**Theory and Practice for the FVA: General Instruments and General Models**

**Presenter: Alexander Antonov: Senior Vice President, Quantitative Research, Numerix**

**Video Lectures:**

**xVA general framework: from replications to adjustments Section 1.1 (Running Time: 32:21)**

**xVA general framework: from replications to adjustments Section 1.2 (Running Time: 44:56)**

**xVA general framework: from replications to adjustments Section 1.3 (Running Time: 43:19)**

**xVA general framework: from replications to adjustments Section 1.4 (Running Time: 48:09)**

**Collateral in Different Currencies Section 1.1 (Running Time: 41:32)**

**Collateral in Different Currencies Section 1.2 (Running Time: 44:53)**

**Approximation for General Instruments (including arbitrary callable instruments) (Running Time: 35.46)**

**Video Lectures:**

**xVA general framework: from replications to adjustments Section 1.1**

**(Running Time: 32:21)**

- Literature
- Introduction

**xVA general framework: from replications to adjustments Section 1.2 (Running Time: 44:56)**

- Classical replication/hedging (before the crisis)
- Modern replication/hedging (after the crisis)

**xVA general framework: from replications to adjustments Section 1.3 (Running Time: 43:19)**

- Pricing equations
- Case studies
- Replication
- Pricing PDE

**xVA general framework: from replications to adjustments Section 1.4 (Running Time: 48:09)**

- Adjustments
- Comparison
- How to avoid double counting and other traps?

**Collateral in Different Currencies Section 1.1 ****(Running Time: 41:32)**

- Foreign currency collateral: pricing PDE via modified replication procedure
- Efficient discount factor for the currency choice option

**Collateral in Different Currencies Section 1.2 (Running Time: 44:53)**

- Analytical methods for the efficient discount factor calculation (two currency case)
- Numerical experiments

**Approximation for General Instruments (including arbitrary callable instruments) (Running Time: 35.46)**

- Main idea
- Theoretical and Numerical Issues
- Numerical experiments for a Bermudan option: comparison of the exact FVA with its different approximations

**Summary:**

1. General multi-currency setup

2. FVA theory framework

- Pricing PDE and its solution by replication
- Probability measure and numeraire
- Generalization

2. Collateral in different currencies

- Replication procedure
- Efficient discount factor for the currency choice option
- Analytical methods for the efficient discount factor calculation (two currency case)
- Numerical experiments

3. Approximation for general instruments (including arbitrary callable instruments)

- Main idea
- Theoretical and Numerical Issues
- Numerical experiments for a Bermudan option: comparison of the exact FVA with its different approximations