Risk Management Stochastic Finance 2004 Autumn School João Duque September 2004.

Risk Management

Stochastic Finance 2004Autumn School

João Duque

September 2004

Risk Management

Risk (latu sensus) = Uncertainty (strictu sensus) = Quantified Uncertainty

Management = Decision Making

Which one is a Stock Price?

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Stock Prices’ PropertiesHistorical prices are useless when forecasting future prices (weak form of efficiency);Prices of financial assets are supposed to be positive or null (never negative);Forecasting financial prices is hard, since they tend to be strongly random;Financial price returns seem random but they seem to respect two empirical governing rules:

i) stock price return is directly proportional to timeii) volatility is directly proportional to the square root of time.

Market Efficiency – Roberts (1967)

Strong FormPrices reflect all price sensitive information

Semistrong FormPrices reflect all publc price sensitive information

Weak FormPrices reflect all past recorded price sensitive information

Stock Prices’ Randomness

Lo, Mamaysky e Wang, Foundations of Technical Analysis: Computational Algorithms, Statistical Inference, and Empirical Implementation, J. of F. (2000)

We find that certain technical patterns, when applied to many stocks over many time periods, do provide incremental information, especially for Nasdaq stocks. Although this does not necessarily imply that technical analysis can be used to generate “excess” trading profits...

Stock Price returns proportional to time

Empirical Experiment:

Collected a data set (time series) of 10 years of daily prices

for some individual stocks, as well as for some stock

indices.

Prices in their original currency were collected on a daily

basis (closing prices) from June 24th 1994, to June 24th 2004.

Prices of individual stocks were adjusted for dividends,

stock splits and other events that may affect stock price

returns.

Stock Price returns proportional to time

Individual Stocks:BCP - the private Portuguese leading bank;IBM, Coca Cola, Pepsico, General Motors - all from the US;BT Group and British Airways - from the UK;BASF - from Germany;Nestle - from Switzerland, and;Nokia – from Finland

Stock Price returns proportional to time

Individual Stocks:PSI 20 – composed of the 20 most liquid stocks of Euronext Lisbon;IBEX 35 – composed of the 35 most liquid stocks of the Madrid Stock Exchange;DJ EuroStoxx – which combines prices from the 50 biggest European companies;FTSE 100 – composed of the top 100 stocks traded in the London Stock Exchange;S&P 500 – composed of the most liquid and significant blue chip stocks of the US market;Nasdaq 100 – composed of the top 100 most promising but risky stocks of the US market.

Stock Price returns proportional to time

Started by computing the daily continuously compounded rate of return

1,

,, ln

ti

tidti P

PR

Also computed the stock price (index) returns for larger time periods: week, month, quarter, semester, year.

Stock Price returns proportional to time

Followed by computing the average stock price (index) return

T

R

R

T

t

kti

ki

1

,

Table 1-A Average stock price returns for individual stocks

estimated on a basis of different time window returns

BCP IBM COCA COLA BT G M PEPSI NESTLE BASF B AIRW. NOKIA

1 Day 0.00007 0.00068 0.00035 -0.00013 0.00007 0.00051 0.00046 0.00039 -0.00015 0.00093

1 Week 0.00037 0.00345 0.00174 -0.00064 0.00029 0.00257 0.00228 0.00197 -0.00074 0.00465 1 Month 0.00147 0.01363 0.00715 -0.00283 0.00138 0.01032 0.00902 0.00774 -0.00320 0.01879

3 Months 0.00480 0.03807 0.02070 -0.01035 0.00136 0.03034 0.02692 0.02278 -0.01065 0.05541 6 Months 0.00543 0.09301 0.04492 -0.01945 0.01525 0.05855 0.05715 0.05296 -0.01620 0.13968

1 Year 0.01085 0.18601 0.08983 -0.03889 0.03049 0.11711 0.11430 0.10592 -0.03241 0.27936

Table 1-B Average stock price returns for stock indices

estimated on a basis of different time window returns

PSI 20 EURO STOXX S&P 500 FTSE 100 IBEX 35 NASDAQ 100

1 Day 0.00025 0.00026 0.00036 0.00017 0.00036 0.00056

1 Week 0.00123 0.00131 0.00180 0.00085 0.00179 0.00278 1 Month 0.00493 0.00523 0.00725 0.00341 0.00713 0.01106

3 Months 0.01470 0.01499 0.02107 0.01010 0.02074 0.03212 6 Months 0.02988 0.03495 0.04735 0.02194 0.04670 0.07354

1 Year 0.05976 0.06990 0.09470 0.04387 0.09340 0.14709

Table 2-A Relative size of the average rate of return of

stock i estimated on a basis of different time window returns BCP IBM COCA COLA BT G M PEPSI NESTLE BASF B AIRW. NOKIA

1 Week 5.00000 5.09778 5.00000 5.00000 4.20758 5.00000 5.00000 5.00000 5.00000 5.00000

1 Month 20.07692 20.12996 20.47836 21.96552 19.75022 20.08236 19.78526 19.59248 21.49011 20.21066

3 Months 65.68198 56.21543 59.31379 80.47608 19.42843 59.03298 59.03995 57.68847 71.60592 59.60138

6 Months 74.24584 137.33108 128.70501 151.18058 218.56934 113.94676 125.33520 134.11031 108.93128 150.23677

1 Year 148.49167 274.66217 257.41002 302.36115 437.13869 227.89353 250.67039 268.22063 217.86256 300.47354

Table 2-B Relative size of the average rate of return of index i estimated on a basis of different time window returns

PSI 20 EURO STOXX S&P 500 FTSE 100 IBEX 35 NASDAQ 100

1 Week 5.00000 5.00000 5.00000 5.00000 5.00000 5.00000

1 Month 20.11813 19.88444 20.11526 19.97577 19.92242 19.91782

3 Months 59.96258 57.02685 58.45653 59.10633 57.91350 57.84876

6 Months 121.88232 132.95919 131.36788 128.32846 130.39680 132.45173

1 Year 243.76464 265.91837 262.73576 256.65691 260.79360 264.90345

Table 3 - Relative size of average rates of return estimated on a basis of different time window returns

EFFECTIVE INCREASE

[1]

INDIVIDUAL STOCKS

[2] INDICES

[3]

TOTAL SAMPLE

[4] 1 Week 5 4.93054 5.00000 4.95658

1 Month 20 20.35618 19.98897 20.21848 3 Months 60 58.80844 58.38576 58.64993

6 Months 125 134.25922 129.56440 132.49866 1 Year 250 268.51843 259.12879 264.99732

That is, the average rate of return of a stock or an index tends to be directly proportional to the time used to estimate the rates of return.

Volatility proportional to the square root of time.

We also computed different standard deviations

based on different time window returns.

The standard deviation for each series of stock

(index) returns was computed using the

following equation:

1

1

2,

,

T

RRT

t

ki

kti

kti

k denotes the type of data used (daily, weekly, monthly, etc.)

Table 4-A Historical volatility for individual stocks

estimated on a basis of different time window returns

BCP IBM COCA COLA BT G M PEPSI NESTLE BASF B AIRW. NOKIA

1 Day 0.02680 0.02191 0.01775 0.02601 0.02092 0.01967 0.01374 0.01860 0.02336 0.03336

1 Week 0.05034 0.04289 0.03794 0.04788 0.04331 0.03733 0.02842 0.04013 0.04793 0.06982 1 Month 0.09477 0.09731 0.06690 0.08675 0.09344 0.06378 0.05014 0.07273 0.08560 0.15346

3 Months 0.15251 0.14984 0.11167 0.15825 0.17245 0.09381 0.08093 0.11543 0.12136 0.28441 6 Months 0.27697 0.14407 0.16196 0.22781 0.21165 0.15974 0.12832 0.17616 0.23022 0.42209

1 Year 0.40776 0.24687 0.25860 0.41383 0.26857 0.23992 0.19030 0.23288 0.25107 0.67416

Table 4-B Historical volatility for individual stock indices

estimated on a basis of different time window returns

PSI 20 EURO STOXX S&P 500 FTSE 100 IBEX 35 NASDAQ 100

1 Day 0.01064 0.01281 0.01125 0.01126 0.01405 0.02232

1 Week 0.02652 0.02741 0.02371 0.02247 0.02943 0.04494 1 Month 0.06170 0.05593 0.04573 0.04288 0.06413 0.09151

3 Months 0.10737 0.09971 0.07397 0.06418 0.10321 0.16075 6 Months 0.21757 0.17059 0.12547 0.12056 0.17359 0.29736

1 Year 0.34443 0.28082 0.21622 0.19171 0.27673 0.51707

Table 5-A Relative size of historical volatility of stock i

estimated on a basis of different time window returns

BCP IBM COCA COLA BT G M PEPSI NESTLE BASF B AIRW. NOKIA

1 Week 1.87848 1.95773 2.13664 1.84055 2.07067 1.89763 2.06815 2.15766 2.05143 2.09287

1 Month 3.53619 4.44213 3.76797 3.33498 4.46676 3.24244 3.64847 3.91084 3.66384 4.59989

3 Months 5.69075 6.83995 6.28979 6.08362 8.24394 4.76937 5.88866 6.20696 5.19437 8.52522

6 Months 10.33482 6.57677 9.12227 8.75794 10.11818 8.12088 9.33633 9.47239 9.85316 12.65208

1 Year 15.21521 11.26905 14.56522 15.90914 12.83946 12.19724 13.84605 12.52264 10.74593 20.20779

Table 5-B Relative size of historical volatility of index i

estimated on a basis of different time window returns

PSI 20 EURO STOXX S&P 500 FTSE 100 IBEX 35 NASDAQ 100

1 Week 2.49389 2.13975 2.10752 1.99576 2.09417 2.01371

1 Month 5.80129 4.36526 4.06583 3.80781 4.56356 4.10010

3 Months 10.09524 7.78314 6.57633 5.69927 7.34485 7.20296

6 Months 20.45739 13.31537 11.15494 10.70623 12.35394 13.32376

1 Year 32.38553 21.91929 19.22355 17.02553 19.69399 23.16849

Table 6 - Relative size of average historical volatility relative to its daily

historical volatility estimated on a basis of different time window returns

EFFECTIVE INCREASE

[1]

SQUARE ROOT OF EFFECTIVE INCREASE

[2]

INDIVIDUAL STOCKS

[3]

INDICES

[4]

TOTAL SAMPLE

[5] 1 Week 5 2.23607 2.01518 2.14080 2.06229

1 Month 20 4.47214 3.86135 4.45064 4.08233 3 Months 60 7.74597 6.37326 7.45030 6.77715

6 Months 125 11.18034 9.43448 13.55194 10.97853 1 Year 250 15.81139 13.93177 22.23606 17.04588

Financial price returns seem random but they seem to respect two empirical governing rules:

i) stock price return is directly proportional to timeii) volatility is directly proportional to the square root of time.

A First Attempt to Model the Empirical Findings

Assume that a variable z starts at time t with the value z(t) and that it will change as a result of the elapsing of time.

Changes on variable z are equal to the differences on that variable taken at two

consecutive moments of time.

t

tzdttzdz

A First Attempt to Model the Empirical Findings

A first attempt to model changes in variable z can be given by the following equation:

dtdz 1,0~ NProperties:

0dzE dtdzVar Values of dz for any two different non-overlapping periods of time are independent (as a result of being independent and identically distributed for any concretisation)

Variable z is said to follow a Wiener Process

A First Attempt to Model the Empirical Findings

If instead of a continuous time process where we continuously observe variable z we assume that observations of z occur regularly every t (time interval)

tzttzz

tz 1,0~ N

Let’s go model it!

A First Attempt to Model the Empirical Findings

Fulfilling Properties:

Property I -

Property II - ?

Property III -

Property IV - ?

The model doesn’t fit!

00 zTzE TzTzVar 0

?001

N

nnzzTz

A Second Attempt to Model the Empirical Findings

Generalized Wiener ProcessIt adds two characteristics to the Wiener Process:

It adds a trend to the random part of the processIt includes an amplifier / reducer within the random part of the process in order to adjust the process to the specific properties of each stock

ttS 1,0~ N

Let’s go model it!

A Second Attempt to Model the Empirical Findings

Fulfilling Properties:

Property I -

Property II - ?

Property III -

Property IV -

The model doesn’t fit!

TzTzE 0 TzTzVar 0

?001

N

nnzzTz

A Third Attempt to Model the Empirical Findings

Ito Process

1,0~ N

Let’s go model it!

ttS

S

tStSeSS 01

0

1lnS

S

S

S

A Third Attempt to Model the Empirical Findings

Fulfilling Properties:

Property I -

Property II -

Property III -

Property IV -

The model fits!

S

dS=> 001 eSS

ttNS

S ,~

Plausible Questions

1) Why do we need models that do not help to forecast prices and the only thing they do is to increase the feeling of price uncertainty? 2) Can we make any money out of these models? How?3) How is possible to profit from some piece of knowledge (model) that only re-emphasises our ignorance about price formation?

Good Applications

Pricing Derivative instruments

Developing arbitrage strategies

Is the Model Right?No!

SkewnessKurtosisVolatility clustersOther stochastic variables

• Volatility• Interest rates• Dividends• Jumps

...

Table 7 – Kurtosis and skewness of daily continuously compounded rates

of return for a database collected from June 24th 1994 to June 25th 2004 Kurtosis Skewness

Minimum Daily Return

BCP 13.002850 0.139014 -0.19591 IBM 5.057391 0.081163 -0.16757 COCA COLA 3.900571 -0.087800 -0.12600 BT GROUP 4.122833 -0.142500 -0.22548 GENERAL MOTORS 4.281433 -0.047520 -0.15769 PEPSICO 3.476316 -0.192710 -0.14188 NESTLE 4.956332 -0.034860 -0.08123 BASF 3.280887 0.318892 -0.08250 B AIRW. 70.70624 1.248128 -0.19684 NOKIA 6.378719 -0.343400 -0.25813 PSI – 20 7.078355 -0.574440 -0.09590 DJ EURO STOXX 2.968146 -0.196200 -0.06599 S&P 500 COMP 3.370042 -0.112950 -0.07113 FTSE 100 2.703326 -0.172010 -0.05885 IBEX 35 2.477150 -0.181340 -0.07339 NASDAQ 100 3.461790 0.117607 -0.10378

AVERAGE 8.826399 -0.011310 -0.13139

20-days rolling historical volatility

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

Jul-94 Jul-95 Jul-96 Jul-97 Jul-98 Jul-99 Jul-00 Jul-01 Jul-02 Jul-03

S&P500

BCP

BCP

S&P500

20-days rolling historical volatility

Standard Deviation

BCP 0.013745 IBM 0.007887 COCA COLA 0.006103 BT GROUP 0.010235 GENERAL MOTORS 0.007050 PEPSICO 0.007361 NESTLE 0.006019 BASF 0.007153 B AIRW. 0.012437 NOKIA 0.013667 PSI – 20 0.005038 DJ EURO STOXX 0.005957 S&P 500 COMP 0.004707 FTSE 100 0.004972 IBEX 35 0.005817 NASDAQ 100 0.009878

Average for individual Stocks 0.009166 Average for Stock Indices 0.006061

Overall Average 0.008002

A Simple Binomial Model

A stock price is currently $20In three months it will be either $22 or $18

Stock Price = $22

Stock Price = $18

Stock price = $20

Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call OptionA 3-month call option on the stock has a strike price

of 21.

Consider the Portfolio: long sharesshort 1 call option

Portfolio is riskless when 22– 1 = 18 or = 0.25

22– 1

18

Setting Up a Riskless Portfolio

Valuing the Portfolio (Risk-Free Rate is 12%)

The riskless portfolio is: long 0.25 sharesshort 1 call option

The value of the portfolio in 3 months is 220.25 – 1 = 4.50The value of the portfolio today is

4.5e – 0.120.25 = 4.3670

Valuing the OptionThe portfolio that is

long 0.25 sharesshort 1 option

is worth 4.367The value of the shares is

5.000 (= 0.2520 )The value of the option is therefore 0.633 (= 5.000 – 4.367 )

Generalization

A derivative lasts for time T and is dependent on a stock

S0u ƒu

S0d ƒd

S0

ƒ0

GeneralizationConsider the portfolio that is long shares and short 1 derivative

The portfolio is riskless when S0u– ƒu = S0d – ƒd or

dSuS

fdu

00

ƒ

S0 u– ƒu

S0d– ƒd

S0– f0

Generalization

Value of the portfolio at time T is S0u – ƒu

Value of the portfolio today is (S0u – ƒu )e–rT

Another expression for the portfolio value today is S0– f0

Hence ƒ0 = S0– (S0u – ƒu )e–rT

GeneralizationSubstituting for we obtain

ƒ = [ p ƒu + (1 – p )ƒd ]e–rT

where

pe d

u d

rT

Risk-Neutral Valuation

ƒ0 = [ p ƒu + (1 – p )ƒd ]e-rT

The variables p and (1– p ) can be interpreted as the risk-neutral probabilities of up and down movementsThe value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate S0u

ƒu

S0d ƒd

S0

ƒ0

p

(1– p )

Irrelevance of Stock’s Expected Return

When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant

ft= value of the derivative instrument at time t.

St= value of the underlying instrument at time t.

K = exercise price of the derivative instrument.

Using Monte Carlo Simulation

Tr fEef 0

KSMaxf TT ;0

0;TT SKMaxf

Call Option

Put Option