Fast swaption pricing in Gaussian term structure models
FFAST SWAPTION PRICING IN GAUSSIAN TERM STRUCTURE MODELS
JAEHYUK CHOI
Peking University HSBC Business School
SUNGCHAN SHIN
Korea Advanced Institute of Science and Technology
Abstract.
We propose a fast and accurate numerical method for pricing European swaptionsin multi-factor Gaussian term structure models. Our method can be used to accelerate thecalibration of such models to the volatility surface. The pricing of an interest rate option insuch a model involves evaluating a multi-dimensional integral of the payoff of the claim on adomain where the payoff is positive. In our method, we approximate the exercise boundary ofthe state space by a hyperplane tangent to the maximum probability point on the boundary andsimplify the multi-dimensional integration into an analytical form. The maximum probabilitypoint can be determined using the gradient descent method. We demonstrate that our methodis superior to previous methods by comparing the results to the price obtained by numericalintegration. Introduction
Swaptions, which are options on interest rate swaps, are the simplest and most liquid optionproducts traded in fixed income markets. From practical and theoretical perspectives, swaptionsare important building blocks for more complicated claims, such as Bermudan callable swaps.Swaptions are traded to hedge the volatility risk of such exotic claims. Therefore, the parametersof a term structure model must be calibrated to exactly reproduce the prices of the swaptionsobserved in the market before they are used to price exotic claims. However, the calibrationprocess is typically a nonlinear multi-dimensional root solving problem for which parametersmust be found using iterative methods. Therefore, it is critical to have a fast and reliablemethod to price swaptions given a set of parameters for a term structure model.
E-mail addresses : [email protected], [email protected] . Date : March 14, 2018.
Key words and phrases.
Gaussian term structure model, volatility surface calibration, fast swaption pricing,swaption analytics.Address correspondence to Sungchan Shin, Department of Mathematical Sciences, KAIST, 335 Gwahangno,Yuseong-gu, Daejeon, Republic of Korea. a r X i v : . [ q -f i n . M F ] M a r J. CHOI AND S. SHIN
The most relevant studies on this topic are by Singleton and Umantsev [2002] and Schragerand Pelsser [2006]. Both studies provide a fast pricing method for the class of affine termstructure models (ATSM). Singleton and Umantsev [2002] observe that the non-linear exerciseboundary for swaptions can be approximated by a hyperplane. They compute the probabil-ity over the approximated domain using the transform inversion method developed by Duffieet al. [2000] and Bakshi and Madan [2000]. Schrager and Pelsser [2006] derive an approximatedstochastic differential equation (SDE) for the underlying swap rate from full interest rate dy-namics, from which the swaption price is easily obtained. They assume that the low variancemartingale (LVM), which is typically the ratio of the discount factors, is constant as time-zerovalue. Andersen and Piterbarg [2010c] further refine the method by improving the estimationof LVMs. Because of its easy and intuitive implementation, the Schrager and Pelsser [2006]method has been favored by practitioners. Considering that the method of freezing LVMs isinspired by a similar method for pricing swaptions in the LIBOR Market Model (LMM), theirmethod is arguably the dominant swaption pricing method for all classes of interest rates termstructure models. Although the Singleton and Umantsev [2002] method appears to be equallypromising, it suffers from several drawbacks. First, because it lacks explicit guidance in selectingthe hyperplane, it fails to provide the best hyperplane to minimize the error. Second, even fora given hyperplane, the probability over the region must be computed under different forwardmeasures; there are as many measures as the number of cash flows of the underlying swaptions.This study demonstrates that the hyperplane approximation can be significantly improvedfor the class of Gaussian term structure models (GTSM). Using the analytical tractability of theGTSM, we can overcome the two drawbacks mentioned above. In the GTSM, the probabilitydensity function of the state is simply a multivariate Gaussian. In other words, the GTSMis similar to the ATSM, where the transform inversion is analytically solved. The knowledgeof the density function enables us to find the best hyperplane to approximate the non-linearboundary. We identify the point on the boundary with the maximum probability density anddetermine the hyperplane tangent at that point.The accuracy of our approximation is better than the accuracy of previous methods byseveral orders of magnitude, regardless of the moneyness, expiry and tenor of the swaptions.Moreover, our method does not sacrifice computational cost. The computational cost growsat most linearly with the number of factors of the GTSM. Although our method is limited tothe GTSM, which is a subset of the ATSM, it is still a significant improvement for swaptioncalibration given the indisputable importance of the GTSM among all term structure models.Several previous term structure models are special cases of the GTSM, e.g., Ho and Lee [1986],Hull and White [1990] and Vasicek [1977]. These models are still used by practitioners intheir extended forms. In the GTSM, we have the added benefit of being able to compareour approximation to the exact swaption price. In contrast to the general ATSM, where it is Other original approaches have been proposed (e.g., Munk [1999] and Collin-Dufresne and Goldstein [2002]).These alternatives are dominated in terms of accuracy and computational cost. See Singleton and Umantsev[2002] and Schrager and Pelsser [2006] for details.
AST SWAPTION PRICING 3 necessary to resort to Monte Carlo simulations, we can obtain the exact price by combining theanalytical result with numerical integration. Thus, we can provide an accurate error analysis.The remainder of the paper is organized as follows. In Section 2, we briefly review theGaussian term structure model. Section 3 describes the hyperplane approximation method andthe exact swaption pricing. Section 4 demonstrates the accuracy of our method and comparesit to previous methods.2.
Multi-factor Gaussian term structure model
In this section, we review the important results of the GTSM. We will define the scope ofthe GTSM and describe the preconditions for which our approximation method is valid. Tosimplify notation, define an element-wise multiplication operator, ◦ , between vectors or betweena vector and a matrix by a ◦ b = b ◦ a = (cid:2) a j b j (cid:3) j , (2.1) M ◦ a = a ◦ M = (cid:2) M jk a j (cid:3) j, k , (2.2) and M ◦ a (cid:62) = a (cid:62) ◦ M = (cid:2) M jk a k (cid:3) j, k , (2.3)where a = [ a j ] j and b = [ b j ] j are d × M jk ] j, k is a d × d matrix.The GTSM in this study is a subclass of the Heath-Jarrow-Morton (HJM) model class [Heathet al., 1992]. A general d -dimension HJM model starts with the dynamics of the price of a zero-coupon bond. Let P ( t, T ) be the time- t price of a zero-coupon bond maturing at T and let theSDE be defined as follows:(2.4) dP ( t, T ) P ( t, T ) = r ( t ) dt − σ P ( t, T ) (cid:62) d W β ( t ) , where r ( t ) is the short rate process; − σ P ( t, T ) is the volatility vector; and W β ( t ) is a d -dimensional Brownian motion under the risk-neutral measure Q β . The components of W β ( t )are correlated with a correlation matrix R( t ) = [ ρ jk ( t ) ] j, k where ρ kk ( t ) = 1. If f ( t, T ) isthe instantaneous forward rate (IFR) for time T observed at the current time t , we can write P ( t, T ) = exp (cid:16) − (cid:82) Tt f ( t, s ) ds (cid:17) . An important result of the HJM model is obtained by insertingthis equation into the SDE for P ( t, T ), to show that(2.5) df ( t, T ) = σ f ( t, T ) (cid:62) σ P ( t, T ) dt + σ f ( t, T ) (cid:62) d W β ( t ) , where the volatility of IFR σ f ( t, T ) is as follows:(2.6) σ f ( t, T ) = ∂∂T σ P ( t, T ) . Furthermore, the short rate process r ( t ) is(2.7) r ( t ) = f ( t, t ) = f (0 , t ) + (cid:62) x ( t ) , for a d × x ( t ) with x (0) = 0 and(2.8) x ( t ) = (cid:90) t σ f ( s, t ) ◦ (cid:90) ts σ f ( s, u ) du ds + (cid:90) t σ f ( s, t ) ◦ d W β ( s ) . J. CHOI AND S. SHIN
We can further simplify the result under the t -forward measure Q t . Using the Girsanov theorem,we obtain(2.9) d W β ( s ) = d W t ( s ) − σ P ( s, t ) ds = d W t ( s ) − (cid:90) ts σ f ( s, u ) du ds, where W t ( · ) is the Brownian motion under the Q t measure. The processes for f ( s, t ) and x ( s )with respect to the time s become driftless(2.10) df ( s, t ) = σ f ( s, t ) (cid:62) d W t ( s ) and x ( t ) = (cid:90) t σ f ( s, t ) ◦ d W t ( s ) . This result is consistent with the intuitive observation that f ( s, t ) is a Martingale under the Q t measure with respect to time s .An important result from Heath et al. [1992] is that, given the interest rate curve f (0 , T ) asan input to the model, the diffusion of the interest rate curve is fully defined by specifying thevolatility σ f ( t, T ). However, an HJM model typically imposes restrictions on σ f ( t, T ) becausethe process is generally path-dependent or non-Markovian.It is known that an HJM model is Markovian if and only if the σ f ( t, T ) is deterministic andseparable in the form of G( T ) h ( t ) for a d × d matrix G( T ) and a d × h ( t ) [Andersenand Piterbarg, 2010a]. Based on this assumption, the IFR f ( t, T ) and the state vector x ( t ) arealso Gaussian.Although it is not a necessary condition for this study, a popular choice for a separable formof σ f is one that causes the short rate r ( t ) to follow a mean-reverting Ornstein-Uhlenbeckprocess,(2.11) d x ( t ) = (Π( t ) − λ ( t ) ◦ x ( t )) dt + σ r ( t ) (cid:62) d W β ( t )for a deterministic mean reversion coefficient λ ( t ), a deterministic short rate volatility vector σ r ( t ) and a drift matrix Π( t ). This choice is equivalent to setting(2.12) σ f ( t, T ) = σ r ( t ) ◦ m ( t, T ) , where m ( t, T ) is the exponential decay factor between time t and T , defined by(2.13) m ( t, T ) = (cid:104) e − (cid:82) Tt λ k ( s ) ds (cid:105) k for λ ( t ) = (cid:104) λ k ( t ) (cid:105) k . The state x ( t ) is multivariate Gaussian. The drift matrix Π( t ) is the covariance matrix of x ( t ).From Eq. (2.10), we obtainΠ( t ) = (cid:90) t ( σ f ( s, t ) ◦ d W t ( s )) ( σ f ( s, t ) ◦ d W t ( s )) (cid:62) (2.14) = (cid:90) t σ r ( s ) ◦ m ( s, t ) ◦ R( s ) ◦ σ r ( s ) (cid:62) ◦ m ( s, t ) (cid:62) ds = (cid:20)(cid:90) t ρ jk ( t ) σ rj ( s ) σ rk ( s ) m j ( s, t ) m k ( s, t ) ds (cid:21) j, k . AST SWAPTION PRICING 5
Finally, we can reconstruct the zero-coupon bond price at the future time t using the Mar-kovian state x ( t ) as(2.15) P ( t, T ) = P (0 , T ) P (0 , t ) exp (cid:18) − g ( t, T ) (cid:62) x ( t ) − g ( t, T ) (cid:62) Π( t ) g ( t, T ) (cid:19) , where g ( t, T ) = (cid:82) Tt m ( t, s ) ds . The volatility of P ( t, T ) is conveniently given as σ P ( t, T ) = − σ r ( t ) ◦ g ( t, T ); thus, g ( t, T ) is the risk loading of the zero-coupon bond with respect to theshort rate volatility. It should be noted that under the t -forward measure, x ( t ) has a zero mean,and E Q t { P ( t, T ) } = P (0 , T ) /P (0 , t ). This result is consistent with the fact that P ( s, T ) /P ( s, t )is a Martingale with respect to time s under the Q t measure.3. Swaption pricing method
Here, we derive the price of a swaption using the result from the previous section. Let usassume that the underlying swap of a swaption begins at a forward time T , pays the fixedcoupon K (payer swap) on a payment schedule { T , · · · T m } , and receives the floating interestrate, typically LIBOR. The value of this underlying swap, at time t , is(3.1) V( t ) = P ( t, T ) − P ( t, T m ) − m (cid:88) k =1 KP ( t, T k )∆ k , where ∆ k is the day count fraction for the k -th period, ∆ k = T k − T k − . In general,(3.2) V( t ) = m (cid:88) k =0 P ( t, T k ) CF( T k ) , for the cash flow series CF( T k ) at time T k .Let T e be the expiry of the swaption to enter into the underlying swap paying the fixedcoupon K (the expiry T e is typically two business days before the start of the swap T ). Usingthe reconstruction formula Eq. (2.15), we can express the future value of the swap as a functionof the state x ( T e ).We first decorrelate and normalize the state variable x ( t ) into z ( t ) by x ( t ) = C ( t ) z ( t ) for aCholesky decomposition, C( t ), of the covariance matrix Π( t ) that satisfies Π( t ) = C( t ) C( t ) (cid:62) .Then, the reconstruction formula becomes(3.3) P ( t, T ) = P (0 , T ) P (0 , t ) exp (cid:18) − a ( t, T ) (cid:62) z ( t ) − | a ( t, T ) | (cid:19) . where a ( t, T ) = C( t ) (cid:62) g ( t, T ).We now have the value of the swap at the expiry T e as a function of the state z = z ( T e ) V ( z ) = 1 P (0 , T e ) m (cid:88) k =0 DCF k exp (cid:18) − a (cid:62) k z − | a k | (cid:19) (3.4)where DCF k is the discounted cash flow CF( T k ) P (0 , T k ), and a k = a ( T e , T k ). The price of thepayer swaption is the expectation of V ( z ) under the T e -forward measure,(3.5) C = P (0 , T e ) E Q Te { max( V ( z ) , } . J. CHOI AND S. SHIN
Similarly, the T e -forward price of the receiver swaption is given by(3.6) P = P (0 , T e ) E Q Te { max( − V ( z ) , } . The remainder of this study focuses on the payer swaption. The receiver swaption can bedetermined from the put-call parity relation.3.1.
Hyperplane approximation.
The evaluation of Eq. (3.5) involves a d -dimensional inte-gration. The difficulty lies in identifying the integration domain Ω, where the underlying swaphas a positive value and the boundary ∂ Ω can be given as(3.7) Ω = { z : V ( z ) ≥ } , ∂ Ω = { z : V ( z ) = 0 } . As described by Singleton and Umantsev [2002], we simplify the integration by approximatingthe boundary ∂ Ω as a hyperplane. However, in this study, we substantially refine the originalidea by providing a systematic way to determine the best hyperplane to use for this approxi-mation.At the exercise boundary, we identify the state z ∗ with the maximum probability density.Then, for the approximation, we use the tangent plane to the boundary at z ∗ , which is a system-atic way of choosing the hyperplane without ad-hoc rules based on experience. This techniquecan be applied to any moneyness of swaptions and any dimension of the GTSM. Because z hasan uncorrelated multivariate normal distribution, the probability density decreases as a functionof | z | . Thus, z ∗ is also the point with the shortest distance to the origin among the points onthe boundary ∂ Ω. Furthermore, it follows that z ∗ should be a scalar multiple of ∇ V ( z ∗ ). Wewill use this property to find z ∗ . See Figure 1 for a geometric illustration.The point z ∗ can be found numerically using the following iterative method. One iterationconsists of the following two steps: from z ( i ) to z ( i + ) and from z ( i + ) to z ( i +1) . (1): First, apply a step of steepest descent method from z ( i ) :(3.8) z ( i + ) = z ( i ) − V ( z ( i ) ) | ∇ V ( z ( i ) ) | ∇ V ( z ( i ) ) (2): Then, project z ( i + ) onto the gradient direction ∇ V ( z ( i + ) ) by(3.9) z ( i +1) = ∇ V ( z ( i + ) ) (cid:62) z ( i + ) | ∇ V ( z ( i + ) ) | ∇ V ( z ( i + ) ) . The convergence to the root is satisfied when the error V ( z ( i +1) ) is below a certain threshold;we use 10 − in our study.It is difficult to prove with mathematical rigor that the iterative scheme converges to aroot for all possible parameterizations. However, our method works without failure for anyreasonable parameterization. In the GTSM, the state variable is proportional to the interestrate as a leading order, even allowing negative interest rates. Thus, it is possible to find a state z for which the swap rate is equal to any given (even negative) strike, which ensures that aboundary ∂ Ω always exists and is close to a hyperplane. This iterative root-finding processuses the most time in the entire computation of the swaption price because the rest of the
AST SWAPTION PRICING 7 computation is analytical. A reasonably calibrated GTSM converges to the root z ∗ quickly,typically within 7 iterations starting from the origin. It should be noted that the number ofiterations does not increase with the dimension d , although the computational cost may increasedue to the increasing number of components. Overall, the computation cost increases linearly,not exponentially, with the dimension d .Now, we can simplify the evaluation of Eq. (3.5). Let q = ∇ V ( z ∗ ) / | ∇ V ( z ∗ ) | be the unitvector in the direction of ∇ V ( z ∗ ). We express the state vector as follows:(3.10) z = y q + · · · + y d q d in a Cartesian coordinate y = ( y , · · · , y d ) with the standard basis { q , · · · , q d } . The unspecifiedunit vectors q , · · · , q d can be chosen arbitrarily as long as the matrix of unit vectors Q = (cid:2) q , · · · , q d (cid:3) forms an orthogonal matrix.Using the above property, we select a scalar d ∗ and vectors b k such that(3.11) z ∗ = d ∗ q , a k = Q b k The approximated domain ˜Ω is(3.12) ˜Ω = { z : q (cid:62) ( z − z ∗ ) ≥ } = { y : y ≥ d ∗ } and the value of the swap becomes(3.13) V ( z ) = 1 P (0 , T e ) m (cid:88) k =0 DCF k exp (cid:18) − | b k | − b (cid:62) k y (cid:19) When integrating on the y coordinate, y is the only axis on which the integration is non-trivial.The rest of the dimensions integrate to unity. We obtain the swaption price in analytic form asfollows: C = P (0 , T e ) (cid:90) Ω V ( z ) f ( z ) d z ≈ P (0 , T e ) (cid:90) ˜Ω V ( y ) f ( y ) d y (3.14) = m (cid:88) k =0 DCF k (cid:90) ∞−∞ dy · · · dy d (cid:90) ∞ d ∗ dy e − | b k | − (cid:80) dj =1 b kj y j n ( y ) · · · n ( y d )= m (cid:88) k =0 DCF k (cid:90) ∞ d ∗ dy e − b k − b k y n ( y )= m (cid:88) k =0 DCF k N ( − b k − d ∗ ) = m (cid:88) k =0 DCF k N ( − ( a k + z ∗ ) (cid:62) q ) , where f ( · ) is the probability density function of the multivariate normal distribution, and n ( · )and N ( · ) are the probability and cumulative density functions of the univariate normal distri-bution, respectively.3.2. Exact pricing method.
Although it is computationally demanding, we can combinenumerical integration and analysis to price the swaption precisely and measure the accuracyof the hyperplane approximation. The integration is performed on the y coordinate, i.e., inthe hyperplane approximation. However, in the exact method, we numerically determine the J. CHOI AND S. SHIN (cid:2868) → Ω 𝑉 𝑧 = 0 { V ( z ) } ˜⌦ Figure 1.
Schematic of the hyperplane approximation method. The domain ofstate variables, Ω, includes the area where the swaption payoff is in the money.We approximate the boundary of Ω, ∂ Ω, as a hyperplane tangent ( y = 0) to thepoint z ∗ where the probability density is maximal on ∂ Ω and integrate the payoffover the domain above the hyperplane ( y ≥ z ∗ .distance to the boundary d for each given ( d − y , · · · , y d ):(3.15) Ω = { y : y ≥ d ( y , · · · , y d ) } . The root finding for d can be determined using the Newton-Raphson method in one-dimension.The integration is performed analytically for y and numerically for the rest of the dimensions: C = P (0 , T e ) (cid:90) Ω V ( z ) f ( z ) d z = P (0 , T e ) (cid:90) Ω V ( y ) f ( y ) d y (3.16) = m (cid:88) k =0 DCF k (cid:90) ∞−∞ dy · · · dy d (cid:90) ∞ d dy e − | b k | − (cid:80) dj =1 b kj y j n ( y ) · · · n ( y d )= m (cid:88) k =0 DCF k (cid:90) ∞−∞ dy · · · dy d N ( − b k − d ( y , · · · , y d )) n ( y + b k ) · · · n ( y d + b kd )We can use the finite difference method for the numerical integration. AST SWAPTION PRICING 9
It should be noted that the error from the hyperplane approximation is due to the differencebetween the integrands of Eq. (3.14) and Eq. (3.16):(3.17) E ( y , · · · , y d ) = m (cid:88) k =0 ( N ( − b k − d ∗ ) − N ( − b k − d ( · · · ))) n ( y + b k ) · · · n ( y d + b kd )We will examine this error through examples in the next section. It is interpreted as the errordensity because the error in the swaption price is(3.18) Price Error = (cid:90) ∞−∞ dy · · · dy d E ( y , · · · , y d )4. Approximation quality and comparison to other methods
To examine the quality of the proposed hyperplane approximation method for swaptionpricing, we apply it to three sets of examples, shown in Table 1 to Table 3. The first twoexamples use different parameter sets in a two-factor GTSM calibrated to realistic swaptionvolatility surfaces in the least-square sense. We select two contrasting market conditions in theshapes of the yield curve and the volatility surface to test our approximation in diverse marketenvironments. In the first example, the market sees high uncertainty in the short-term interestrate, and the yield curve is flat at 5% at time 0. In the second example, the market sees highuncertainty in the long-term interest rate, and the interest rate curve increases steeply from the0% short-term interest rate, most likely because of monetary policies.To calibrate the surface as closely as possible, we use a piece-wise-constant term structurefor volatility and a mean reversion structure for the first factor with Parameter Sets 1 and 2.The parameters for the second factor are specified through the constant volatility ratio σ /σ and the constant mean reversion difference λ − λ . This structure allows the instantaneouscorrelation between f ( t, T ) and f ( t, T ) to be stationary [Andersen and Piterbarg, 2010b].For the third example, we reuse the three-factor GTSM parameter set from Collin-Dufresneand Goldstein [2002]. This parameter set was also used by Schrager and Pelsser [2006] tocompare their result to those of Collin-Dufresne and Goldstein [2002]. Table 1.
Parameter Set 1: A two-factor Gaussian model calibrated to a volatil-ity surface, where the swaptions on the shorter tenor swaps are relatively expen-sive. The current forward rate curve is assumed to be flat at 5%.Time(year) 0 ∼ ∼ ∼ ∼ ∼ ∼ Volatility( σ ) 0.030 0.024 0.024 0.022 0.018 0.012Time(year) 0 ∼ ∼ ∼ Mean reversion( λ ) 0.115 0.073 0.029 σ /σ λ − λ ρ -77% f (0 , t ) 5%The swaption pricing errors are shown in Tables 4 to 11. For each example, we first presentthe price and its error in basis points for a 5 × Table 2.
Parameter Set 2: A two-factor Gaussian model calibrated to a volatil-ity surface where the swaptions on the longer tenor swaps are relatively expen-sive. The current forward rate curve is assumed to increase steeply from 0% to6%. Time(year) 0 ∼ ∼ ∼ ∼ ∼ ∼ Volatility( σ ) 0.020 0.014 0.013 0.012 0.01 0.009Time(year) 0 ∼ ∼ ∼ Mean reversion( λ ) -0.051 0.059 0.017 σ /σ λ − λ ρ -77% f (0 , t ) 6% × (1 − e − t/ ) Table 3.
Parameter Set 3: A three-factor Gaussian model from Collin-Dufresneand Goldstein [2002] and Schrager and Pelsser [2006] σ σ σ λ λ λ ρ ρ ρ f (0 , t )0.010 0.005 0.002 1.0 0.2 0.5 -20% -10% 30% 5.5%normal volatility and its error. The normal volatility is the volatility under the Bachelierprocess, i.e., normal diffusion. For our study, we assume that the normal volatility is morerelevant than the Black-Scholes (or log-normal) volatility. First, the normal volatility is widelyused among practitioners in the fixed income area [Choi et al., 2009]. Second, the short rate orIFR in the GTSM follows the Bachelier process, and the same holds nearly true for the swaprate (in fact, this is the key assumption of Schrager and Pelsser [2006], and we will discussits accuracy shortly). Therefore, the normal volatility is nearly constant across options withdifferent strikes, which makes it a better measure of error than the price. The price of optionscan change drastically as moneyness changes; thus, pricing errors, both relative and absolute,can be misleading, whereas the normal volatility is a consistent measure of error regardless ofthe moneyness.We further convert the normal volatility to daily basis point (DBP) units by multiplying itby 10 / √ K = F ± n σ ATM (cid:112) T e for n = 0 . , , or 2where F is the forward swap rate, and σ ATM is the normal volatility for ATM.The accuracy of the hyperplane approximation is uniformly good across the volatility surfacefor all three examples. The maximum volatility error across all examples is of the order of 10 − DBP. This level of error does not require further correction for practical purposes.
AST SWAPTION PRICING 11
Table 4.
Prices and errors of the hyperplane approximation with Parameter Set1 in basis point units. Relative pricing errors, calculated as fractions of exactprices, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − σ ATM √ T e K = F + σ ATM √ T e Table 5.
Implied normal volatilities and errors of the hyperplane approxima-tion with Parameter Set 1 in daily basis point units. Relative volatility errors,calculated as fractions of exact volatilities, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − σ ATM √ T e K = F + σ ATM √ T e Table 6.
Prices and errors of the hyperplane approximation with Parameter Set2 in basis point units. Relative pricing errors, calculated as fractions of exactprices, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − . σ ATM √ T e K = F + 0 . σ ATM √ T e Table 7.
Implied normal volatilities and errors of the hyperplane approxima-tion with Parameter Set 2 in daily basis point units. Relative volatility errors,calculated as fractions of exact volatilities, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − . σ ATM √ T e K = F + 0 . σ ATM √ T e AST SWAPTION PRICING 13
Table 8.
Prices and errors of the hyperplane approximation with Parameter Set3 in basis point units. Relative pricing errors, calculated as fractions of exactprices, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − σ ATM √ T e K = F + 2 σ ATM √ T e Table 9.
Implied normal volatilities and errors of the hyperplane approxima-tion with Parameter Set 3 in daily basis point units. Relative volatility errors,calculated as fractions of exact volatilities, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − σ ATM √ T e K = F + 2 σ ATM √ T e Table 10.
Prices and errors of the Schrager and Pelsser [2006] method with Pa-rameter Set 3 in basis point units. Relative pricing errors, calculated as fractionsof exact prices, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − σ ATM √ T e K = F + 2 σ ATM √ T e Table 11.
Implied normal volatilities and errors of the Schrager and Pelsser[2006] method with Parameter Set 3 in daily basis point units. Relative volatilityerrors, calculated as fractions of exact volatilities, are in parentheses. option swap maturityexpiry 1 2 5 10 30ATM K = F K = F − σ ATM √ T e K = F + 2 σ ATM √ T e AST SWAPTION PRICING 15
In particular, our method gives results superior to those from the method of Schrager andPelsser [2006] because it accurately captures the skew in the normal volatility. For comparison,we reproduce the results of Schrager and Pelsser [2006] for Parameter Set 3; compare Tables 10and 11 to Tables 8 and 9. The error in Schrager and Pelsser [2006]’s method is primarily causedby the condition that the normal implied volatility is constant across strikes, whereas the GTSMhas a slightly upward sloping volatility skew, as indicated by our hyperplane approximation andexact methods. This tendency arises because LVMs are assumed to be constant at their time-zero values in deriving the SDE for the swap rate in Schrager and Pelsser [2006]. It shouldbe mentioned that Andersen and Piterbarg [2010c] further refine the swap rate SDE in thebroader context of the linear local volatility Gaussian model. In their improved SDE, the swaprate follows a displaced log-normal diffusion, thus exhibiting the volatility skew. We do notimplement their method here and leave the performance comparison for future study.We further analyze the error using a particular example: a 2 ×
10 swaption on ParameterSet 1. First, we present the exact exercise boundary for this case in Fig. 2(a). The boundarylines for different strikes are slightly convex upward but are close to flat lines. In our method,by approximating the boundary with a flat line, we incorrectly exercise the swaption when thestate falls into the area between the boundaries where the underlying swap has a negative value.Thus, we have a negative pricing error of − . × − for ATM. In Fig. 2(b), we provide theerror density as defined in Eq. (3.17). Finally, we plot the DBP volatility error as a function ofthe strike in Fig. 3. For all three parameter sets, the error tends to increase for a higher strike.This increase is most likely because each term in Eq. (3.3) becomes more convex as the statebecomes larger; this increases the deviation between the exercise boundary and the flat line. References
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Figure 2. (a) The exact exercise boundary of the 2 ×
10 swaptions in ParameterSet 1. The strikes for ATM, ITM and OTM are 5.06%, 3.51% and 6.62%,respectively. Our method approximates this boundary as the horizontal axis d ( y ) = d ∗ . Both the x and y axes are normalized by the standard deviationof the state variables. Although the distance between the exact boundary andthe hyperplane grows quadratically from the tangent point, the probability ofthe normal distribution decays significantly faster. (b) The error density definedin Eq. (3.17) for the same swaptions and parameter sets. It is the probability-weighted swaption payoff integrated over the area between the exact boundary ∂ Ω and the approximated hyperplane ∂ ˜Ω (shaded area in Fig. 1) in the directionof y . The error peaks at approximately two standard deviations and quicklydecays because of the normally distributed probability density. AST SWAPTION PRICING 17 −7 Strike(%) E rr o r s Parameter Set 1Parameter Set 2Parameter Set 3
Figure 3.
The implied volatility errors of the hyperplane approximation forvarying strikes in daily basis point units. We use a 2 ×
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