qf11_lecture09

8/13/2019 qf11_lecture09

1/33

8/13/2019 qf11_lecture09

2/33

Chapter 6: Implied volatility

1 Chapter 6: Implied volatilityA preparation: solving a nonlinear equationComputing the implied volatility

Prof. Dr. Erich Walter Farkas Quantitative Finance 2011: Lecture 9 2 / 31
http://find/http://goback/

8/13/2019 qf11_lecture09

3/33

Chapter 6: Implied volatility

Introduction

Previous chapters: introduction to the theory of options

non-linear instruments

put-call parity

fundamentals of option valuation

pricing by replicationrisk-neutral pricingBlack-Scholes PDE and formulas

option sensitivities (the Greeks)

valuing options by numerical methods

http://find/

8/13/2019 qf11_lecture09

4/33

Chapter 6: Implied volatility A preparation: solving a nonlinear equation

Computing the implied volatility

1 Chapter 6: Implied volatilityA preparation: solving a nonlinear equationComputing the implied volatility


A i l i li i

8/13/2019 qf11_lecture09

5/33



Motivation and setup

the goal of this chapter is to treat the implied volatility which

requires an algorithm for solving a nonlinear equation

the general problem is

given a function F : R R, find an x R suchthat F(x) = 0

in general, of course, we cannot find an x analytically, and musttherefore content ourselves with an approximation via acomputational method

it is worth keeping in mind that, depending on the nature ofF,

there may be no suitable x

, exactly one x

or many x

values

we introduce two algorithms for solving a nonlinear equation

the bisection methodNewtons method (also called Newton-Raphson method)


A ti l i li ti
http://find/

8/13/2019 qf11_lecture09

6/33



The bisection method

is based on the observation that if a continuous function changes

sign, then it must pass through zero, that is

for continuous functions F, ifxa

8/13/2019 qf11_lecture09

7/33



The bisection method: algorithm

Step 1: find xa and xbwith xa0 for our stoppingcriterion xb xa< it is easy to see that the value (xa+xb)/2 on termination is no

more than a distance /2 from a solution x

(hence controls theaccuracy of the process)

because the bisection method halves the length of theinterval [xa, xb] on each iteration, we may bound the error at thekth iteration by L/2k+1 where L is the length of the original

interval, xb xaProf. Dr. Erich Walter Farkas Quantitative Finance 2011: Lecture 9 7 / 31

A preparation: solving a nonlinear equation

8/13/2019 qf11_lecture09

8/33



Newtons method

is faster than the bisection method

can be derived in a number of ways: here we will use a Taylor seriesapproach

suppose we wish to compute a sequence x0, x1, x2,... that convergesto a solution x

we may expand F(x+) for small byF(xn+) =F(xn) +F

(xn) +O(2)

ignoring O(2) and setting F(xn) +F(xn) = 0

gives = F(xn)/F(xn)it follows that ifxn is close to a solution x

then

xn+1=xn F(xn)F(xn)

should be even closer

given a starting value,x0, the last iterationdefinesNewtonsmethodProf. Dr. Erich Walter Farkas Quantitative Finance 2011: Lecture 9 8 / 31

A preparation: solving a nonlinear equation
http://find/

8/13/2019 qf11_lecture09

9/33



Newtons method

since we discarded an O(2) term in Taylors approximation we may

expect that the error xn x squares as n increases to n+ 1: that isifxn x =O() then xn+1 x =O(2)to see this more clearly, note that using F(x) = 0 andassuming F(xn) = 0 a Taylor series gives

xn+1 x = xn x F(xn) F(x)F(xn)

= xn x

(xn x)F(xn) +O((xn x)2)

F(xn)

= O((xn x)2)

this type of analysis can be formalised in a theorem


Ch 6 I li d l iliA preparation: solving a nonlinear equation
http://find/

8/13/2019 qf11_lecture09

10/33



Newtons method: Theorem

Suppose

Fhas a continuous second derivative

x R satisfies F(x) = 0 and F(x) = 0Then

there exists a >0 such that for| x0 x

|< the sequence givenby

xn+1=xn F(xn)F(xn)

is well-defined for all n>0,

withlim

n| xn x |= 0,

and there exists a constant C>0 such that

|xn+1

x

|C

|xn

x

|2 .

http://find/

8/13/2019 qf11_lecture09

11/33


8/13/2019 qf11_lecture09

12/33

Chapter 6: Implied volatilityp p g q


Newtons method: computational example

suppose we wish to find the value ofx such that P(X

x) = 23

where X N(0, 1)essentially we want to solve F(x) = 0, where F(x) :=N(x) 23 with

N(x) = 1

2

x

es

2

2 ds

it follows from the definition ofN that F is an increasing functionand F(0) = 12 23 0

hence we may immediately conclude that the equation F(x) = 0 has

a unique solution 0< x

<


http://find/

8/13/2019 qf11_lecture09

13/33

Chapter 6: Implied volatilityComputing the implied volatility

Newtons method: computational example (contd)

we may apply the bisection method with xa = 0 and with xb

sufficiently large such that F(xb)> 0

for the choice xb= 10 and a tolerance of= 105 in the stopping

criterion the bisection method needs 20 iterations

setting x0 = 1 and stopping with Newtons method

when| xn+1 xn|

8/13/2019 qf11_lecture09

14/33


Motivation

the Black-Scholes call and put values depend on S, K, r, T

t

and 2

of these five quantities, only the asset volatility cannot be observeddirectly; how do we find a suitable value for ?

approach: extract the volatility from the observed market data -

given a quoted option value, and knowing S, t, K, r and T findthe that leads to this value

having found , we may use Black-Scholes formula to value otheroptions on the same asset

a computed this way is known as an implied volatility; the nameindicated that is implied by option value data in the market

this is a totally different way to get compared with the historicalvolatility


Chapter 6: Implied volatility A preparation: solving a nonlinear equationC i h i li d l ili
http://find/

8/13/2019 qf11_lecture09

15/33


Option value as a function of volatility

we focus here on the case of extracting from a European call

option quote

an analogous treatment can be given for a put, or alternatively, theput quote could be converted into a call quote via put-call parity

we assume that the parameters K, r and Tand the asset price S

and time tare knownin practice we will typically be interested in the time-zerocase, t= 0 and S=S0

we thus treat the option value as function of only, and, from nowon, denote it by C()

given a quoted value C, our task is to find the implied volatility

that solves C() =C

it is possible to exploit the special form of the nonlinear equationarising in this context


Chapter 6: Implied volatility A preparation: solving a nonlinear equationC ti th i li d l tilit
http://find/

8/13/2019 qf11_lecture09

16/33

p p yComputing the implied volatility

Option value as a function of volatility: since volatility is non-negative, only values

[0,

) are of interest

let us look at C() in the case of large or small volatility

first assume recall

d1 =

log(S/K) + (r+ 12 2)(T

t)

T tso that d1 and hence N(d1) 1similarlyd2 =d1

T tso that d2 and

hence N(d2) 0using Black-Scholes formula

C() =S N(d1) K er(Tt) N(d2)it follows that

lim

C() =S


Chapter 6: Implied volatility A preparation: solving a nonlinear equationComputing the implied volatility
http://find/

8/13/2019 qf11_lecture09

17/33

p p yComputing the implied volatility

Option value as a function of volatility: 0+

next we look at

0+ and distinguish three cases

1 S Ker(Tt) >0; in this case log(S/K) +r(T t)> 0 sothat if 0+ we have d1 , N(d1) 1, d2 andN(d2) 1.Hence, C S Ker(Tt).

2 S

Ker(Tt)

8/13/2019 qf11_lecture09

18/33


Bounds for the option value as a function of volatility

now we recall from Chapter 10 (see Lecture 7) that the derivative

ofCwith respect to , that is the vega, is given by

vega =S

T t N(d1)and in particular we know that C/ >0

since C=C() is continuous with a positive first derivative, we

conclude that C is monotonically increasing on [0, )from

lim0+

C() = max(S Ker(Tt), 0)and from

limC() =S

the values ofC() must lie between max(S Ker(Tt), 0) and Sconsequently the equation C() =C has a solution if, and only if,

max(S

Ker(Tt), 0)

C

S


http://find/

8/13/2019 qf11_lecture09

19/33


The second derivative ofC()

for later use we will calculate the second derivative

differentiating

vega :=C

S

T t N(d1)

we get2C

2 = S

T

t

2e

12d

21 d

1

d1

using

d1 =log(S/K) + (r+ 12

2)(T t)

T twe have

d1

= log(S/K) +r(T t)2

T t +

1

2

T t

= log(S/K) + (r 2/2)(T t)

2

T

t

= d2


http://goforward/http://find/

8/13/2019 qf11_lecture09

20/33


The second derivative ofC() (contd)

consequently

2C

2 =

S

T t2

e12 d

21 d1d2

=

d1d2

C

from the last equation it follows that C/ has its maximum

over [0, ) at= given by:=

2 | log(S/K) +r(T t)

T t |

Exercise L09.1. Prove that C/ has a unique maximum over [0,

)at= defined above.

moreover2C

2 =

T t43

(

4 4) C


http://find/

8/13/2019 qf11_lecture09

21/33

p g p y

Bisection for computing the implied volatility

we will write our nonlinear equation for in the form F() = 0

where F() =C() Cto apply the bisection method, we require an interval [a, b] overwhich F() changes its sign

it follows from

limC() =S

and fromlim

0+C() = max(S Ker(Tt), 0)

and the monotonicity ofC() that this can be done by fixing K

(say K= 0.05) and trying [0, K], [K, 2K], [2K, 3K],...


http://find/

8/13/2019 qf11_lecture09

22/33

p g p y

Newtons method for computing the implied volatility

Newtons method takes the form

n+1=n F(n)F(n)

where F() = C

is given above

using F() = 0 and the mean value theorem, we have

n+1 = n F(n) F()

F(n)

= n (n )F(n)

F(n)

for some n between n and

hencen+1

n = 1 F(n)

F(n)


http://find/

8/13/2019 qf11_lecture09

23/33


we know that F() is positive and takes its maximum at the

point from abovehence, using the starting value 0 = we musthave 0< F(0)< F

() so that the last equality implies

0 and 1 lies between 1and

hence 0< F(1)< F(1) and

0< 2

1

8/13/2019 qf11_lecture09

25/33


overall we conclude that with the choice 0 = the error will

always decrease monotonically as n increases

it follows that the error must tend to zero and the previous theoryshows that the convergence must be quadratic

therefore using 0 =

: this is our method for computing the

implied volatility


http://find/

8/13/2019 qf11_lecture09

26/33

Implied volatility with real data

we now look at the implied volatility for call options traded at the

London International Financial Futures and Options Exchange(LIFFE) as reported in the Financial Timeson Wednesday, 22August 2001

the data is for the FTSE 100 index, which is an average of100 equity shares quoted on the London Stock Exchange

Exercise price Option price

5125 4755225 405

5325 3405425 280.55525 2265625 179.55725 1395825 105


http://find/

8/13/2019 qf11_lecture09

27/33


the expiry date for these options was December 2001

the initial price (on 22 August 2001) was 5420.3

we take values ofr= 0.05 for the interest rate and T= 4/12 forthe duration of the option

the implied volatility computed for the eight different exercise prices

is decreasing (from approx. 0.19 to 0.174)

of course, if Black-Scholes formula would be valid, the volatilitywould be the same for each exercise price

however in this example the implied volatility varies by around 10%


http://find/

8/13/2019 qf11_lecture09

28/33


note: implied volatility is higher for options that start

out-of-the-money than for options that starting in-the-money

this behaviour is typical for data arising after the stock market crashof October 1987

pre-crash plots of implied volatility against exercise price would

often produce a convex smileshape; more recent data tends toproduce more of a frown


http://find/

8/13/2019 qf11_lecture09

29/33

Implied volatility: some final comments

the widely reported phenomenon that the implied volatility is not

constant as other parameters are varied does, of course, imply thatthe Black-Scholes formulas fail to describe the option values thatarise in the marketplace

this should be no surprise, given that the theory is based on anumber of simplifying assumptions

despite the disparities, the Black-Scholes theory, and the insightsthat it provides, continue to be regarded highly by both academicsand market traders

it is common for option values to be quoted in terms ofvol; ratherthan giving C, the such that C() =C in the Black-Scholes

formula is used to describe the value

many attempts have been made to fix the nonconstant volatilitydiscrepancy in the Black-Scholes theory; a few of these have metwith some success but none lead to the simple formulas and cleaninterpretation of the original work: see Chapter17 ofHull(2000)


http://find/

8/13/2019 qf11_lecture09

30/33

Answers to the Exercises

Exercise L09.1 Hint: Discuss the monotonicity ofC/

analysing the sign of 2C2


http://find/

8/13/2019 qf11_lecture09

31/33

If you are tired following the implied volatility:

Recall


http://find/

8/13/2019 qf11_lecture09

32/33


Recall

Easter holidays:

starts on Thursday April 21, 2011 at 16.00 oclock (4 pm)ends on Sunday, May 1, 2011

consequently:

no Quantitative Finance lecture on Thursday, April 28, 2011

Remember

next Quantitative Finance lecture will take place on

Thursday, May 5, 2011



f f
http://find/

8/13/2019 qf11_lecture09

33/33


Recall

Easter holidays:

starts on Thursday April 21, 2011 at 16.00 oclock (4 pm)ends on Sunday, May 1, 2011

consequently:

no Quantitative Finance lecture on Thursday, April 28, 2011

Remember

next Quantitative Finance lecture will take place on

Thursday, May 5, 2011

Enjoy the holidays!

http://find/

Date post:	04-Jun-2018
Category:	Documents
Upload:	abhishek-puri
View:	221 times
Download:	0 times

qf11_lecture09

Documents