diff --git a/ql/legacy/libormarketmodels/lfmcovarparam.hpp b/ql/legacy/libormarketmodels/lfmcovarparam.hpp
index 2367d26ccfd..f2f534e2a43 100644
--- a/ql/legacy/libormarketmodels/lfmcovarparam.hpp
+++ b/ql/legacy/libormarketmodels/lfmcovarparam.hpp
@@ -45,9 +45,9 @@ namespace QuantLib {
         Size size() const { return size_; }
         Size factors() const { return factors_; }
 
-        virtual Matrix diffusion(Time t, const Array& x = Null<Array>()) const = 0;
-        virtual Matrix covariance(Time t, const Array& x = Null<Array>()) const;
-        virtual Matrix integratedCovariance(Time t, const Array& x = Null<Array>()) const;
+        virtual Matrix diffusion(Time t, const Array& x = {}) const = 0;
+        virtual Matrix covariance(Time t, const Array& x = {}) const;
+        virtual Matrix integratedCovariance(Time t, const Array& x = {}) const;
 
       protected:
         const Size size_;
diff --git a/ql/legacy/libormarketmodels/lfmcovarproxy.hpp b/ql/legacy/libormarketmodels/lfmcovarproxy.hpp
index c4c6fd67d36..fc7438898bd 100644
--- a/ql/legacy/libormarketmodels/lfmcovarproxy.hpp
+++ b/ql/legacy/libormarketmodels/lfmcovarproxy.hpp
@@ -40,12 +40,11 @@ namespace QuantLib {
         ext::shared_ptr<LmVolatilityModel>  volatilityModel() const;
         ext::shared_ptr<LmCorrelationModel> correlationModel() const;
 
-        Matrix diffusion(Time t, const Array& x = Null<Array>()) const override;
-        Matrix covariance(Time t, const Array& x = Null<Array>()) const override;
+        Matrix diffusion(Time t, const Array& x = {}) const override;
+        Matrix covariance(Time t, const Array& x = {}) const override;
 
         using LfmCovarianceParameterization::integratedCovariance;
-        virtual Real integratedCovariance(
-            Size i, Size j, Time t, const Array& x = Null<Array>()) const;
+        virtual Real integratedCovariance(Size i, Size j, Time t, const Array& x = {}) const;
 
       protected:
         const ext::shared_ptr<LmVolatilityModel>  volaModel_;
diff --git a/ql/legacy/libormarketmodels/lfmhullwhiteparam.hpp b/ql/legacy/libormarketmodels/lfmhullwhiteparam.hpp
index d874ad55483..2256d4d11f8 100644
--- a/ql/legacy/libormarketmodels/lfmhullwhiteparam.hpp
+++ b/ql/legacy/libormarketmodels/lfmhullwhiteparam.hpp
@@ -45,9 +45,9 @@ namespace QuantLib {
             const ext::shared_ptr<OptionletVolatilityStructure> & capletVol,
             const Matrix& correlation = Matrix(), Size factors = 1);
 
-        Matrix diffusion(Time t, const Array& x = Null<Array>()) const override;
-        Matrix covariance(Time t, const Array& x = Null<Array>()) const override;
-        Matrix integratedCovariance(Time t, const Array& x = Null<Array>()) const override;
+        Matrix diffusion(Time t, const Array& x = {}) const override;
+        Matrix covariance(Time t, const Array& x = {}) const override;
+        Matrix integratedCovariance(Time t, const Array& x = {}) const override;
 
       protected:
         Size nextIndexReset(Time t) const;
diff --git a/ql/legacy/libormarketmodels/lmconstwrappercorrmodel.hpp b/ql/legacy/libormarketmodels/lmconstwrappercorrmodel.hpp
index 44a10999bc9..02a7244bedd 100644
--- a/ql/legacy/libormarketmodels/lmconstwrappercorrmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmconstwrappercorrmodel.hpp
@@ -38,13 +38,13 @@ namespace QuantLib {
 
         Size factors() const override { return corrModel_->factors(); }
 
-        Matrix correlation(Time t, const Array& x = Null<Array>()) const override {
+        Matrix correlation(Time t, const Array& x = {}) const override {
             return corrModel_->correlation(t, x);
         }
-        Matrix pseudoSqrt(Time t, const Array& x = Null<Array>()) const override {
+        Matrix pseudoSqrt(Time t, const Array& x = {}) const override {
             return corrModel_->pseudoSqrt(t, x);
         }
-        Real correlation(Size i, Size j, Time t, const Array& x = Null<Array>()) const override {
+        Real correlation(Size i, Size j, Time t, const Array& x = {}) const override {
             return corrModel_->correlation(i, j, t, x);
         }
         bool isTimeIndependent() const override { return corrModel_->isTimeIndependent(); }
diff --git a/ql/legacy/libormarketmodels/lmconstwrappervolmodel.hpp b/ql/legacy/libormarketmodels/lmconstwrappervolmodel.hpp
index 11c747d01e6..6280d769093 100644
--- a/ql/legacy/libormarketmodels/lmconstwrappervolmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmconstwrappervolmodel.hpp
@@ -37,15 +37,14 @@ namespace QuantLib {
           volaModel_(volaModel) {
         }
 
-        Array volatility(Time t, const Array& x = Null<Array>()) const override {
+        Array volatility(Time t, const Array& x = {}) const override {
             return volaModel_->volatility(t, x);
         }
         Volatility volatility(
-            Size i, Time t, const Array& x = Null<Array>()) {
+            Size i, Time t, const Array& x = {}) {
             return volaModel_->volatility(i, t, x);
         }
-        Real
-        integratedVariance(Size i, Size j, Time u, const Array& x = Null<Array>()) const override {
+        Real integratedVariance(Size i, Size j, Time u, const Array& x = {}) const override {
             return volaModel_->integratedVariance(i, j, u, x);
         }
 
diff --git a/ql/legacy/libormarketmodels/lmcorrmodel.hpp b/ql/legacy/libormarketmodels/lmcorrmodel.hpp
index 209ddc0840c..26973e31aa9 100644
--- a/ql/legacy/libormarketmodels/lmcorrmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmcorrmodel.hpp
@@ -43,9 +43,9 @@ namespace QuantLib {
         std::vector<Parameter>& params();
         void setParams(const std::vector<Parameter> & arguments);
 
-        virtual Matrix correlation(Time t, const Array& x = Null<Array>()) const = 0;
-        virtual Matrix pseudoSqrt(Time t, const Array& x = Null<Array>()) const;
-        virtual Real correlation(Size i, Size j, Time t, const Array& x = Null<Array>()) const;
+        virtual Matrix correlation(Time t, const Array& x = {}) const = 0;
+        virtual Matrix pseudoSqrt(Time t, const Array& x = {}) const;
+        virtual Real correlation(Size i, Size j, Time t, const Array& x = {}) const;
         virtual bool isTimeIndependent() const;
 
     protected:
diff --git a/ql/legacy/libormarketmodels/lmexpcorrmodel.hpp b/ql/legacy/libormarketmodels/lmexpcorrmodel.hpp
index bfd259c1fa1..3706f3859d8 100644
--- a/ql/legacy/libormarketmodels/lmexpcorrmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmexpcorrmodel.hpp
@@ -46,8 +46,8 @@ namespace QuantLib {
       public:
         LmExponentialCorrelationModel(Size size, Real rho);
 
-        Matrix correlation(Time t, const Array& x = Null<Array>()) const override;
-        Matrix pseudoSqrt(Time t, const Array& x = Null<Array>()) const override;
+        Matrix correlation(Time t, const Array& x = {}) const override;
+        Matrix pseudoSqrt(Time t, const Array& x = {}) const override;
         Real correlation(Size i, Size j, Time t, const Array& x) const override;
         bool isTimeIndependent() const override;
 
diff --git a/ql/legacy/libormarketmodels/lmextlinexpvolmodel.hpp b/ql/legacy/libormarketmodels/lmextlinexpvolmodel.hpp
index 61f653afd43..70eac4ab44a 100644
--- a/ql/legacy/libormarketmodels/lmextlinexpvolmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmextlinexpvolmodel.hpp
@@ -49,11 +49,10 @@ namespace QuantLib {
         LmExtLinearExponentialVolModel(const std::vector<Time>& fixingTimes,
                                        Real a, Real b, Real c, Real d);
 
-        Array volatility(Time t, const Array& x = Null<Array>()) const override;
-        Volatility volatility(Size i, Time t, const Array& x = Null<Array>()) const override;
+        Array volatility(Time t, const Array& x = {}) const override;
+        Volatility volatility(Size i, Time t, const Array& x = {}) const override;
 
-        Real
-        integratedVariance(Size i, Size j, Time u, const Array& x = Null<Array>()) const override;
+        Real integratedVariance(Size i, Size j, Time u, const Array& x = {}) const override;
     };
 
 }
diff --git a/ql/legacy/libormarketmodels/lmfixedvolmodel.hpp b/ql/legacy/libormarketmodels/lmfixedvolmodel.hpp
index 043cabb8d69..173b00a6e87 100644
--- a/ql/legacy/libormarketmodels/lmfixedvolmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmfixedvolmodel.hpp
@@ -32,7 +32,7 @@ namespace QuantLib {
       public:
         LmFixedVolatilityModel(Array volatilities, const std::vector<Time>& startTimes);
 
-        Array volatility(Time t, const Array& x = Null<Array>()) const override;
+        Array volatility(Time t, const Array& x = {}) const override;
         Volatility volatility(Size i, Time t, const Array& x) const override;
 
       private:
diff --git a/ql/legacy/libormarketmodels/lmlinexpcorrmodel.hpp b/ql/legacy/libormarketmodels/lmlinexpcorrmodel.hpp
index 6019462fcc1..b08272de844 100644
--- a/ql/legacy/libormarketmodels/lmlinexpcorrmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmlinexpcorrmodel.hpp
@@ -47,8 +47,8 @@ namespace QuantLib {
         LmLinearExponentialCorrelationModel(Size size, Real rho, Real beta,
                                             Size factors = Null<Size>());
 
-        Matrix correlation(Time t, const Array& x = Null<Array>()) const override;
-        Matrix pseudoSqrt(Time t, const Array& x = Null<Array>()) const override;
+        Matrix correlation(Time t, const Array& x = {}) const override;
+        Matrix pseudoSqrt(Time t, const Array& x = {}) const override;
         Real correlation(Size i, Size j, Time t, const Array& x) const override;
 
         Size factors() const override;
diff --git a/ql/legacy/libormarketmodels/lmlinexpvolmodel.hpp b/ql/legacy/libormarketmodels/lmlinexpvolmodel.hpp
index 7fcf0b618ce..d71d3eb53f4 100644
--- a/ql/legacy/libormarketmodels/lmlinexpvolmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmlinexpvolmodel.hpp
@@ -45,15 +45,13 @@ namespace QuantLib {
 
     class LmLinearExponentialVolatilityModel : public LmVolatilityModel {
       public:
-        LmLinearExponentialVolatilityModel(
-                                         const std::vector<Time>& fixingTimes,
-                                         Real a, Real b, Real c, Real d);
+        LmLinearExponentialVolatilityModel(const std::vector<Time>& fixingTimes,
+                                           Real a, Real b, Real c, Real d);
 
-        Array volatility(Time t, const Array& x = Null<Array>()) const override;
-        Volatility volatility(Size i, Time t, const Array& x = Null<Array>()) const override;
+        Array volatility(Time t, const Array& x = {}) const override;
+        Volatility volatility(Size i, Time t, const Array& x = {}) const override;
 
-        Real
-        integratedVariance(Size i, Size j, Time u, const Array& x = Null<Array>()) const override;
+        Real integratedVariance(Size i, Size j, Time u, const Array& x = {}) const override;
 
       private:
         void generateArguments() override;
diff --git a/ql/legacy/libormarketmodels/lmvolmodel.hpp b/ql/legacy/libormarketmodels/lmvolmodel.hpp
index d6108cd7bc6..3f36825ca44 100644
--- a/ql/legacy/libormarketmodels/lmvolmodel.hpp
+++ b/ql/legacy/libormarketmodels/lmvolmodel.hpp
@@ -40,10 +40,9 @@ namespace QuantLib {
         std::vector<Parameter> & params();
         void setParams(const std::vector<Parameter> & arguments);
 
-        virtual Array volatility(Time t, const Array& x = Null<Array>()) const = 0;
-        virtual Volatility volatility(Size i, Time t, const Array& x = Null<Array>()) const;
-        virtual Real integratedVariance(Size i, Size j, Time u,
-                                        const Array& x = Null<Array>()) const;
+        virtual Array volatility(Time t, const Array& x = {}) const = 0;
+        virtual Volatility volatility(Size i, Time t, const Array& x = {}) const;
+        virtual Real integratedVariance(Size i, Size j, Time u, const Array& x = {}) const;
 
       protected:
         const Size size_;
diff --git a/test-suite/libormarketmodelprocess.cpp b/test-suite/libormarketmodelprocess.cpp
index 03df6d510c7..63ccbceb4a0 100644
--- a/test-suite/libormarketmodelprocess.cpp
+++ b/test-suite/libormarketmodelprocess.cpp
@@ -155,7 +155,7 @@ BOOST_AUTO_TEST_CASE(testLambdaBootstrapping) {
 
     ext::shared_ptr<LiborForwardModelProcess> process = makeProcess();
 
-    Matrix covar = process->covariance(0.0, Null<Array>(), 1.0);
+    Matrix covar = process->covariance(0.0, {}, 1.0);
 
     for (Size i=0; i<9; ++i) {
         const Real calculated = std::sqrt(covar[i+1][i+1]);