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.

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.

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.

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 $$F^d(0,t) = F^f(0,t) \exp\left( -\int_0^t \rho_{S,x} \sigma_X \sigma_S(u) du \right)$$

• This is a clear answer. Several questions: Do you confirm there's no need to simulate X ? $\rho$(X,$\nu$) could be approximated with $\rho$(S,X).$\rho$(S,$\nu$) isn't it ? Testing the 2 adjustments on a specific worst of 2 basket makes the put premium move by a few bps, say from 8% to 7.95%. Shouldn't the gap be much bigger ($\rho$(S,X) around -10%) ? How long should be the observation window for the historical correl ? – Cedric_W Jun 27 '18 at 15:14
• 1) Yes, no need to simulate $X_t$; 2) Yes this is a good approximation as far as I can remember (to test it for yourself check for instance the historical correlation VIX-USD/EUR to the product of the historical correlations (VIX-SP500, SP500-USD/EUR)), 3) no idea, sorry. It's always the same with quantos, there is no straightforward answer. Note also that because the volatility is stochastic you need to be careful as to the instantaneous vs. terminal correlations problem. I think there are several questions on this site concerning quantos, maybe these will help you? – Quantuple Jun 27 '18 at 15:21
• – Quantuple Jun 27 '18 at 15:23
• I literally read all topics on quanto on this exchange forum but still no clue why premiums are barely changing. Regarding the VIX correlation I read a paper from Giese 2012 saying that this could be a good approximation but I wanted a confirmation. Still hoping for someone to explain this weird behavior. – Cedric_W Jun 27 '18 at 15:31
• Two additional questions. While simulating asset paths, do you confirm the adjustments have to be done at each time steps ? And where is the rates differential taken into account ? I think this is why my premiums are barely moving. I might have a problem regarding the forward computation. @Quantuple – Cedric_W Jun 28 '18 at 15:06