1. Linear and Non-linear Derivatives

Linear derivatives has a linear relationship between underlying risk factors(底层风险因子) and derivative instruments. The transmission parameter(δ\deltaδ) needs to be constant(e.g. forward, futures).

ΔPLinear Derivative=δ×ΔSRisk Factor\Delta P_{\text{Linear Derivative}}=\delta\times\Delta S_{\text{Risk Factor}}ΔPLinear Derivative=δ×ΔSRisk Factor

VaRLinear Derivative=δ×VaRRisk Factor\text{VaR}_{\text{Linear Derivative}}=\delta \times \text{VaR}_{\text{Risk Factor}}VaRLinear Derivative=δ×VaRRisk Factor

Example: Eric has a long forward position on an underlying stock maturing in 3 months. The daily 95%VaR95\%\;\text{VaR}95%VaR of the stock is 2 million. What is the daily 95%VaR95\%\;\text{VaR}95%VaR of the forward position?

远期的 δ\deltaδ 为111

VaRForward=1×VaRUnderlying Stock=2million\text{VaR}_{\text{Forward}}=1\times \text{VaR}_{\text{Underlying Stock}}=2\;\text{million}VaRForward=1×VaRUnderlying Stock=2million

Nonlinear derivatives has a changing relationship between underlying risk factors and derivative instruments depending on the state of underlying asset. The transmission parameter(δ\deltaδ) is inconstant(e.g. option, bond).

ΔPNon-linear Derivative=δ×ΔSRisk Factor\Delta P_{\text{Non-linear Derivative}}=\delta\times\Delta S_{\text{Risk Factor}}ΔPNon-linear Derivative=δ×ΔSRisk Factor

VaRNon-linear Derivative=δ×VaRRisk Factor\text{VaR}_{\text{Non-linear Derivative}}=\delta \times \text{VaR}_{\text{Risk Factor}}VaRNon-linear Derivative=δ×VaRRisk Factor

2. Historical Simulation

Historical simulation is a non-parametric method, where the future behavior of the underlying market variable is determined in a very direct way from their past behavior.
These scenarios can be evaluated using full revaluation(全局定价法).

Steps of Historical Simulation

Identify risk factors (equity price, interest rate etc.) on which the value of the portfolio under consideration depends.
Collect daily data on the behavior of the risk factors over a period in the past.
Create scenario by assuming that the change in each risk factor over the next day corresponds to a change observed during one of the previous days.
Sort the loss based on the scenarios and calculate VaR or ES for any given confidence level.

Identify risk factors (Two categories)
Percentage change(equity price, foreign exchange)
Actual change(interest rate, credit spread)
Collect date

Day	Stock Price(USD)	Int. Rate	…	Port.Val.(USD million)
0	50	2.52%	…	72.1
1	52	2.54%	…	72.5
2	46	2.55%	…	70.4
…	…	…	…	…
498	60	2.30%	…	…
498	60	2.32%	…	75.3
500	63	2.36%	…	76.3

Create Scenario

Scenario	Stock Price(USD)	Int. Rate	…	Port.Val.(USD million)	Loss(USD millions)
1	65.52	2.38%	…	76.8	-0.5
2	55.73	2.37%	…	71.7	4.6
…	…	…	…	…	…
499	63.00	2.38%	…	75.3	1
500	66.15	2.40%	…	76.7	-0.4

Sort the scenarios and calculate VaR and ES

Scenario	210	195	2	23	48	367	235	…	…	…
Loss(USD millions)	7.8	6.5	4.6	4.3	3.9	3.7	3.5	…	…	…

ESdaily(1%)=(7.8+6.5+4.6+4.3)/4=5.8million\text{ES}_{\text{daily}}(1\%)=(7.8+6.5+4.6+4.3)/4=5.8\;\text{million}ESdaily(1%)=(7.8+6.5+4.6+4.3)/4=5.8million

VaRdaily(1%)=3.9million\text{VaR}_{\text{daily}}(1\%)=3.9 \;\text{million}VaRdaily(1%)=3.9million

Advantages:

Free of model risk
Actual market prices are accumulated for full revaluation
All correlations risk factors are included in the prices.

Disadvantages:

It gives equal weight(等权重) to all the observations.
Historical data may not be a good forecast for future.
It cannot reflect the new volatility and correlation changing in the market condition instantly.

Question: Jackson, FRM, a junior risk analyst of Fürstentum Bank (a bank with operations in Germany, Switzerland and Austria), is using historical simulation approach to calculating VaR of DAX 30 index, He has accumulated 501 days data and created 500 scenarios. The points of the DAX 30 on Days 0, 1, 2, 498, 499, and 500 is 13300, 13510, 13230, 15400, 15750 and 15925, respectively. What is the point of the stock index in the corresponding Day 501 under scenarios 500P

Scenario500=(15925−15750)/15750=11.11%\text{Scenario}\;500=(15925-15750)/15750=11.11\%Scenario500=(15925−15750)/15750=11.11%

The point of the DAX 30 under scenario 500 would be: 15925×(1+11.11%)=1610215925\times(1+11.11\%)=1610215925×(1+11.11%)=16102

3. Delta Normal Approach

Delta normal approach(Delta approximation) is based on the risk factor’s deltas of a portfolio and assumes normal distributions for the risk factors.

For standalone non-linear derivatives:
VaRoption=∣Δ∣×VaRstock\text{VaR}_{\text{option}}=|\Delta|\times \text{VaR}_{\text{stock}}VaRoption=∣Δ∣×VaRstock

VaRbond=∣−D×P∣×VaRyield\text{VaR}_{\text{bond}}=|-D \times P|\times \text{VaR}_{\text{yield}}VaRbond=∣−D×P∣×VaRyield

Step 1: Calculate the VaR\text{VaR}VaR of the underlying risk factor.
Step 2: Use delta with respect to the underlying to transmit the risk factor VaR\text{VaR}VaR to the nonlinear derivatives VaR\text{VaR}VaR.

Example: Louise wants to use delta normal approach to estimate the VaR\text{VaR}VaR of a call position. The call option is at-the-money and the underlying stock is trading at USD 232323 with daily volatility 2.5%2.5\%2.5%. Please calculate the daily dollar VaR\text{VaR}VaR of this call option at 99%99\%99% confidence level.

VaRstock=∣0−2.33×2.5%∣×23=1.3398\text{VaR}_{\text{stock}}=|0-2.33\times2.5\%|\times 23= 1.3398VaRstock=∣0−2.33×2.5%∣×23=1.3398

When a call call option is at the money , Δ\DeltaΔ is 0.5

VaRoption=∣Δ∣×VaRstock=0.5×1.3398=0.6699\text{VaR}_{\text{option}}=|\Delta|\times \text{VaR}_{\text{stock}}=0.5 \times 1.3398=0.6699VaRoption=∣Δ∣×VaRstock=0.5×1.3398=0.6699

For a portfolio:

Calculate individual risk factors’ mean and standard deviation and their correlation between each other.
Calculate the means and volatility of the portfolio
Calculate the portfolio’s VaR assuming normality of value change of the portfolio.

VaR(X%)=∣μportfolio−ZX%×σportfolio∣\text{VaR}(X\%)=|\mu_{\text{portfolio}}-Z_{X\%}\times\sigma_{\text{portfolio}}|VaR(X%)=∣μportfolio−ZX%×σportfolio∣

Example: Many pension fund are using VaR\text{VaR}VaR to measure their portfolio risk. Tao, FRM, is a risk analyst of Pluto Pension Fund. In his routine work, he will send risk report of large portfolios to his manager on daily basis. Consider a portfolio consisting of $20 million invested in Stock A and $40 million invested in Stock B. Assuming that returns are normally distributed. Daily volatilities of A and B are 0.5%0.5\%0.5% and 2%2\%2% and daily returns are 0%0\%0%. The correlation between two stocks is 0.250.250.25. What is the daily dollar VaR\text{VaR}VaR with 95%95\%95% confidence level of the portfolio?

σp2=w12σ12+w22σ22+2w1w2σ1σ2ρ1,2\sigma_p^2=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\sigma_1\sigma_2 \rho_{1,2}σp2=w12σ12+w22σ22+2w1w2σ1σ2ρ1,2

用资产价值作为替代w1w_1w1，w2w_2w2作为权重

σp2=202×0.5%2+402×2%2+2×20×40×0.5%×2%×0.25→σp=0.83million\sigma_p^2=20^2\times0.5\%^2+40^2\times2\%^2+2\times20\times40\times0.5\%\times2\%\times0.25\to\sigma_p=0.83\;\text{million}σp2=202×0.5%2+402×2%2+2×20×40×0.5%×2%×0.25→σp=0.83million

VaR(X%)=∣μportfolio−ZX%×σportfolio∣=1.65×0.83=1.37million\text{VaR}(X\%)=|\mu_{\text{portfolio}}-Z_{X\%}\times\sigma_{\text{portfolio}}|=1.65\times0.83=1.37\;\text{million}VaR(X%)=∣μportfolio−ZX%×σportfolio∣=1.65×0.83=1.37million

Taylor Series approximation(Delta-gamma approximation): the change in the derivative value is approximated by slope and curvature. The first derivative is delta linear approximation and second derivative is the gamma correction.

VaRbond=∣−D×P∣×VaRyield−12×C×P×VaRyield2\text{VaR}_{\text{bond}}=|-D\times P|\times \text{VaR}_{\text{yield}}-\frac{1}{2}\times C\times P\times \text{VaR}_{\text{yield}}^2VaRbond=∣−D×P∣×VaRyield−21×C×P×VaRyield2

VaRoption=∣Δ∣×VaRstock−12×Γ×VaRstock2\text{VaR}_{\text{option}}=|\Delta|\times \text{VaR}_{\text{stock}}-\frac{1}{2}\times\Gamma\times \text{VaR}_{\text{stock}}^2VaRoption=∣Δ∣×VaRstock−21×Γ×VaRstock2

注意是减去 gamma correction

Limitations
Cannot provide accurate approximation for portfolio with non-linear derivatives such as MBS, barrier options, as the normal distribution assumption for the change of risk factors does not translate into the normal distribution assumption for the change of portfolio value.

Adding gamma can work much better and the result is a quadratic model(二阶项模型). But there are no easy-to-use analytic results for quadratic model.

Question 1 Stephen Ross, a trader of Neptune Security, has a call option position on CAC-40 index. The index is 534553455345 currently. Stephen uses BSM model to analyze the option, which has N(d1)=0.67N(d1)=0.67N(d1)=0.67, N(d2)=0.576N(d2)=0.576N(d2)=0.576. In this call option contract, every point of index values €5€5€5. The daily volatility of the underlying index is 0.45%0.45\%0.45%. What should be the 95%95\%95% daily dollar VaR\text{VaR}VaR of this call option under delta-normal approach?

VaRstock=1.65×0.45%×5345×5=198.433\text{VaR}_{\text{stock}}=1.65\times0.45\%\times5345\times5=198.433VaRstock=1.65×0.45%×5345×5=198.433

Δ=N(d1)=0.67\Delta=N(d1)=0.67Δ=N(d1)=0.67

VaRoption=∣Δ∣×VaRstock=0.67×198.433=132.95\text{VaR}_{\text{option}}=|\Delta|\times \text{VaR}_{\text{stock}}=0.67\times198.433=132.95VaRoption=∣Δ∣×VaRstock=0.67×198.433=132.95

Question 2 Jean-Paul, FRM, is using delta-gamma approach to supplement the VaR\text{VaR}VaR result of delta-approximation. The 99%99\%99% daily VaR\text{VaR}VaR of the underlying stock is 6.77%6.77\%6.77%. He has an at-the- money put option on stock of Air Solid Inc. The value of the underlying stock is €350,000€350,000€350,000. What could be the 99%99\%99% VaR\text{VaR}VaR of the put option under delta-gamma framework?

A:€11847.5A: €11847.5A:€11847.5
B:€10084B: €10084B:€10084
C:€55209C: €55209C:€55209
D:€27605D: €27605D:€27605

VaRstock=6.77%×350000=23695\text{VaR}_{\text{stock}}=6.77\%\times350000=23695VaRstock=6.77%×350000=23695
VaRoption=∣Δ∣×VaRstock=∣=0.5∣×23695=11847.5\text{VaR}_{\text{option}}=|\Delta|\times \text{VaR}_{\text{stock}}=|=0.5|\times23695=11847.5VaRoption=∣Δ∣×VaRstock=∣=0.5∣×23695=11847.5

The result under delta-gamma approach should be less than the delta-approximation. The only answer is B.

4. Monte Carlo Simulation

Monte Carlo simulations generate scenarios by taking random samples from the distributions assumed for the risk factors (rather than using historical data). Monte Carlo simulation is a full revaluation method.

Steps in Monte Carlo:

Value the portfolio today using the current values of the risk factors.
Sample once from the multivariate normal probability distribution for the change of risk factors. (e. G. Stock price interest rate change)
Use the sampled values of the change of risk factor to determine the values of the risk factors at the end of the period under consideration (usually one day).
Revalue the portfolio using these risk factor values. Subtract this portfolio value from the current value to determine the loss.
Repeat steps 2 to 5 many times to determine a probability distribution for the loss.

Advantages:

Can generate correlated scenarios and model the correlations among different risk factors based on a statistical distribution.
Work for both linear and non-linear portfolio.

Disadvantages:

There is model risk for the generation of statistical distribution on the risk factor.
Computationally intensive and thus quite slow
The correlation and standard deviation for the risk factor in the past may not be a good indicator of the future.

Summary

Category	Historical Simulation	Delta Normal Approach	Monte Carlo Simulation
Full Revaluation	Yes	No	Yes
Model Risk Distribution assumption	No	Yes	Yes
Computationally intensive	Yes	No	Yes

Correlation Breakdown
Definition: Correlations and volatility can be quite different in stressed market conditions from those in normal market conditions.

Correlations usually increase in stressed market conditions: During the 2007-2008 crisis, default rates on mortgages in all parts of the United States increased together.

Worst case analysis
Worst case analysis: an analyst will calculate statistics for worst-case results.

E.g., Monte Carlo simulation can be used to calculate the expected worst-case result over 52 weeks, the 95th percentile of the worst-case result.
Should not be regarded as an alternative to VaR and ES.

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4.3.2 Calculating and Applying VaR

1. Linear and Non-linear Derivatives

2. Historical Simulation

3. Delta Normal Approach

4. Monte Carlo Simulation

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