Quanto Drift Adjustments for Heston Equity Models

# Quanto pricing explanation


# Quanto pricing explanation












I have paths generated from Heston, correlation Eq/FX, FX ATM vol but then I'm struggling to find the correct methodology.

I tried to adjust the dividend in asset paths from my Heston Monte Carlo by q'=q + rho.sigmaFx.sigmaEquity but the price of my option barely moves - which seams fair because 1e-2 < adjustment < 1e-3 and therefore q is dominant - (in my test, quanto in EUR, stock US) so I guess this is not how things should be done. Should I change my discount rate ? For the moment I'm just searching for the Spot adjustment, I'll see later for the vol. Any help would be appreciated.

Edit: additional question

Do I have to correlate the dynamics of Equity and FX (using Black scholars model for the FX) or just apply the adjustment above to the asset dividend and thus using Monte Carlo only for the asset ? I hope this is clear.

## Answer by Quantuple (score 2, accepted)

https://quant.stackexchange.com/a/40530

Assuming the FX spot exchange rate follows a GBM, under the domestic risk-neutral measure $\Bbb{Q}_{\text{DOM}}$ the Heston dynamics of an equity underlying denominated in the foreign currency $\text{FOR}$ should read: \begin{align} \frac{dS_t}{S_t} = \color{blue}{\tilde{\mu}_t} dt + \sqrt{v_t} dW_S(t),\,\,\, S(0) = S_0 \\ dv_t = \kappa(\color{blue}{\tilde{\theta}}-v_t)dt + \xi \sqrt{v_t} dW_v(t),\,\,\ v(0) = v_0 \\ \frac{dX_t}{X_t} = (r_t^d - r_t^f) dt + \sigma_X dW_X(t),\,\,\ X(0) = X_0 \end{align}

$$ d\langle W_S, W_v\rangle_t = \rho_{S,v} dt,\,\,\, d\langle W_S, W_X \rangle_t = \rho_{S,X} dt,\,\,\, d\langle W_v, W_X \rangle_t = \rho_{v,X} dt$$

In the above, $X_t$ represents the $\text{FOR/DOM}$ exchange rate (i.e. 1 unit of foreign currency equals X units of domestic currency at time $t$), with in your particular case $\text{FOR}$=USD, $\text{DOM}$=EUR.

As such $\rho_{S,X}$ then represents the correlation between the equity underlying $S$ and $X$ the $\text{FOR/DOM}$ exchange rate, which is the opposite of that of the $\text{DOM/FOR}$ rate, so make sure you have this right.

The quanto drift adjustments on the other hand read $$\color{blue}{\tilde{\mu}_t} = \mu_t - \rho_{S,X} \sigma_{X} \sqrt{v_t}$$ $$\color{blue}{\tilde{\theta}} = \theta - \frac{\rho_{v,X} \sigma_X \xi \sqrt{v_t} }{\kappa } $$

So back to your original question and writing $\mu_t = r^f_t - q_t$ you could indeed keep the same money market rates and adjust the "dividend yield" by writing $\tilde{\mu}_t = r^f_t - \tilde{q}_t$ with $$ \tilde{q}_t = q_t + \rho_{S,X} \sigma_X \sqrt{v_t} $$ but note how this adjustment is stochastic.

[Additional info]

Applying Itô's lemma to the SDE describing the evolution of the equity spot price under the quanto measure one gets $$ d\ln(S_t) = \left( \tilde{\mu}_t - \frac{1}{2}v_t \right) dt + \sqrt{v_t} dW_S(t) $$ Integrating over $[0,t]$ then yields \begin{align} S_t &= S_0 \exp\left( \int_0^t \tilde{\mu}_u du \right) \underbrace{ \exp \left( \int_0^t \sqrt{v_u} dW_S(u) - \frac{1}{2} \int_0^t v_u du \right)}_{ := \mathcal{E}\left( \int_0^t \sqrt{v_u} dW_S(u) \right) } \\ &= \underbrace{S_0 \exp\left( \int_0^t \mu_u du\right)}_{ := F^f(0,t)} \exp\left(-\int_0^t \rho_{S,x} \sigma_X \sqrt{v_u} du\right) \mathcal{E}\left( \int_0^t \sqrt{v_u} dW_S(u) \right) \end{align} where $\mathcal{E}(X_t)$ denotes the stochastic exponential of a stochastic process (Doléans-Dade exponential) i.e. $$ \mathcal{E}(X_t) = \exp\left( X_t - \frac{1}{2}\langle X \rangle_t \right) $$

Now taking the conditional expectation under the quanto measure one gets that $$ F^d(0,t) = F^f(0,t) \Bbb{E}_0^d \left[ \underbrace{\exp\left( -\int_0^t \rho_{S,x} \sigma_X \sqrt{v_u} du \right)}_{A_t} \underbrace{\mathcal{E}\left( \int_0^t \sqrt{v_u} dW_S(u) \right)}_{B_t} \right] $$ where $F^d(0,t)$ represents the quanto forward and $F^f(0,t)$ the forward price of the equity.

By the properties of the Doléans-Dade exponential, we know that $\Bbb{E}_0[B_t] = 1$. The question now is whether $A_t$ and $B_t$ are independent so that we can write

$$\Bbb{E}_0[A_t B_t] = \Bbb{E}_0[A_t] \Bbb{E}_0[B_t] = \Bbb{E}_0[A_t]$$

For instance this is the case if $v(u) = \sigma^2_S(u)$ is deterministic, this degenerates to the usual Black-Scholes forward quanto price $$ F^d(0,t) = F^f(0,t) \exp\left( -\int_0^t \rho_{S,x} \sigma_X \sigma_S(u) du \right) $$