market_risk.rmd

---
title: "FRM Market Risk Management"
author: "Alex Dou"
date: "October 26, 2019"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## VaR and Risk Measurement

* Profit/Loss Data

$$ P/L_t = P_t+D_t-P_{t-1}  $$

* Arithmetic return data

$$ r_t=\frac{P_t+D_t-P_{t-1}}{P_{t-1}} = \frac{P_t+D_t}{P_{t-1}} -1  $$

* Geometric return data

$$ R_t = ln(\frac{P_t+D_t}{P_{t-1}}) = ln(1+r_t)$$

### VaR

#### Parametric Approach
##### Normal VaR

$$ -(u_{PL}-Z_{\alpha} \sigma_{PL}) $$

##### Lognormal VaR

$$ (1-e^{u_R-z_{\alpha} \sigma_R})P_{t-1}$$

##### Standard Error of VaR

* Standard Error: The standard deviation of a random variable.

$$ \frac{\sqrt{\frac{p(1-p)}{n}}}{f(q)}  $$

##### Quantile-Quantile plots
![QQ Plot](images/qqplot.JPG)

#### Non-parametric Approach
* Sample Number

$$ n \times \alpha +1  $$

##### Bootstrap
* 有放回重复抽样


##### Age-weighted historical simulation (BRW)



$$ \omega_1 = \frac{1-\lambda}{1-\lambda^n}  $$

$$ \omega_1 = \frac{\lambda^{i-1}(1-\lambda)}{1-\lambda^n}  $$

$$ 0 \le \lambda  \le 1  $$


##### Volatility-weighted historical simulation (HW)

$$r^*_{t,i} = (\frac{\sigma_{T,i}}{\sigma_{t,i}})r_{t_i}  $$


### Backtesting VaR

#### Failure Rate

$$ \frac{N_{exception}}{T}  $$

#### Classic testing framework

$$z = \frac{X-u}{\sigma} = \frac{x-np}{\sqrt{p(1-p)n}}$$

* x, the number of exceptions

#### Unconditionial coverage model
* H0: The model is correctly calibrated
* Reject H0 with 95% confidence level
$$ LR_{uc} > 3.84  $$


#### Conditional coverage model
* H0: The model is correctly calibrated
* Reject H0 with 95% confidence level
$$ LR_{cc} = LR_{uc} + LR_{ind}  $$
$$ LR_{cc} > 5.991  $$

#### Type 1 and Type 2 Error

$$ Type_1 = \alpha$$

$$ Type_2 = \beta $$
$$ PowerOfTest = 1-\beta  $$


## Correlations

#### Correlations in financial investment

$$ r_p = w_1 r_1 + w_2 r_2  $$

$$ \rho_p = \sqrt{w_1^2 \sigma_1^2+ w_2^2 \sigma_2^2 + 2w_1 w_2Cov_{12}}  $$

$$ Cov_{xy} = \frac{\sum{(x_i-u_x)(y_i-u_y)}}{n-1}  $$

$$ \rho_{xy} = \frac{Cov_{xy}}{\sigma_x \sigma_y}  $$

#### Multi-assets options/Rainbow options
* Options on the better of two

$$ max(s_1, s_2)  $$

* Options on the worse of two

$$ min(s_1, s_2)  $$

* Call on the max of two

$$ max[0, max(s_1, s_2) - k]  $$

* Exchange option

$$ max(0, s_2 - s_1) $$

* Spread call option

$$ max[0, (s_2-s_1) - k]  $$

#### Correlation swap

$$ Amount_{national}(\rho_{realized} - \rho_{fixed})  $$

$$ \rho_{realized} = \frac{2}{n^2-n} \sum_{i>j} {\rho_{i,j}}  $$


#### Joint probability of default

$$ P(AB) = \rho_{AB} \sqrt{PD_A(1-PD_A)PD_B(1-PD_B)} + PD_A PD_B  $$

#### Mean reversion and autocorrelation

##### n-period lag Autocorrelation

$$ AC(\rho_t, \rho_{t-n}) = \frac{conv(\rho_t, rho_{t-n})}{\sigma(\rho_t) \sigma{\rho_{t-n}}}  $$

##### Mean reversion

$$ S_t-S_{t-1} = a(u-S_{t-1}) \Delta t + \sigma_s \epsilon \sqrt{\Delta t} \approx a(u-S_{t-1})  $$


#### Correlation and Correlation volatility
TBD

### Statistical Correlation Models
#### The Pearson correlation

$$ Cov_{xy} = \frac{\sum{(x_i-u_x)(y_i-u_y)}}{n-1}  $$

$$ \rho_{xy} = \frac{Cov_{xy}}{\sigma_x \sigma_y}  $$

#### The Spearman rank correlation

$$ \rho_s = 1- \frac{6 \sum{ d_i^2}}{n(n^2-1)}  $$

$$ d_i = rank(x_1) -rank(y_i)  $$

#### The Kendall's tau

$$ \tau = \frac{n_c-n_d}{n(n-1)/2}  $$

* Concordant pairs

$$ sign(x_i - x_j) = sign(y_i - y_j)  $$

* Discordant pairs

$$ sign(x_i - x_j) = -sign(y_i - y_j)  $$

#### Gaussian Copula

$$ Probability = N(quantile)$$

$$ Quantile = N^{-1}(Probability)  $$

## Term structure models of interest rates

## Other related topics