Finance - Financial Models in Excel - Volatility

Financial Models in Excel

Lecture 7

Volatility Prediction

Literature: The exercise!

Financial models in Excel, (c) 2003 Peter Raahauge – p. 1/21

Motivation

What are we looking for? {Excel-slide}

Purpose of predicting volatility:

• Stock pricing according to, e.g., CAPM• Long horizon (yearly)

• Risk management, e.g., Value at Risk• J.P. Morgan/Risk MetricsTM (weekly)

• Option pricing• Exercise 6. (Daily to yearly)


Approaches to Volatility Prediction

1) Case oriented predictions, (ex: likelihood of war)

, Forward looking

/ Using past information informally

/ Shaky (any?) theoretical foundation ...


Approaches to Volatility Prediction (2)

2) Implied volatilities via option prices

, Forward looking

, Solid theoretical foundation

/ Cannot be used to price options

/ Depend on option pricing model


Approaches to Volatility Prediction (3)

3) Time series statistics: Use historical information fromthe series itself.

, Solid theoretical foundation

, Very popular in finance

/ Only backward looking

In practice: Combination of 2) and 3) (Datastream)


Exercise 7: Overview

7.1 Data is S&P 500 index

7.2 Moving average st. dev. or Historical volatility (Ex 6)

7.3 Measure of fit introduced => Choice of horizon

7.4 Exponentially weighted• Used by JP Morgans Risk-Metrics• Estimation/calibration of free parameter

7.5 GARCH (widely recognized)• More free parameters and Dynamic predictions

7.6 Maximum likelihood => Estimation and TEST


Exercise 7.1: Data

Data• 10 years of daily S&P 500 index values• Download Turnover figures to remove “dead”

observations: Holidays, post Sep. 11 etc.• Use {Data | Sort }

Our model: Rt ∼ N(R̄, σ2t )

• N not important before Ex 7.6. (Max. likelihood)

• R̄ is constant, (often = 0!)

• σ2t vary over time: The topic of the exercise.

We work with excess return rt = Rt − R̄ ∼ N(0, σ2t )


Exercise 7.2: Historical volatility

Historical volatility / Moving average

σ̂2t =

n∑

i=1

(rt−i − 0)2

n

=1

n

n∑

i=1

r2t−i

=1

nr2t−1 +

1

nr2t−2 +

1

nr2t−3 + · · · +

1

nr2t−n

• Often called “equally weighted moving average”• Try 2 months, 1 month, 2 weeks, 1 week.

(What do you expect?)


Exercise 7.3: RMSE

Q: What is best: 1 month or 2 weeks?

A: We need a measure of fit!

Root Mean Squared Error (RMSE), (RiskMetrics)

RMSE =

√

√

√

√

1

T

T∑

t=1

(

r2t − σ̂2

t

)2

Intuition: Average distance between predicted andrealized volatility

Exercise: Calculate RMSE for the 4 horizons andcompare


Exercise 7.4: EWMA

Exponentially Weighted Moving Average (EWMA)• Motivation• Used by JP Morgan/RiskMetrics for everything!

σ̂2t = (1 − λ)

∞∑

i=1

λi−1r2t−i

= (1 − λ)(

λ0r2t−1 + λ1r2

t−2 + λ2r2t−3 + · · ·

• Weights sum to one• {blackboard} Different λ values, (RiskMetrics: 0.94)• Cut-off problems => Correction methods


Exercise 7.4 (2)

Very nice updating formula:

σ̂2t = (1 − λ)r2

t−1 + λσ̂2t−1

Exercise:• Calculate EWMA for S&P 500 and compare• Determine RMSE and compare• Use Solver to find optimal λ and compare with 0.94


Exercise 7.4 (3)

How to predict volatility beyond t

• Our information: {rt−1, rt−2, rt−3, · · · , r1}.

• σ̂2t = (1 − λ)r2

t−1 + λσ̂2t−1 standard

• σ̂2t+1 = (1 − λ)r2

t + λσ̂2t don’t work: rt missing

Solution: Best guess on r2t is σ̂2

t

• σ̂2t+1 = (1 − λ)σ̂2

t + λσ̂2t = σ̂2

t

• And so on: σ̂2t+2 = σ̂2

t+1

Constant volatility predictions, no dynamicsFinancial models in Excel, (c) 2003 Peter Raahauge – p. 12/21

Exercise 7.4 (4)

Remember:

V ar(rweekly) = V ar(r1) + V ar(r2) + · · · + V ar(r5)

= 5V ar(rdaily) If constant variance

Fits EWMA predictions


Exercise 7.5: ARCH and GARCH

Autoregressive Conditional Heteroscedasticity ARCH(n):

σ̂2t = α1r

2t−1 + α2r

2t−2 + · · · + αnr2

t−n

All α’s are free• Special cases: Historical vol. and EWMA• Too hard to estimate• (Robert Engle awarded Nobel price this month)


Exercise 7.5: GARCH

Generalized ARCH: GARCH(n,q)

σ̂2t = α1r

2t−1 + α2r

2t−2 + · · · + αnr2

t−n

+β1σ̂2t−1 + β2σ̂

2t−2 + · · · + βqσ̂

2t−q

• Not a generalization from a theoretical point of view

• More parameters!, but ...

• In practice GARCH(1,1) is enough:σ̂2

t = α1r2t−1 + β1σ̂

2t−1

• Two parameters only. Determine with RMSE


Exercise 7.5: GARCH, (2)

Special case of GARCH(1,1)• Impose “sum-to-one” restriction:

α1 = 1 − β1

⇓

σ̂2t = (1 − β1)r

2t−1 + β1σ̂

2t−1

Which is EWMA (and Historical volatility)


Exercise 7.5: GARCH, (3)

Generalization of GARCH(1,1):

• Add long term level of σ2: V

• Motivation {Blackboard}

σ̂2t = γV + α1r

2t−1 + β1σ̂

2t−1

• Consider γ + α1 + β1 = 1 (interpretation)

• “Sum-to-one” good for the Solver during estimation


Exercise 7.6: Maximum likelihood

Motivation: RMSE is not the way to proceed• We need statistical tests of, e.g.

• “Sum-to-one” restriction• γ = 0

Maximum likelihood idea: What is likelihood of data givenour model:

• Our model: rt ∼ N(0, σ2t ) plus volatility model

• Density plot on blackboard• Estimation of parameters maximizes

likelihood/density

Maximum likelihood is crown jewel of statistics


Exercise 7.6: Maximum likelihood (2)

Likelihood of individual rt in Excel:• L(rt) = NORMDIST(rt,0,σ̂t, FALSE)• Note L(rt) depend on γ, α, and β via σ̂t

Likelihood of all data:• L(r) = L(r1) · L(r2) · L(r3) · · ·L(rT )

Numerical problems => Log-likelihood: ℓ(r) = ln(L(r))

• ℓ(r) = ℓ(r1) + ℓ(r2) + ℓ(r3) + · · · + ℓ(rT )

Maximizing ℓ(r) or L(r) gives the same parameters


Exercise 7.6: Maximum likelihood (3)

How to test a restriction (say γ + α1 + β1 = 1):

1 Estimate parameters without restrictions=> Unrestricted loglikelihood value ℓ(r)U

2 Estimate parameters with restriction imposed=> Restricted loglikelihood value ℓ(r)R

3 Two times the difference is χ2 distributed=> 2(ℓU − ℓR) ∼ χ2(n)where n is number of restrictions (here n = 1)

4 If 2(ℓU − ℓR) > 3.84 we reject restrictionwhere 3.84 is 5% level {Blackboard}

We can test e.g. γ = 0 exactly the same way.


Literature

Exercise 13 pages

Optional:• Alexander’s Market models: The place to start...• Hull’s option book: Quick and precise introduction


Finance - Financial Models in Excel - Volatility

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