A Well-Posed Algorithm to Recover Implied Volatility

IT12 006

Examensarbete 30 hp

Februari 2012

A Well-Posed Algorithm to Recover

Implied Volatility

Yan Wang

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

A Well-Posed Algorithm to Recover Implied Volatility

Yan Wang

Abstract:Implied volatility plays a very important role in financial sector. In the assets of trading market,everyone wants to know the implied volatility in the future.

However,it is difficult to predict it. In this paper,we use a new well-posed algorithm to recover implied volatility under the Black-Scholes theoretical framework. I reproduce this algorithm at first,then prove its stability and give some examples to test. The results show that this algorithm can work and the error is small. We can use it in practice.

Tryckt av: Reprocentralen ITC IT 12 006

Table of Contents

Chapter 1 Introduction………1

1.1Volatility………...1

1.1.1Types of volatility………...1

Chapter 2 General Information About Calculating Volatility…………...4

2.1 Implied volatility estimation………..4

2.2 Calculating implied volatility……….………...5

Chapter 3 The Original Problem………....8

3.1 The implied volatility as a constant………..………8

3.2 Volatility smile and volatility skew………8

3.3 Some reasonable assumptions and a modified model………….…10

3.4 Original problem…...12

Chapter 4 Dupire’s Equation……….13

4.1 Dupire’s formula………...13

4.2 Duality problem………...15

Chapter 5 The Regularization Method……….19

5.1 Regularization idea………...19

5.2 The regularized version of the original problem…………...…….20

Chapter 6 The Stability of the Volatility Function………...…23

Chapter 7 Calculation of the Implied Volatility……….……..30

7.1 Calculating the option price..…………...………....30

7.2 Calculating the local implied volatility……...……….31

7.3 The error in the implied volatility………..……..33

7.4 The error in the option price………...34

7.5 The implied volatility is a function of the stock price………36

Chapter 8 Conclusion………..39

Reference………...41

Appendix………...43

Chapter 1 Introduction

An option is an agreement that gives the right to the holder to trade in the future at a precisely agreed price. There are many kinds of options. We will introduce two main options (call options and put options). A call option is a right to buy a special asset for an agreed amount at a specified time in the future. A put option is a right to sell a special asset for an agreed amount at a specified time in the future. In this paper, we will focus on European call option.

1.1 Volatility

In financial sector, volatility is an important concept. There are many applications of volatility but a few people really understand it. Volatility is one important parameter in Black-Scholes formula. It is sensitive to the changes in option price. For most people, it is understood from their intuition. Actually it is a measure of price changes in the value of financial products over a time period. Generally speaking, it is difficult for people to predict what the volatility will be in the future. However, the option markets exist and they “know” something about the volatility.

1.1.1 Types of volatility

There are many types of volatility, such as the seasonal volatility, the expect volatility, the realized volatility, the historical volatility, the implied volatility and so on. The realized volatility, the historical volatility and the implied volatility are the most useful and common volatilities. We will introduce them as follows.

Realized volatility

In order to apply majority of the financial models, being able to use the empirical data to measure the degree of variability of asset prices is necessary. Suppose that S _t denotes the price of an asset at time t. The realized volatility of the asset in a period

2 2 1 252 1 n i i r n    



. Here 1 ln i i i S r S_

 fori1, 2,3 n, and 252 is an animalization factor corresponding to the typical number of trading days in a year.

Historical volatility

The formula for historical volatility is very similar as realized volatility and it is defined as follows ^ 2 1 252 ( ) 1 n i i r r n      



, where ln ₁ i i i S r S_  , i1, 2,3 n, and 1 1 n i i r r n   



(it is the mean return).

Here 252 is an animalization factor corresponding to the typical number of trading days in a year.

If the returns are supposed to be drawn independently from the same probability distribution, and then r



is the sample mean. The historical volatility is simply the

annualized sample standard deviation. We can see that 2 2 2

1 1 ( ) n n i i i i r r r n r       





. And

then we can get that

^ 2 2 252 1 n r n      

 . It means that the realized volatility is equal to

the historical volatility when the sample means approach to zero. Both types of the volatility can be used as predictors of the future volatility.

Implied volatility

Implied volatility is the value that we get after taking the market price of options into the Black-Scholes model. The Black-Scholes model [6] gives the relationship between the option prices and the five basic parameters (underlying stock priceS , strike _t priceK, interest rater, maturity timeT_t, implied volatility ). So the only unknown parameter (implied volatility ) can be solved when take the first four basic parameters and the actual market price of options into the option pricing model. Implied volatility also can be regarded as an expectation of the actual market volatility. Therefore, implied volatility plays an increasingly important role in the option prices. It is not only assisted with how to measure the price changes in the whole period, but also provides the consistency of a future risk of the current market level to the traders or analysts.

4

Chapter 2 General Information about Calculating Volatility

2.1 Implied volatility estimation

The former researchers propose the two segmentation method [3] which is a relatively simple, fast and accurate iterative method.

The iterative method [3] formula is

0 ( 0 ). H L L L H L C C C C         ,

where₀is the estimated volatility in the next iteration,_Lis the lowest volatility estimate,_His the highest volatility estimate,C₀is the actual market price option,C is _L the low value of the option.

The two segmentation method is suitable for all types of the options contract. The implied volatility is the assumption of unbiased estimate volatility. It means that the volatility will be the same at the same maturity time. But in practical terms, when the strike price is different at the same maturity time, the implied volatility will also be different.

(VSI). We can use it to estimate the future implied volatility.

2.2 Calculating implied volatility

In the previous part, it is mentioned that implied volatility can be solved by taking the first four basic parameters and the actual market price of options into the option pricing model. But actually, it is quite tough to solve the volatility by the deformation of the model. We often solve the volatility using some numerical methods. Like, Manaster [5] proposed the Newton-Raphson fast interior point search algorithm, but the estimate value was strongly dependent on the initial value; Liu Yang [15] put forward to use the optimal method to get the implied volatility based on the average options price. It is known that the interior point search algorithm needs the helps with the computer in order to get the approximation.

Due to the continuity in the Black-Scholes equation, the researchers brought forward to an easy algorithm which can be directly used by the market investors. Brenner [4] made the Taylor expansion to the standard normal distribution and it was given the formula of the implied volatility in the parity option. We can get

* * 2 t t t C T t S     , here * t

C is the market value in the parity option. However, it cannot calculate the implied volatility when the option is in or out of the option price. Chance [7] improved the Brenner’s model. He thought that the error between the in (out) option price and the average option price was caused by the strike price and the volatility, which wasC K*( *,*). Here we make two order-Taylor expansions, and then we can get

6 * * * * 2 * * ( ) ( ) 2 K K K C K q C C C K   .

Here *is the estimated value according to the Brenner model.

Chance model just considered the effect of the strike price and make the maturity time as the constant. So we cannot get the implied volatility at different maturity time. It also maybe has some influence on the accuracy.

Kelly [8] put the volatility as an implicit function of the option market price, and then rose computing the implied volatility algorithm which was based on the implicit function.

There is also another method called the implied volatility surface model [2], which is, the volatility is the implicit function of the maturity time and the strike price. In this model, it uses the parity option (S K e* r T( *t)). It is believed that the deviation between the parity option and non-parity option is aroused by the volatility, strike price and the maturity time, that isC K_t*( *,*,T*). Firstly, we take the two-order Taylor expansion toC_t*and then we can get

* * * * * ( ) * 1 * ( ) r T t C C _ K T te  n d      , * 3 2 * * * 2 ( ) * * 1 *2 ( ) ( ) 4 r T t C K T t e C_  n d          , * 2 * * ( ) * * 1 * * ( ) 2 r T t X C T te C n d K  _         , * * *2 _* 2 * * ( ) * ( ) * 2 2 * _{* *} ( )( ) 4 ( ) ( ) r T t r T t XT r T t C C re N d e n d K T _{T t}              _ , * * * *2 * * 2 * * * ( ) * ( ) * 1 1 * _* ( ) 4 ( ) ( ) 2 2 r T t r T t T K r T t C K C e n d e n d T _{T t}                _ .

Substitute them into【2.2.1】, we can get that:

* 2 * ( t) t 0 a   b   q , where * * * / 2 aC_{ } , * * * * * * * * ( ) K T bC_ C_ K C_  , * * * * * * 2 * ( ) * * * ( ) 2 K K K K T C K qC K   C    K  C .

By using the surface model, the numerical calculation shows that it has improved the estimated precision of volatility. At the same time, the implied volatility surface model can give the characters of volatility sneer and term structure of volatility.

8

Chapter 3 The Original Problem

3.1 The implied volatility as a constant

Black-Scholes [6] model leads an important role in option pricing and it is widely applied both in the theory and in practice. The volatility is the important part in the Black-Scholes. One of the assumptions of Black-Scholes is that we always treat the volatility  as the constant. According to the Black-Scholes formula, we can express the option value asV V S t( , ; , K T, ). From the option market, we can know whentt₀ ,SS₀ ,K K₀, T T₀ and the option value is V . ₀

We take all of them into the Black-Scholes formula; then derive an equation of :

0 ( 0, ; ,0 0, 0)

V V S t  K T . 【3.1】

Since V 0

  _

 , the implied volatility   0 is the only solution in the equation

【3.1】. Thus, from the price of the options with the strike price K and maturity ₀ timeT , we can derived the implied volatility as ₀   0.

Based on the Black-Scholes formula assumption, when the implied volatility  is a constant, the implied volatility 0 should not be related to the strike price K and 0

the maturity timeT . It seems that, according to the Black-Scholes pricing theory, the ₀ same assets with the same maturity time and different strike prices should have the same volatility. However, the market and the empirical test show that the volatility is not the constant. The volatility  is the function ofK T, . it can be expressed as  ( , )K T .

3.2 Volatility smile and volatility skew

will become larger. As the shape looks like smiling, it is known as “the smile volatility”. For “the smile volatility” phenomenon, so many scholars have done some significant researches in previous. Heston [17] pointed that the reason of the “the smile volatility” may be the non-continuous process of the stock price change in 1993. In 2003, Xiaorong Zhang [9] said that the main cause of “the smile volatility” was the assets price process assumptions, and the factor of the market mechanism which brought the additional risk and hedging costs to the option sellers. For a given maturity timeT, whentt₀, the asset price isS , the implied volatility ₀  varies with the strike price K . “The smile volatility” shows in picture

0

/ K S

The implied volatility curve of the stock option also can appear skewed; we call it “the fake smile”. “The fake smile” is related to the expectation of calibration of the future asset price movements. “The fake smile” graph is following

0

/ K S .

10

These graphs shows that treating the implied volatility as a constant can not reflect the reality problem. In one word, there is a certain error in assuming the implied volatility as a constant.

3.3 Some reasonable assumptions and a modified model

The reasonable assumption should be that the implied volatility  is the function of the time T and the asset priceS, that is, in the sense of the risk-neutral measure, we change the random process into the model

( ) ( )( ) ( ) ( , ( )) ( )

dS t S t rq dtS t  t S t dW t .

Then we can get the corresponding Black-Scholes equation

2 2 2 ( , ) 1 ( , ) ( , ) ( , ) ( ) ( , ) 0 2 V t S V t S V t S S t S r q S rV t S t  S S  _  _{ }  _{ }    .

It can be proved as follows Original Black-Scholes equation

( ) ( )

dB t rB t dt, 【3.3.1】

( ) ( ) ( , ( )) ( ) ( , ( )) ( )

dS t S t  t S t dtS t  t S t dW t . 【3.3.2】 We suppose that trading in the market and their price has the form

( )t V t S t( , ( ))

2 2 2 2 ( , ) ( , ) 1 ( , ) ( , ) 2 V t S V t S V t S rS S rV t S t S  S          . 【3.3.4】

After changing it, we have that

( ) ( )( ) ( ) ( , ( )) ( )

dS t S t rq dtS t  t S t dW t . 【3.3.5】 So now, we can get the equation of the Black-Scholes

2 2 2 ( , ) 1 ( , ) ( , ) ( , ) ( ) ( , ) 0 2 V t S V t S V t S S t S r q S rV t S t  S S  _  _{ }  _{ }    . 【3.3.6】

Here we use the call option V S T( , )(SK), 【3.3.7】 in order to get 【3.3.6】, we can apply Ito’s lemma to 【3.3.3】 and 【3.3.5】given

( )

(

)

( )

( ) ( )

( )

d



t

 

r

q

_



t dt





_

t



t dW t

, 【3.3.8】 where 2 2 1 ( ) 2 ( ) t S SS V r q SV S V r q V        , SFS F     .

The function of  , , , ,V V V V_{t S}, _SSare evaluated. We build a portfolio from the stock and the derivative. If (u u_S, _) denotes the relative portfolio and V denotes the value process, then

















( ) ( ) ( ) ( ) S S S dV V u r q dt dW u r q dt dW V u r q u r q dt V u u dW                        .

So, if we choose u and u_{S }, that u_S u_{ } 0. 【3.3.9】 Then we can make the dW term vanish. Knowing【3.3.9】and u_S u_ 1,

we can see thatuS ,u

               .

At last, we can get that S ,

S S S SV V u u SV V  SV V      .

So now we can get 【3.3.6】.

12

3.4 Original problem

After changing the model of the underlying asset, we question that “how can we determine the volatility of an underlying asset price from its option price quotes in the options market? Mathematically,tt S₀, S₀ , we know V S t( ₀, ; ,₀  K T_k, )_l V_{k l,} , (k=1,…,m, l=1,…,n), and how to get the volatility   ( , )S t ? ”

From the call-put parity, we know that we will get the same volatility   ( , )S t by using the call option or the put option. In this paper, we take the call option as the example. Here we show the problem P at first.

Problem P Let V V S t( , ; , K T, )be the price of the call option, and it satisfies the equation: 2 2 2 2 1 ( , ) ( ) 0 2 V V V S t S r q S rV t  S S            (0    S , 0 t T), 【3.4.1】 ( , ) ( ) V S T  SK , (0  S ). 【3.4.2】 Suppose attt*, (0 t* T₁),S S*, we know that

* *

( , ; , , ) ( , )

V S t  K T F K T , (0K ,T₁ T T₂).

Chapter 4 Dupire’s Equation [16]

4.1 Dupire’s formula

In this paper, we use the Dupire’s Method to get the volatility . Assume that the European call option price isV V S t K T( , ; , ).

Based on the Black-Scholes formula

( ) 0

( , ; , ) r T t ( , ; , )( )

V S t K T e 



p S t  T K d,

wherep S t( , ; , ) T is the transition probability density of stochastic process S . _t Let T 0 andK 0, the price at time 0 of a European call option expiring at time

T with the strike price K is

( , ) rT ( ) ( , )

K

V T K e



 SK p T S dS . And here we can calculate that

( ) ( , ) rT rT K K K V e SpdS K pdS e p T S dS K K             







, 【4.1.1】 2 2 ( , ) rT V e p T K K     . 【4.1.2】

So we can know that

2 ( ) 2 ( , ; , ) r T t V p S t T e K  _    . 【4.1.3】

Now, in order to get the volatility, we use the formula

( , ) rT ( ) ( , ) K V T K e



 SK p T S dS , ( , ) ( ) ( , ) ( ) rT rT K K V p T S re S K p T S dS e S K dS T T            





 . 【4.1.4】 In 【4.1.4】, we take the ₀ ( ) ( , ) K p T S I S K dS T     



, 【4.1.5】

we can get that the equation 【4.1.4】then it can be expressed as

0 rT V rV e I T   _{  }  . 【4.1.6】

For 【4.1.2】, we use the Kolmogorov’s Equation and we can get the formula:

14 2 2 2 2 1 ( ) ( ( , ) ( , ) 2 K S K _S  S T S p T S dS     



. 【4.1.7】 Here we take 1 ( )( ) ( ( , )) K I S K r q Sp T S dS S       



, and that 2 2 2 2 2 1 ( ) ( ( , ) ( , ) 2 K I S K S T S p T S dS S      



.

We can get I₁ and I using integration by parts ₂

1 _K( )( ) ( ( , )) I S K r q Sp T S dS S       





( )( ) ( , ) |



_K ( ) ( , ) K S K r q Sp T S  r q Sp T S dS      



( ) ( ) ( , ) ( , ) K K r q  S K p T S dS rK p T S dS  



 



,

where we have to assume thatlim ( , ) 2 0

Sp T S S  . So we can get that

( ) ( ) ( , ) ( ) rT

K

rq



 SK p T S dS  r q Ve . And then we can get the expression of the

1

I , ₁ ( ) rT ( , )

K

I  r q Ve rK



p T S dS . 【4.1.8】 Now we will compute the I ₂

2 2 2 2 2 1 ( ) ( ( , ) ( , )) 2 K I S K S T S p T S dS S      



2 2 2 2 1 1 ( ) ( ( , ) ( , ) | ( , ) ( , ) 2 S K _S  S T S p T S K 2 K _S  S T S p T S dS       _{ }  _  _  _{ }    



2 2 1 ( , ) ( , ) 2 K T K p T K  . 【4.1.9】 We take the 【4.1.8】and 【4.1.9】into the 【4.1.7】we can get the I at first, then 0

we take the I into the equation 【4.1.6】, we can get that ₀

2 2 1 ( ) ( , ) ( , ) ( , ) 2 rT rT K V rV e r q Ve rK p T S dS K T K p T K T     _{  }  _{  }     



. 【4.1.10】

Here we plug 【4.1.1】and 【4.1.3】into 【4.1.10】, we can get that:

2 2 2 2 1 ( ) ( ) ( , ) 2 V V V rV r q V r q K K T K T K  K  _{     }  _     .

2 2 2 2 1 ( , ) ( ) , (0 , ) 2 |_{T t} ( ) , (0 ) V V V K T K r q K qV K t T T K K V S K K                     _{    }  . 【4.1.11】

Then we solve the equation 【4.11】, we can get the volatility 

2 2 2 ( ) ( , ) 1 2 V V r q K qV T K K T V K K   _{ }  _      . 【4.1.12】

Therefore, attt*, (0 t* T₁),S S*, we know that

* *

( , ; , , ) ( , )

V S t  K T F K T , (0K ,T₁ T T₂).

Then we can easily calculate the ( , )K T by 【4.1.12】. From the equation 【4.1.12】 we can see that to get ( , )K T , we must compute the derivatives F_KK,F and _K F_T at first. But a small error in F can result in a big changes in its derivatives, especially in its second derivatives. So the result is overly sensitive and the algorithm is unstable. Then, we can say that the algorithm to calculate( , )K T is ill-posed.

We must point out that F K T( , ) is given on a set of discrete

points



(K T_k, ) (_l



k1,..., ,m l1,... )n . But in the domain (0K ,

1 2)

T  T T , F K T( , ) is got from the discrete points using interpolation or extrapolation technique and it will occur the error. This will also make the value of

( , )K T

 lose its distortion. So Dupire’s method cannot easily be applied to practical problems.

4.2 Duality problem

16

Using the Dupire’s dual method, we can make the original problem P into the following inverse problem of parabolic equation(q0).

QuestionD: Suppose that att 0,S S₀, we know that all the maturity time are in the domain



T T , the price of the European option asset 1, 2



V K T0( , ) with the strike

priceK(0, ) , we need to find the implied volatility ( , ) K T

 : 0 1 1 2 ( ), (0 , 0 ) ( , ) ( , ), ( , 0 ) K T T K K T K T T T T K             _{    }  , 【4.2.1】 subject to V K T S( , ; ₀, 0)V K T₀( , ), (0  K ,T₁ T T₂). 【4.2.2】 It satisfies the equation:

2 2 2 2 2 0 0 1 ( , ) , (0 , 0 ) 2 |_T ( ) , (0 ) V V V K K T rK T T K T K K V S K K     _  _  _{    }       _{    }  . 【4.2.3】

Problem D can be divided into two problems: ProblemD₁: GivenV K T , find( , )₁ ₀( )K .

Subject toV K T( , )₁ V K T S( , ;_{1 0}, 0;₀( ))K .

ProblemD : Given₂ V K T( , ),(T₁ T T₂)and ₀( )K , find ( , )K T .

In order to simplify the problem D , we need to set and change the variables 0 ln K y S  ,  T, 【4.2.4】 * 0 0 0 1 ( , ) ( y, ) v y V S e S    , 【4.2.5】 0 0 1 ( , ) ( y, ) v y V S e S    , 【4.2.6】 0 1 1 2 ( ), ( , 0 ) ( , ) ( , ), ( , ) a y y T a y a y y T T            _{  }  . 【4.2.7】

And we must point out that

2 0 0 0 1 ( ) ( ) 2 y a y   S e , 【4.2.8】 2 0 1 ( , ) ( , ) 2 y a y   S e  . 【4.2.9】 Now we will get the new problems in new variables:

1

P : Find a y₀( ) in the domain



y  , 0  T₁



where v y( , ; a₀) at time T₁,

*

1 0 1

( , ; ) ( , )

v y T a v y T satisfies the equations

18

Question P is a typical terminal problem, which is, we know the solution in last ₁ time T₁, and then we need to find the first factora y₀( ).

2

P : Find a y( , ) in the domain(y,T₁  T₂), where a y T( , )₁ a y₀( ) and

*

( , ; ) ( , )

v y a v y satisfies the equations

2 2 1 0 ( , )( ) ( , ) ( , ; ) v v v v a y r y y y v y T v y T a                 _  . 【4.2.11】

The question P is different from the question₂ P . For this problem, the given value 1 *

( , )

v y is in the whole domain and it needs to find the first factor a y( , ) which is the function of yand .

Question P and ₁ P have the same difficulties, that is, the ill-posed in the inverse ₂ problem. It means that a y₀( ) and a y( , ) have no continuous dependence on the given function * 1 ( , ) v y T and * ( , )

v y  _{. In another word,} a y0( ) _and a y( , ) _are

Chapter 5 The Regularization Method

5.1 Regularization idea

To solve the ill-posed problem, we take the regularization method in this paper which is proposed by A. N. Tikhonov in 1950s. Its main idea is as follows.

Let U , F be given metric spaces, A is an operator defined on F , that isA F: U.

The original problem: GivenV₀U, find ₀F ,

satisfied A(₀)V₀. 【5.1.1】 There are two possibilities

1.We know that AFU, but AF U, according to the given V₀U, 【5.1.1】may be have no solution in F .

2. According to the given V , 【5.1.1】may be have solution ₀ ₀, but it is unstable. That is,₀ has no continuous dependence on V .We can say that the small change in ₀

0

V in Umay lead to a big change 0inF.

The main idea of the regularization method is to use a cluster of well-pose problems to take place of the original problem. Although it is just an approximation solution to the original problem 【5.1.1】, the process of getting the approximation solution is quite stable. We can achieve the approximation solution in computing and use this solution to instead of the original true solution. We call the well-pose problems as the “regularization problem”. It is often from the operator A and takes the parameter

 to implement it.

Regularization Problem: GivenV₀U, find _F(0),

20

 ( )---the regularization operator,

 ---the regularization factor.

The regularization method is that solving the well-posed problem【5.1.2】 by using the solution _ instead of ₀. The  can be any number. When0,_ ₀ we can get the true solution, but the algorithm is unstable. When becomes bigger, _ will get more far from the true solution₀. So in practice, we try our best to take the

 smaller then we can make the process more stable.

5.2 The regularized version of the original problem

Now we go back to the problem P and₁ P . Firstly we take the regularization method ₂ into the problem P and we call it problem₁ Q₀.

ProblemQ₀: At T₁, finda y₀( ) F   , which satisfies 0 0 0 0 0 ( ) min ( ) a F J a J a    , where 0 * 2 0 2 0 1 0 1 1 ( ) | ( , ; ) ( , ) | | | 2 2 da J a v y T a v y T dy dy dy     

_

 

_

, 0 ( , ; )

v y a is the solution to the Cauchy problem P 【4.2.10】, ₁

2 0 0 2 0 1 { ( ) | 0 ( ) , |da | } F a y a a y a dy dy     

_

  .

We take a and₁ a as the given positive constant. We often call ₂ Fas admissible set. This set make the Cauchy problem 【4.2.9】to have the exactly solution. J0( )a is called the cost function. a₀ a y₀( )is called control variable. a y₀( )



is named as optimal control or minimizer. We call this variation problem Q₀ as the optimal control problem.

There exists a minimizer a y0( ) F



 in variation problem Q0 .We can prove it by

Proof: let ( ,V a be a minimizing sequence. Since _{n n}) J a( _n)Cand due to the special structure ofJ,we infer that

2_{( )}

n L R

a C

  (Cis dependent of n ).

And now we use the imbedding theorem we can get that

1 2_{( )}

n C R

a C.

Thus, we know that

1/2,1/4_{( )} ( , ) n C Q V y   C_{, (Cis dependent of n)} 2 1/2,1 1/4_{( )} ( , ) n _{C w} V y    C, for anywQ, hereQ  0,*_.

Thence we can choose a subsequence of a and_n V , again denoted by _n a and _n V ,such _n that 1 2 0 ( ) ( ) ( ) n a y a y C     , uniformly in C( ) (0 1 2    ), 0 ( , ) ( , ) n V y V y   , uniformly in C , /2( )Q C_loc2 ,1/2( )Q     .

We can check easily that (a y V y0( ), 0( , ))

 

satisfy these two equations

( )( _{yy y}) ( ) _y 0

LV V_ a y V V  r q V  , y,(0,*),

( , 0) (1 y)

V y  e , y.

By the Lebegue control convergence theorem and the weak semi continuity of the L 2 norm we get that

0 0

( ) lim inf ( _n) min ( )

n a F

J a J a J a



 

  .

Hence, we can know that ( ₀) min ( ₀)

a F

J a J a

 

 .

So we can know that there exists a minimizer a y₀( ) F



22

h 1 (T₂ T₁) N

  ,

_n  T₁ n h , (n0,1,... )N .

From the problemQ₀, when  ₀ we can get thea y₀( )



.And then we use the induction method, then we can geta y_n( ) a y( ,_n)



 when  _n.

ProblemQ_n: Suppose we know a y₀( ),...a_{n 1}( )y

 

 and ( , ;v y ak)



(k 1,...n1)which is the solution to the equation

2 2 ( )( ) k v v v v a y r y y y              , (y,k1  k ), 【5.2.1】 1 1 1 ( , _k ) ( , _k ; _k ) v y v y a      , (y). .【5.2.2】 Find a y_n( ) F   in the domain



y,_n1   _n



,

which satisfies ( ) min ( )

n n n n n a F J_ a J_ a    . We define * 2 2 2 1 1 1 ( ) | ( , ; ) ( , ) | | ( ) ( ) | | | 2 2 n n n n n n n n da J a v y a v y dx a y a y dy dy h h dy     _         _   _  







where v y( ,_n;a_n)is the solution to the equations 【5.2.1】and 【5.2.2】.

We also can prove that there exists a minimizer a y_n( ) F



 in variation problem Qn[13][14].

Now we consider whenh0, how the

 

a y_n( )



will show. We fixh, and define the

We can know that whenh0,



ah( , ),y v yh( , )



convergences to the limit function

24

Chapter 6 The Stability of the Volatility Function [1]

Our main reference in this chapter is [1]. In this part, we need to check whether the volatility functions a y₀( )



and a y( , ) is stable or not. Firstly, we establish a necessary condition for the minimizer a y( )



. Set a y₀( )



is a minimizer, and Fis a convex set, thus for a arbitrary given

^ ( ) h y F, ^ 0 ( ) (1 ) a_ y  a h F      (

 

0,1 ).

And here we define the function j( )

0 0 ( ) ((1 ) ) j  J_  a h     ,

then we can know that

2 2 * 0 1 0 1 1 ( ) ( , ;(1 ) ) ( , ) 2 2 da j v y T a h v y T dy dy        

_

   

_

.

Now we attain the minimum at0, thus

^ ' * 2 2 1 1 0 0 (0) d | ( , ) ( , ) | d | d ((1 ) ) | j v y T v y T dy a h dy d  d dy   _    _   _     _ 





 .【6.1】 We know that j'(0)0,

and herev y_( , ) which is satisfied the equation

2 2 ( ) ( , 0) (1 y) v v v v a y r y y y v y e                 _ _{ }  _     _{ }  . 【6.2】 Here we make ( , )y dv d     

 and calculate 【6.2】,we can get that

2 2 _^ 0 2 2 ( ) ( ) ( , 0) 0 v v a y r h a y y y y y y                 _  _ _  _ _  _ _  _{ }  _  _{ }        _  . 【6.3】

So we can express 【6.1】as

Here v y( , )



is the solution to the problemP . And ₁ a y( ) a y₀( ). ( , )₀ y 



 is the

solution to 【6.3】.

When0, we have the following equation

2 2 _^ 0 0 0 0 0 0 2 2 0 0 ( ) ( ) ( , 0) 0 v v L a y r h a y y y y y y              _{    } _{ }   _{    } _  _  _ _{ }  _{    }    _{ }    . 【6.5】 Let  ( , )y 

be the solution to the ad-joint problem of the problem 【6.5】i.e

* 1 * 1 1 1 0, ( , 0 ) ( , ) ( , ) ( , ), ( ) L y T y T v y T v y T y                _{  }  , 【6.6】

where the differential operator *

L



is the ad-joint operator of L

 and 2 * 0 0 2( ) ( ) L a a r y y y           _{ }   _   _          .

From the Green formula,

1 _* 0 0 0 ( ) T L L dyd         

 

1 _{0 0 0} 0 0 0 0 0 0 ( ) ( ) ( ( ) ) ( ( ( ) )) ( ( ) ) T a y a y a y r dyd y y y y y y   _  _{   }   _           _{      }     _     _           

 

1 0 0 0 (  )_{ T} (  )_ dy         _  _  



.

From the equations 【6.5】 and【6.6】,we can derive that

1 2 _^ * 0 0 1 1 1 2 0 ( ) ( , ) ( , ) ( , ) T v v h a dyd y T v y T v y T dy y y           _{ }  _{ }  _  _{  }          

 



. 【6.7】

Now we take【6.7】into 【6.4】, we can get that

26 Now we let 1 2 2 0 1 ( ; , ) T ( v v) f y v d y y          _{ }    



, 【6.9】

then we can see that the equation 【6.1.8】can change into the form of

^ ^ ^ 0 ( 0) ( ; , )( 0) 0, d d a h a f y v h a dy h F dx dx         _{  }  _{  }    



. We know that ^ ( )

h y is arbitrary, so the equation above is equal to

2 0 2 ( ; , ) 0 d a f y v dy        , a₂ a y₀( ) a₁    , 【6.10】 2 0 2 ( ; , ) 0 d a f y v dy        , a y₀( ) a₁   , 【6.11】 2 0 2 ( ; , ) 0 d a f y v dy        , a y₀( ) a₂   . 【6.12】

These equations are the double obstacles elliptic variation inequality problem. So now we finish proving the necessary condition of the variation problem Q₀ Then, we start to talk about the sufficient condition of the variation problemQ₀.

If a y0( ) F



 is the minimizer of the variation problemQ0, there exists a triplet



a y v y0( ), ( , ), ( , )  y



  

, which is a solution to the following PDE problem in the

domain



y  , 0  T₁



2 0( ) 2 0( ) 0 ( , 0) (1 y) v v v v a y a y r y y y v y e               _{   }          , 【6.13】 2 0 0 2 * 1 1 1 ( ( ) ) ( ( ) ) 0 ( , ) ( , ) ( , ) a y a y r y y y y T v y T v y T  _{ }                                . 【6.14】

* 0 0 0 1 ( , ) ( y, ) v y V S e S    , 1 2 2 0 1 ( ; , ) T ( v v) f y v d y y          _{ }    



.

From these equations, we can see that 【6.13】is a Cauchy problem to a forward parabolic equation; 【6.14】is a Cauchy problem to a backward parabolic equation; 【6.10】【6.11】【6.12】are a variation inequality to a second order ODE. So this is a series of forward-backward parabolic equations coupled with an elliptic variation inequality. Here we will prove the implied volatility a y₀( )



uniqueness [10][12].

Proof: supposea y₁( ),a y₂( ) be the two minimizes of the problem

( ) min ( ) F J_{ } J_      .

When ha₂, we take _ a₁.When ha₁,we use _ a₂ in the equation【6.8】. Then we have the following equations

1 1 1 1 2 1 1 2 1 0 ( ) ( ) 0 T yy y V V a a dyd a a a dy                

 



, 【6.15】 1 2 2 2 1 2 2 1 2 0 ( ) ( ) 0 T yy y V V a a dyd a a a dy                

 



, 【6.16】

where



V_i,_i



(i1, 2)are the solutions of the equations 【6.13】and 【6.14】.All of them are witha₀ a i_i( 1, 2)



  .

From the equations 【6.15】and 【6.16】,we can get that

28





1 1 1 0 1 1 ( ) T yy y A V V dyd a   

_{ }

 









1 2 2 1 1 1 2 0 0 1 T T yy y yy y A V V dyd A V V dyd a  _{  }   

 

 

 

  .

From the assumption, there exist a point y₀  



,



and it satisfies that

0 1 0 2 0

( ) ( ) ( ) 0

A y a y a y  .

Here we can prove that the implied volatility a y₀( )



uniqueness.

Now we look into the other variation problem Q_n.The limit function a y( , ) is determined whenh0,



ah( , ),y v yh( , )



convergences to the limit function



a y( , ), ( , ) v y



.

Suppose a y( , ) is the volatility determined by the limit function. And it exist a function



a y( , ), ( , ) v y



in the domain



y,T1   T2



which is the solution of

the following partial differential equations:

2 2 ( , ) ( , ) 0 v v v v a y a y r y y y     _  _  _  _     , 【6.17】 2 2 * 2 2 1 ( , ) 0 2 a a v v v y y  y    _ _{   } _      , a2 a y( , ) a1,【6.18】 2 2 * 2 2 1 ( , ) 0 2 a a v v v y y  y     _{ }   _  _     , a y( , ) a1, 【6.19】 2 2 * 2 2 1 ( , ) 0 2 a a v v v y y  y    _ _{   } _      , a y( , ) a2, 【6.20】 1 1 ( , ) ( , ) v y T v y T   , 【6.21】 1 ( , ) ( ) a y T a y   . 【6.22】 We find out that the v y( , )



and a y₀( )



30

Chapter 7 Calculation of the Implied Volatility

In order to get the implied volatility in the time domain



0,T , we must divide it ₂



into two parts. Firstly we must get a y₀( )



from the problemQ₀, and then use the

0( )

a y



as the initial value to solve the series of the variation problem Q_n in the time domain



T T . Or we can directly solve the Euler equations 【6.17】₁, ₂



【6.18】【6.19】 【6.20】【6.21】【6.22】. Now we can get the valuea y( , ) .

There are two ways to get the numerical solution to the problemQ₀.One is starting from the variation problemQ₀, discrete it, and make it into optimization problem which has a convex constraint, find its approximate solution at last. The other is starting from the necessary condition of the variation problem, it means that find the triplet



a y v y( ), ( , ), ( , )  y



  

,that is satisfied the forward-backward partial differential

equation problem coupled with an elliptic variation inequality【6.12】【6.13】【6.14】

【6.10】【6.11】.

In this paper, we focus on using the second method, that is solving the problem【6.12】 【6.13】【6.14】【6.10】【6.11】.

7.1 Calculating the option price

Suppose that whent0, *

S S , and we can get the different strike price Kat the same maturity time, the option price V S( *, 0;K)F K( ).Then we define the function that: * * * 1 ( ) ( y) v y F S e S  , here y ln K_* S  .

For testing, the initial implied volatility 0.4. At the same time, assuming

*

1

S S  ,K_ max10,r0.2,T₁ 1,T₂ 5.We divide the strike price K, the time

 

0,T and 1



T T into 1, 2



N30 steps andM1 300,M2 1000,respectively.

As we know, when the implied volatility is constant, we can use the Black-Scholes formula directly. The Black-Scholes formula is following

( ) 1 2 ( , ) ( ) r T t ( ) V S t SN d Ke  N d , here 2 1 ln ( )( ) 2 S r T t K d T t        , d₂  d₁  T t .

We take all the parameters into the Black-Scholes formula, we can get the option price

( )

F K . We can show F K( )in the picture.

7.2 Calculating the local implied volatility

Now we use the option price F K( ) which is got form the part 7.1. Due to the strict

monotonic of the option priceF K( ), we can know that the Black-Scholes equation

32

(1) At first, we can directly use the defined function

2 * 0 0 1 ( ) ( ) 2 y a y   S e ,

We can get that ₀ 1 0.42 0.08

2

a    .

(2)Now we start to solve the Cauchy problem 【6.13】

2 0( ) 2 0( ) 0 ( , 0) (1 y) v v v v a y a y r y y y v y e               _{   }          , where we take a₀ a₀ 

 , we can get the solution v y₀( , ) now.

(3)We start to get the ₀ from the Cauchy problem to the backward parabolic equation【6.14】 2 0 0 2 * 1 1 1 ( ( ) ) ( ( ) ) 0 ( , ) ( , ) ( , ) a y a y r y y y y T v y T v y T  _{ }                                , whicha₀ a₀   , * 1 0 1 ( , )y T v y T( , ) v y( )    , we can get * ( ) v y it from the 7.1.

(4)From the definition of the 1

2 2 0 1 ( ; , ) T ( v v) f y v d y y               



, we can get 1 2 0 0 0 0 ₀ 0 2 1 ( ; , ) T ( , ) v v f y v y d y y          _  _    



.

(5)Now we start to solve the variation inequality equation

2 0 2 ( ; , ) 0 d a f y v dy        , a2 a y0( ) a 1    , 2 0 2 ( ; , ) 0 d a f y v dy        , a y₀( ) a₁   , 2 0 2 ( ; , ) 0 d a f y v dy        , a y₀( ) a₂   , where the f y v( ; , )  

is defined in the step (4), therefore we can get the solution

1( )( )

( )

n aa y .

(6)At last, we get all of them back to the original variables and compute the local implied volatilityn( )K . * ( ) 2 ( ) 2 (ln ) n n n K K a y a S    .

In the time domain



T T , we use the same method as in1, 2



 

0,T .But we must point out 1

that the initial value is at the timetT₁.

We also show the local implied volatility _n in the picture.

7.3 The error in the implied volatility

We can express the error term between the original implied volatility and the local implied volatility as_err   _local.We can see it in the picture clearly.

34

From the figure, for the price part, we can see that the error of local implied volatility is large when the strike price is more different from the initial option price. That is, in the cases, deeply in the money and out of the money, the local volatility needs to be used. For the maturity data part, the local implied volatility is more accurate than the original one when the maturity data is small. And when it is large, the error will become smaller.

7.4 The error in the option price

We can also show the option price error. It is the error between the option price which is derived by the given volatility and the option price which is calculated by the local volatility.

At last, we will show the option price error pictures. When the implied volatility 0.4

36

7.5 The implied volatility is a function of the stock price

The same assumptions as below but the implied volatility  is not a constant. We use the smile curve to take the place of it. The original volatility can show in the picture:

Now we should apply the Black-Scholes equation to get the option price F K( ).The

Black-Scholes equation is as follows

2 2 2 2 1 0 2 V V V S rS rV t  S S           .

We also give the picture ofF K( ).

38

The error of the implied volatility is showing.

The error of the option price is following.

Chapter 8 Conclusion

As we know that the implied volatility has been playing an important role both in judging the futures market and application of strategic investment options. So the analysts need to know that how the implied volatility varies. In fact, we cannot be prophets to forecast what  will be in the future. But we can be interpreters and translate all the information for the option markets into the volatility  of the underlying asset. The only thing for us is that we need to take the Today’s observed market option pricing into the Black-Scholes equations and solve it by the numerical methods.

40

the implied volatility.

In this paper, we use a new method to calculate the implied volatility [1]. The numerical methods what we introduced in this paper is stable. On one hand, we prove the stability of this method. On the other hand, we show the method of how to calculate the implied volatility in this paper. The best way to check whether this method is useful is to see the error term of the implied volatility. Now we will show the error of the implied volatility.

The picture of the error is following.

From above, we can see that the error is very small. We can say that the algorithm we proposed in this paper is stable. So we can use this numerical algorithm in nowadays option markets. We just need to put all the information into this method and then use the numerical method to calculate the implied volatility. At last, we can get the implied local volatility t ( , )S tt _{.Using this method, the implied volatility}

( , )

t S tt

 

References

[1]. Lishang Jiang, Mathematical Modeling and Methods of Option Pricing,2003.

[2]. Ailing Zhang, Chongfeng Wu, An Improved Differential Approach to Computing Improved Volatility—Implied Volatility Surface Model, Dec 2007.

[3]. Jiangming Zhu, The Estimated Volatility in Option Pricing, Sep 2005.

[4] Brenner M, Subrahmanyam M G, A Simple Formula to Compute the implied Standard Deviation, May 1988.

[5]. Manaster S, Koehler G, The Calculation of Implied Variances from the Black-Scholes Model, 1982.

[6]. Black F, Scholes M, The Pricing of Options and Corporate Liabilities, 1973.

[7]. Chance D M, A Generalized Simple Formula to Compute the Implied Volatility, 1996.

[8]. Kelly M A, Faster Implied Volatilities Via the Implicit Function Theorem, 2006.

[9]. Xiaorong Zhang, The Analysis of the “Smile Implied Volatility” in options, 2003.

42

Assets from Option Prices, Oct 2001.

[11]. Bouchouev I and Isakov V , The Inverse Problem of Option Pricing, 1997.

[12]. Bouchouev I and Isakov V, Uniqueness, Stability and Numerical Methods for the Inverse Problem that Arises in Financial Markets, 1999.

[13]. Lishang Jiang, B J Bian, Identifying the Principal Coefficient of Parabolic Equations with Non-divergent Form, Sep 2005.

[14]. Lishang Jiang, B J Bian, The Regularized Implied Local Volatility Equations- A New Model to Recover the Volatility of the Underlying Asset from Option Pricing, 2005.

[15]. Liu Yang, Jianning Yu, Zuicha Deng, The Optimization Approach for the Determination of the Implied Volatility Through the Average Option Price, 2006.

[16]. Dupire, Pricing with a Smile, 1994.

Appendix

The calculation Results

When the implied volatility is 0.4, we can get the local volatility data:

44

The error form is:

46

The error form is: