# S. Bossu's Correlation Swaps Model

I am reading Sebastien Bossu's "A new Approach For Modelling and Pricing Correlation Swaps" (link). I am recalling some of the definitions from the paper and would like to understand how to prove one of the very first claims made. Here we go.

The universe $$S = (S_i) \quad i = 1..N$$. The weight vector $$w = (w_i) \quad i = 1..N$$.

1. Denote $$S_i(t)$$ the price of stock $$S_i$$ at the time $$t$$, with convention $$S(0) = 1$$, and we define their geometric as: $$I(t) = \prod_{i=1}^N S_i(t)^{w_i}$$.

Under a probability space $$(\Omega, E, P)$$ with $$P$$-filtration $$F$$, and assuming that the vector $$S$$ of stock prices is an $$F$$-adapted, positive Ito process.

Given a time period $$T$$ and a positive Ito process $$X$$, we define: $$|\tau| = \int_\tau ds$$

$$\sigma^X(\tau) = \sqrt{ |\tau|^{-1} \int_\tau (d \ln X_s)^2}$$

$$\overline{\sigma}^S(\tau) = \sqrt{\sum_1^n w_i (\sigma^{S_i} (\tau))^2}$$

$$\epsilon(\tau) = \sqrt{\sum_1^N w^2_i (\sigma^{S_i}(\tau))^2}$$

Now that we have defined the terms, the claim is that, $$\overline{\sigma}^S(\tau) >= \sigma^I(\tau)$$. I tried to prove this identity in vain. I am not sure what I am missing. Here are my attempts.

$$\ln (I) = \ln (\prod_{i=1}^N S_i(t)^{w_i}) = \sum w_i \ln S_i(t)$$

$$d \ln (I) = \sum w_i d \ln S_i(t)$$

$$\overline{\sigma}^S(\tau) = \sqrt{\sum_1^N w_i \frac{1}{\tau} \int_\tau (d \ln S_i)^2} = \sqrt{\sum_1^N w_i \sigma_i^2}$$

$$\sigma^I(\tau) = \sqrt{|\tau|^{-1} \int_\tau (d \ln I)^2}$$

$$=\sqrt{|\tau|^{-1} \int_\tau (\sum w_i d \ln S_i(t))^2 } = \sum_{i,j =1}^N w_i w_j \rho_{ij} \sigma_i \sigma_j$$

$$\sigma^I(\tau)^2 = \sum_{i,j} \rho_{ij} w_i w_j \sigma_i \sigma_j \le \sum_{i,j} w_i w_j \sigma_i \sigma_j$$

I am thinking that inductive proof should work well, but it goes nowhere; Here it is, for $$N = 1$$ it is true, now if I assume it is true for $$N$$ we have to show that,

$$w_{N+1} \sigma_{N+1}^2 \ge \sum_{i=1}^{N+1} w_i w_{N+1} \rho_{i{N+1}} \sigma_i \sigma_{N+1}$$

Clearly, $$\sum_{i=1}^{N+1} w_i w_{N+1} \rho_{i{N+1}} \sigma_i \sigma_{N+1} \le \sum_{i=1}^{N+1} w_i w_{N+1} \sigma_i \sigma_{N+1}$$ $$\le \sum_{i=1}^{N+1} w_{N+1} \sigma_i \sigma_{N+1}$$

Now I am stuck because the Identity I chose clearly $$\ge$$ the left hand side. I have tried several other ways to go about it but with similar results. I am not sure what I am missing.

What you seem to be missing is $$\sum_{i,j} w_i w_j \sigma_i \sigma_j = \left(\sum_i w_i\sigma_i\right)^2$$
Now apply Jensen's inequality to get $$\left(\sum_i w_i\sigma_i\right)^2 \leq \sum_i w_i\sigma_i^2$$