Tutorial questions - week 12: Option Markets: Testing Black-Scholes

Learning Objectives.

Read the Lecture Notes from Week 11.

1. To reproduce and extend some of the lecture exTutorial Questions. amples in Week 11.

2. Show how to test the Black-Scholes model for un- The following questions are to be completed in the

Tutorial of Week 12. Question 11.1. focusses on testbiasedness. ing the Black-Scholes model for biasedness, Ques3. Show how to test the Black-Scholes model for tion 11.2. focusses on testing for heteroskedasticity heteroskedasticity. and Question 11.3. focusses on testing for a volatility smile. All solutions will be placed on the website of the

4. Show how to test for a volatility smile. course at the end of Week 12.

Background material.

Eviews files: stock option.wf1.

Question 11.1. Testing the Black-Scholes Model for Unbiasedness.

This question is based on the EViews file (i) Estimate the parameters of the model and comstock option.wf1 which contains N = 27, 695 option pute the value of the log-likelihood function contracts on April 4, 1995 written on the S&P 500 inbased on the maximum likelihood parameter esdex. The variables are: timates. call p k h r

Call option price written on the S&P 500

S&P 500 stock index

Strike price

Time-to-Maturity

Risk-free interest rate

Consider the following empirical option price model: ci = cBS i ( ) + ui

⇣

⌘ ui ⇠ iidN 0, !2

(ii) The empirical option price model is extended to allow for biasedness by respecifying the model as: ci = 0 + cBS i ( ) + ui where 0 is an unknown intercept. Estimate the parameters of the model ✓ = { 2 , !2 , 0 } and compute the value of the log-likelihood function based on the maximum likelihood parameter estimates.

where ✓ = { 2 , !2 } are the unknown parameters and cBS ( ) is the Black-Scholes call price on stock options (iii) A test of the unbiasedness property that the interi cept is zero is based on the hypothesis: defined as: p cBS ki exp( ri hi ) (di hi ),

0 =0 i = pi (di ) with di defined as: log(pi /ki ) + (ri + 0.5 2 )hi di =

.

p hi • Test for unbiasedness using a Wald test.

• Test for unbiasedness using a likelihood ratio test. Question 11.2. Testing the Black-Scholes Model for Heteroskedasticity.

This question is based on the same data Ole used in (iii) Reconsider the heteroskedastic empirical option

Question 1. price model where the form of heteroskedasticity is based on moneyness of the option:

(i) Consider the following empirical option price model that allows for heteroskedasticity: ci = 0 + cBS i ( ) + ui ci = 0 + cBS

(

)

+

u i i

2

!i = exp (↵0 + ↵1 dum ati + ↵2 dum ini )

2

⇣

⌘

i = exp (↵0 + ↵1 dum mayi + ↵2 dum junei )

2

⇣

⌘

u

⇠

iidN

0,

! i i ui ⇠ iidN 0, !2i where ✓ = { 2 , 0 , ↵0 , ↵1 , ↵2 } are the unknown parameters, cBS

( ) is the Black-Scholes call price as i defined in Question 1 and dum mayi and dum junei are dummy corresponding to May and June option contracts respectively according to:

(

1 : hi = 0.123288 dum mayi =

0 : otherwise (

1 : hi = 0.2 dum junei =

0 : otherwise

Estimate the parameters of the model and compute the value of the log-likelihood function based on the maximum likelihood parameter estimates.

(ii) A test of the heteroskedasticity is based on the restrictions:

↵1 = ↵2 = 0

• Test for homoskedasticity using a Wald test.

• Test for homoskedasticity using a likelihood ratio test where the log-likelihood function of

the