lecture 1: the financial econometrics of option markets · 2. basic pricing ideas 3. econometric...

Lecture 1: The Financial Econometricsof Option Markets

Professor Vance L. Martin

October 7th, 2013

Professor Vance L. Martin ()Lecture 1: The Financial Econometrics

of Option Markets October 7th, 2013 1 / 67

Outline of Workshop

Day 1:

1. Introduction to options

2. Basic pricing ideas

3. Econometric interpretation to pricing

4. Speci�cation of price dynamics

5. The Black-Scholes model

6. Eyeballing options data

7. Estimation

(a) Using returns data(b) Using options data(c) Using returns and options data



Outline of Workshop

Day 2:

1. Testing the Black-Scholes model

(a) Biasedness(b) Heteroskedasticity(c) Smiles and smirks

2. Relaxing the assumptions of Black-Scholes

(a) Distribution(b) Pricing skewness and kurtosis(c) Functional forms and nonparametrics(d) Time-varying volatility

(GARCH, stochastic volatility, forward looking)(e) Jumps

(Poisson, Hawkes)

3. Monte Carlo pricing methods

4. Application to Exchange options and contagion



Introduction

Options were �rst traded on April 26, 1973, at the Chicago BoardOptions Exchange (CBOE) - the number of traded contracts was 911.Since then the options market has grown enormously. The Figureshows the number of contracts traded daily on options written on theS&P500 index, increased from 87,286 in 2000 to 783,768 in 2011.

0

100,000

200,000

300,000

400,000

500,000

600,000

700,000

800,000

00 01 02 03 04 05 06 07 08 09 10 11

Volume

Reason: option contracts provide a more �exible way to manage risksthan more conventional assets including stocks and bonds.



Introduction

De�nitionAn option is the right to buy (call option) or sell (put option) an asset inthe future at a speci�c date for a particular price known as the exercise orstrike price. The time period from when the option contract is written tothe time when the option is exercised represents the maturity of thecontract.

Key Determinants1. The exercise price (k) .2. The maturity of the contract (h) .3. The discount rate (r) .4. The model used to explain the evolution of stock prices over time.

The speci�cation of the model of the underlying asset price the optioncontract is written on is fundamental to option pricing.The simplest model speci�cation (Black-Scholes) is that stock returnsare normally distributed with constant volatility σ. This is equivalentto stock prices being lognormally distributed.



Introduction

For the Black-Scholes model σ is the only unknown parameter thatneeds to be estimated.

1. Historical methods: use past returns of the underlying asset that theoption contract is written on.

2. Implicit methods: use observed option prices.3. Combined methods: use both returns data and option price data.

Use an iterative maximum likelihood estimator as the log-likelihood isa nonlinear function of the unknown parameter σ.

As models of price dynamics become more involved, so do therequirements placed on the estimation methods.



Basics and De�nitions

Type De�nitionShort-term (less than 1 year)

European Exercised on the expiration day. Payo¤ is pt+h � k.American Can exercise before expiration day.

Bermudan Exercise option on speci�c dates.

Asian The payo¤ is based on the average of p over the contract.

Barrier Exercise when p exceeds some �barrier�.

Binary The payo¤ is the price of the underlying asset at maturity.Long-term (between 1 and 3 years)

LEAPS Available for equities and indexes.

Options Written on More than a Single AssetExchange Payo¤ is p1,t+h � p2,t+h.




In determining an option contract at time t, the key features are:

1. Strike or exercise price (k)The agreed price that an investor can purchase (call) or sell (put) theunderlying asset until the contract expires.

2. Maturity (h)The time period of the contract, expressed as a proportion of a year.

3. Price of asset (p)The price of the asset the contract is written on at the time the optionis written.

The underlying asset that an option is written on can cover a broadarray of asset markets including stocks, bonds, futures and foreignexchange markets.




Option prices on March 6th, 2013, written on S&P500 index.

Strike (k) May Call Option Prices (c) June Call Option Prices (c)Bid Ask Bid Ask

1, 400 141.40 144.60 145.50 148.901, 425 118.80 120.40 123.80 126.901, 450 97.10 100.10 103.00 106.001, 475 76.20 79.00 83.20 86.001, 500 58.00 59.30 65.30 67.301, 525 39.90 41.70 48.30 50.201, 550 25.20 26.70 34.00 35.401, 575 13.90 15.20 21.50 23.301, 600 6.70 7.00 12.80 14.101, 625 2.90 3.40 6.80 7.601, 650 1.15 1.50 3.60 4.101, 675 0.35 0.90 1.85 2.201, 700 0.10 0.50 1.05 1.35

Source: NASDAQ Website, live quotes on March 6th, 2013.Professor Vance L. Martin ()Lecture 1: The Financial Econometrics



When dealing with options data, maturity (h) and volatility (σ) areannualised. In the case of maturity1. May contracts

h (May) =25+ 30+ 18

365= 0.2

2. For June contracts

h (June) =25+ 30+ 18+ 35

365= 0.29589

The value of the S&P500 index on March 6th in 2013 is

pt = 1, 541.46

�moneyness�of the contract is de�ned as

pt = 1, 541.46 > k : in-the-money optionspt = 1, 541.46 = k : at-the-money optionspt = 1, 541.46 < k : out�of-the-money options




Table shows that option prices increase as �moneyness�of thecontract increases. This is a re�ection of the payo¤ of the contract

PAYOFF =�pt+h � k : pt+h � k � 0,

0 : pt+h � k < 0

Interpretation:

1. When pt+h � k > 0, net gain from exercising the option contract inthe future as the investor can purchase the S&P500 index at the pricek and immediately sell the index on the spot market at the higher priceof pt+h .

2. When pt+h � k < 0, no gain to be made from exercising the contract.

But what is the future price of the asset pt+h?




A comparison of the quotes for the May and June options shows thatoption prices are higher for the longer maturity options.This feature of option prices is to be expected given that the S&P500index has a positive trend suggesting that for a given strike price k,the longer the time period, the greater will be the expected (positive)di¤erence between the future price pt+h and k.

0

400

800

1,200

1,600

2,000

55 60 65 70 75 80 85 90 95 00 05 10

Figure: Daily S&P500 stock index, 3/1/1950 to 27/9/2013.Professor Vance L. Martin ()

Lecture 1: The Financial Econometricsof Option Markets October 7th, 2013 12 / 67

Pricing Options

An investor at time t wants to have the option to purchase an assetin the stock market in the future at t + h, at a price of k. How muchshould the investor be prepared to pay for the option contract now?To work out the price of this contract it is necessary to understandthe evolution of the stock price pt .The following Figure gives 5 potential sample paths of pt over a 6month time horizon assuming 250 trading days in a year (h = 0.5),beginning at the initial spot price p0 = 5.

4.0

4.5

5.0

5.5

6.0

6.5

10 20 30 40 50 60 70 80 90 100 110 120

P1 P2 P3P4 P5 K



Pricing Options

Contracts Finishing in the Money

If k = 4.8, cases 1, 2 and 5 are in-the-money at t + h

p1,125 = 5.2104 > 4.8

p2,125 = 6.1121 > 4.8

p5,125 = 5.6535 > 4.8

Investor will exercise the option contract in either of these three caseswith payo¤s

PAYOFF1 = 5.2104� 4.80 = 0.4104PAYOFF2 = 6.1121� 4.80 = 1.3121PAYOFF5 = 5.6535� 4.80 = 0.8535



Pricing Options

Contracts Finishing out of the Money

For cases 3 and 4 the contracts are out-of-the-money at t + h

p3,125 = 4.69 < 4.8

p4,125 = 4.75 < 4.8

Investor will not exercise the option contract is these two cases, sopayo¤s are

PAYOFF3 = 0PAYOFF4 = 0



Pricing Options

The price that the investor is prepared to pay for the option contractat the time the contract matures is the average of the payo¤s

PAYOFF =15

5

∑i=1PAYOFFi

=0.4104+ 1.3121+ 0.00+ 0.00+ 0.8535

5= 0.5152

As the contract is written at t and not at t + h, PAYOFF at t + h isdiscounted back to t at the rate r

ct = exp (�rh)15

5

∑i=1PAYOFFi

If r = 0.05, the price of the 6 month call option is

ct = exp (�rh)15

5

∑i=1PAYOFFi = exp (�0.05� 0.5)� 0.5152 = 0.5025

so $0.5025 is the price the investor will pay for the option contract.Professor Vance L. Martin ()


Pricing Options

Allowing for N time paths the option price is

ct = exp (�rh)1N

N

∑i=1PAYOFFi

As the number of time paths approached in�nity, N ! ∞, then thesample average approaches its conditional expectation in which casethe option price is given by

ct = exp (�rh)Et [PAYOFF ] = exp (�rh)Et [pt+h � k ]

The role of Monte Carlo methods to evaluate option prices arediscussed on Day 2.



Pricing OptionsSpeci�cation of the Asset Price Distribution

Important ingredients from an econometrics perspective.1. ForecastingAs options are concerned with purchasing an asset in the future at timet + h, it is necessary to derive the forecast distribution of the asset price

f (pt+h j pt )based on information available at time t.

2. Conditional MeanThe price of the asset that an investor is prepared to pay for in thefuture is equal to the conditional mean of f (pt+h j pt )

µ t+hjt =Z ∞

0pt+hf (pt+h j pt ) dpt+h

3. TruncationAs only value positive payo¤s in the future, pt+h � k > 0, theconditional mean of an option is a truncated conditional mean

ct = exp (�rh)Z ∞

k(pt+h � k) f (pt+h j pt ) dpt+h




To derive the forecast distribution f (pt+h j pt ) , it is necessary tospecify the asset price dynamics.The standard model is

log pt+h � log pt =

�r � 1

2σ2�h+

pσ2hzt+h

zt+h � N (0, 1)

where h represents the pertinent time horizon.The distribution of the asset price at t + h conditional on informationat time t, is a conditional lognormal distribution

f (pt+h j pt ) =1

pt+hp2πσ2h

exp

"� (log pt+h � µt )

2

2σ2h

#where

µt = log pt +�r � 1

2σ2�h




The �gure gives plots of the forecast distribution for selected timehorizons of 1 month (h = 1/12), 3 months (h = 3/12), and 6months (h = 6/12), assuming p0 = 5, r = 0.05, and σ = 0.2.The 1-month forecast distribution is the more compact distribution,with dispersion increasing for longer time horizons.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10 11

P

F_1MTHF_3MTHF_6MTH

Figure: Forecast distributions of the asset price for selected time horizons of1-month (h = 1/12) , 3-months (h = 3/12) and 6-months (h = 6/12). Basedon a log-normal distribution Ln

�µt , σ

2h�, where µt = log pt +

�r � 1

2σ2�h, the

intial price is p0 = 5, the interest rate is 5 = 0.05, and volatility is σ = 0.2.



Pricing OptionsFormal Derivation

The call option price is

ct = exp (�rh)Z ∞

k(pt+h � k) f (pt+h j pt ) dpt+h

= exp (�rh)Z ∞

kpt+hf (pt+h j pt ) dpt+h

� exp (�rh)Z ∞

kkf (pt+h j pt ) dpt+h

= exp (�rh) I1 � exp (�rh) I2

For this model, analytical expressions of I1 and I2 are available.




Consider the �rst integral

I1 =

∞Zk

pt+hf (pt+h j pt ) dpt+h

=

∞Zk

pt+h �1

pt+hp2πσ2h

exp


2

2σ2h

#dpt+h

=

∞Zk

1p2πσ2h

exp


2

2σ2h

#dpt+h

Using a change of variable y = log pt+h, then

I1 =

∞Zlog k

1p2πσ2h

exp

"� (y � µt )

2

2σ2h

#exp [y ] dy




Upon simplifying (completing the square, change of variable again...)

I1 = exp�

µt +σ2h2

� ∞Zlog k

1p2πσ2h

exp

"��y �

�µt + σ2h

��22σ2h

#dy

= exp�

µt +σ2h2

�Φ�

µt + σ2h� log kσph

�where Φ (a) is the normal cdf.

0.0

0.2

0.4

0.6

0.8

1.0

4 3 2 1 0 1 2 3 4

Z

F

Figure:Professor Vance L. Martin ()



As

µt = log pt +�r � 1

2σ2�h

the �rst integral simpli�es to

I1 = exp�log pt +

�r � 1

2σ2�h+

σ2h2

��Φ

log (pt/k) +

�r + 1

2σ2�h

σph

!

= pt exp [rh]Φ

log (pt/k) +

�r + 1

2σ2�h

σph

!




For the second integral

I2 =

∞Zk

f (pt+h j pt ) dpt+h

=

∞Zk

1

pt+hp2πσ2h

exp


2

2σ2h

#dpt+h

=

∞Zk

1

yp2πσ2h

exp

"� (log y � µt )

2

2σ2h

#dy

where y = pt+h is used just to simplify the notation.




Using a change of variable z = (log y � µt ) /σph, the integral

becomes

I2 =

∞Zk

1

yp2πσ2h

exp

"� (y � µt )

2

2σ2h

#dy

=

µt�log kσphZ

�∞

1p2π

exp��z

2

2

�dz

= Φ�

µt � log kσph

�= Φ

log (pt/k) +

�r � 1

2σ2�h

σph

!by using µt = log pt +

�r � 1

2σ2�h in the last step.




Using the expressions for I1 and I2 gives

ct = exp [�rh] I1 � exp [�rh] kI2

= exp [�rh] pt exp [rh]Φ log (pt/k) +

�r + 1

2σ2�h

σph

!

� exp [�rh] kΦ

log (pt/k) +

�r � 1

2σ2�h

σph

!

= ptΦ

log (pt/k) +

�r + 1

2σ2�h

σph

!

� exp [�rh] kΦ

log (pt/k) +

�r � 1

2σ2�h

σph

!




A more compact expression for the call option is

ct = ptΦ (d)� ke�rhΦ(d � σph)

where

d =log (pt/k) +

�r + 1

2σ2�h

σph

This is the Black-Scholes price of a European call option contract.



The Black-Scholes Model

Black and Scholes (1973) derived a model of �nancial marketsallowing for European derivative securities.

Various methods exist to solve for the Black-Scholes price, includingmethods based on stochastic calculus with the price given as thesolution of a partial di¤erential equation.

The solution of the price given above is based on solving for the meanof a truncated distribution where truncation is based on discontinuouspayo¤ function at k.

Advantages

1. Highlights that the price is a function of a (truncated) conditionalmean of the asset price distribution at the time of maturity.

2. Suggests how the model can be expanded.



The Black-Scholes ModelCall Options

An option price is equal to the expected value of its discounted payo¤.

For a European call option expiring at t + h, the price is

ct = e�rhEt [max (pt+h � k, 0)]

Assuming that pt is lognormally distributed gives

ct = ptΦ(d)� ke�rthΦ(d � σph)

where

d =log(pt/k) + (r + 1

2σ2)h

σph

and Φ(z) is the standard normal cumulative distribution.




Interpretation

1. The term Φ(d � σph) represents the probability the option is

exercised (in a risk neutral world) so kΦ(d � σph) is the strike price

times the probability that the strike price is paid.

2. The term ptΦ(d) is the discounted expected value that the price atmaturity is pt+h , provided that pt+h > k.

3. The plus sign in front of ptΦ(d), and a negative sign in front of,suggests the investor takes a long position in holding the underlyingasset and a short position in holding the option.




Special Cases

1. If the distribution is not truncated so k = 0, then d ! ∞, whereby

limk!0

Φ(d)! 1

resulting inct = pt

2. Alternatively, if k = 0, the conditional mean of the price at t + h for alognormal random variable is simply pt exp (rh) , which reduces to ptwhen discounted back to t, when the contract is written usingexp (�rh) as the discount factor.




Example (Pricing a Call Option)

Consider an option contract that expires in 3 months, with k = $6.00. Ifr = 0.059 and σ = 0.15, the price of the equity at t is pt = $6.20, then

d =log(pt/k) + (r + 1

2σ2)h

σph

= 0.67136

Now

Φ(d) = Φ(0.67136) = 0.7490, Φ(d � σph) = Φ(0.5964) = 0.7245

so the price of a call option is

ct = ptΦ(d)� ke�rhΦ(d � σph)

= 6.20� 0.7490� 6.00� exp (�0.0509� 0.25)� 0.7245= 0.3518



The Black-Scholes ModelPut Options

For a European put option expiring at t + h, the price is

putt = e�rhEt [max (k � pt+h, 0)]

Assuming that pt is lognormally distributed gives

putt = ke�rthΦ(�d + σph)� ptΦ(�d)

where

d =log(pt/k) + (r + 1

2σ2)h

σph

and Φ(z) is the standard normal cumulative distribution.



The Black-Scholes ModelStock Options

To allow for options written on stocks earning dividends (equal to acontinuous dividend stream at the rate q per annum), the asset priceequation becomes

log pt+h � log pt =�r � q � 1

2σ2�h+

pσ2hzt+h

zt+h � N (0, 1)

The Black-Scholes price for a European call option paying acontinuous dividend yield, is

ct = pte�qhΦ(d)� ke�rhΦ(d � σph)

where now

d =log(pt/k) + (r � q + 1

2σ2)h

σph



The Black-Scholes ModelCurrency Options

Let pt represent the exchange rate between two currencies.The asset exchange rate equation becomes

log pt+h � log pt =�r � i � 1

2σ2�h+

pσ2hzt+h

zt+h � N (0, 1)

where r is the domestic interest rate, and it is the foreign interest rate.Equation represents uncovered interest rate parity asEt [log pt+h � log pt ] measures the expected depreciation of theexchange rate, which equals r � i , the interest rate interest ratedi¤erential between two countries.The Black-Scholes price for a European currency call option is

ct = pte�ihΦ(d)� ke�rhΦ(d � σph),

where now

d =log(pt/k) + (r � i + 1

2σ2)h

σph



The Black-Scholes ModelExchange Options

An exchange option provides the right to exchange one asset foranother asset - taken as exhcange asset 2 for asset 1.

The holder of the exchange option is betting that the price of asset 1will rise relative to asset 2.

For a European exchange option which gives the holder the right toexchange asset 2 for asset 1 when the contract expires at time t + h,the price of an exchange option (ct ) is given by the expected value ofits discounted payo¤

ct = e�rhEt [max (p1,t+h � p2,t+h, 0)]

where pi ,t is the price of the i th asset.

As both prices are assumed to be stochastic, the call option ise¤ectively a model with a stochastic strike price.



The Black-Scholes ModelExchange Options

The asset price dynamics are

log p1,t+h � log p1,t =�r � q1 � 1

2σ21�h+

qσ21hz1,t+h

log p2,t+h � log p2,t =�r � q2 � 1

2σ22�h+

qσ22hz2,t+h

where �z1,tz2,t

�� N

��00

�,

�1 ρρ 1

��The Black-Scholes price is

ct = p1,te�q1hN (d)� p2,te�q2hN�d � σ

ph�,

where

d =ln (p1,t/p2,t ) +

�q2 � q1 + σ2/2

�(T � t)

σpT � t

and σ =q

σ21 + σ22 � 2ρσ1σ2.



A First Look at the Data

The data consist of N = 27, 695 contracts on European call optionswritten on the S&P500 stock index on the 4th of April, 1995.

In selecting the options data, three �lters are applied.

1. Option prices satisfy the no arbitrage lower bound condition

LB = max�pte�qh � ke�rh , 0

�2. Include only contracts conducted between 9am and 3pm to minimizenonsynchronicity problems.

3. Duplicate observations are removed to ensure that observations areunique.




The Table summarizes the key characteristics of European call optiondata.

Statistic Option Price Strike Price Maturity Stock Price(ct ) (k) (h) (pt )

Number 27, 695 30 3Minimum $1.0350 $350.0000 0.123288 $496.4390Maximum $156.2500 $550.0000 0.457534 $502.9450Mean $60.36025 $449.2538 0.270635 $499.8471St. Dev. $30.97777 $33.8460 0.141486 $1.9290




The call option prices (ct ) are computed as the average of thebid�ask prices. The average price is $60.3602 and a standarddeviation of $30.9778.There are 30 di¤erent strike prices (k) with 3 di¤erent maturitiesresulting in a total of 30� 3 = 90 unique contracts for the day. Strikeprice ranges from $350 to $550.The 3 di¤erent maturities are May, June and September.The stock price (pt ) is the synchronously recorded price of the indexat the time the quote is recorded.The stock price is adjusted for dividends by multiplying the price ofthe S&P500 stock index by e�qh, where q is computed as the averageannual rate of dividends paid on the S&P500 index over 1995. Thedividend adjusted stock price varies over the day with an averageprice of $499.8471 ranging from $496.4390 to $502.9450.The interest rate (r) is the annualized 3-month Treasury bill rate onApril 4, which is set at 5.61% over the trading day.




The following table gives the �rst 13 contracts in the data set.

Contract Option Strike Maturity Stock InterestNumber Price (ct ) Price (k) (h) Price (pt ) Rate (r)1 1.7550 550 September 496.8740 0.05912 4.9400 535 September 496.7260 0.05913 4.9400 535 September 497.0910 0.05914 4.8800 535 September 497.1010 0.05915 5.0650 535 September 498.2770 0.05916 6.3800 530 September 496.9830 0.05917 6.3150 530 September 497.0220 0.05918 6.3150 530 September 497.0520 0.05919 6.8750 530 September 498.5040 0.059110 6.8150 530 September 498.5040 0.059111 1.0950 525 May 500.9720 0.059112 2.5650 525 June 499.9930 0.059113 1.0650 525 May 501.0410 0.0591



Estimation

For the Black-Scholes option price model the only unknown parameteris the volatility parameter (σ) , corresponding to the volatility of thereturns on the stock price that the option contract is written on.

All remaining components used to price an option are speci�ed by thecontract.

There are three broad methods to estimate σ.

1. Historical methods: use past returns of the underlying asset that theoption contract is written on.

2. Implicit methods: use observed option prices.3. Combined methods: use both returns data and option data.



EstimationStock Price Data: Historical Approach

As σ in the Black-Scholes European call options model corresponds tothe standard deviation of stock returns, a natural estimator of σ isthen obtained by computing the standard deviation of stock returns.

As σ needs to be expressed in annual terms, this estimate needs to bescaled.

This approach is commonly referred to as the historical method.

Suppose that stock prices (pt ) are measured daily.




The steps to compute σ, are as follows.1. First, compute daily returns from the stock prices as

yt = log (pt )� log (pt�1)For a sample of T daily observations on returns, now compute anestimate of the variance of the daily stock returns as

bσ2y = 1T

T

∑t=1(yt � y)2

where y is the sample mean of yt .2. An estimate of the annualized variance is thenbσ2 = 250bσ2y ,where 250 is chosen to represent the number of trading days that existwithin a calendar year.

3. An estimate of the annualized volatility parameter isbσ = p250bσyProfessor Vance L. Martin ()



Example (Estimating Volatility by Stock Prices)Daily data on the S&P500 stock index for 1995, from January 2 toDecember 31, are given in the Figure.

440

480

520

560

600

640

I II III IV1995

(a) P_SP500

.02

.01

.00

.01

.02

I II III IV1995

(b) R_SP500

Daily log-returns computed as yt = log pt � log pt�1, which begin January3, resulting in a sample size of T = 259.




Example (Estimating Volatility by Stock Prices continued)The mean of daily log-returns is r = 0.001133 while the estimate of thevariance is

bσ2y = 1T � 1

T

∑t=1(yt � y)2 =

0.006067259� 1 = 2.3516� 10�5

so bσy = p2.3516� 10�5 = 4.8493� 10�3The annualized estimate of volatility in stock returns is

bσ = q250� bσ2y = p250� 2.3516� 10�5 = 7.6675� 10�2or 7.6675% per annum.




Example (Estimating Volatility by Stock Prices continued)Alternatively, expressing the annualized volatility in terms of calendar daysthe estimate becomes

p365� 2.3516� 10�5 = 9.2646� 10�2

Or, using just returns in May

bσy = 0.006572yielding an annualized estimate of volatility of based on calendar days gives

bσ = p365� bσy = p365� 0.006572 = 0.12556or 12.556% per annum.




Constant Mean Model with Constant Volatility

An alternative approach is to estimate the �constant mean�model ofreturns given by

log pt � log pt�1 = γ+ utut � iid N

�0, σ2y

�where γ is mean return and σy is the daily �historical� volatility.The log-likelihood function is

log L = �T2log�2πσ2y

�� 12

T

∑t=1

�log pt � log pt�1 � γ

σy

�2where the unknown parameters are

θ =�

γ, σ2y




Constant Mean Model with Constant Volatility

Maximising the log-likelihood function is achieved in one step withthe maximum likelihood estimators given by

bγ =1T

T

∑t=1(log pt � log pt�1)

bσ2y =1T

T

∑t=1(log pt � log pt�1 � bγ)2

The MLEs of γ and σ2y are also the OLS estimators as MLE and OLSare equivalent in this case.




Example (Estimating Volatility by Stock Prices)Using the T = 259 observations on the S&P500 index, the samplebecomes 258 from computing returns. The log-likelihood is

log L (θ) = �T2log�2πσ2y

�� 12

T

∑t=1

�log pt � log pt�1 � γ

σy

�2where the unknown parameters are θ =

�γ, σ2y

. The maximum likelihood

estimates are bγ = 0.001133, bσ2y = 0.0048492In which case the estimate of volatility is

bσ = q250bσ2y = p250� 0.0048492 = 7.6669� 10�2which is the same value as based on descriptive statistics.




Constant Mean Model with GARCH Volatility

An alternative approach is to estimate the �constant mean�modelwith a GARCH conditional variance

log pt � log pt�1 = γ+ utσ2y ,t = α0 + α1u2t�1 + β1σ

2y ,t�1

ut � iid N�0, σ2y ,t

�where γ is mean return and σ2y ,t is the GARCH conditional variance.

The variance is now time-varying with the �memory� controlled bythe parameters α1 and β1.




The log-likelihood function is

log L = �T2log (2π)� 1

2

T

∑t=1

σ2y ,t �12

T

∑t=1

(log pt � log pt�1 � γ)2

σ2y ,t

= �T2log (2π)� 1

2

T

∑t=1

�α0 + α1u2t�1 + β1σ

2y ,t�1

��12

T

∑t=1


α0 + α1u2t�1 + β1σ2y ,t�1

where the unknown parameters are

θ = fγ, α0, α1, β1g

Maximising the log-likelihood function requires an iterative solution asthe log-likelihood is a nonlinear function of the parameters.




Example (Estimating Volatility by Stock Prices)Using the T = 259 observations on the S&P500 index, the samplebecomes 258 from computing returns. The log-likelihood is

log L = �T2log (2π)� 1

2

T

∑t=1

�α0 + α1u2t�1 + β1σ

2y ,t�1

��12

T

∑t=1


α0 + α1u2t�1 + β1σ2y ,t�1

where the unknown parameters are θ = fγ, α0, α1, β1g .




Example (Estimating Volatility by Stock Prices continued)The maximum likelihood estimates are

bγ = 0.001144, bα0 = 1.29� 10�6, bα1 = 0.0128, bβ1 = 0.9357An estimate of the long-run variance from the GARCH estimates is

bσ2y = bα01� bα1 � bβ1 = 1.29� 10�6

1� 0.0128� 0.9357 = 2.5049� 10�5

The annualised volatility based on the long-run GARCH estimate is

bσ = p250� 2.5049� 10�5 = 3.9606� 10�4This estimate is smaller than the estimate based on the descriptivestatistics.




Example (Estimating Volatility by Stock Prices continued)The following �gure highlights the reason why the volatility estimatesbased on GARCH are smaller as the estimates of the volatility arerelatively low in the �rst part of the period, which eventually climb toabout 0.005, an estimate consistent with the constant variance models.

.0020

.0025

.0030

.0035

.0040

.0045

.0050

.0055

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12

1995

Conditional standard deviation



EstimationOptions Price Data: Implied Approach

Given i = 1, 2, � � � ,N observations on the observed market prices ofoption contracts

fc1, c2 � � � , cNgand the corresponding time-stamped price

fp1, p2 � � � , pNg

of the underlying asset the option is written on, it is possible toback-out an estimate of the volatility σ.

This estimate represents an implied estimate of σ as it is the impliedvalue of volatility that must have been used to price the option in the�rst place.




The empirical model is

ci = cBSi (σ) + ui

where ci represents the observed price of the i th option and cBSi (σ) isthe Black-Scholes price of the i th contract

cBSi = piΦ(di )� kie�rihi Φ(di � σphi )

with di de�ned as

di =log(pi/ki ) + (ri + 1

2σ2)hiσphi

The subscript i on fpi , ki , hi , rig allows for di¤erent option contracts.




The disturbance term

ui = ci � cBSi (σ)

represents the pricing error which is assumed to satisfy

ui � iid N�0,ω2�




To estimate the parameter

θ =�

σ2,ω2of the empirical model by maximum likelihood, the logarithm of thelikelihood is given by

log L = �N2log�2πω2�� 1

2

N

∑i=1

�ci � cBSi (σ)

ω

�2where N is the number of option contracts in the sample.

As cBSi (σ) is a nonlinear function of σ an iterative algorithm isneeded to estimate the parameters θ.




Example (Estimating Volatility Using Option Prices)Using the N = 27, 695 option contracts on April 4, 1995, thelog-likelihood is

log L = �N2log�2πω2�� 1

2

N

∑i=1

�ci � cBSi (σ)

ω

�2where the unknown parameters are θ =

�σ2,ω2

cBSi = piΦ(di )� kie�rhi Φ(di � σ

phi )

di =log(pi/ki ) + (r + 1

2σ2)hiσphi

The starting estimates are θ(0) =n

σ2(0) = 0.1,ω2(0) = 0.1

o.




Example (Estimating Volatility Using Option Prices continued)The number of iterations for the algorithm to converge is 59 yielding alog-likelihood value of

log L�bθ�� 38217.69

The parameter estimates are

bθ = nbσ2 = 0.017024, bω2 = 0.924947o

with standard errors of se(bσ2) = 2.98� 10�5 andse(bω2) = 0.7975� 10�2 respectively. The (implied) volatility estimate isthen bσ = p0.017024 = 0.1305or 13.05%. This estimate is roughly twice the size of the volatility estimatebased on the stock price data given in previous example.



EstimationCombining Returns and Options Price Data

An alternative approach is to use information on both the returns ofthe underlying asset that the option is written on as well as the pricesof the option.There are two ways to include option prices.1. Direct Approach: use data on observed option prices, ct , in the case ofcall options.

2. Indirect Approach: use data on the VIX.

The use of the VIX is motivated by the property that it approximatesthe 30-day variance swap rate on the S&P500 index, therebyproviding a measure of the risk-neutral expectation of integratedvariance over a month (Bollerslev, Gibson and Zhou (2011))�

VIXt100

�2=36530E �t

30

∑j=1ht+j

where E �t is the expectation under the risk neutral measure.




Consider the following variation of the constant mean model withGARCH conditional variance

log pt � log pt�1 = r + λσy ,t �12

σ2y ,t +ωtzt

σ2y ,t = α0 +�α1z2t�1 + β1

�σ2y ,t�1

zt � iid N (0, 1)

where r is the risk-free rate of interest.The return dynamics expressed under the risk-neutral measure is

log pt � log pt�1 = r � 12

σ2y ,t +ωtz�t

σ2y ,t = α0 +�

α1 (z�t�1 � λ)2 + β1

�σ2y ,t�1

z�t � iid N (0, 1)

where z�t = zt + λ is the risk-neutral probability measure.




For the variance to be weakly stationary

ψ = α1�1+ λ2

�+ β1 < 1

which represents the persistence under the risk-neutral probabilitymeasure.

Given initial values for the parameters and an initial value σ2y ,0, fromthe expression of the GARCH variance it is possible to deriveestimates of the conditional variance from the returns data as

σ2y ,t+1 = α0 +�α1z2t + β1

�σ2y ,t

where zt are the standardised returns given by

zt =log pt � log pt�1 �

�r + λσy ,t � 1

2σ2y ,t�

σy ,t




Now

E �t σ2y ,t+n = σ2y ,t+1ψn�1 + α0

n�1∑i=1

ψi�1

whereψ = α1

�1+ λ2

�+ β1

In which case

VIXt = 100�

vuut36530

30

∑j=1

�σ2y ,t+1ψ

n�1 + α01� ψj�1

1� ψ

�This expression provides an alternative estimate of the variance basedon VIXt given by

σ�2y ,t =1

∑30j=1 ψj�1

30365

�VIXt100

�2� α0

30

∑j=1

1� ψj�1

1� ψ

!




The two alternative expressions for the conditional variances suggeststhat for T observations on returns and the VIX, the parameters of themodel can be estimated by solving

bθ = argminθ

1T

T

∑t=1

�σ�2y ,t � σ2y ,t

�2where the unknown parameters are

θ = fα0, α1, β1,λg



lecture 1: the financial econometrics of option markets · 2. basic pricing ideas 3. econometric...

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