Derivations of the pricing PDE for the Heston-Hull-White or Heston-CIR models

Question

Consider the hybrid model given by $$dS=(r-q) S dt + \sqrt{v} S dZ_1$$ $$dv = \kappa_v (\theta_v - v) dt + \sigma_v \sqrt{v} dZ_2$$ $$dr = \kappa_r (\theta_r - r) dt + \sigma_r r^p dZ_3$$

with correlation values $\rho_{S,v}$, $\rho_{S,r}$ and $\rho_{v,r}$, where $p=0$ for the Heston-Hull-White (HHW) and $p=1/2$ for the Heston-CIR (HCIR) models.

I am trying to find the derivation of the pricing PDE given in the paper

$$0 = \frac{\partial f}{\partial t} + \frac{1}{2} v S^2 \frac{\partial^2 f}{\partial S^2} +\frac{1}{2} \sigma_v^2 v \frac{\partial^2 f}{\partial v^2} +\frac{1}{2} \sigma_r^2 r^{2p} \frac{\partial^2 f}{\partial r^2} + \rho_{S,v} \sigma_v \sqrt{v} r^p S \frac{\partial^2 f}{\partial S \partial v} + \rho_{S,r} \sigma_r r S \frac{\partial^2 f}{\partial S \partial r} + \rho_{v,r} \sigma_v \sigma_r \sqrt{v} r^p \frac{\partial^2 f}{\partial v \partial r} + (r - q) S \frac{\partial f}{\partial S} + \kappa_v (\theta_v - v) \frac{\partial f}{\partial v} + \kappa_r (\theta_r - r) \frac{\partial f}{\partial r} -rf$$

This is equation 2.16 without the log-transform of $S$.

If there is another derivation for the HHW or the HCIR pricing PDE's (preferably in terms of a delta heding portfolio), please could you point me to such a paper. I can get the derivations for the Heston model pricing PDE easily, but not for the hybrid models. Maybe I am looking in the wrong places?

Answer 1

Let's consider the following dynamics under a risk-neutral measure:\begin{align} \text{d}S&=(r-q) S \text{d}t + \sqrt{v} S \text{d}Z_S,\\ \text{d}v &= \kappa_v (\theta_v - v) \text{d}t + \sigma_v \sqrt{v} \text{d}Z_v,\\ \text{d}r &= \kappa_r (\theta_r - r) \text{d}t + \sigma_r r^p \text{d}Z_r. \end{align} with correlations $\text{d}Z_S\text{d}Z_v=\rho_{Sv}\text{d}t$, $\text{d}Z_S\text{d}Z_r=\rho_{Sr}\text{d}t$ and $\text{d}Z_v\text{d}Z_r=\rho_{vr}\text{d}t$.

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Itô's Lemma suggests \begin{align} \text{d}f =& f_t\text{d}t + f_S\text{d}S+f_v\text{d}v+f_r\text{d}r+\frac{1}{2}f_{SS}(\text{d}S)^2+\frac{1}{2}f_{vv}(\text{d}v)^2+\frac{1}{2}f_{rr}(\text{d}r)^2 \\ &+ f_{Sv}\text{d}S\text{d}v+ f_{Sr}\text{d}S\text{d}r+ f_{vr}\text{d}v\text{d}r. \end{align}

Note that \begin{align} (\text{d}S)^2 &= vS^2\text{d}t, \\ (\text{d}v)^2 &= \sigma_v^2v\text{d}t,\\ (\text{d}r)^2 &= \sigma_r^2r^{2p}\text{d}t,\\ \text{d}S\text{d}v &= vS\sigma_v\rho_{Sv}\text{d}t,\\ \text{d}S\text{d}r &= \sqrt{v}S\sigma_rr^p\rho_{Sr}\text{d}t,\\ \text{d}v\text{d}r &= \sigma_v\sqrt{v}\sigma_rr^p\rho_{vr}\text{d}t. \end{align} Going back to Itô's Lemma yields \begin{align} \text{d}f =& f_t\text{d}t + (r-q) Sf_S \text{d}t +\kappa_v (\theta_v-v)f_v \text{d}t +\kappa_r (\theta_r - r)f_r \text{d}t \\ &+ \sqrt{v} Sf_S \text{d}Z_S+ \sigma_v \sqrt{v} f_v\text{d}Z_v + \sigma_r r^p f_r\text{d}Z_r\\ &+\frac{1}{2}f_{SS}vS^2\text{d}t+\frac{1}{2}f_{vv}\sigma_v^2v\text{d}t+\frac{1}{2}f_{rr}\sigma_r^2r^{2p}\text{d}t \\ &+ f_{Sv}vS\sigma_v\rho_{Sv}\text{d}t+ f_{Sr}\sqrt{v}S\sigma_rr^p\rho_{Sr}\text{d}t+ f_{vr}\sigma_v\sqrt{v}\sigma_rr^p\rho_{vr}\text{d}t. \end{align} Because of their martingale property, Itô integrals have zero expectation. Thus, \begin{align} \mathbb{E}^\mathbb{Q}[\text{d}f] =& f_t\text{d}t + (r-q) Sf_S \text{d}t +\kappa_v (\theta_v-v)f_v \text{d}t +\kappa_r (\theta_r - r)f_r \text{d}t \\ &+\frac{1}{2}f_{SS}vS^2\text{d}t+\frac{1}{2}f_{vv}\sigma_v^2v\text{d}t+\frac{1}{2}f_{rr}\sigma_r^2r^{2p}\text{d}t \\ &+ f_{Sv}vS\sigma_v\rho_{Sv}\text{d}t+ f_{Sr}\sqrt{v}S\sigma_rr^p\rho_{Sr}\text{d}t+ f_{vr}\sigma_v\sqrt{v}\sigma_rr^p\rho_{vr}\text{d}t. \end{align} Finally, due to the absence of arbitrage, we have $\mathbb{E}^\mathbb{Q}[\text{d}f]=rf\text{d}t$. Thus, the pricing PDE is \begin{align} f_t &+ (r-q) Sf_S+\kappa_v (\theta_v-v)f_v +\kappa_r (\theta_r - r)f_r \\ &+\frac{1}{2}f_{SS}vS^2+\frac{1}{2}f_{vv}\sigma_v^2v+\frac{1}{2}f_{rr}\sigma_r^2r^{2p} \\ &+ f_{Sv}vS\sigma_v\rho_{Sv}+ f_{Sr}\sqrt{v}S\sigma_rr^p\rho_{Sr}+ f_{vr}\sigma_v\sqrt{v}\sigma_rr^p\rho_{vr}-rf=0. \end{align}

Setting $x=\ln(S)$, we have $\text{d}S=S\text{d}x$ which gives $Sf_S=f_x$ and $S^2f_{SS}=f_{xx}-f_x$. Thus, the pricing PDE turns into \begin{align} f_t &+ \left(r-q-\frac{1}{2}v\right) f_x+\kappa_v (\theta_v-v)f_v +\kappa_r (\theta_r - r)f_r \\ &+\frac{1}{2}vf_{xx}+\frac{1}{2}\sigma_v^2vf_{vv}+\frac{1}{2}\sigma_r^2r^{2p}f_{rr} \\ &+ \rho_{Sv}\sigma_vvf_{xv}+ \rho_{Sr}\sigma_r\sqrt{v}r^pf_{xr}+ \rho_{vr}\sigma_v\sigma_r\sqrt{v}r^pf_{vr}-rf=0. \end{align}

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Regarding a hedging argument: You can't use delta hedging. You have three sources of uncertainty and thus you need three tradable assets to hedge the risks. Only using the stock to hedge delta is not sufficient. If you have enough tradable assets, you can set up a portfolio $\Pi$ and eliminate all risks such that $\text{d}\Pi=...\text{d}t$ (without any $\text{d}Z$ terms). Then, you can equate $\text{d}\Pi=r\Pi\text{d}t$ and get a PDE. Alternatively, assuming that your SDEs are given under a risk-neutral measure, you can directly take an expectation of $\text{d}f$ and thereby removing the $\text{d}Z$ terms. Equating this conditional expectation with $rf\text{d}t$ then gives you the pricing PDE. Both approaches are equivalent in the end.