Instantaneous Versus Terminal Correlation in Stochastic Models

# Clarification on Instantaneous vs Terminal Correlation in Stochastic Models


# Clarification on Instantaneous vs Terminal Correlation in Stochastic Models












I'm studying Correlation Parameterization and Calibration for the LIBOR Market Model (page 22) and have some questions regarding the concepts of instantaneous correlation and terminal correlation. My questions are as follows:

- Instantaneous Correlation: My understanding is that instantaneous correlation refers to the correlation between infinitesimal increments of a stochastic process as defined by its stochastic differential equation (SDE). For example, consider a process $X(t)$ with the SDE $$ \frac{dX(t)}{X(t)} = \mu(t)\, dt + \sigma(t)\, dW(t), $$ where $dW(t)$ is the increment of a Brownian motion. Can we assume that the instantaneous correlation between the increments $dX(t)$ is identical to the correlation between the increments $dW(t)$? That is, if we have two processes $$ \frac{dX(t)}{X(t)} = \mu_X(t)\, dt + \sigma_X(t)\, dW_X(t) \quad \text{and} \quad \frac{dY(t)}{Y(t)} = \mu_Y(t)\, dt + \sigma_Y(t)\, dW_Y(t), $$ with $$ \text{Corr}(dW_X(t), dW_Y(t)) = \rho(t), $$ is it always valid to state that the instantaneous correlation of $dX(t)$ and $dY(t)$ is also $\rho(t)$?



- Instrument Sensitivity: The thesis states: "Prices of correlation-sensitive products depend on terminal correlation, and thus instantaneous correlation and instantaneous volatility. It is important to note that there is no instrument that is sensitive solely to instantaneous correlation." Could someone explain why there is no financial instrument that isolates the effect of instantaneous correlation alone? How do instantaneous volatility and instantaneous correlation jointly determine the terminal correlation that impacts the pricing of these instruments?

Any insights or corrections would be greatly appreciated.

## Answer by KT8 (score 2, accepted)

https://quant.stackexchange.com/a/82239

First of all, I agree with your definition of both correlations, I think you got the ideas quite right.

Regarding question 1. the answer is Yes, just do the $dX(t) dY(t)$ calculation. Note the extra factors, but what we'll call correlation is indeed $\rho(t)$.

The answer to question 2 is yes!

Now, moving to point 3. Imagine a derivative contract that is european and has a payoff that depends on some combination of $S_1(T)$ and $S_2(T)$. You observe the market and see a price $p$ for it and you are determined to use it to calibrate your correlation (assume you already have a perfect volatility surface). What the quote you selected from the thesis claims is that, the price $p$ does depend on the whole thing, the whole continuum of instantaneous correlations and volatilities and that there exists no contract (at least a standard contract that its actively traded in real markets) that allows you to have sensitivity to the instantaneous correlation of say time $t'$ (with $0 < t' < T$). It will always rely on the whole curve, $\rho(t)$, with $t\in[0, T]$. The sensitivity to the different values ot $t$ may vary depending on how you structure the contract, but you cannot isolate the value of $\rho$ at a single time and calibrate it.

If that would be the case, such products would be ideal to calibrate correlation curves point by point, but unfortunately thats not happening in correlation markets.

PS: The book Volatility and Correlation: The Perfect Hedger and the Fox from Fabio Mercurio has a nice discussion on correlation, I'd really recommend that section.