The Gauss2++ Model -- A Comparison of Different Measure Change Specifications for a Consistent Risk Neutral and Real World Calibration
TThe Gauss2++ Model – A Comparison of DifferentMeasure Change Specifications for a Consistent RiskNeutral and Real World Calibration
Christoph Berninger a,b , Julian Pfeiffer b a Department of Statistics, LMU Mnchen b ROKOCO GmbH, Ludwig-Ganghofer-Str. 6, 82031 Grnwald
June 16, 2020
Abstract
Especially in the insurance industry interest rate models play a crucial role e.g. to cal-culate the insurance company’s liabilities, performance scenarios or risk measures. Aprominant candidate is the – ina different representation also known as the . In this paper,we propose a framework to estimate the model such that it can be applied under the riskneutral and the real world measure in a consistent manner. We first show that any pro-gressive and square-integrable function can be used to specify the change of measurewithout loosing the analytic tractability of e.g. zero-coupon bond prices in both worlds.We further propose two time dependent candidates, which are easy to calibrate: a stepand a linear function. They represent two variants of our framework and distinguishbetween a short and a long term risk premium, which allows to regularize the interestrates in the long horizon. We apply both variants to historical data and show that theyindeed produce realistic and much more stable long term interest rate forecast than theusage of a constant function. This stability over time would translate to performancescenarios of e.g. interest rate sensitive fonds and risk measures.
Keywords:
Preprint submitted to Elsevier June 16, 2020 a r X i v : . [ q -f i n . M F ] J un . Introduction Two prominent approaches to model the term structure of interest rates are theclasses of equilibrium and no-arbitrage models. Most equilibrium models concentrateon the dynamic of the short-rate – the instantaneous interest rate and derive interestrates with longer maturities from it. Prominent candidates of this model class include[3], [8] and [15]. No-arbitrage models focus on exactly fitting the term structure at aspecific point in time to prevent arbitrage possibilities. Representatives of this class areintroduced by [10] and [11].Applications of these models often relate to pricing interest rate derivatives, which isthe reason why they are directly defined under the risk neutral measure most of thetime. A general form of a one-factor short-rate model under the risk neutral measureis, e.g., given by dr ( t ) = µ ( t, r ) dt + σ ( t, r ) dW ( t ) , where µ and σ are two functions, which can depend on time point t and the short-rate r , and W is a Brownian motion. A lot of advances in theoretic models and theirestimation have been conducted in the last 30 years, but only in connection to pricing[6]. Regarding these models little attention has been given to forecasting and riskmanagement purposes [6]. For these applications the corresponding model needs tobe regarded under the real world measure. Under this measure the corresponding onefactor short-rate model has the following dynamic dr ( t ) = (cid:20) µ ( t, r ) + λ ( t, r ) σ ( t, r ) (cid:21) dt + σ ( t, r ) d (cid:102) W ( t ) , where λ is the market price of risk and can also depend on t and r . (cid:102) W is a Brownianmotion under the real world measure. The exact functional choice for λ completes themodel specification under the real world measure. Dai and Singleton [5] as well asJong [13] use a fixed multiple of the model’s variance for the market price of risk andinvestigate the in sample fit of specific short-rate models, but do not focus on forecast-ing. Duffee [7] concludes that the class of term structure models analysed in [5] failin forecasting. He argues that a restriction for the market price of risk to be a fixedmultiple of the variance reduces the flexibility of the model. Hull et al. [12] stress thatthe market price of risk for a model with few factors should be time dependent. Thisresults not from an economic interpretation but from a modelling issue because of aninsufficient number of factors [12]. They estimated the market price of risk based onhistorical 3-month and 6-month interest rates and came to a similar result as [1], [4]and [14]. But they argue that this value is only valid in the short horizon. Keeping thismarket price of risk constant could lead to extreme risk premiums and interest rates inthe long horizon.In this paper we tackle exactly this problem. Instead of assuming a constant, we as-sume a time-varying function for the market price of risk. In contrast to Hull et al. [12],who estimate the market price of risk for each forecasting horizon individually, we pro-pose two parametric functions. The step function is the easiest non-constant function,which allows to model a market price of risk valid in the short and one valid in the longhorizon. The linear function assumes that the market price of risk in the short horizon2onverges linearly to a long-term level. With these simplified time dependent functionsit is possible to account for the problem mentioned by Hull et al. [12] and the functionscan still be easily estimated by historical data or calibrated in a forward looking mannerto interest rate forecasts.The structure of the paper is as follows. In Section 2 we introduce the Gauss2++model under the risk neutral and the real world measure in a very general framework.In Section 3 we propose the constant function for comparison reason as well as the stepand the linear function to specify the change of measure and explain how they can beestimated. All three variants of the Gauss2++ model are applied to data and backtestedfor the last 3 years in Section 4. In the final section the results are summarized andconcluded.
2. The Gauss2++ Model in the Risk Neutral and the Real World
Throughout this section a filtered probability space (Ω , F , ( F t ) t ∈ [0 , T ] , M ) is given,where M is either the risk neutral measure Q with respect to the bank account or the realworld measure P . T represents an appropriate modelling horizon. The bank account ( B ( t )) t ∈ [0 , T ] is given by dB ( t ) = r ( t ) B ( t ) dt, B (0) = 1 . A challenge of modelling the yield curve is the multivariate setting as each interestrate with a specific maturity represents a dimension. Instead of modelling all maturitiessimultaniously, short-rate models just model the short-rate and derive interest rates withlonger maturities via pricing zero-coupon bonds. Given the price of a zero-couponbond, the corresponding interest rate can be calculated by r ( t, T ) = − ln ( P ( t, T )) T − t , (1)where r ( t, T ) and P ( t, T ) represent the interest rate and the price of a zero-couponbond at time t and a maturity of T , respectively.For pricing zero-coupon bonds the financial mathematical method of risk neutral valu-ation can be applied. The risk neutral interest rates generated in this way can be usedin a Monte Carlo simulation to price interest rate derivatives or bonds. This is the mainapplication of short-rate models and the reason why they are often defined directly un-der the risk neutral measure.The method of risk neutral valuation is a general concept in financial mathematicsand uses the property, that price processes of any security in the market discountedby the bank account are martingales under Q . Therefore, the risk neutral price of azero-coupon bond at time point t is obtained by P ( t, T ) B ( t ) = E Q (cid:20) P ( T, T ) B ( T ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) .
3s the value of the bank account at time point t is given by B ( t ) = e (cid:82) t r ( s ) ds and thepayoff of a zero-coupon bond is one amount of currency at time T this leads to P ( t, T ) = E Q (cid:20) e − (cid:82) Tt r ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , where r ( s ) is the short-rate at time point s . If the distribution of r ( s ) is known andsuch, that the conditional distribution of e − (cid:82) Tt r ( s ) ds can be determined, zero-couponbond prices of different maturities at different time points can be analytically calcu-lated. From bond prices interest rates are available using (1), so that indeed the wholeinterest rate curve is characterized in terms of distributional properties of r .If one is not interested in pricing interest rate derivatives or bonds but in risk mea-sures or performance scenarios, interest rates under the real world measure are needed.The challenge in the real world is that every financial product has a different drift inits process depending on its risk the (in general risk averse) investor wants to be com-pensated for. To get a martingale as in the risk neutral world such that we can usethe conditional expectation to price a security in the market, we have to discount theprice process with a cash flow, which is product specific and different from the riskneutral bank account. This cash flow is in general not known, which is the reason whyone switches to the risk neutral world if interested in pricing and valuation. But byknowing the dynamics of the processes under the risk neutral measure and defining thechange of measure, we implicitly define this cash flow for every security in the marketand therefore we can calculate the price of a zero-coupon bond analogously with theconditional expectation P ( t, T ) X P ( t,T ) ( t ) = E P (cid:20) P ( T, T ) X P ( t,T ) ( T ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , where X P ( t,T ) ( t ) is the value of the cash flow at time point t , with which we have todiscount P ( t, T ) such that P ( t,T ) X P ( t,T ) ( t ) is a martingale under P . Note that we take theexpectation under the real world measure P . As P ( T, T ) is one amount of currency theconditional expectation reduces to P ( t, T ) = E P (cid:20) X P ( t,T ) ( t ) X P ( t,T ) ( T ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . (2)We will show in Section 2.3 that if we define the change of measure in the Gauss2++model in a specific way, X P ( t,T ) ( t ) can be easily extracted and a closed form solutionfor the price of a zero-coupon bond or interest rates can still be obtained. Short-rate models differ in the underlying process for the short-rate. The Gauss2++model assumes that the short-rate is given by a sum of two correlated normally dis-tributed processes, ( x ( t )) t ∈ [0 , T ] and ( y ( t )) t ∈ [0 , T ] , and a deterministic function ϕ , whichis well defined on the time interval [0 , T ] : r ( t ) = x ( t ) + y ( t ) + ϕ ( t ) , r (0) = r , r is the short-rate at time point . The processes ( x ( t )) t ∈ [0 , T ] and ( y ( t )) t ∈ [0 , T ] satisfy under the risk neutral measure Q the following stochastic differ-ential equations dx ( t ) = − ax ( t ) dt + σdW ( t ) , x (0) = 0 ,dy ( t ) = − by ( t ) dt + ηdW ( t ) , y (0) = 0 ,ρdt = dW ( t ) dW ( t ) , where a , b , σ , η are non-negative constants and − ≤ ρ ≤ is the instantaneouscorrelation between the two Brownian motions W and W .The short-rate is therefore normally distributed and it can be shown that (cid:82) Tt r ( s ) ds isalso normally distributed with mean M ( t, T ) = (cid:90) Tt ϕ ( s ) ds + B ( a, t, T ) x ( t ) + B ( b, t, T ) y ( t ) and variance V ( t, T ) = σ a (cid:20) ( T − t ) + 2 a e − a ( T − t ) − a e − a ( T − t ) − a (cid:21) + η b (cid:20) ( T − t ) + 2 b e − b ( T − t ) − b e − b ( T − t ) − b (cid:21) + 2 ρ σηab (cid:34) ( T − t ) + e − a ( T − t ) − a + e − b ( T − t ) − b − e − ( a + b )( T − t ) − a + b (cid:35) , where B ( z, t, T ) = 1 − e − z ( T − t ) z . A derivation of the mean and the variance can be found in [2].The expression e − (cid:82) Tt r ( s ) ds is therefore log-normally distributed and the zero-couponbond price P ( t, T ) , which is the conditional expectation of this expression, is given by P ( t, T ) = E Q (cid:20) e − (cid:82) Tt r ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = e − M ( t,T )+ V ( t,T )2 = e − (cid:82) Tt ϕ ( s ) ds − B ( a,t,T ) x ( t ) − B ( b,t,T ) y ( t )+ V ( t,T ) . (3)With this closed form solution for the conditional expectation zero-coupon bond prices under the risk neutral measure are readily defined and interestrates can be directly derived.The financial market we actually model consists of a bank account and a set of zero-coupon bonds, P ( t, T ) , which differ in the maturity T . The dynamic of a zero-couponbond price can be derived from the bond price formula in (3) by applying Ito’s formulaand is given by dP ( t, T ) = P ( t, T ) (cid:20) r ( t ) dt − σB ( a, t, T ) dW ( t ) − ηB ( b, t, T ) dW ( t ) (cid:21) . To calculate performance scenarios and risk indicators the Gauss2++ model mustbe regarded under the real world measure P . By specifying the Gauss2++ model under the risk neutral measure, we implicitlyassume an arbitrage free market. Therefore, we can make the transition to a real worldmeasure P by defining the change of measure according to Girsanov, who states that aprogressive and square-integrable process ( Φ ( t )) t ∈ [0 , T ] = (cid:0) Φ ( t ) , Φ ( t ) , ..., Φ d ( t ) (cid:1) t ∈ [0 , T ] determines a new probability mea-sure P such that if ( (cid:99) W ( t )) t ∈ [0 , T ] is a standard d -dimensional ( F t ) t ∈ [0 , T ] -Brownianmotion under Q , then ˘ W ( t ) := (cid:99) W ( t ) + (cid:90) t Φ ( s ) ds defines a standard d -dimensional ( F t ) t ∈ [0 , T ] -Brownian motion under P [9]. We canchoose any Φ , which fullfills the conditions in the Girsanov theorem, to specify thechange of measure.The Gauss2++ model is a two-factor model and Φ is therefore 2-dimensional. Itscomponents can be interpreted as the market price of risk for each factor in the model.We will represent Φ as follows to simplify calculations Φ ( t ) = (cid:18) Φ ( t )Φ ( t ) (cid:19) = (cid:32) − ad x ( t ) σ − bd y ( t ) η √ − ρ + ρad x ( t ) σ √ − ρ (cid:33) . (4)Note that we have not restricted the set of functions by this representation. The condi-tions for the Girsanov theorem translate directly to the functions d x ( t ) and d y ( t ) . In thefollowing we will specify the change of measure via d x ( t ) and d y ( t ) . An appropriateinterpretation of these functions will be given in Section 2.3.2. P With the representation of Φ as in (4) the dynamics of the processes x and y in theGauss2++ model change according to Girsanov to dx ( t ) = a ( d x ( t ) − x ( t )) dt + σd (cid:102) W ( t ) , x (0) = 0 , (5) dy ( t ) = b ( d y ( t ) − y ( t )) dt + ηd (cid:102) W ( t ) , y (0) = 0 , (6)where (cid:102) W and (cid:102) W are two correlated Brownian motions under P . The derivationcan be found in Appendix B. We observe that x and y are still Ornstein-Uhlenbeck6rocesses with the solutions x ( t ) = (cid:90) t e − a ( t − u ) ad x ( u ) du + σ (cid:90) t e − a ( t − u ) d (cid:102) W ( u ) , (7) y ( t ) = (cid:90) t e − b ( t − u ) bd y ( u ) du + η (cid:90) t e − b ( t − u ) d (cid:102) W ( u ) . (8)The mean reversion level of each process at time point t amounts to d x ( t ) and d y ( t ) ,respectively. Recall that the sum of x ( t ) and y ( t ) and a deterministic function ϕ ( t ) under the risk neutral measure adds up to the instantaneous return rate r ( t ) of a riskfree investment. Changing the measure changes the mean reversion level at time point t from to d x ( t ) for the process x and to d y ( t ) for the process y . Therefore, d x ( t ) + d y ( t ) can be interpreted as the local long run risk premium of the short-rate – theamount, which is added in the real world to the risk neutral short-rate in the long run, if d x ( t ) + d y ( t ) would stay constant over time. If this amount is negative, future bondprices increase in expectation compared to the risk neutral world and a risk averseinvestor, therefore, gets compensated for the risk of investing in a risky bond. Thismeans in contrast to equity prices, in a market where investors are risk averse, futureinterest rates tend to be lower in the real world than in the risk neutral world [12].Therefore, d x ( t ) and d y ( t ) can be interpreted as the local long run risk premium thecorresponding risk factor is mean reverting to at time point t .In the following we will specify the change of measure by these two functions insteadof the market prices of risk. The market price of risk of each risk factor is then directlydefined by these two functions.Market price of risk of risk factor 1: − ad x ( t ) σ Market price of risk of risk factor 2: − bd y ( t ) η (cid:112) − ρ + ρad x ( t ) σ (cid:112) − ρ . If we assume a step or a piecewise linear function for d x ( t ) and d y ( t ) the functionalform of the individual market prices of risk are the same.The dynamics of a zero-coupon bond with maturity T under P has the following form dP ( t, T ) = P ( t, T ) [ r ( t ) − B ( a, t, T ) ad x ( t ) − B ( b, t, T ) bd y ( t )] dt − P ( t, T ) B ( a, t, T ) σd (cid:102) W ( t ) − P ( t, T ) B ( b, t, T ) ηd (cid:102) W ( t ) (9)The derivation can be found in Appendix C. To calculate the price of a zero-coupon bond under the real world measure with theconditional expectation in (2), the cash flow X P ( t,T ) , with which we have to discountthe zero-coupon bond such that the discounted price process is a martingale under P ,needs to be determined. The dynamic of X P ( t,T ) coincides with the deterministic partof the zero-coupon bond price dynamic in (9) and is therefore specified by the changeof measure: 7 X P ( t,T ) ( t ) = X P ( t,T ) ( t ) [ r ( t ) − B ( a, t, T ) ad x ( t ) − B ( b, t, T ) bd y ( t )] dt, X P ( t,T ) (0) = 1 . A short proof can be found in Appendix D. The solution of this dynamic is given by X P ( t,T ) ( t ) = e (cid:82) t ( r ( u ) − B ( a,u,T ) ad x ( u ) − B ( b,u,T ) bd y ( u )) du . As P ( t,T ) X P ( t,T ) ( t ) is a martingale we can use the conditional expectation in (2) to calculatethe price of a zero-coupon bond at time point t : P ( t, T ) = E P (cid:20) X P ( t,T ) ( t ) X P ( t,T ) ( T ) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . The ratio in the expectation amounts to X P ( t,T ) ( t ) X P ( t,T ) ( T ) = e − (cid:82) Tt ( r ( u ) − B ( a,u,T ) ad x ( u ) − B ( b,u,T ) bd y ( u )) du . To determine the distribution of this ratio, we first derive the distribution of the integralin the exponent, i.e., I ( t, T ) := (cid:90) Tt ( r ( u ) − B ( a, u, T ) ad x ( u ) − B ( b, u, T ) bd y ( u )) du. It can be shown that I ( t, T ) is normally distributed with mean M ( t, T ) = (cid:90) Tt ϕ ( u ) du + 1 − e − a ( T − t ) a x ( t ) + 1 − e − b ( T − t ) b y ( t ) (10)and variance V ( t, T ) = σ a (cid:20) ( T − t ) + 2 a e − a ( T − t ) − a e − a ( T − t ) − a (cid:21) + η b (cid:20) ( T − t ) + 2 b e − b ( T − t ) − b e − b ( T − t ) − b (cid:21) + 2 ρ σηab (cid:34) ( T − t ) + e − a ( T − t ) − a + e − b ( T − t ) − b − e − ( a + b )( T − t ) − a + b (cid:35) . (11) The variance is the same as in the risk neutral world as the change of measure does notinfluence the variance of the processes. Note that also the mean has the same form asin the risk neutral case as the terms B ( a, u, T ) ad x ( u ) and B ( b, u, T ) bd y ( u ) in I ( t, T ) cancel out in the calculations. The derivations can be found in Appendix E.The expression e − I ( t,T ) is therefore log-normally distributed and the zero-coupon bondprice under P is given by P ( t, T ) = E P (cid:104) e − (cid:82) Tt r ( u ) − B ( a,u,T ) ad x ( u ) − B ( b,u,T ) bd y ( u ) du | F t (cid:105) = e − M ( t,T )+ V ( t,T ) = e − (cid:82) Tt ϕ ( u ) du − − e − a ( T − t ) a x ( t ) − − e − b ( T − t ) b y ( t )+ V ( t,T ) . x ( t ) and y ( t ) are now the values at time point t of thecorresponding processes under the real world measure P .
3. Local Long Run Risk Premium Functions – Specification and Calibration
In the following three different types of functions for d x ( t ) and d y ( t ) are intro-duced: the constant, the step and the linear function. Following the interpretation inSection 2.3.2 these functions represent the long run risk premium for each risk factorat a specific time point t in the Gauss2++ model. The functional equations of the threetypes areConstant: d x ( t ) = d x d y ( t ) = d y Step: d x ( t ) = t ≤ τ d x + t>τ l x d y ( t ) = t ≤ τ d y + t>τ l y Linear: d x ( t ) = t ≤ τ (1 − m x t ) d x + t>τ l x d y ( t ) = t ≤ τ (1 − m y t ) d y + t>τ l y where d x , l x , m x and d y , l y , m y are real valued constants and A represents the indi-cator function of a subset A .The constant function assumes that the local long run risk premium is constant for thewhole modelling horizon. The latter two functions distinguish between a local longrun risk premium valid in the short and in the long horizon, seperated at time point τ .As mentioned in Section 2.3.2 the same holds for the market price of risk, respectively.Hull et al. [12] argue that a time varying market price of risk is necessary to accountfor unobserved risk factors and to prevent unrealistic interest rate forecasts in the longhorizon. They therefore estimate an individual market price of risk for each forecastinghorizon. We use a more parsimonious function with regard to the number of parame-ters. The step function we propose is the simplest time varying function that expectsthat the local long run risk premium differs in the short and the long horizon but is stillconstant in each period. The linear function implements the property that the local longrun risk premium in the short horizon approaches the long term level linearly. The sim-plicity of these functions allows a straight forward calibration to interest rate forecasts.Because of the distributional properties of the Gauss2++ model the expected valuesfor interest rates under the real world measure P for any future time point can be cal-culated: E P [ r ( t, T )] = E Q [ r ( t, T )] + B ( a, t, T ) T − t RP x ( t ) + B ( b, t, T ) T − t RP y ( t ) , (12)where RP x ( t ) and RP y ( t ) represent the actual risk premium of the short-rate at time9oint t for each risk factor and are given by the first integral in (7) and (8) RP x ( t ) := (cid:90) t e − a ( t − u ) ad x ( u ) du,RP y ( t ) := (cid:90) t e − b ( t − u ) bd y ( u ) du. For the constant, the step and the linear function these integrals can be easily calcu-lated. To get the risk premium for longer maturities the functions RP x ( t ) and RP y ( t ) are weighted by a loading function, which accounts for the different riskiness of thecorresponding zero-coupon bonds B ( a, t, T ) T − t and B ( b, t, T ) T − t . To calibrate the local long run risk premium functions, d x ( t ) and d y ( t ) , the parametersof the functions are chosen in such a way that the model meets specific interest rateforecasts in expectation. For the constant type two interest rate forecasts are needed.For the other two types four interest rate forecasts are necessary – two short term andtwo long term forecasts. The time parameter τ , which determines the separation be-tween the short and the long term local long run risk premium must lie between theforecasting horizons of the two short and the two long term forecasts.In Figure 1 the three types of local long run risk premium functions have been ex-emplary calibrated. τ has been set to months, which is the forecasting horizon ofthe short term interest rate forecasts. (a) (b) (c)Figure 1: Local long run risk premium functions In the following subsections the calibration procedures for all three types of local longrun risk premium functions, which are applied in this paper, are described.
The constant functions represented in Figure 1 (a) implement a constant local longrun risk premium for the whole modelling horizon, which can amount to up to years for actual applications in the insurance industry, e.g., to classify certified pension10ontracts into risk classes. The absolute risk premiums, RP x ( t ) and RP y ( t ) , are givenby: RP x ( t ) = (1 − e − at ) d x ,RP y ( t ) = (1 − e − bt ) d y . Note that if t → ∞ , RP x ( t ) and RP y ( t ) indeed converge to d x and d y , the long runrisk premiums, respectively. To calibrate the parameters of the constant functions twointerest rate forecasts, ˆ r ( t , T ) and ˆ r ( t , T ) , are used. Plugging the absolute riskpremium functions, RP x ( t ) and RP y ( t ) , into (12) and setting the expectations equalto the interest rate forecasts results in the following two equations (I) ˆ r ( t , T ) ! = E Q [ r ( t , T )] + B ( a,t ,T )( T − t ) (1 − e − at ) d x + B ( b,t ,T )( T − t ) (1 − e − bt ) d y ,(II) ˆ r ( t , T ) ! = E Q [ r ( t , T )] + B ( a,t ,T )( T − t ) (1 − e − at ) d x + B ( b,t ,T )( T − t ) (1 − e − bt ) d y . As the expectations are linear functions in d x and d y , the two parameters can be easilydetermined.The constant function for the local long run risk premium in the Gauss2++ modeland this calibration procedure is a standard approach in the insurance industry. As thevalues for d x and d y determine the risk premium for the whole modelling horizon, theircalibration is crucial for the model’s interest rate distribution. Especially if the interestrate forecasts used for the calibration have a short forecasting horizon, the resultingdistribution in the long horizon is very sensitive to these forecasts. For example if theinterest rate forecasts and the forward rates – calculated from the current yield curve –are very different, to reach the forecasts a huge risk premium is necessary, which mightbe valid in the short horizon, but produces extreme interest rates in the long horizon.The next two functions account for this problem by representing a time varying locallong run risk premium. The step functions represented in Figure 1 (b) take the same value as the corre-sponding constant function up to time τ as the same interest rate forecasts have beenused for the short horizon, but then they jump to a different level to account for therisk premium in the long horizon. Similar to the constant function the absolute riskpremium functions can easily be calculated and amount to RP x ( t ) = (cid:16) e − a ( t − min( t,τ )) − e − at (cid:17) d x + (cid:16) − e − a ( t − min( t,τ )) (cid:17) l x ,RP y ( t ) = (cid:16) e − b ( t − min( t,τ )) − e − bt (cid:17) d y + (cid:16) − e − b ( t − min( t,τ )) (cid:17) l y . Note that if t → ∞ , RP x ( t ) and RP y ( t ) now converge to l x and l y , respectively. Tocalibrate the four parameters of the step function two short term and two long terminterest rate forecasts are used resulting in the following equations:11 I) ˆ r ( t , T ) ! = E Q [ r ( t , T )] + B ( a,t ,T )( T − t ) RP x ( t ) + B ( b,t ,T )( T − t ) RP y ( t ) ,(II) ˆ r ( t , T ) ! = E Q [ r ( t , T )] + B ( a,t ,T )( T − t ) RP x ( t ) + B ( b,t ,T )( T − t ) RP y ( t ) ,(III) ˆ r ( t , T ) ! = E Q [ r ( t , T )] + B ( a,t ,T )( T − t ) RP x ( t ) + B ( b,t ,T )( T − t ) RP y ( t ) ,(IV) ˆ r ( t , T ) ! = E Q [ r ( t , T )] + B ( a,t ,T )( T − t ) RP x ( t ) + B ( b,t ,T )( T − t ) RP y ( t ) , where t ≤ t < t ≤ t . τ must lie between t and t , i.e. t ≤ τ < t .Instead of interest rate forecasts direct forecasts of the absolute risk premium of theshort-rate can be used. This approach is applied by Hull et al. [12], who estimate riskpremiums for each forecasting horizon from historical data, but they also scale theirresult to a long term short-rate forecast. The linear functions represented in Figure 1 (c) avoid the sudden jump as it is thecase in the step functions and converge in the short term linearly to a long term level.The absolute risk premiums at time point t can be calculated as before and amount to RP x ( t ) = (cid:16)(cid:16) e − a ( t − min( t,τ )) − e − at (cid:17) (cid:16) m x a (cid:17) + e − a ( t − min( t,τ )) m x min( t, τ ) (cid:17) d x + (cid:16) − e − a ( t − min( t,τ )) (cid:17) l x ,RP y ( t ) = (cid:16)(cid:16) e − b ( t − min( t,τ )) − e − bt (cid:17) (cid:16) m y b (cid:17) + e − b ( t − min( t,τ )) m y min( t, τ ) (cid:17) d y + (cid:16) − e − b ( t − min( t,τ )) (cid:17) l y . Note again that if t → ∞ , RP x ( t ) and RP y ( t ) converge to l x and l y , the long termrisk premiums, respectively. To calibrate d x , l x , d y and l y four interest rate forecasts asfor the step function are used. By imposing that the absolute risk premium functions, RP x ( t ) and RP y ( t ) , are differentiable at the forecasting horizon τ to prevent a kink inthe absolute risk premium function, two further conditions are incorporated to specify m x and m y : (V) RP (cid:48) x ( t ) (cid:12)(cid:12) t = τ − = RP (cid:48) x ( t ) (cid:12)(cid:12) t = τ + ,(VI) RP (cid:48) y ( t ) (cid:12)(cid:12) t = τ − = RP (cid:48) y ( t ) (cid:12)(cid:12) t = τ + .Solving the equations for m x and m y leads to the following closed form solutionsreducing the number of free parameters to four: m x = d x − l x d x τ ,m y = d y − l y d y τ . Note that with this condition the same number of interest rate forecasts as for the stepfunction are needed to calibrate d x ( t ) and d y ( t ) .12 b σ η ρ . . . . − . Table 1: Parameters of the Gauss2++ model calibrated at 31.12.2019
4. Results
In this Section the calibration results of three variants of our framework for theGauss2++ model are presented. The variants differ in the assumption about the locallong run risk premium functions, which determine the change from the risk neutral tothe real world measure. Variant 1 assumes a constant, variant 2 a step and variant 3a linear local long run risk premium function for the risk factors. In the first Subsec-tion the three variants of the Gauss2++ model are compared if calibrated at the samevaluation date. In Subsection 4.2 we show with a backtest over the last three yearsthat variant 2 and 3 produce much more stable interest rate scenarios for the long fore-casting horizon over this time period. This stability would transfer to performancescenarios and risk measures of e.g. an interest rate sensitive fonds.
The calibration process of the Gauss2++ model can be split into two steps. In thefirst step the model is calibrated under the risk neutral measure. This step does notdepend on the choice of the local long run risk premium function and is therefore thesame for all modelling cases. In the second step the change of measure is calibrated.The choice of the local long run risk premium function plays an important role andleads to different interest rate scenarios, performance measures and risk indicators.To calibrate the model at a specific valuation date under the risk neutral measure theterm structure of interest rate swaps and swaption volatilities at this date are used. TheGauss2++ model presumes a specific dynamic for the short-rate and with it for inter-est rates with longer maturities. The parameters of the model are chosen in such away, that the current term structure is met in expectation and that the model prices ofthe swaptions coincide with the market prices. In this way market consistency of themodel is ensured. As ϕ is a deterministic function of time, a perfect fit in expecta-tion to the current term structure of interest rates can be achieved, i.e. the function ϕ is implicitly given by the current interest rate curve. Later in the modelling processwe use the term structure of german government bond yields with the assumption thatthe dynamic of this term structure is the same as for the term structure of interest rateswaps. For the calibration of the five parameters a, b, σ, η and ρ the downhill simplexalgorithm is used to find the parameter set, which replicates the market swaption pricesbest. Table 1 shows the results of a calibration at the 31.12.2019. We use swaptionswith a maturity and tenor combination of { , , , , , } x { , , , , , } ,i.e. in total swaption prices. These parameters together with the current interest ratecurve determine the dynamics of the Gauss2++ model under the risk neutral measure.13n the second step the local long run risk premium functions, which determine thechange of measure, are calibrated to interest rate forecasts as described in Section 3.1-3.3. For the short term interest rate forecasts we use forecasts published by the OECDfor a 3-month and a 10-year interest rate. The latest forecasts regarding the 31.12.2019for the longest horizon, which is the fourth quarter of 2021, amount to − . and . , respectively . For the long term interest rate forecasts, which are needed to cal-ibrate the step and the linear function, we take the average of monthly 3-month and10-year interest rates over the last 15 years also published by the OECD. This is a validapproach if interest rates follow a stationary process, because in this case historical datacan be considered as a random sample from the corresponding interest rate distribution.Hull et al. [12] point out that this approach is questionable if monetary and fiscal poli-cies are expected to be materially different from those in the past. Nevertheless anyother model based on historical data would be questionable and the user of the modelcan alternatively provide personal estimates or an expert judgment. The historical av-erage amounts to . for the 3-month and . for the 10-year interest rate and aswe assume these forecasts to be a long run average we set the forecasting horizon to years – the modelling horizon. We further set τ to 24 months, which is the forecastinghorizon of the short term OECD forecasts.Table 2 shows the calibration results for the three local long run risk premium func-tion types. d x d y l x l y Constant Function − . . Step Function − . . − . − . Linear Function − . . − . − . Table 2: Parameters of the local long run risk premium functions
The values of d x and d y coincide for the constant and the step function as the sameinterest rate forecasts have been used in the calibration process. But in contrast to thestep function, which takes the values of l x and l y after months, the constant functionstays constant for the whole modelling horizon. It also appears that the step and thelinear function take the same values for l x and l y . But there is a slight difference astheir functional forms differ in the first two years, which influences the absolute riskpremium in future time points. This influence decreases in time, such that the differ-ence is negligible as we calibrated l x and l y to forecasts with an forecasting horizon of years. https://stats.oecd.org -year and the -year interest ratefor forecasting horizons of up to 40 years. The solid line represents the expectationunder the risk neutral measure, the dashed line shows the expected values under thereal world measure. (a) (b) (c)Figure 2: Constant Function(a) (b) (c)Figure 3: Step Function(a) (b) (c)Figure 4: Linear Function For the variant of the Gauss2++ model, which uses the constant function as the locallong run risk premium function, the expected real world interest rates lie above the15 igure 5: Absolute risk premium function for the variants of the Gauss2++ model risk neutral expectation. This means, that a risk seeking behaviour of the investors isassumed for the whole modelling period, because an investor accepts a lower expectedreturn for a corresponding bond if the interest rates are expected to be higher in the realworld compared to the risk neutral world. Ahmad and Wilmott [1] show that there havebeen time periods where investors seem to have historically behaved in this way. Butin general investors are assumed to be risk averse and therefore interest rates should belower in the real world than in the risk neutral world, which is an opposite behaviourto equity prices [12]. For the other two variants of the Gauss2++ model the expectedreal world interest rates lie also above the risk neutral interest rates in the short hori-zon but below in the long horizon. This assumption of risk seeking behaviour in theshort horizon stems from the quite high forecasts of the OECD for the short horizon,but it might be valid in the current market situation. In contrast to the constant case,which keeps this risk seeking behaviour assumption for the whole modelling horizon,in the long run the other two variants of the Gauss2++ model assume in this calibra-tion a risk averse behaviour. Furthermore, the absolute difference in the risk neutraland real world expectations decreases for interest rates with longer maturities. Thisresults from the less variation of interest rates with longer maturities, which is an im-plicit model characteristic of the Gauss2++ model and is supported by historical dataas well. A risk premium is therefore higher (less negative) for a risk averse and lower(less positive) for a risk seeking investor in an arbitrage free market.Figure 5 shows the absolute risk premium functions of the short-rate for all three mod-elling types. It can be observed that for the constant and the step function the absoluterisk premium is the same up to year . After that year the Gauss2++ variant with thestep function has a kink in the absolute risk premium as the local long run risk premiumchanges to a different level, while the modelling case with the constant function con-tinuous to apporach the long term risk premium determined by the short term interestrate forecasts. The modelling case with the linear function results in a different riskpremium for the first years, but approaches – without a kink – the same long term16 ate Short Term Interest Rate Forecasts Historical Average Forecasting Horizon -m IR -y IR -m IR -y IR(in months) (in %) (in %) (in %) (in %)30.09.2019 − . . .
13 1 . − . . .
18 1 . − . . .
22 2 . − . . .
26 2 . − . . .
31 2 . − . . .
35 2 . − . . .
39 2 . − . . .
44 2 . − . . .
48 2 . − . . .
52 2 . − . . .
57 2 . − . . .
63 2 . Table 3: Interest rate forecasts of the OECD and historical average of the 3-month and the 10-year interestrate risk premium as the step function. All three functions intersect after years as this isthe forecasting horizon of the short term interest rate forecasts, which were used forthe calibration. The absolute risk premium at this time point must be the same for allmodelling cases such that the expected interested rates of the model coincide with theforecasts. In this Subsection the different variants of the Gauss2++ model calibrated on aquarterly basis over the last years are compared.As in Section 4.1 interest rate swaps and swaption volatilities have been used for therisk neutral calibration of the Gauss2++ model. To calibrate the parameters of the locallong run risk premium functions in the second calibration step short term interest rateforecasts published by the OECD and a long term average have been used. The fore-casts are shown in table (3). The calibration results of the parameters of the Gauss2++model under the risk neutral measure and of the local long run risk premium functionfor each variant of the Gauss2++ model can be found in table (F.4)-(F.7) in Appendix F.For each calibration the absolut risk premium function of the short-rate and the de-velopment of the expected 10-year interest rate have been calculated and visualised inFigure 6 and 7. https://stats.oecd.org a) (b) (c)Figure 6: Absolute risk premium functions(a) (b) (c)Figure 7: Development of the expectation of the 10-year interest rate over the modelling horizon for all threevariants of the Gauss2++ model The absolute risk premium function of the short-rate for the Gauss2++ model, whichuses the constant function for the local long run risk premium, depends highly on therisk neutral calibration results and the forecasts of the OECD. An unfavorable combi-nation of market data and interest rate forecasts can lead to a high value for the locallong run risk premium. This value might be reasonable to meet the short term forecastsused for the calibration, but as it stays constant over time it is the value the absolute riskpremium is converging to. Therefore, this problem can strike through if the modellinghorizon is much longer than the forecasting horizon of the interest rates used for thecalibration. In this case a time-varying local long run risk premium function, whichcan be calibrated to a short and a long term forecast, is more convenient to regularizethe risk premium. As it can be seen in Figure 6 the variants of the Gauss2++ model,which use the step or the linear function for the local long run risk premium, producemore stable risk premiums in the long horizon. In each calibration the absolute riskpremium is positive in the first years, which presumes a risk seeking behaviour of theinvestors, but in the long horizon the absolute risk premium lies between − . and − . representing a risk averse market. Also the interest rate distribution in the longhorizon is more stable. Figure 7 (b) and (c) show that the expectation of the 10-yearinterest rate in the long horizon change only little in each calibration according to thehistorical average, which was used for the long term interest rate forecast.18 . Conclusion As the Gauss2++ model is often used for pricing purposes, the focus in the literaturelies on the evolution of interest rates under the risk neutral measure Q . But regardingrisk management and forecasting applications the model under the real world measureis needed. In this paper we introduced a framework to apply the model under bothmeasures in a consistent manner. This framework first conducts a calibration under therisk neutral measure and then determines the change of measure such that it is possibleto switch between the risk neutral and the real world. We showed that according to Gir-sanov this change of measure can be specified by any progressive and square-integrablefunction without loosing the analytic tractability for e.g. zero-coupon bond prices. Hullet al. [12] argue that because of unobserved risk factors, which are not included in themodel, a time-varying function should be used, because otherwise unrealistic interestrates in the long forecasting horizon could be reached. We therefore compared a vari-ant of our framework, which uses constant functions to model the change of measure,with two variants, which use either a step or a linear functions. These functions arethe simplest extensions of the constant function to a time varying function without in-creasing the computational effort much. By accounting for different risk premiums inthe short and in the long horizon the time varying functions result in much more stableinterest rate forecasts in the long run if calibrated at different valuation dates. From amacroeconomical point of view it makes sense that current market fluctuations shouldnot influence interest rate forecasts in the long horizon, e.g. in years, much. Thiswould also imply that risk measures calculated with the Gauss2++ model, which usesone of the time-varying functions for the change of measure, would be more consistentif estimated at different valuation time points. Acknowledgement
This research was supported by
ROKOCO predictive analytics GmbH . We thankour colleagues from
ROKOCO predictive analytics GmbH who provided insight andexpertise that greatly assisted the research.19 eferences [1] Riaz Ahmad and Paul Wilmott. The market price of interest-rate risk: Measuringand modelling fear and greed in the fixed-income markets.
Wilmott magazine ,pages 64–70, 2006.[2] Damiano Brigo and Fabio Mercurio.
Interest rate models – theory and practice:with smile, inflation and credit . Springer Science & Business Media, 2007.[3] John C Cox, Jonathan E Ingersoll Jr, and Stephen A Ross. An intertemporal gen-eral equilibrium model of asset prices.
Econometrica: Journal of the EconometricSociety , pages 363–384, 1985.[4] Samuel H Cox and Hal W Pedersen. Nonparameteric estimation of interest rateterm structure and insurance applications. In
Proceedings of the 1999 ASTINColloquium, Tokyo, Japan (to appear) , 1999.[5] Qiang Dai and Kenneth J Singleton. Specification analysis of affine term structuremodels.
The journal of finance , 55(5):1943–1978, 2000.[6] Francis X Diebold and Canlin Li. Forecasting the term structure of governmentbond yields.
Journal of econometrics , 130(2):337–364, 2006.[7] Gregory R Duffee. Term premia and interest rate forecasts in affine models.
TheJournal of Finance , 57(1):405–443, 2002.[8] Darrell Duffie and Rui Kan. A yield-factor model of interest rates.
Mathematicalfinance , 6(4):379–406, 1996.[9] Igor Vladimirovich Girsanov. On transforming a certain class of stochastic pro-cesses by absolutely continuous substitution of measures.
Theory of Probability& Its Applications , 5(3):285–301, 1960.[10] David Heath, Robert Jarrow, and Andrew Morton. Bond pricing and the termstructure of interest rates: A new methodology for contingent claims valuation.
Econometrica: Journal of the Econometric Society , pages 77–105, 1992.[11] John Hull and Alan White. Pricing interest rate derivative securities.
The reviewof financial studies , 3(4):573–592, 1990.[12] John Hull, Alexander Sokol, and Alan White. Short rate joint measure models.
Risk , 10:59–63, 2014.[13] Frank de Jong. Time series and cross-section information in affine term-structuremodels.
Journal of Business & Economic Statistics , 18(3):300–314, 2000.[14] Richard Stanton. A nonparametric model of term structure dynamics and themarket price of interest rate risk.
The Journal of Finance , 52(5):1973–2002, 1997.[15] Oldrich Vasicek. An equilibrium characterization of the term structure.
Journalof financial economics , 5(2):177–188, 1977.20 ppendix A. Bond Price Dynamic under the Risk Neutral Measure
By defining A ( t, T ) = − (cid:90) Tt ϕ ( s ) ds + 12 V ( t, T ) , the price of a zero-coupon bond P ( t, T ) at time point t and maturity T can be calculatedfor the Gauss2++ model under the risk neutral measure Q by P ( t, T ) = e A ( t,T ) − B ( a,t,T ) x ( t ) − B ( b,t,T ) y ( t ) . (A.1)A proof of this formula can be found in [2]. The derivatives of A ( t, T ) and V ( t, T ) with respect to the first entry and of B ( z, t, T ) with respect to the second entry aregiven by A (cid:48) ( t, T ) = ϕ ( t ) + 12 V (cid:48) ( t, T ) ,V (cid:48) ( t, T ) = − σ B ( a, t, T ) − η B ( b, t, T ) − σηρB ( a, t, T ) B ( b, t, T ) ,B (cid:48) ( z, t, T ) = − e − z ( T − t ) . Furthermore, it holds B ( z, t, T ) z − B (cid:48) ( z, t, T ) = 1 . To calculate the zero-coupon bond price dynamic, we apply It’s formula to (A.1), i.e., dP ( t, T ) = P ( t, T ) [ A ( t, T ) − B ( a, t, T ) x ( t ) − B ( b, t, T ) y ( t )] (cid:48) dt + P ( t, T )( − B ( a, t, T )) dx ( t )+ P ( t, T )( − B ( b, t, T )) dy ( t )+ 12 P ( t, T ) B ( a, t, T ) σ dt + 12 P ( t, T ) B ( b, t, T ) η dt + P ( t, T ) B ( a, t, T ) B ( b, t, T ) σηρdt = P ( t, T ) (cid:20) A (cid:48) ( t, T ) − B (cid:48) ( a, t, T ) x ( t ) − B (cid:48) ( b, t, T ) y ( t ) + B ( a, t, T ) ax ( t ) + B ( b, t, T ) by ( t )+ 12 B ( a, t, T ) σ + 12 B ( b, t, T ) η + B ( a, t, T ) B ( b, t, T ) σηρ (cid:21) dt − B ( a, t, T ) P ( t, T ) σdW ( t ) − B ( b, t, T ) P ( t, T ) ηdW ( t )= P ( t, T )[ ϕ ( t ) + x ( t ) + y ( t )] dt − B ( a, t, T ) P ( t, T ) σdW ( t ) − B ( b, t, T ) P ( t, T ) ηdW ( t )= P ( t, T ) r ( t ) dt − B ( a, t, T ) P ( t, T ) σdW ( t ) − B ( b, t, T ) P ( t, T ) ηdW ( t ) . ppendix B. The Dynamics of the Gauss2++ Factors x and y under the RealWorld Measure The dynamics of the two processes x and y under the risk neutral measure Q canbe expressed in terms of two independent Brownian motions (cid:99) W and (cid:99) W , i.e. dx ( t ) = − ax ( t ) dt + σd (cid:99) W ( t ) ,dy ( t ) = − by ( t ) dt + ηρd (cid:99) W ( t ) + η (cid:112) (1 − ρ ) d (cid:99) W ( t ) , where dW ( t ) = d (cid:99) W ( t ) ,dW ( t ) = ρd (cid:99) W ( t ) + (cid:112) (1 − ρ ) d (cid:99) W ( t ) . According to Girsanov’s theorem , as (cid:99) W = ( (cid:99) W , (cid:99) W ) is a standard 2-dimensionalBrownian motion and let ( Φ ( t )) t ∈ [0 , T ] = (Φ ( t ) , Φ ( t )) t ∈ [0 , T ] be a progressive andsquare-integrable process, the process ˘ W defined by ˘ W ( t ) := (cid:99) W ( t ) + (cid:90) t Φ ( s ) ds is a standard 2-dimensional Brownian motion under a new measure, which we call P and declare to be the real world measure. This means that the dynamic of the twoBrownian motion (cid:99) W and (cid:99) W under the real world measure P is given by d (cid:99) W ( t ) = d ˘ W ( t ) − Φ ( t ) dt,d (cid:99) W ( t ) = d ˘ W ( t ) − Φ ( t ) dt. Therefore, the dynamics of the two processes x and y under the real world measure arethen given by dx ( t ) = (cid:20) − Φ ( t ) σ − ax ( t ) (cid:21) dt + σd ˘ W ( t ) ,dy ( t ) = (cid:20) − Φ ( t ) ηρ − Φ ( t ) η (cid:112) (1 − ρ ) − by ( t ) (cid:21) dt + ηρd ˘ W ( t )+ η (cid:112) (1 − ρ ) d ˘ W ( t ) . If we specify Φ( t ) as in (4) this simplifies to dx ( t ) = a ( d x ( t ) − x ( t )) dt + σd ˘ W ( t ) ,dy ( t ) = b ( d y ( t ) − y ( t )) dt + ηρd ˘ W ( t ) + η (cid:112) (1 − ρ ) d ˘ W ( t ) . Representing the dynamics by two correlated Brownian motions (cid:102) W and (cid:102) W resultsin the equations given in (5) and (6). 22 ppendix C. Bond Price Dynamic under the Real World Measure The dynamic of a zero-coupon bond price P ( t, T ) under the risk neutral measure Q expressed by the two independent Brownian motions (cid:99) W and (cid:99) W is given by dP ( t, T ) = P ( t, T ) r ( t ) dt − P ( t, T ) B τ ( a ) σd (cid:99) W ( t ) − P ( t, T ) B τ ( b ) ηρd (cid:99) W ( t ) − P ( t, T ) B τ ( b ) η (cid:112) (1 − ρ ) d (cid:99) W ( t ) , = P ( t, T ) r ( t ) dt − (cid:20) P ( t, T ) B τ ( a ) σ + P ( t, T ) B τ ( b ) ηρ (cid:21) d (cid:99) W ( t ) − P ( t, T ) B τ ( b ) η (cid:112) (1 − ρ ) d (cid:99) W ( t ) . Applying Girsanov’s theorem as in appendix Appendix B the dynamic under the realworld measure P amounts to dP ( t, T ) = P ( t, T ) r ( t ) dt − (cid:20) P ( t, T ) B τ ( a ) σ + P ( t, T ) B τ ( b ) ηρ (cid:21) d (cid:99) W ( t ) − P ( t, T ) B τ ( b ) η (cid:113) (1 − ρ ) d (cid:99) W ( t )= P ( t, T ) (cid:20) r ( t ) + (cid:18) B τ ( a ) σ + B τ ( b ) ηρ (cid:19) (cid:18) − ad x ( t ) σ (cid:19) + B τ ( b ) η (cid:113) (1 − ρ ) (cid:32) − bd y ( t ) η (cid:112) (1 − ρ ) + ρad x ( t ) σ (cid:112) (1 − ρ ) (cid:33) (cid:21) dt − (cid:20) P ( t, T ) B τ ( a ) σ + P ( t, T ) B τ ( b ) ηρ (cid:21) d ˘ W ( t ) − P ( t, T ) B τ ( b ) η (cid:113) (1 − ρ ) d ˘ W ( t )= P ( t, T ) (cid:20) r ( t ) − B τ ( a ) ad x ( t ) − B τ ( b ) bd y ( t ) (cid:21) dt − (cid:20) P ( t, T ) B τ ( a ) σ + P ( t, T ) B τ ( b ) ηρ (cid:21) d ˘ W ( t ) − P ( t, T ) B τ ( b ) η (cid:113) (1 − ρ ) d ˘ W ( t ) . Representing the dynamic by two correlated Brownian motions (cid:102) W and (cid:102) W results inthe equation given in (9). 23 ppendix D. Individual Discount Rate for the Zero-Coupon Bonds in the RealWorld Proof.
To proof that P ( t,T ) X ( t,T ) is indeed a martingale we calculate the dynamic of thediscounted price process. d P ( t, T ) X ( t ) = d (cid:0) X ( t ) · P ( t, T ) (cid:1) = 1 X ( t ) dP ( t, T ) + P ( t, T ) d X ( t ) + d (cid:28) P ( t, T ) , X ( t ) (cid:29) = 1 X ( t ) dP ( t, T ) − P ( t, T ) X ( t ) [ r ( t ) − B ( a, t, T ) ad x ( t ) − B ( b, t, T ) bd y ( t )] dt = P ( t, T ) X ( t ) (cid:2) r ( t ) − B ( a, t, T ) ad x ( t ) − B ( b, t, T ) bd y ( t ) (cid:3) dt − P ( t, T ) X ( t ) B ( a, t, T ) σd (cid:102) W ( t ) − P ( t, T ) X ( t ) B ( b, t, T ) ηd (cid:102) W ( t ) − P ( t, T ) X ( t ) [ r ( t ) − B ( a, t, T ) ad x ( t ) − B ( b, t, T ) bd y ( t )] dt = − P ( t, T ) X ( t ) B ( a, t, T ) σd (cid:102) W ( t ) − P ( t, T ) X ( t ) B ( b, t, T ) ηd (cid:102) W ( t ) Appendix E. Bond Price Formula under the Real World Measure
To calculate the price of a zero-coupon bond under the real world measure P , thedistribution of exp (cid:32) − (cid:90) Tt ( r ( u ) − B ( a, u, T ) ad x ( u ) − B ( b, u, T ) bd y ( u )) du (cid:33) has to be determined. In the following we show, that the integral in the exponent isnormaly distributed and calculate the mean and the variance of I ( t, T ) := (cid:90) Tt ( r ( u ) − B ( a, u, T ) ad x ( u ) − B ( b, u, T ) bd y ( u )) du. (E.1)We first concentrate on the integral over the short-rate r ( s ) , which is a sum of the x -and the y -process and a deterministic function r ( s ) = x ( s ) + y ( s ) + ϕ ( s ) . x is given by (cid:90) Tt x ( u ) du = (cid:90) Tt (cid:18) x ( t ) e − a ( u − t ) + (cid:90) ut ae − a ( u − s ) d x ( s ) ds + (cid:90) ut σe − a ( u − s ) d (cid:102) W ( s ) (cid:19) du = (cid:90) Tt x ( t ) e − a ( u − t ) du (cid:124) (cid:123)(cid:122) (cid:125) (cid:13) + (cid:90) Tt (cid:90) ut ae − a ( u − s ) d x ( s ) dsdu (cid:124) (cid:123)(cid:122) (cid:125) (cid:13) + (cid:90) Tt (cid:90) ut σe − a ( u − s ) d (cid:102) W ( s ) du. (cid:124) (cid:123)(cid:122) (cid:125) (cid:13) The first integral amounts to (cid:13) = x ( t ) (cid:90) Tt e − a ( u − t ) du = x ( t ) (cid:20) − a e − a ( u − t ) (cid:21) Tt = x ( t ) 1 − e − a ( T − t ) a . For the second integral we use the integration by parts formula (cid:13) = (cid:90) Tt (cid:18)(cid:90) ut e as d x ( s ) ds (cid:19) ae − au du = a (cid:90) Tt (cid:18)(cid:90) ut e as d x ( s ) ds (cid:19) d u (cid:18)(cid:90) ut e − av dv (cid:19) = a (cid:34)(cid:32)(cid:90) Tt e au d x ( u ) du (cid:33) (cid:32)(cid:90) Tt e − av dv (cid:33) − (cid:90) Tt (cid:18)(cid:90) ut e − av dv (cid:19) e au d x ( u ) du (cid:35) = a (cid:34)(cid:90) Tt (cid:32)(cid:90) Tu e − av dv (cid:33) e au d x ( u ) du (cid:35) = (cid:90) Tt (cid:16) − e − a ( T − u ) (cid:17) d x ( u ) du = (cid:90) Tt aB ( a, u, T ) d x ( u ) du. (cid:13) = σ (cid:90) Tt (cid:18)(cid:90) ut e as d (cid:102) W ( s ) (cid:19) ae − au du = σ (cid:90) Tt (cid:18)(cid:90) ut e as d (cid:102) W ( s ) (cid:19) d u (cid:18)(cid:90) ut e − av dv (cid:19) = σ (cid:34)(cid:32)(cid:90) Tt e au d (cid:102) W ( u ) (cid:33) (cid:32)(cid:90) Tt e − av dv (cid:33) − (cid:90) Tt (cid:18)(cid:90) ut e − av dv (cid:19) e au d (cid:102) W ( u ) (cid:35) = σ (cid:34)(cid:90) Tt (cid:32)(cid:90) Tu e − av dv (cid:33) e au d (cid:102) W ( u ) (cid:35) = σ (cid:90) Tt (cid:20) − e − av a (cid:21) Tu e au d (cid:102) W ( u )= σa (cid:90) Tt (cid:16) − e − a ( T − u ) (cid:17) d (cid:102) W ( u )= σa (cid:90) Tt (cid:16) − e − a ( T − u ) (cid:17) d (cid:102) W ( u ) . The corresponding expressions for (cid:82) Tt y ( u ) du can be obtained analogously. We ob-serve that the results of integral (cid:13) for (cid:82) Tt x ( u ) du and (cid:82) Tt y ( u ) du cancel out with thelast two terms in equation (E.1). Therefore it remains I ( t, T ) = (cid:90) Tt ϕ ( u ) du + 1 − e − a ( T − t ) a x ( t ) + 1 − e − b ( T − t ) b y ( t )+ σa (cid:90) Tt (cid:16) − e − a ( T − u ) (cid:17) d (cid:102) W ( u ) + ηb (cid:90) Tt (cid:16) − e − b ( T − u ) (cid:17) d (cid:102) W ( u ) . As (cid:102) W = ( (cid:102) W , (cid:102) W ) is a 2-dimensional Brownian motion under P , I ( t, T ) is normallydistributed and the mean and the variance can be easily retrieved resulting in (10) and(11). 26 ppendix F. Tables of Backtest Results Date a b σ η ρ . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . . . . . − . Table F.4: Calibration results of the risk neutral calibration on a quarterly basis from 31.12.2016 until30.09.2019
Date d x d y − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . Table F.5: Quarterly calibration results for the constant local long run risk premium functions from31.12.2016 to 30.09.2019. d x d y l x l y − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . − . . − . − . Table F.6: Quarterly calibration results for the step local long run risk premium functions from 31.12.2016to 30.09.2019.