Fx Smile adjustment to Fit Devaluation
The quanto ratio term structure can be overlayed on a FX uncondition distribution by introducing devaluation/revaluation terms that have an expectation of 1 and whose realisation are independent of the FX.
Given time partition $\{t_0, \cdots t_k, \cdots t_n\}$ for default intervals, the conditional FX at a given $[t_{n-1}, t_n]$ is assumed to be lognormal or a smiled distribution independently of credit, there are $n+1$ possible default states corresponding to default on intervals $t_{k-1} < \tau < t_k, k \leq n$ and survival $t_n<\tau$. In another article, we see that if the conditional distribution is assumed to be independent of that state, the credit quanto curve gives us devaluation/revaluations rates $r_k$ such that: \begin{eqnarray} \sum p_k &=& 1, 0 \leq p_k \leq 1 \\ \sum p_k r_k &=& 1, 0 < r_k \end{eqnarray}
Our goal is to define a spline adjustment to the fx call prices that ensures we can calibrate the devaluation model to the unconditional FX smile for several strikes, and also obtain the cumulative function adjustment so that a properly adjusted condition fx distribution can be used in the Monte-Carlo simulation.
Model background: from conditional price to model implied price
We denote $F(K)$ the market implied cumulative function of FX, $\tilde{F}$ the FX cumulative function conditional on default state (default and its timing), and $\bar{F}(K)$ the devaluation model implied cumulative function.
We have the property linking the model cumulative $\bar{F}$, and the call price $\bar{c}$ to the function conditional on default state $\tilde{F}$ and $\tilde{c}$ : \begin{eqnarray} \bar{F}(X) &:=& \Pr(x>X) = \sum_k p_k \Pr(x r_k > X) = \sum_k p_k \tilde{F}(X / r_k) \\ \bar{c}(X,K) &:=& \int_{X>K} (X-K) d\bar{F} = \sum_k p_k \int_{X r_k>K} (X r_k -K) d\tilde{F} = \sum_k p_k r_k \tilde{c}(X, K/r_k) \end{eqnarray} In this section we established the relation between conditional distribution $\tilde{c}$ and model implied price $\hat{c}$.
Market Calibration
In this section, will see how to calibrate the model price $\bar{c}$ to the fx option implied market price $c$.
Conditional Price as a Perturbation of Market Price
We expect $F$ and $\tilde{F}$ to be relatively similar as we saw in practical cases that devaluation typically affects fx volatility by less than 1%. We can express the conditional function $\tilde{F}$ as a perturbation $\hat{F}$ from the market implied cumulative $F$, same respectively for $\tilde{c}$, $c$ and $\hat{c}$: \begin{eqnarray} \bar{F}(X) &:=& F(X) + \hat{F}(X) \\ \bar{c}(X,K) &:=& c(X,K) + \hat{c}(X,K) \\ \end{eqnarray}
The calibration constraints are: \begin{eqnarray} F(X) = \bar{F}(X) &:=& F(X) + \hat{F}(X) = \sum_k p_k F(X / r_k)+ \sum_k p_k \hat{F}(X / r_k)\\ c(X,K) = \bar{c}(X,K) &:=& c(X,K) + \hat{c}(X,K) = \sum_k p_k r_k c(X, K/r_k) + \sum_k p_k r_k \hat{c}(X, K/r_k) \end{eqnarray}
Boundary Conditions
The boundary conditions on $\hat{F}$ and $F$ are: \begin{eqnarray} {F}(0) &=& 0, {F}(\infty) = 1, \int X dF = E(X) \\ \hat{F}(0) &=& 0, \hat{F}(\infty) = 0, \int X d\hat{F} = 0 \end{eqnarray} The boundary conditions on $\hat{c}$ are integrated due to $\frac{\partial \hat{c} }{\partial K} = -\hat{F} $, they are: \begin{eqnarray} \hat{c}(0) &=& 0, \hat{c}(\infty) = 0 \\ \frac{\partial \hat{c} }{\partial K}(0) &=& 0, \frac{\partial \hat{c} }{\partial K}(\infty) = 0 \end{eqnarray}
We could either have a piecewise linear adjustment to $F$ or a cubic spline $c$ adjustment, with derivatives imposed to be 0 on the bounds. The adjustment on $F$ would need to be made such that a 0 integral constraint needs to be imposed. The constraints on $c$ are easier to impose as values and derivatives of spline at boundaries can easily be set by adding points.
Calibration at Market Strikes
The calibration equations if we want to calibrate market options at strikes $K_i, i\in {1, \cdots p}$. For instance, we can use $p=1$ to recalibrate ATM, and $p=3$ to recalibrate RR and Butterfly using atm, as well as the 25% delta and call and put.
\begin{eqnarray} \sum_k p_k r_k \hat{c}(X, K_i/r_k) &=& c(X,K_i) - \sum_k p_k r_k c(X, K_i/r_k) \end{eqnarray}
We see that we will need in general to evaluate the spline $\hat{c}$ at $np$ different strikes and as we have $p$ constraints besides the boundary constraints, we shall have $p$ points for the spline at $K_i$ and use a non linear newton solver to fit prices.
Once solution $\hat{c}$ is given, $\hat{F}$ is given by $\frac{\partial \hat{c} }{\partial K} = -\hat{F}$, which allows us to fill KOT perturbation information, which is defined by a dense set of $K_i, \hat{F}_i$.