantaneous proxy of the overnight rate. We define for t ≤ τ R ( t ) = E τ t [ R ] , (1)where E τ t [ · ] denotes the F t -conditional expectation with respect to the τ -forward measure Q τ .In other words, a swap exchanging a floating payment R at τ against a fixed payment K willhave zero value at time t if K = R ( t ). Note that R ( τ ) = R , since the random variable R is F τ -measurable.In a post-Libor world, there will possibly be two types of caplets. The first type, which we referto as the backward-looking caplet, pays at time τ V bcpl ( τ ) = ( R ( τ ) − K ) + , where K is the strike rate. This payoff is only known at payment time τ . The second type,which we refer to as the forward-looking caplet, pays at time τ V fcpl ( τ ) = ( R ( τ ) − K ) + . The payoff of the forward-looking caplet is already known at time τ , but only paid at time τ .The forward-looking caplet is therefore very similar to a traditional Libor caplet and equivalentto a single-period OIS swaption (see Remark 2.1). The value at time t ≤ τ of the backwardand forward-looking caplets are respectively given by: V bcpl ( t ) = P ( t, τ ) E τ t [( R ( τ ) − K ) + ] ,V fcpl ( t ) = P ( t, τ ) E τ t [( R ( τ ) − K ) + ] , where P ( t, τ ) denotes the price at time t of a zero-coupon bond with maturity τ .From Jensen’s inequality, we get for t ≤ τ V fcpl ( t ) ≤ P ( t, τ ) E τ t (cid:2) E τ τ (cid:2) ( R ( τ ) − K ) + (cid:3)(cid:3) = V bcpl ( t ) . (2)Therefore, the backward-looking caplet is always worth at least as much as the forward-lookingcaplet, provided we have not yet entered the accrual period. The above inequality has beenhighlighted by several authors, e.g., Lyashenko and Mercurio (2019), Dorn et al. (2019), andPiterbarg (2020). Note that we have equality in (2) if and only if interest rates are deterministicin the accrual period. Remark 2.1.
By changing to the τ -forward measure, we can express the value at time t ≤ τ of the forward-looking caplet as V fcpl ( t ) = P ( t, τ ) E τ t [( R ( τ ) − K ) + ]= P ( t, τ ) E τ t [ P ( τ , τ )( R ( τ ) − K ) + ] . In practice, overnight rates are compounded discretely on a daily basis, rather than continuously. For thepurpose of this paper, however, this distinction is only of minor importance. ote that P ( τ , τ )( R ( τ ) − K ) corresponds to the value at time τ of a spot starting single-periodOIS exchanging at time τ a floating payment R = R ( τ ) against a fixed payment K . Therefore,the forward-looking caplet and the single-period OIS swaption have identical present values attime t ≤ τ . Note that this is not true for the backward-looking caplet. From its definition in (1), we require R ( t ) to be a Q τ -martingale. Within the accrual period,i.e., for t ∈ [ τ , τ ], R ( t ) contains an increasing number of realized rates and this should bereflected in its volatility. With these considerations in mind, we propose the following dynamicsfor R ( t ): d R ( t ) = ψ ( t ) σ ( t ) R ( t ) β d B ( t ) , ψ ( t ) = min (cid:18) , τ − tτ − τ (cid:19) q , (3)d σ ( t ) = νσ ( t ) d W ( t ) , σ (0) = α, (4)with q > β ∈ [0 , α, ν >
0, and B ( t ) , W ( t ) two Q τ -Brownian motions with d B ( t ) d W ( t ) = ρ d t , ρ ∈ [ − , For t ≤ τ , R ( t ) follows standard SABR dynamics. For t ≥ τ , the stochasticvolatility process is scaled down so that R ( t ) becomes increasingly deterministic for t → τ . Thenewly introduced parameter q controls how fast the volatility is reduced. Larger values lead toa faster reduction, see Figure 1 for an illustration. We show in Appendix A that q − R ( τ ), can be done easily using the impliedvolatility approximations of Hagan et al. (2002): V fcpl (0) = P (0 , τ ) π (cid:16) τ , K, R (0) , σ fIV (cid:17) , (5) σ fIV ≈ σ hagan ( τ , K, R (0) , α, β, ρ, ν ) , (6)where π ( · · · ) denotes Black (1976)’s formula and σ hagan ( · · · ) denotes the Black implied volatil-ity approximation of Hagan et al. (2002), see Appendix B for explicit expressions. Pricingbackward-looking caplets, i.e., options on R ( τ ), is more involved because of the time depen-dence through the function ψ ( t ). In the next section we derive effective SABR parameters ˆ α , ˆ ρ ,ˆ ν such that V bcpl (0) = P (0 , τ ) π (cid:16) τ , K, R (0) , σ bIV (cid:17) , (7) σ bIV ≈ σ hagan ( τ , K, R (0) , ˆ α, β, ˆ ρ, ˆ ν ) . (8)We leave β unchanged, since the local volatility component in the backward-looking SABRmodel has not changed compared to the standard SABR model. Note the difference between(5)-(6) and (7)-(8) in the time-to-exercise parameter, which reflects the fact that the forward-looking rate fixes at τ while the backward-looking rate fixes at τ . Also note that no additionaltreatment is required for pricing a backward-looking caplet with τ <
0, i.e., when some of therate fixings in the payoff have already realized. We assume a zero lower bound for R ( t ), however it is straightforward to accommodate an arbitrary lowerbound through a displacement. Remark that (5)-(6) only makes sense for τ >
0. For τ ≤
0, we have V fcpl (0) = P (0 , τ )( R ( τ ) − K ) + . .0 0.2 0.4 0.6 0.8 1.0t0.00.20.40.60.81.0 p s i ( t ) q = 0.5q = 1.0q = 1.5 Figure 1: Function ψ ( t ) for different values of q with τ = 0 . τ = 1.We see two practical use cases for the backward-looking SABR model. Suppose first that weobserve liquid prices for both backward and forward-looking caplets at a range of strikes. Onepossibility is to fix q to an exogenous value (e.g., based on historical data), and then use themodel to separately mark two sets of parameters ( α, ρ, ν ) to backward and forward-looking capletprices. The forward-looking parameter markings are just the standard SABR parameters.The backward-looking parameter markings represent the prices of the corresponding forward-looking caplets in the standard SABR model. The purpose of the backward-looking SABRmodel extension is in this case to make the two sets of marked parameters comparable, sinceit takes care of the additional volatility in the accrual period under the hood of the model.This allows a trader to easily compare how expensive or cheap the backward-looking caplet isversus the forward-looking one. Alternatively, we can opt to use a single set of parameters forboth backward and forward-looking caplets. Suppose for example that prices of forward-lookingcaplets are more liquid. In this case, we could mark ( α, ρ, ν ) to forward-looking caplets, andmark q to hit the at-the-money backward-looking caplet. Remark 3.1.
We choose to derive effective SABR parameters with respect to a time-to-exerciseparameter τ , since this is the only choice that allows to price backward-looking caplets through-out the entire accrual period. We note however that it is straightforward to transform theseparameters to correspond to a different time-to-exercise parameter T > using the relationship π ( τ , K, R (0) , σ hagan ( τ , ˆ α, β, ˆ ρ, ˆ ν )) = π (cid:18) T, K, R (0) , σ hagan (cid:18) T, ˆ α r τ T , β, ˆ ρ, ˆ ν r τ T (cid:19)(cid:19) . The parameter β is typically not calibrated to market prices, but rather to historical data or expert judgement. This relationship follows from a simple time-change argument in the SABR model. One can directly verifythat it also holds in the implied volatility approximation of Hagan et al. (2002). or example, when τ > we can express the effective parameters in terms of a time-to-exerciseparameter τ by simply multiplying ˆ α and ˆ ν by q τ τ . Remark 3.2.
The dynamics in (3) - (4) are in fact a special case of the general FMM modelintroduced by Lyashenko and Mercurio (2019) for once specific tenor. Our results can thereforebe used to construct a SABR style FMM model with analytic prices for both backward andforward-looking caplets. We apply the results of Hagan et al. (2018b) to find effective SABR parameters ˆ α , ˆ ρ , ˆ ν forthe model in (3)-(4). Their results, based on singular perturbation techniques and effectivemedium theory, are not exact but have the same accuracy as the original SABR approximationof Hagan et al. (2002). We distinguish two cases, based on the sign of τ .If τ ≤
0, then part of the backward-looking caplet’s payoff has realized already. In this case ψ ( t )is strictly decreasing towards zero as time moves forward, which decreases the volatility of R ( t ).The following theorem presents effective SABR parameters incorporating this time-dependentvolatility scaling: Theorem 4.1. If τ ≤ , then the backward-looking effective SABR parameters are ˆ ρ = 2 ρ √ ζ (3 q + 2) , ˆ ν = ν ζ (2 q + 1) , ˆ α = α q + 1 (cid:18) τ τ − τ (cid:19) q e ( ν q +1 − ˆ ν ) τ , where we have defined ζ = 34 q + 3 (cid:18) q + 1 + ρ q (3 q + 2) (cid:19) . Proof.
See Appendix.Remark that the effective smile parameters ˆ ρ and ˆ ν in Theorem 4.1 do not depend on τ and τ , which is linked to a particular scaling property of the backward-looking SABR model when τ ≤
0. To see this, define the processes R ′ ( t ) and σ ′ ( t ) as R ′ ( t ) = R ( τ t ) , σ ′ ( t ) = √ τ (cid:18) τ τ − τ (cid:19) q σ ( τ t ) . Using Itˆo’s lemma and time-scaling properties of the Brownian motion we get for t ≥ τ d R ′ ( t ) = R ′ ( t ) β (1 − t ) q σ ′ ( t ) d B ( t ) , d σ ′ ( t ) = ν √ τ σ ′ ( t ) d W ( t ) . Hence, if τ ≤
0, then the time-0 price of a backward-looking caplet with accrual period[ τ , τ ] is equivalent to one with canonical accrual period [0 ,
1] together with the following re-parameterization α → α √ τ (cid:18) τ τ − τ (cid:19) q , ν → ν √ τ . (9)In other words, starting from effective parameters for the accrual period [0 , τ , τ ], τ ≤ < τ , by first making the substitutions69) and then adjusting for the change in time-to-exercise from 1 to τ (see Remark 3.1). One caneasily verify that this scaling property is indeed satisfied for the effective parameters in Theorem4.1. In particular, since there is no dependence on α or ν in ˆ ρ , the effective correlation doesnot depend on the accrual period. At first sight, the effective volatility-of-volatility may have adependence on τ through the substitution (9), however this is offset by the subsequent changein time-to-exercise from 1 to τ .The only effective parameter that depends on how far we are in and how much is remainingof the accrual period is therefore ˆ α . The exponential factor in ˆ α is a second order correctionand does typically not contribute much. At the start of the accrual period, i.e., when τ = 0,ˆ α is roughly (2 q + 1) times smaller than α . The fact that ˆ α is smaller than α is intuitive.Indeed, by increasing the time-to-exercise from τ to τ we increase the option value, ceterisparibus. However, because of the volatility reduction in the accrual period we have increasedthe option value too much and need to bring it down again with a smaller effective ˆ α . When weprice further in the accrual period, i.e., when we decrease τ and τ , then ˆ α gradually decreasestowards zero, which reflects the fact that an increasing part of the compounded overnight ratehas realized already.If τ ≥
0, then none of the rates referenced in the backward-looking caplet’s payoff have beenrealized yet as of pricing time 0. The rate R ( t ) follows standard SABR dynamics for t ≤ τ , andonly afterwards will the volatility process be scaled down towards zero as t → τ . The followingtheorem provides the effective SABR parameters in this case. Theorem 4.2. If τ ≥ , then the backward-looking effective SABR parameters are ˆ ρ = ρ τ + 2 qτ + τ √ γ (6 q + 4) , ˆ ν = ν γ q + 1 τ τ , ˆ α = α q + 1 ττ e Hτ , where we have defined τ = 2 qτ + τ , H = ν τ + 2 qτ + τ τ τ ( q + 1) − ˆ ν ,γ = τ τ + τ + (4 q − q ) τ + 6 qτ τ (4 q + 3)(2 q + 1) + 3 qρ ( τ − τ ) τ − τ + 5 qτ + 4 τ τ (4 q + 3)(3 q + 2) . Proof.
See Appendix.When pricing at the start of the accrual period, i.e., when τ = 0, the effective parameters inTheorem 4.1 and 4.2 agree. For τ = τ >
0, the backward-looking SABR model reduces to thestandard SABR model and Theorem 4.2 indeed gives in this case ˆ ρ = ρ , ˆ ν = ν , and ˆ α = α .In Figure 2, we set τ − τ = 0 . q = 1, ρ = − ν = 50%, and plot the ratios ˆ ρ/ρ , ˆ ν/ν , andˆ α/α for τ ∈ [ − . , τ and becomes lessnoticeable for larger τ . This makes sense intuitively, since for large τ , the model behaves likethe standard SABR model most of the time. We also observe that the effective correlation doesnot change much, while the effective volatility-of-volatility and the initial volatility are reducedsubstantially for small τ . Note that the scaling property is not guaranteed to hold a priori, since the effective parameters in Theorem4.1 are only approximations. The fact that it does hold is comforting. r h o _ e ff / r h o (a) ˆ ρ/ρ a l p h a _ e ff / a l p h a (b) ˆ α/α nu _ e ff / nu (c) ˆ ν/ν Figure 2: Ratio of effective SABR parameters over original ones for τ ∈ [ − . , τ − τ = 0 . q = 1, ρ = − ν = 50%.For q → ∞ , the volatility of R ( t ) is reduced to zero immediately after the start of the accrualperiod. In this limit case, the backward looking and forward-looking caplet should have thesame price. Taking the limit q → ∞ for the effective parameters in Theorem 4.2 giveslim q →∞ ˆ ρ = ρ, lim q →∞ ˆ ν = ν τ τ , lim q →∞ ˆ α = α τ τ . Recall that the effective parameters ˆ α, ˆ ρ, ˆ ν correspond to a time-to-exercise parameter τ , while α, ρ, ν correspond to a time-to-exercise parameter τ . From the observation in Remark 3.1 weimmediately see that the backward and forward-looking caplet indeed have equal price in thislimit case. Taking the limit q → ∞ for the effective parameters in Theorem 4.1 gives zero valuesfor all effective parameters, which is not surprising since there is no volatility left in the accrualperiod. For q →
0, the volatility decays very slow in the accrual period. Taking the limit q → q → ˆ ρ = ρ, lim q → ˆ ν = ν , lim q → ˆ α = α . The effective parameters therefore coincide with the original ones. However, we note again thatbecause of the difference in the corresponding time-to-exercise parameter, this does not meanthat the backward and forward-looking caplet have the same price. In fact, in the limit case q →
0, the difference between the two is maximized.8 emark 4.3.
Piterbarg (2020) derives for τ > an adjusted initial volatility ˜ α = α r τ − τ τ (10) such that V bcpl (0) = P (0 , τ ) π ( τ , K, R (0) , ˜ σ bIV ) , ˜ σ bIV ≈ σ hagan ( τ , K, R (0) , ˜ α, β, ρ, ν ) . Note that ˜ α corresponds to a time-to-exercise τ . In order to compare ˆ α with ˜ α , we need to scaleit appropriately (see Remark 3.1): ˆ α r τ τ = α r τ (2 q + 1) τ e Hτ = α r τ − τ (2 q + 1) τ e Hτ . (11) Ignoring the exponential factor, which is typically very close to one, and setting q = 1 , weobserve that (10) agrees with (11) . In order to evaluate the approximation quality of the effective SABR parameters derived inthe previous section, we perform a Monte-Carlo simulation of the model as a benchmark. Wearbitrarily choose the following parameter values: β = 1, R (0) = 0 . τ = 0 . τ = 1, ρ = − . ν = 0 . α = 0 .
10, and q = 1. The effective parameters, computed using Theorem4.2, become ˆ α = 0 . , ˆ ρ = − . , ˆ ν = 0 . . We simulate (3)-(4) using a log-Euler discretization scheme with time step 1 /
512 and 10 sim-ulation trajectories. Figure 3 shows an example of three simulated trajectories of R ( t ). Noticehow the volatility starts to decrease for t > τ . We use the simulated trajectories to pricebackward-looking caplets and then compute Black implied volatility by inverting Black (1976)’sformula. Figure 4 shows that Hagan et al. (2002)’s implied volatility approximation with effec-tive parameters ˆ α, ˆ ρ, ˆ ν produces virtually identical results as the Monte-Carlo simulation. Asa reference, we also plot implied volatilities computed using Hagan et al. (2002)’s approxima-tion with parameters α, ρ, ν and ˆ α, ρ, ν . In the first case, we do not apply any adjustments tothe SABR parameters, and Figure 4 clearly shows we are substantially mispricing caplets atall strikes. In the second case, we only apply a correction to the initial volatility parameter.The at-the-money strike is priced correctly in this case, but strikes away from the money arenot. We have presented a natural extension of the SABR model to consistently price backwardand forward-looking caplets. Forward-looking caplets can be priced using the market standardapproximations of Hagan et al. (2002). Building on the results of Hagan et al. (2018b), we havederived closed-form effective SABR parameters for pricing backward-looking caplets. We have9 .0 0.2 0.4 0.6 0.8 1.0t0.0440.0460.0480.0500.0520.0540.0560.058 R _ t Figure 3: Three simulated trajectories of R ( t ). The vertical black line indicates the start of theaccrual period t = τ . B l a c k i m p li e d v o l MC priceAll effectiveOnly alpha effectiveNo effective
Figure 4: Black implied volatility smiles. The blue line corresponds to the Monte-Carlo price,the orange line to the SABR price with ( ˆ α, ˆ ρ, ˆ ν ), the green line to SABR price with ( ˆ α, ρ, ν ),and the red line to the SABR price with ( α, ρ, ν ).10hown the effective SABR approximation to be highly accurate when compared to a Monte-Carlosimulation benchmark. The correction to the initial volatility and the volatility-of-volatilityparameters are most pronounced, while the correction to the correlation parameter is of secondorder importance. Our results allow traders to think of backward and forward-looking capletsin normalized units, which helps to assess their relative values. A similar analysis can be donefor the Heston (1993) model using, for example, the results from Hagan et al. (2018a). We leavesuch extensions for future research. 11 Term rate implied by Hull-White model
In this section we investigate what dynamics are implied for R ( t ) from the Hull and White (1990)short-rate model. The goal of this exercise is to gain insight in what happens to the volatilityof R ( t ) inside the accrual period. Specifically, consider the following risk-neutral dynamics forthe short-rate r ( t ): d r ( t ) = ( θ ( t ) − κr ( t )) d t + ξ ( t ) d B Q ( t ) , where θ ( t ) is a deterministic function chosen such that the initial term-structure T P (0 , T ) isrecovered, ξ ( t ) is a deterministic volatility function, B Q ( t ) is a standard Brownian motion underthe risk-neutral measure Q where the money-market account serves as numeraire, and κ ∈ R isthe speed of mean-reversion. The time- t price of a zero-coupon bond with maturity T ≥ t isgiven by P ( t, T ) = e − C ( t,T ) − D ( t,T ) r ( t ) , D ( t, T ) = 1 − e − κ ( T − t ) κ , and C ( t, T ) is a deterministic function that plays no particular role here. Taking the short-rateas an instantaneous proxy of the overnight rate, we get1 + ( τ − τ ) R ( t ) = 1 P ( t, τ ) E Q t h e − R τ t r ( s ) d s e R τ τ r ( s ) d s i = 1 P ( t, τ ) × ( P ( t, τ ) , t ≤ τ , e R tτ r ( s ) d s , t ≥ τ , where E Q t [ · ] denotes the conditional expectation under Q . Standard arguments show that thedynamics of R ( t ) under the τ -forward measure becomed R ( t ) = (cid:18) τ − τ + R ( t ) (cid:19) × ( ( D ( t, τ ) − D ( t, τ )) d B ( t ) , t ≤ τ ,D ( t, τ ) d B ( t ) , t ≥ τ , (12)where B ( t ) is a standard Brownian motion under the τ -forward measure. Defining˜ ψ ( t ) = min (cid:18) , e − κt − e − κτ e − κτ − e − κτ (cid:19) , ˜ σ ( t ) = e − κ ( τ − t ) − e − κ ( τ − t ) κ ξ ( t ) , we can rewrite (12) as d R ( t ) = (cid:18) τ − τ + R ( t ) (cid:19) ˜ ψ ( t )˜ σ ( t ) d B ( t ) . The function ˜ ψ ( t ) is equal to one for t ∈ [0 , τ ] and decays monotonically towards zero for t ∈ [ τ , τ ]. The speed of mean-reversion parameter κ controls how fast ˜ ψ ( t ) decreases. Thequalitative behaviour of the function ψ ( t ) defined in (3) is very similar to ˜ ψ ( t ), where q − κ . Indeed, ψ ( t ) and ˜ ψ ( t ) are both equal to one for t ∈ [0 , τ ] and decay monotonicallytowards zero for t ∈ [ τ , τ ]. For κ > q >
1, the decay is convex, for κ < < q < κ = 0 and q = 1 it is linear. The reason we did not use ˜ ψ ( t ) as a definitionin the backward-looking SABR model is that the expressions for the effective SABR parametersbecome much more complicated. However, this example shows that q plays a similar role to thespeed of mean-reversion in a short-rate model.12 Black (1976) and Hagan et al. (2002) formulas
In the main text we have used Black (1976)’s formula and the Black implied volatility approx-imation for the SABR model by Hagan et al. (2002). Both are well known and standard toolsin interest rate derivatives markets, however for the sake of completeness we write them outexplicitly in this section.The Black (1976) formula is π ( T, K, R, σ ) = R Φ( d + ) − K Φ( d − ) , d ± = x ± σ Tσ √ T , with x = log (cid:0) RK (cid:1) , Φ( · ) the standard normal cumulative distribution, and R, K, T, σ > σ hagan ( T, K, R, α, β, ρ, ν ) = α ( RK ) (1 − β ) / (cid:18) − β ) x + (1 − β ) (cid:19) − zχ ( z ) (cid:18) (cid:20) (1 − β ) α ( RK ) − β + 14 ρβνα ( RK ) (1 − β ) / + 2 − ρ ν (cid:21) T (cid:19) , with z = να ( RK ) (1 − β ) / x, χ ( z ) = log p − ρz + z + z − ρ − ρ ! . C Proofs
C.1 Proof of Theorem 4.1
We start by summarizing a particular case of the results in Hagan et al. (2018b).
Theorem C.1 (Hagan et al. (2018b)) . Consider for ǫ > the following dynamics d R ǫ ( t ) = ǫφ ( t ) R ǫ ( t ) β σ ǫ ( t ) d B ( t ) (13)d σ ǫ ( t ) = ǫνσ ǫ ( t ) d W ( t ) , σ ǫ (0) = 1 . (14) For a given expiry
T > , Hagan et al. (2018b) show that to within O ( ǫ ) , the implied volatilityof European options under the model in (13) - (14) coincides with the implied volatility under thestandard SABR model with constant parameters ˆ α = ∆e ǫ ∆ GT , ˆ ρ = b √ c , ˆ ν = ∆ √ c, where the following constants are defined ∆ = v ( T ) T , b = 2 ρνv ( T ) Z T ( v ( T ) − v ( s )) φ ( s ) d s, G = 2 ν v ( T ) Z T v ( T ) − v ( s ) d s − c,c = 3 ν v ( T ) Z T ( v ( T ) − v ( s )) d s + 9 v ( T ) Z T w ( s ) φ ( s ) d s − b , with v : u R u φ ( s ) d s , and w : u ρν R u φ ( s ) d s . O ( ǫ ). The accuracy of the effective parameter approximation is therefore of the sameorder as the original approximation.We can apply the above result to the backward-looking SABR model in (3)-(4) by setting ǫ = 1, φ ( t ) = αψ ( t ), and T = τ . For τ <
0, we have therefore φ ( t ) = α (cid:18) τ − tτ − τ (cid:19) q , t ≤ τ . It remains to calculate the integrals appearing in Theorem C.1. We start with the functions v and w : v ( u ) = α ( τ − τ ) q Z u ( τ − s ) q d s = α τ q +11 − ( τ − u ) q +1 (2 q + 1)( τ − τ ) q ,w ( u ) = ρντ − τ Z u ( τ − s ) d s = ρνα τ q +11 − ( τ − u ) q +1 ( q + 1)( τ − τ ) q . In particular, we have v ( τ ) = α τ q +11 (2 q +1)( τ − τ ) q and v ( τ ) − v ( s ) = α τ − s ) q +1 (2 q +1)( τ − τ ) q . Elementary,but rather tedious, calculations give b = νρ (4 q + 2) α (3 q + 2) (cid:18) τ − τ τ (cid:19) q ,c = ν α (cid:18) τ − τ τ (cid:19) q q + 1(3 q + 2) (4 q + 3) (cid:0) (3 q + 2) + ρ (4 q + 2 q ) (cid:1) , ∆ = α q + 1 (cid:18) τ τ − τ (cid:19) q , G = ν ∆ ( q + 1) − c. Putting things together givesˆ ν = ∆ c = ν ζ (2 q + 1) , ˆ ρ = b √ c = 2 ρ √ ζ (3 q + 2) , ˆ α = ∆ e ∆ Gτ = α q + 1 (cid:18) τ τ − τ (cid:19) q e ( ν q +1 − ˆ ν ) τ , with ζ as defined in Theorem 4.1. C.2 Proof of Theorem 4.2
Similarly as in the proof of Theorem 4.1, we again apply Theorem C.1 with ǫ = 1, φ ( t ) = αψ ( t ),and T = τ . For the case τ ≥ φ ( t ) = ( α, ≤ t ≤ τ ,α (cid:16) τ − tτ − τ (cid:17) q , τ ≤ t ≤ τ . The functions v and w become: v ( u ) = ( α u, ≤ u ≤ τ , α q +1 (cid:16) qτ + τ − ( τ − u ) q +1 ( τ − τ ) q (cid:17) , τ ≤ u ≤ τ , ( u ) = ( ρναu, ≤ u ≤ τ , ρναq +1 (cid:16) qτ + τ − ( τ − u ) q +1 ( τ − τ ) q (cid:17) , τ ≤ u ≤ τ . Elementary, but rather tedious, calculations give b = νρα (2 q + 1) 3 τ + 2 qτ + τ τ (3 q + 2) , c = ν α γ (2 q + 1) τ ∆ = α ττ (2 q + 1) , G = ν ∆ τ + 2 qτ + τ τ τ ( q + 1) − c, with τ and γ as defined in Theorem 4.2.Putting things together givesˆ α = ∆ e ∆ Gτ = α ττ (2 q + 1) e ∆ Gτ , ˆ ν = ∆ c = ν γ (2 q + 1) τ τ , ˆ ρ = b √ c = ρ √ γ τ + 2 qτ + τ q + 2) . Substituting H = ∆ G concludes the proof. 15 eferences Black, F. (1976). The pricing of commodity contracts.
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