A realized volatility approach to option pricing with continuous and jump variance components

Abstract

Stochastic and time-varying volatility models typically fail to correctly price out-of-the-money put options at short maturity. We extend realized volatility option pricing models by adding a jump component which provides a rapidly moving volatility factor and improves on the fitting properties under the physical measure. The change of measure is performed by means of an exponentially affine pricing kernel which depends on an equity and two variance risk premia, associated with the continuous and jump components of realized volatility. Our choice preserves analytical tractability and offers a new way of estimating variance risk premia by combining high-frequency returns and option data in a multicomponent pricing model.

Appendices

A MGF under $\varvec{\mathbb {P}}$ measure

The relations which follow are derived for the log-return dynamics specified in Eq. (2). For the ease of computation, expression (4) is rewritten as

$$\begin{aligned} \varTheta ({\mathbf{RV}}^{{\varvec{c}}}_{t},{\mathbf{L}}_{t})=d+\sum \limits _{i=1}^{22}\beta _i\mathrm {RV}_{t+1-i}^c+\sum \limits _{i=1}^{22}\alpha _i\left( \epsilon _{t+1-i}-\gamma \sqrt{\mathrm {RV}^c_{t+1-i}+\mathrm {RV}^j_{t+1-i}}\right) ^2\,, \end{aligned}$$

with

$$\begin{aligned} \beta _i= {\left\{ \begin{array}{ll} \beta _d &{}\text {for}\,i=1\\ \beta _w/4 &{}\text {for}\,2\le i\le 5\\ \beta _w/17 &{}\text {for}\,6\le i\le 22\\ \end{array}\right. } \alpha _i= {\left\{ \begin{array}{ll} \alpha _d &{}\text {for}\,i=1\\ \alpha _w/4 &{}\text {for}\,2\le i\le 5\\ \alpha _w/17 &{}\text {for}\,6\le i\le 22\\ \end{array}\right. }. \end{aligned}$$

(8)

We start showing that JLHARG processes satisfy the affine relation

$$\begin{aligned} {\mathbb {E}}\left[ e^{zy_{s+1}+{\mathbf {b}}\cdot {\mathbf {RV}}_{s+1}+c\ell _{s+1}}|{\mathcal {F}}_s\right] =e^{{\mathcal {A}}\left( z,{\mathbf {b}},c\right) +\sum \limits _{i=1}^p{\varvec{{\mathcal {B}}}}_i\left( z,{\mathbf {b}},c\right) \cdot {\mathbf {RV}}_{s+1-i}+\sum \limits _{j=1}^q{\mathcal {C}}_j\left( z,{\mathbf {b}},c\right) \ell _{s+1-j}}\,, \end{aligned}$$

(9)

for some functions ${\mathcal {A}}\,:\,{\mathbb {R}}\times {\mathbb {R}}^2\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, ${\varvec{{\mathcal {B}}}}_i\,:\,{\mathbb {R}}\times {\mathbb {R}}^2\times {\mathbb {R}}\rightarrow {\mathbb {R}}^2$, ${\varvec{{\mathcal {C}}}}_j\,:\,{\mathbb {R}}\times {\mathbb {R}}^2\times {\mathbb {R}}\rightarrow {\mathbb {R}}$, where ${\mathbf {RV}}_t=(\mathrm {RV}_t^c,\mathrm {RV}_t^j)$, ${\mathbf {b}}\in {\mathbb {R}}^2$, $c\in {\mathbb {R}}$, and $\cdot$ is the scalar product in ${\mathbb {R}}^2$. To derive the explicit form of the functions ${\mathcal {A}},\,{\mathcal {B}}_i,\,{\mathcal {C}}_j$ which allows to characterize the MGF we show that

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{zy_{t}+{\mathbf {b}}\cdot {\mathbf {RV}}_t+c\ell _{t}}|{\mathcal {F}}_{t-1}\right] \nonumber \\&\quad ={\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{z(r+\lambda _c\mathrm {RV}^c_{t}+\lambda _j\mathrm {RV}^j_{t}+\sqrt{\mathrm {RV}^c_{t}+\mathrm {RV}^j_{t}}\epsilon _{t})+{\mathbf {b}}\cdot {\mathbf {RV}}_t+c\ell _{t}}|{\mathcal {F}}_{t-1}\right] \nonumber \\&\quad ={\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{z(r+\lambda _c\mathrm {RV}^c_{t}+\lambda _j\mathrm {RV}^j_{t})+{\mathbf {b}}\cdot {\mathbf {RV}}_t} {\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{z\sqrt{\mathrm {RV}^c_{t}+\mathrm {RV}^j_{t}}\epsilon _{t}+c(\epsilon _{t}-\gamma \sqrt{\mathrm {RV}^c_{t}+\mathrm {RV}^j_{t}})^2}|{\mathbf {RV}}_t\right] |{\mathcal {F}}_{t-1}\right] \nonumber \\&\quad ={\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{z(r+\lambda _c\mathrm {RV}^c_{t}+\lambda _j\mathrm {RV}^j_{t})+b_1\mathrm {RV}_t^c+b_2\mathrm {RV}_t^j-\frac{1}{2}\ln (1-2c)+\left( \frac{\frac{z^2}{2}+\gamma ^2c-2c\gamma z}{1-2c}\right) (\mathrm {RV}^c_{t}+\mathrm {RV}^j_{t})}|{\mathcal {F}}_{t-1}\right] \nonumber \\&\quad ={\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{zr-\frac{1}{2}\ln (1-2c)+\left( z\lambda _c+b_1+\frac{\frac{z^2}{2}+\gamma ^2c-2c\gamma z}{1-2c}\right) \mathrm {RV}^{c}_{t}+\left( z\lambda _j+b_2+\frac{\frac{z^2}{2}+\gamma ^2c-2c\gamma z}{1-2c}\right) \mathrm {RV}^j_{t}}|{\mathcal {F}}_{t-1}\right] \nonumber \\&\quad =\mathrm {e}^{zr-\frac{1}{2}\ln (1-2c)}{\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{\left( z\lambda _c+b_1+\frac{\frac{z^2}{2}+\gamma ^2c-2c\gamma z}{1-2c}\right) \mathrm {RV}^{c}_{t}}|{\mathcal {F}}_{t-1}\right] \nonumber \\&\qquad {\mathbb {E}}^{{\mathbb {P}}}\left[ \mathrm {e}^{\left( z\lambda _j+b_2+\frac{\frac{z^2}{2}+\gamma ^2c-2c\gamma z}{1-2c}\right) \mathrm {RV}^j_{t}}|{\mathcal {F}}_{t-1}\right] . \end{aligned}$$

(10)

In the third line we have used the result that if $Z\sim {\mathcal {N}}(0,1)$ then

$$\begin{aligned} {\mathbb {E}}\left[ \exp \left( x\left( Z+y\right) ^2\right) \right] =\exp \left( -\frac{1}{2}\ln \left( 1-2x\right) +\frac{xy^2}{1-2x}\right) . \end{aligned}$$

For a noncentered gamma random variable, from Gourieroux and Jasiak (2006) we know that

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {P}}}\left[ e^{x_1\mathrm {RV}^{c}_{t}}|{\mathcal {F}}_{t-1}\right] =\exp \left( -\delta {\mathcal {W}}\left( x_1,\theta \right) +{\mathcal {V}}\left( x_1,\theta \right) \left( d+\sum \limits _{i=1}^{p}\beta _i\mathrm {RV}_{s-i}^c+\sum \limits _{j=1}^{q}\alpha _j\ell _{s-j}\right) \right) \,, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {V}}(x_1,\theta )=\frac{\theta x_1}{1-\theta x_1}, \;\;\; {\mathcal {W}}(x_1,\theta )=\ln (1-x_1\theta ), \end{aligned}$$

and

$$\begin{aligned} x_1(z,b_1,c)=z\lambda _c+b_1+\frac{\frac{1}{2}z^2+\gamma ^2c-2c\gamma z}{1-2c}\,. \end{aligned}$$

(11)

For the computation of the last expectation in the final line of (10), we use the property that if $Z_t$ is a compound Poisson process with rate $\omega$ and i.i.d. jump sizes $D_i$, then

$$\begin{aligned} {\mathbb {E}}\left[ e^{xZ_t}|{\mathcal {F}}_{t-1}\right] =\exp \left( \omega \left( M_D(x)-1\right) \right) , \end{aligned}$$

(12)

where $M_D(x)$ is the MGF of the jump size random variable D. Since sizes are distributed according to a gamma distribution, we have

$$\begin{aligned} M_D(x)=\frac{1}{\left( 1-x{\tilde{\theta }}\right) ^{{\tilde{\delta }}} }\,. \end{aligned}$$

(13)

From expressions (12) and (13) we obtain

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {P}}}\left[ e^{x_2 \mathrm {RV}^{j}_{t}}|{\mathcal {F}}_{t-1}\right] =\exp \left( {\tilde{\varTheta }}{\mathcal {J}}\left( x_2,{\tilde{\theta }},{\tilde{\delta }}\right) \right) , \end{aligned}$$

where

$$\begin{aligned} {\mathcal {J}}(x_2,{\tilde{\theta }},{\tilde{\delta }})=\frac{1-(1-{\tilde{\theta }}x_2)^{{\tilde{\delta }}}}{(1-{\tilde{\theta }}x)^{{\tilde{\delta }}}} \ \ \text{ and } \ \ x_2(z,b_2,c)=z\lambda _j+b_2+\frac{\frac{1}{2}z^2+\gamma ^2c-2c\gamma z}{1-2c}\,. \end{aligned}$$

Gathering all the previous results, we finally conclude

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}^{{\mathbb {P}}}\left[ e^{zy_{t}+{\mathbf {b}}\cdot {\mathbf {RV}}_t+c\ell _{t}}|{\mathcal {F}}_{t-1}\right] \\&\quad =\exp \left[ zr-\frac{1}{2}\ln (1-2c)+{\mathcal {V}}(x_1,\theta )\left( d+\sum \limits _{i=1}^{p}\beta _i\mathrm {RV}_{t-i}^c+\sum \limits _{j=1}^{q}\alpha _j\ell _{t-j}\right) \right. \\&\qquad \left. -\delta {\mathcal {W}}(x_1,\theta )+{\tilde{\varTheta }}{\mathcal {J}}(x_2,{\tilde{\theta }},{\tilde{\delta }}) \phantom {{\mathcal {V}}(x_1,\theta )\left( d+\sum \limits _{i=1}^{p}\beta _i\mathrm {RV}_{t-i}^c+\sum \limits _{j=1}^{q}\alpha _j\ell _{t-j}\right) }\right] \,. \end{aligned} \end{aligned}$$

The direct comparison of the last expression with (9) allows to derive the following explicit expressions

$$\begin{aligned}&{\mathcal {A}}(z,{\mathbf {b}},c)=zr-\frac{1}{2}\ln (1-2c)-\delta {\mathcal {W}}(x_1,\theta )+d{\mathcal {V}}(x_1,\theta )+{\tilde{\varTheta }}{\mathcal {J}}(x_2,{\tilde{\theta }},{\tilde{\delta }}) \,, \end{aligned}$$

(14)

$$\begin{aligned}&{\mathcal {B}}_i(z,b_1,c)={\mathcal {V}}(x_1,\theta )\beta _i \,, \end{aligned}$$

(15)

$$\begin{aligned}&{\mathcal {C}}_j(z,b_1,c)={\mathcal {V}}(x_1,\theta )\alpha _j \,. \end{aligned}$$

(16)

As shown in Majewski et al. (2015), once we have above expressions we obtain

$$\begin{aligned} \phi ^{{\mathbb {P}}}\left( t,T,z\right) ={\mathbb {E}}^{{\mathbb {P}}}\left[ e^{zy_{t,T}}|{\mathcal {F}}_t\right] =\exp \left( \mathrm {a}_t+\sum \limits _{i=1}^{p}\mathrm {b}_{t,i}\mathrm {RV}_{t+1-i}^c+\sum \limits _{i=1}^{q}\mathrm {c}_{t,i}\ell _{t+1-i}\right) \end{aligned}$$

where

$$\begin{aligned}&\mathrm {a}_s=\mathrm {a}_{s+1}+zr-\frac{1}{2}\log (1-2\mathrm {c}_{s+1,1})+d{\mathcal {V}}(\mathrm {x}_{s+1}^{c},\theta )-\delta {\mathcal {W}}(\mathrm {x}_{s+1}^{c},\theta )+{\tilde{\varTheta }}{\mathcal {J}}(\mathrm {x}_{s+1}^{j},{\tilde{\theta }}) \nonumber \\&\mathrm {b}_{s,i}= {\left\{ \begin{array}{ll} \mathrm {b}_{s+1,i}+{\mathcal {V}}(\mathrm {x}_{s+1}^{c},\theta )\beta _i &{} \text {for} \, 1\le i \le p-1\\ {\mathcal {V}}(\mathrm {x}_{s+1}^{c},\theta )\beta _i &{}\text {for} \, i=p \end{array}\right. } \nonumber \\&\mathrm {c}_{s,i}= {\left\{ \begin{array}{ll} \mathrm {c}_{s+1,i}+{\mathcal {V}}(\mathrm {x}_{s+1}^{c},\theta )\alpha _i &{}\text {for} \, 1\le i \le q-1\\ {\mathcal {V}}(\mathrm {x}_{s+1}^{c},\theta )\alpha _i &{}\text {for} \, i=q \end{array}\right. } \end{aligned}$$

(17)

with

$$\begin{aligned}&\mathrm {x}_{s+1}^{c}=z\lambda _c+\mathrm {b}_{s+1,1}+\frac{\frac{1}{2}z^2+\gamma ^2\mathrm {c}_{s+1,1}-2\mathrm {c}_{s+1,1}\gamma z}{1-2\mathrm {c}_{s+1,1}}\,, \end{aligned}$$

(18)

$$\begin{aligned}&\mathrm {x}_{s+1}^{j}=z\lambda _j+\frac{\frac{1}{2}z^2+\gamma ^2\mathrm {c}_{s+1,1}-2\mathrm {c}_{s+1,1}\gamma z}{1-2\mathrm {c}_{s+1,1}}\,. \end{aligned}$$

(19)

The functions ${\mathcal {V}}$, ${\mathcal {W}}$ and ${\mathcal {J}}$ are defined as above. The terminal conditions read $\mathrm {a}_T=\mathrm {b}_{T,i}=\mathrm {c}_{T,j}=0$ for $i=1,2,...,p$ and $j=1,2,...,q$.

B No-arbitrage condition

The no-arbitrage conditions are

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {P}}}\left[ M_{s,s+1}|{\mathcal {F}}_s\right]&=1\; \text {for}\; s\in {\mathbb {N}}\,,\nonumber \\ {\mathbb {E}}^{{\mathbb {P}}}\left[ M_{s,s+1}e^{y_{s+1}}|{\mathcal {F}}_s\right]&=\mathrm {e}^r\; \text {for}\; s\in {\mathbb {N}} . \end{aligned}$$

(20)

The first relation is satisfied by definition of $M_{s,s+1}$. From a general result in Majewski et al. (2015), condition (20) is satisfied if, and only if

$$\begin{aligned} \begin{aligned}&{\mathcal {A}} (1-\nu _y,-{\varvec{\nu }},0)=r+{\mathcal {A}} (-\nu _y,-{\varvec{\nu }},0)\,,\\&{\mathcal {B}}_i(1-\nu _y,-{\varvec{\nu }},0)= {\mathcal {B}}_i(-\nu _y,-{\varvec{\nu }},0)\,,\\&{\mathcal {C}}_j(1-\nu _y,-{\varvec{\nu }},0)= {\mathcal {C}}_j(-\nu _y,-{\varvec{\nu }},0)\,, \end{aligned} \end{aligned}$$

with ${\varvec{\nu }}=(\nu _c,\nu _j)$. To conclude, it is sufficient to show under which conditions the following two relations hold true

$$\begin{aligned} x_1(1-\nu _y,-\nu _c,0)=x_1(-\nu _y,-\nu _c,0)\,,\\ x_2(1-\nu _y,-\nu _j,0)=x_2(-\nu _y,-\nu _j,0)\,. \end{aligned}$$

Simple computations show that the latter equations are satisfied if and only if

$$\begin{aligned} \nu _y=\lambda _c +\frac{1}{2} = \lambda _j +\frac{1}{2}. \end{aligned}$$

Remarkably, the only specification for the log-return dynamics in Eq. (2) which ensures consistency with no-arbitrage is the dynamics where the equity premia $\lambda _c$ and $\lambda _j$ are equal and coincide to $\lambda$. Then, we obtain

$$\begin{aligned} \nu _y = \lambda + \frac{1}{2}\,. \end{aligned}$$

It is important to notice that the no-arbitrage condition for the equity premium does not constrain the value of the variance risk premia $\nu _c$ and $\nu _j$.

C Risk-neutral dynamics

JLHARG models imply a risk-neutral MGF for log-returns whose exponential affine terms can be re-parameterized in order to obtain an expression formally equivalent to the physical MGF. Firstly we observe that the risk-neutral MGF can be expressed with a recursive set of expressions, involving a combination of the functions ${\mathcal {A}},\,{\mathcal {B}}_i,\,{\mathcal {C}}_j$. Then, recalling the results given in Majewski et al. (2015), the MGF for JLHARG model under measure ${\mathbb {Q}}$ has the following form

$$\begin{aligned} \phi ^{{\mathbb {Q}}}_{\nu _c\,\nu _j\,\nu _y}\left( t,T,z\right) ={\mathbb {E}}^{{\mathbb {Q}}}\left[ e^{zy_{t,T}}|{\mathcal {F}}_t\right] =\exp \left( \mathrm {a}^*_t+\sum \limits _{i=1}^{p}\mathrm {b}_{t,i}^{*}\mathrm {RV}_{t+1-i}^c+\sum \limits _{i=1}^{q}\mathrm {c}^*_{t,i}\ell _{t+1-i}\right) \,, \end{aligned}$$

where

$$\begin{aligned}&\mathrm {a}^*_s=\mathrm {a}^*_{s+1}+zr-\frac{1}{2}\log (1-2\mathrm {c}^*_{s+1,1})+d{\mathcal {V}}(\mathrm {x}_{s+1}^{c\,*},\theta )-d{\mathcal {V}}(\mathrm {y}_{s+1}^{c\,*},\theta ) \nonumber \\&\qquad -\delta {\mathcal {W}}(\mathrm {x}_{s+1}^{c\,*},\theta )+\delta {\mathcal {W}}(\mathrm {y}_{s+1}^{c\,*},\theta )+{\tilde{\varTheta }}{\mathcal {J}}(\mathrm {x}_{s+1}^{j\,*},{\tilde{\theta }})-{\tilde{\varTheta }}{\mathcal {J}}(\mathrm {y}_{s+1}^{j\,*},{\tilde{\theta }}) \nonumber \\&\mathrm {b}_{s,i}^{*}= {\left\{ \begin{array}{ll} \mathrm {b}_{s+1,i}^{*}+\left( {\mathcal {V}}(\mathrm {x}_{s+1}^{c\,*},\theta )-{\mathcal {V}}(\mathrm {y}_{s+1}^{c\,*},\theta )\right) \beta _i &{}\text {for} \, 1\le i \le p-1\\ \left( {\mathcal {V}}(\mathrm {x}_{s+1}^{c\,*},\theta )-{\mathcal {V}}(\mathrm {y}_{s+1}^{c\,*},\theta )\right) \beta _i &{}\text {for} \, i=p \end{array}\right. } \nonumber \\&\mathrm {c}^*_{s,j}= {\left\{ \begin{array}{ll} \mathrm {c}_{s+1,j}^{*}+\left( {\mathcal {V}}(\mathrm {x}_{s+1}^{c\,*},\theta )-{\mathcal {V}}(\mathrm {y}_{s+1}^{c\,*},\theta )\right) \alpha _j &{}\text {for} \, 1\le j \le q-1\\ \left( {\mathcal {V}}(\mathrm {x}_{s+1}^{c\,*},\theta )-{\mathcal {V}}(\mathrm {y}_{s+1}^{c\,*},\theta )\right) \alpha _j &{}\text {for} \, j=q \end{array}\right. } \end{aligned}$$

(21)

where

$$\begin{aligned}&x^{c\,*}_{s+1}=(z-\nu _y)\lambda +\mathrm {b}_{s+1,1}^{*}-\nu _c+\frac{\frac{1}{2}(z-\nu _y)^2+\gamma ^2 \mathrm {c}^*_{s+1,1}-2\mathrm {c}^*_{s+1,1}\gamma (z-\nu _y)}{1-2c^*_{s+1,1}}\\&x^{j\,*}_{s+1}=(z-\nu _y)\lambda -\nu _j+\frac{\frac{1}{2}(z-\nu _y)^2+\gamma ^2 \mathrm {c}^*_{s+1,1}-2\mathrm {c}^*_{s+1,1}\gamma (z-\nu _y)}{1-2c^*_{s+1,1}}\\&y^{l\,*}_{s+1}=-\nu _y\lambda -\nu _l+\frac{1}{2}\nu _y^2, \end{aligned}$$

with $l=c,j$ and the terminal conditions are $\mathrm {a}^*_T=\mathrm {b}_{T,i}^{*}=\mathrm {c}^*_{T,j}=0$ for $i=1,2,...,p$ and $j=1,2,...,q$.

The first passage consists in comparing expression (21) with (17). We have to find a set of new parameters for which the recursive expressions for $\mathrm {a}^*_t,\mathrm {b}^{*}_t,\mathrm {c}^*_t$ under ${\mathbb {Q}}$ correspond to the expressions under ${\mathbb {P}}$. We start defining

$$\begin{aligned}&x^{c\,**}_{s+1,i}=z\lambda ^*+b^{*}_{s+1,1}+\frac{\frac{1}{2}z^2+(\gamma ^*)^2 c^*_{s+1,1}-2c^*_{s+1,1}\gamma ^* z}{1-2c^*_{s+1,1}}\,,\\&x^{j\,**}_{s+1,i}=z\lambda ^*+\frac{\frac{1}{2}z^2+(\gamma ^*)^2 c^*_{s+1,1}-2c^*_{s+1,1}\gamma ^* z}{1-2c^*_{s+1,1}}\,. \end{aligned}$$

Then, the following relations have to hold

$$\begin{aligned} \delta \left( {\mathcal {W}}\left( \mathrm {x}_{s+1}^{c\,*},\theta \right) -{\mathcal {W}}\left( \mathrm {y}^{c\,*},\theta \right) \right)&=\delta ^*{\mathcal {W}}\left( \mathrm {x}_{s+1}^{c\,**},\theta ^*\right) \end{aligned}$$

(22)

$$\begin{aligned} \beta _i\left( {\mathcal {V}}\left( \mathrm {x}_{s+1}^{c\,*},\theta \right) -{\mathcal {V}}\left( \mathrm {y}^{c\,*},\theta \right) \right)&=\beta ^*_i{\mathcal {V}}\left( \mathrm {x}_{s+1}^{c\,**},\theta ^*\right) \end{aligned}$$

(23)

$$\begin{aligned} \alpha _j\left( {\mathcal {V}}\left( \mathrm {x}_{s+1}^{c\,*},\theta \right) -{\mathcal {V}}\left( \mathrm {y}^{c\,*},\theta \right) \right)&=\alpha ^*_j{\mathcal {V}}\left( \mathrm {x}_{s+1}^{c\,**},\theta ^*\right) \end{aligned}$$

(24)

$$\begin{aligned} {\tilde{\varTheta }}\left( {\mathcal {J}}\left( \mathrm {x}_{s+1}^{j\,*},{\tilde{\theta }}\right) -{\mathcal {J}}\left( \mathrm {y}^{j\,*},{\tilde{\theta }}\right) \right)&={\tilde{\varTheta }}^*{\mathcal {J}}\left( \mathrm {x}_{s+1}^{j\,**},{\tilde{\theta }}^*\right) \end{aligned}$$

(25)

with $\mathrm {y}^{c\,*}=-\lambda ^2/2-\nu _c+\frac{1}{8}$ and $\mathrm {y}^{j\,*}=-\lambda ^2/2-\nu _j+\frac{1}{8}$.

Equation (22) can be explicitly written as

$$\begin{aligned} \delta \log \left[ 1-\frac{\theta }{1-\theta \mathrm {y}^{c\,*}}\left( \mathrm {x}_{s+1}^{c\,*}-\mathrm {y}^{c\,*}\right) \right] =\delta ^*\log \left( 1-\theta ^*\mathrm {x}_{s+1}^{c\,**}\right) , \end{aligned}$$

which implies the following three sufficient conditions

$$\begin{aligned} \delta ^*&=\delta \nonumber \\ \theta ^*&=\frac{\theta }{1-\theta \mathrm {y}^{c\,*}} \nonumber \\ \mathrm {x}_{s+1}^{c\,**}&=\mathrm {x}_{s+1}^{c\,*}-\mathrm {y}^{c\,*} . \end{aligned}$$

(26)

It can be easily verified that last condition (26) is satisfied by substituting

$$\begin{aligned} \lambda _c^*&=-\frac{1}{2} \,,\\ \gamma ^*&=\gamma +\lambda +\frac{1}{2} . \end{aligned}$$

Equation (23) can be equivalently expressed in the form

$$\begin{aligned} \frac{\beta _i}{1-\theta \mathrm {y}^{c\,*}}\frac{\theta }{1-\theta \mathrm {y}^{c\,*}}\frac{\mathrm {x}_{s+1}^{c\,*}-\mathrm {y}^{c\,*}}{1-\theta /(1-\theta \mathrm {y}^{c\,*})\left( \mathrm {x}_{s+1}^{c\,*}-\mathrm {y}^{c\,*}\right) }=\beta _i^*\frac{\theta ^*\mathrm {x}_{s+1}^{c\,**}}{1-\theta ^*\mathrm {x}_{s+1}^{c\,**}} \end{aligned}$$

which gives another sufficient condition for the mapping

$$\begin{aligned} \beta _i^*=\frac{\beta _i}{1-\theta \mathrm {y}^{c\,*}}\,. \end{aligned}$$

An analogous consideration about third condition (24) allows to obtain the condition on $\alpha _i^*$,

$$\begin{aligned} \alpha _i^*=\frac{\alpha _i}{1-\theta \mathrm {y}^{c\,*}}\,. \end{aligned}$$

Relation (8) gives us the expressions for $\beta _d^*$, $\beta _w^*$, $\beta _m^*$, $\alpha _d^*$, $\alpha _w^*$ and $\alpha _m^*$. Finally, equation (25) provides the last sufficient condition

$$\begin{aligned} \frac{{\tilde{\varTheta }}}{\left( 1-{\tilde{\theta }}\mathrm {y}^{j\,*}\right) ^{{\tilde{\delta }}}}\frac{1-\left( \left( 1-{\tilde{\theta }}\mathrm {x}^{j\,*}_{s+1}\right) /\left( 1-{\tilde{\theta }}\mathrm {y}^{j\,*}\right) \right) ^{{\tilde{\delta }}}}{\left( \left( 1-{\tilde{\theta }}\mathrm {x}^{j\,*}_{s+1}\right) /\left( 1-{\tilde{\theta }}\mathrm {y}^{j\,*}\right) \right) ^{{\tilde{\delta }}}}={\tilde{\varTheta }}^*\frac{1-(1-{\tilde{\theta }}^*\mathrm {x}_{s+1}^{j\,**})^{{\tilde{\delta }}^*}}{(1-{\tilde{\theta }}^*\mathrm {x}_{s+1}^{j\,**})^{{\tilde{\delta }}^*}}, \end{aligned}$$

which is satisfied if

$$\begin{aligned} {\tilde{\delta }}^*&={\tilde{\delta }}\,,\nonumber \\ {\tilde{\varTheta }}^*&=\frac{{\tilde{\varTheta }}}{\left( 1-{\tilde{\theta }}\mathrm {y}^{j\,*}\right) ^{{\tilde{\delta }}}}\,,\nonumber \\ {\tilde{\theta }}^*&=\frac{{\tilde{\theta }}}{1-{\tilde{\theta }}\mathrm {y}^{j\,*}}\,,\nonumber \\ \mathrm {x}_{s+1}^{j\,**}&=\mathrm {x}_{s+1}^{j\,*}-\mathrm {y}^{j\,*} \,. \end{aligned}$$

(27)

As it can be seen last condition (27) is redundant when compared to condition (26).