Valuation of portfolios under uncertain volatility: Black-Scholes …hh.diva-portal.org/smash/get/diva2:238852/FULLTEXT01.pdf · 2009-09-26 · The Uncertain Volatility Method (UVM)

Technical report, IDE0739 , November 14, 2007

Valuation of portfolios underuncertain volatility:

Black-Scholes-Barenblattequations and the static

hedging

Master’s Thesis in Financial Mathematics

Anna Kolesnichenko & Galina Shopina

School of Information Science, Computer and Electrical EngineeringHalmstad University

Valuation of portfolios under uncertainvolatility: Black-Scholes-Barenblattequations and the static hedging

Anna Kolesnichenko & Galina Shopina

Halmstad University

Project Report IDE0739

Master’s thesis in Financial Mathematics, 15 ECTS credits

Supervisor: Prof. Ljudmila A. BordagExaminer: Prof. Ljudmila A. Bordag

External referee: Prof. Vladimir Roubtsov

November 14, 2007

Department of Mathematics, Physics and Electrical engineeringSchool of Information Science, Computer and Electrical Engineering

Halmstad University

Preface

We would like to thank Prof. Ljudmila A. Bordag for a very interestingsubject of investigations, set us on the right track, carefully reading themanuscript and her useful comments. We was glad to work with so famousscientist, interesting women, provident supervisor. Big thanks for possibilityto study at Department of Mathematics, Physics and Electrical Engineering

We would also like to thank:

External referees for attentiveness, honesty, a valuable advice ;

Associated professor Vasilieva Tatiana for her help us in Volgograd StateUniversity, benevolence and comprehension ;

Our parents for their love and help us ;

Best Thanks to Halmstad University, which was our real home during thetime we have been working with Master’s Theses.

ii

AbstractThe famous Black-Scholes (BS) model used in the option pricing theorycontains two parameters - a volatility and an interest rate. Bothparameters should be determined before the price evaluation procedurestarts. Usually one use the historical data to guess the value of theseparameters. For short lifetime options the interest rate can be estimatedin proper way, but the volatility estimation is, as well in this case,more demanding. It turns out that the volatility should be consideredas a function of the asset prices and time to make the valuation selfconsistent. One of the approaches to this problem is the method ofuncertain volatility and the static hedging. In this case the envelopesfor the maximal and minimal estimated option price will be introduced.The envelopes will be described by the Black - Scholes - Barenblatt(BSB) equations. The existence of the upper and lower bounds for theoption price makes it possible to develop the worse and the best casesscenario for the given portfolio. These estimations will be financiallyrelevant if the upper and lower envelopes lie relatively narrow to eachother. One of the ideas to converge envelopes to an unknown solutionis the possibility to introduce an optimal static hedged portfolio.

iii

iv

Contents

1 Introduction 1

2 The Uncertain Volatility Method (UVM) 32.1 The uncertain volatility model . . . . . . . . . . . . . . . . . . . . 32.2 Upper and Lower bonds estimations for the sensitivity parameter

delta VS(S, t) for a Call option . . . . . . . . . . . . . . . . . . . . 10

3 The Static Hedging 153.1 The narrowing procedure for the envelopes with the static hedging 163.2 Numerical examples for the static hedging . . . . . . . . . . . . . 18

4 The numerical approach 214.1 The Finite Difference Method . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Application of the explicit finite difference method to thevalues of the option V (S, t) . . . . . . . . . . . . . . . . . 22

4.1.2 Application of the explicit finite difference method to thedelta function . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Stability of the finite difference method . . . . . . . . . . . . . . . 254.2.1 Stability of the finite difference method by valuating of the

option V (S, t) . . . . . . . . . . . . . . . . . . . . . . . . . 254.2.2 Stability of the finite difference method by the delta func-

tion valuation . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Implicit method for the option price V (S, t) . . . . . . . . . . . . 284.4 Implicit method for the function δ . . . . . . . . . . . . . . . . . . 30

5 Conclusions 33

Notation 35

Glossary 37

Bibliography 39

6 Appendix 41

v

vi

Chapter 1

Introduction

In option pricing literature we can found many models for the determinationof the volatility. There are different methods of determination volatility value.The most simple model assumes constant volatility. It was offered by Black,Scholes and Merton in 1973. There was a famous paper, but in real world marketprices of options cannot be described by constant volatility in proper way. Morecomplicated is the implied volatility model, where volatility is constructed bythe Black-Scholes equation for a currently traded contracts. A next possibilityto introduce the volatility is the stochastic volatility model. In that case thevolatility is assumed to follow some random process.

In our paper we will consider uncertain volatility model, numerical meth-ods for the solving BSB equation and the static hedging. We will achieve someimprovements of previous results.

In detail, Chapter 2 presents uncertain volatility model for option prices andfor the sensitivity parameter delta. We will represent some examples where σ0

and σ1 are functions depending on time t.In Chapter 3 will illustrated using the static hedging narrow the envelop. We

improved spread between upper and lower bounds of the option price.In Chapter 4 we will represent explicit and implicit finite difference methods

for valuation of an option price. We study the stability problem for the explicitmethod. Our algorithms will be applied to different types of European andAmerican options such as straddle, butterfly, double barrier and others.

Chapter 5 include some concluding remarks.Finally, Chapter 6 contains most important programmes, which were used for

computing .

1

2 Chapter 1. Introduction

Chapter 2

The Uncertain Volatility Method(UVM)

2.1 The uncertain volatility model

In this chapter we consider the paper of Meyer [1]. He studied a portfolio witha non-convex and a non-monotone value function. The author proved that aBlack-Scholes-Barenblatt (BSB) equation deeply connected to the Black-Scholesequation. The author consider three equivalent forms of Black-Scholes-Barenblattequation and define and analyze the idea of the static hedging.

In [1] are considered a simple Up-and-Out European Call V (S; t) with anmaturity date T , a strike price K and a Up-and-Out barrier in S = X > K .There is a nonlinear problem because the payoff of the Up-and-Out EuropeanCall is non-convex.

For 0 < S < X and 0 < t ≤ T the following differential operator is introduced

L(σ; r)V ≡ 1

2σ2(S; t)S2VSS + r(S; t)SVS − r(S; t)V − Vt = 0, (2.1)

where t = T − τ (T is the maturity date and τ is current time). It is assumedthat an uncertain volatility σ and an interest rate r are functions of time t andthe asset price S.

The boundary conditions for the Up-and-Out Call are

V (0, t) = V (X, t) = 0, (2.2)

and initial the condition is

V (S, 0) = (S −K)+. (2.3)

The author of [1] used in his reasoning the standard maximum principle fora partial differential equation (PDE) of the parabolic type. Let us now look atthe value of the Up-and-Out Barrier option.

3

4 Chapter 2. The Uncertain Volatility Method (UVM)

To use the maximum principle we shall avoid a discontinuity at (X, 0).According to equations (2.2),(2.3) we can see, that

limS→X

V (S, 0) 6= limt→0

V (X, t).

The plot of the function V (S, t) is presented in Figure 2.1

Figure 2.1: Payoff for the Up-and-Out

Call at t = 0, K = 10, S ∈ [0; 140],X = 140.

Figure 2.2: The piecewise linear approxi-

mation for the Up-and-Out Call at t = 0,

K = 10, S ∈ [0; 140], X = 140, 0 ≤ ε < 1.

In [1] the initial condition was replaced by a piecewise linear function, whichapproximate the original condition (see Figure 2.2). Let 0 ≤ ε < 1,

V (S, 0) =

0, S < KS −K, K ≤ S ≤ (X − ε)(X − ε−K)X−S

ε, X − ε < S < X.

As laid down by [1], we assume that the functions σ(S, t) and r(S, t) arebounded above and belove

σ0(S, t) ≤ σ(S, t) ≤ σ1(S, t), (2.4)

where σ0(S, t) ≥ c > 0 and

r0(S, t) ≤ r(S, t) ≤ r1(S, t), (2.5)

where r0(S, t) ≥ 0.The author of [1] introduced functions V0(S, t) and V1(S; t) such that

V0(S, t) ≤ V (S, t) ≤ V1(S, t). (2.6)

They are the lower and upper bounds of V (S, t) for all S ∈ (0; X) and t ∈ (0; T ],for all σ and r which are within the above bounds (2.4), (2.5), respectively. In[1] is assumed throughout that BS equation has a solution for all such σ and r.

Valuation of portfolios under uncertain volatility:Black-Scholes-Barenblatt equations and static hedging 5

To find the lower and upper bounds V0 and V1 the author used Black-Scholes-Barenblatt equation

LBSB0 V0 ≡

1

2f0(σ)2S2V0SS + g0(r)(SV0S − V0)− V0t = 0, (2.7)

where the function g0(r) is chosen in the following way

g0(r) =

{r0(S, t), (SV0S − V0) ≥ 0,r1(S, t), (SV0S − V0) < 0,

(2.8)

and, correspondingly, the function f0(σ) is represented by

f0(σ) =

{σ0(S, t), V0SS ≥ 0,σ1(S, t), V0SS < 0.

(2.9)

Author of [1] to produce proofs that

V0(S, t) ≤ V (S, t) ≤ V1(S, t). (2.10)

We will not present full proof in our work (see [1]). But we prove, that V0 is thelower bound in case simple European Call option. Let us investigate the equation(2.7) for European Call option at time t = 0.1 with strike price K = 50 and timeof expiry T = 1.

Figure 2.3, Figure 2.4 shows that in case of the European Call option g0(r) =r0(S, t), i.e., (SV0S − V0) ≥ 0 for all S and f0(σ) = σ0(S, t), i.e., V0SS ≥ 0 for allS.

Hence the solution V (S, t) for the European Call is convex for all values Sand t and it means that the BSB equation can be reduced to the standard Black-Scholes equation, i.e. to the equation

L(σ0, r0)V0 = 0.

The equation (2.7) for the function V0(S, t) can be rewritten in another form,which is more convenient for further investigations. We represent a as a suma = a+ + a−, consequently LBSB

0 V0 takes the form

LBSB0 V0 ≡ 1

2σ2

0S2V +

0SS + 12σ2

1S2V −0SS

+r0(SV0S − V0)+ + r1(SV0S − V0)

− − V0t = 0.(2.11)

Let us add and subtract L(σ, r)V0 (see (2.1)), to the above expression (2.11).Then the right-hand expression in the previous formula take the form

LBSB0 V0 ≡ 1

2σ2

0S2V +

0SS +1

2σ2

1S2V −0SS + r0(SV0S − V0)

+ + r1(SV0S − V0)− − V0t

+1

2σ2S2V +

0SS +1

2σ2S2V −0SS + r(SV0S − V0)

+ + r(SV0S − V0)− − V0t

− 1

2σ2S2V +

0SS −1

2σ2S2V −0SS − r(SV0S − V0)

+ − r(SV0S − V0)− + V0t = 0.


Figure 2.3: Plot of V0SS for the Call op-

tion V0 with r = const = 0. The solid

line corresponds to σ = 0.6, long dashed

line to σ = 0.2 and short dashed line to

σ = 0.4.

Figure 2.4: Plot of (SV0S − V0) for Call

option with σ = const = 0.2. The solid

line corresponds to r = 0.8, long dashed

line to r = 0.2 and short dashed line r =0.4.

Combine the similar terms we obtain the equation for the lower bond V0(S, t)

L(σ, r)V0 ≡ 12S2((σ2 − σ2

0)V+0SS + (σ2 − σ2

1)V−0SS

)+ ((r − r0)(SV0S − V0)

+ + (r − r1)(SV0 − V0)−) .

(2.12)

In [1] the author give proof that the function V0(S, t) is a lower bound for theoption value V (S, t) for any σ and r between the imposed limits (2.4), (2.5). Wewill not go in detail on this fact.

Analogously, we have for the upper bound V1(S, t) the following equation

LBSB1 V1 ≡

1

2f1(σ)2S2V1SS + g1(r)(SV1S − V1)− V1t = 0,

where the function g1(r) and f1(σ) are given by

g1(r) =

{r0(S, t), (SV1S − V1) ≥ 0,r1(S, t), (SV1S − V1) < 0,

(2.13)

and

f1(σ) =

{σ0(S, t), V1SS ≥ 0,σ1(S, t), V1SS < 0.

(2.14)

The equation (2.12) in this case can be rewritten in the form

L(σ, r)V1 ≡ 12S2((σ2 − σ2

0)V+1SS + (σ2 − σ2

1)V−1SS

)+ ((r − r0)(SV1S − V1)

+ + (r − r1)(SV1 − V1)−) .

(2.15)


We will use the following identities for obtaining a symmetric form of theseequations

a+ =a + |a|

2, a− =

a− |a|2

.

Let σ2 and r are

σ2 =σ2

0 + σ21

2, r =

r0 + r1

2.

Insert the last expression in (2.12) and obtain

12S2((

σ20+σ2

1

2− σ2

0)V+0SS + (

σ20+σ12

2− σ2

1)V−0SS

)+(( r0+r1

2− r0)(SV0S − V0)

+ + ( r0+r1

2− r1)(SV0 − V0)

−) .

Let us collect the similar terms

1

2S2σ2

1 + σ20

2

(V +

0SS − V −0SS

)+

r1 − r0

2

((SV0S − V0)

+ − (SV0 − V0)−) ,

so that introducing the function

F (S, V, VS, |VSS|) =1

2

σ21 − σ2

0

2S2|VSS|+

r1 − r0

2|SVS − V |, (2.16)

equations (2.12), (2.13) can be rewrite in another forms

L(σ, r)V0 = F (S, V0, V0S, |V0SS|), (2.17)

L(σ, r)V1 = −F (S, V1, V1S, |V1SS|). (2.18)

The initial and boundary conditions for equations (2.17),(2.18) are

V0(S, 0) = V1(S, 0) = V (S, 0) = 0,

V0(0, t) = V1(0, t) = V (0, t) = 0,

V0(X, t) = V1(X, t) = V (X, t) = 0.

The equations (2.12),(2.7),(2.17) and (2.15),(2.13),(2.18) are equivalent toeach other. They are be called Black-Scholes-Barenblatt equations.

Example 2.1.1 We repeat calculations given in [1]. At first step we constructthe envelopes of a Double-Barrier-European-Straddle V (S, t) for t = T = 0.1which is defined by

L(σ, r)V (S, t) = 0,V (S, 0) = (S − 100)+ + (100− S)+,

V (80, t) = V (120, t) = 0,(2.19)

where

0.1 < σ < 0.2,

0.05 < r < 0.06.

The results of calculations we are given in Figure 2.5.


Figure 2.5: Envelopes V0(S, t) and

V1(S, t) (dashed lines) and Black-Scholes

solution V (S, t) (solid line) for the

European-Double-Barrier-Straddle (2.19),

with σ = 0.1, r = 0.55, K = 100, S ∈[80, 120], t = T = 0.1, σ0 = 0.1, r0 = 0.05,

σ1 = 0.2, r1 = 0.06.



solution V (S, t) (solid line) for the

Double-Barrier-European-Butterfly-

Spread (2.20), with σ = 0.2, r = 0.1,

K1 = 90, K2 = 110, S ∈ [80, 120],t = T = 0.25, σ0 = 0.15, r0 = 0.1,

σ1 = 0.25, r1 = 0.1.

Example 2.1.2 Now we represent Double-Barrier-European-Butterfly-Spread V (S, t)at T = 0.1 with K1 = 90, K2 = 110 . It was considered in the paper of D.M.Pooey,P.A.Forsyth and K.R.Vetzal (2003) [8].

Option price is described by:

V (S, 0) = (S − 90)+ − 2(S − (110 + 90)/2)+ + (S − 110)+,V (80, t) = V (120, t) = 0,

0.15 < σ < 0.25,r = 0.1.

(2.20)

In Figure 2.6 we represent results of calculations. It is similar to the examplewhich was given in [8].

Example 2.1.3 The BSB equations can be also applied to an American option.We again repeat calculations given in [1]. From the beginning we consider Strad-dle similar to (2.19) with early exercise boundary s(t) for S < K. The optionprice V (S, t) at t = T = 0.1 is described by

L(σ, r)V (S, t) = 0,V (S, 0) = (S − 100)+ + (100− S)+,

V (120, t) = 0,V (s(t), t) = 100− s(t),

VS(s(t), t) = −1,

(2.21)


where

0.1 < σ < 0.4,

0.04 < r < 0.06.

The results of calculations are represented in Figure 2.7. The curve between theenvelopes is the solution of the Straddle at time t = T for the fixed values σ = 0.2and r = 0.05.



solution V (S, t) (solid line) for the Amer-

ican Straddle (2.21), with σ = 0.2, r =0.05, K = 100, S ∈ [70, 120], t = T = 0.1,

σ0 = 0.1, r0 = 0.04, σ1 = 0.4, r1 = 0.06.



solution V (S, t) (solid line) for the Amer-

ican Straddle (2.22), with σ = 0.2, r =0.05, K = 100, S ∈ [70, 120], t = T = 0.1,

σ0 = 0.1, r0 = 0.04, σ1 = 0.4, r1 = 0.06.

Further we consider the Straddle similar to (2.21) with early exercise boundarys(t) for all S

L(σ, r)V (S, t) = 0,V (S, 0) = (S − 100)+ + (100− S)+,

V (120, t) = 0,V (s(t), t) = 100− s(t),

VS(s(t), t) = ±1,

(2.22)

where

0.1 < σ < 0.4,

0.04 < r < 0.06.

The results of calculations we present in Figure 2.8.


We can now see that envelopes V0 and V1 locally approach a constant volatilitysolution of the Black-Scholes equation and they have jumps when the convexityof the option value changes.

We used implicit and explicit methods for solving this problems. They givethe similar to each other results.

2.2 Upper and Lower bonds estimations for the

sensitivity parameter delta VS(S, t) for a Call

option

We introduce the sensitivity parameter delta (δ) as usual like first derivativeoption price V with respect to the asset price S.

To simplify the reasoning in [1] author assumed that σ(S, t) and r(S, t) areconstant or function of t only. We will repeat Meyer’s argumentation in thissection and consider new more advanced examples.

After differentiating the Black-Scholes equation (2.1) with respect to S weobtain

1

2σ2S2VSSS + σ2SVSS + rSVSS + rVS − rVS − VSt = 0,

1

2σ2S2VSSS + (σ2 + r)VSSS − VSt = 0.

And now we can write down the following differential equation for the deltaVS ≡ δ(S, t).

M(σ, r)δ ≡ 1

2S2δSS + (σ2 + r)SδS − δt = 0. (2.23)

We study the Up-and-Out Call. In this case we have following initial conditions

δ(S, 0) =

{0, S < K,1, S > K.

It is known that for σ and r in (2.4), (2.5) all solutions of the Black-Scholesequation have to lee between the envelopes V0 and V1. As long as V0(0, t) =V1(0, t) and V0(X, t) = V1(X, t) we can write down following inequalities forδ(0, t) and δ(X, t)

V0S(0, t) ≤ δ(0, t) ≤ V1S(0, t),V1S(X, t) ≤ δ(X, t) ≤ V0S(X, t).

(2.24)


Similar to previous results for the option price V (S, t) we transform of BSBmodel (2.23) to equation on a lower bond for δ(S, t)

M(σ, r)δ0 = 12S2(σ2 − σ2

0)δ+0SS + (σ2 − σ2

1)δ−0SS

+(σ2 − σ20)Sδ+

0S + (σ2 − σ21)Sδ−0S

+(r − r0)Sδ+0S + (r − r1)Sδ−0S,

(2.25)

Suppose thatδ−0S + δ+

0S = δ0S,

then we can rewrite previous expression in the following form

−1

2σ2

0S2δ+

0SS −1

2σ2

1S2δ−0SS − σ2

0Sδ+0S − σ2

1Sδ−0S − r0Sδ+0S − r1Sδ−0S + δ0t = 0,

or1

2σ2

0S2δ+

0SS +1

2σ2

1S2δ−0SS + (σ2

0 + r0)Sδ+0S + (σ2

1 + r1)Sδ−0S − δ0t = 0.

We obtain equation for the lower bond on δ(S, t) in the following form

1

2f0(σ)2S2δ0SS +

(k0(σ)2) + g0(r)

)Sδ0S − δ0t = 0. (2.26)

Here we denoted by f0(σ), k0(σ), g0(σ) the following functions

f0(σ) =

{σ0, δ0SS ≥ 0,σ1, δ0SS < 0,

(2.27)

k0(σ) =

{σ0, δ0S ≥ 0,σ1, δ0S < 0,

(2.28)

g0(σ) =

{r0, δ0S ≥ 0,r1, δ0S < 0.

(2.29)

A symmetric form of these equations we obtain if we use the identities

δ+0SS =

δ0SS + |δ0SS|2

,

δ−0SS =δ0SS − |δ0SS|

2,

σ2 =σ2

0 + σ21

2,

r =r0 + r1

2.


Insert this identities in (2.23) we obtain the representation

M(σ, r)δ0 =1

2σ2S2δ0SS + (σ2 + r)Sδ0S − δ0t =

1

2(σ2 − σ2

0)S2 δ0SS + |δ0SS|

2+

1

2(σ2 − σ2

1)S2 δ0SS − |δ0SS|

2

+1

2(σ2 − σ2

0)Sδ0 + |δ0S|

2+

1

2(σ2 − σ2

1)Sδ0 − |δ0S|

2

+(r − r0)Sδ0S + |δ0S|

2+ (r − r1)S

δ0S − |δ0S|2

=

1

2S2

(σ2δ−0SS +

δ0SS

2(σ2

1 − σ20)

)+ S

(σ2δ0SS −

δ0SS

2(σ2

0 + σ21) +

σ21 − σ2

0

2

)+S

(rδ0S −

r0 + r1

2δ0S +

r1 − r0

2|δ0S|

)=

1

2

σ21 − σ2

0

2S2|δ0SS|+

σ21 − σ2

0

2S|δ0S|+ S

r1 − r0

2|δ0S|

Finally the symmetric form of the BSB equation for the lower bond of δ(S, t)is

M(σ, r)δ0 =1

2S2σ2

1 − σ20

2|δ0SS|+ S

(σ2

1 − σ20

2+

r1 − r0

2

)|δ0S|. (2.30)

Example 2.2.1 The valuation of δ for the Up-and-Out Call (2.31) at t = T =0.1 is represented in Figure 2.9.

L(σ, r)V (S, t) = 0, (2.31)

V (S, 0) = (S − 100)+,

V (0, t) = V (120, t) = 0.

For σ and r we have constraints

0.2 = σ0 ≤ σ ≤ σ1 = 0.4,

0.04 = r0 ≤ r ≤ r1 = 0.06,

where for the Black-Scholes (2.1) solution we use σ = 0.3 and r = 0.05.

In Figure 2.9 you can see two different envelopes for the delta function. Theresults of calculation with the explicit method is represented by dash-dot curvesand with the implicit method by dashed curves. The bounding dash-dot curvesare not envelopes. It repeats the results of [1]. In our paper we have more preciseresults (dashed curves) obtained because we used the implicit finite differencemethod and apply new IT-decision.


Figure 2.9: The bounds for function

Delta δ0 and δ1 (dash-dot lines stays for

explicit method and dashed lines for im-

plicit method), function δ (solid line) for

the Up-and-Out European Call (2.31).

Figure 2.10: The bounds for function

Delta δ0 and δ1: dashed lines for σ and r

like in Example 2.2.2 and solid lines for σ

and r like in Example 2.2.1.

It many papers about the uncertain volatility were stated that the boundaryfunctions σ0 and σ1 dependent on S and t. But in all examples the boundaryfunctions σ0 and σ1 were constant. We will represent an example where σ0 and σ1

are functions dependent on time t. We will use the implicit method to calculatevalue of (2.31) in this example.

Example 2.2.2 Now we consider quite the same Up-and-Out Call, but for arange of σ we take a linear function with respect to time t. This case was notconsidered before in our knowledge. We have

L(σ, r)V (S, t) = 0, (2.32)

V (S, 0) = (S − 100)+,

V (0, t) = V (120, t) = 0.

σ and r lee in the following boundaries

0.2 + t = σ0 ≤ σ ≤ σ1 = 0.4− t,

0.04 = r0 ≤ r ≤ r1 = 0.06,

where for a Black-Scholes solution we use σ = 0.3 and r = 0.05.

In Figure 2.10 two different envelopes for delta are represented. Solid linesrepresents results of previous Example 2.2.1. Dashed lines represents Example2.2.2. Functions σ0 and σ1 constrict the boundary of the uncertain volatility withincreasing t. For Call options it leads to narrowing of the envelops.


Chapter 3

The Static Hedging

It is known, that hedging is a method of an insurance a real transaction from amarket risk. It is realized usually with help of special option portfolio. Hedgersminimize the risk by buying or selling other options.

The modern hedging was possible after three important events only. All thisevents happened around 1973.

• The Breton-Wood agreement was reversed, as a result of it appear a possi-bility of the market risk initiation.

• The Chicago Board Options Exchange (CBOE) was opened. There is oneof the biggest places where options are traded. The option trading give usa real possibilities to manage the market risk.

• The famous paper of Black, Scholes & Merton about the option pricingtheory and the risk managment was published.

There are two different types of hedging: Static hedging and Dynamic hedg-ing.

Static hedging. We buy necessary options for the hedging purposes and holdthem during all the hedging horizon. We can (for example) apply this strategy ifall options in our hedge portfolio are based on the same asset. For this methodis important: the Prices of all our options have to equally change during all thehedging horizon.

Dynamic hedging. We permanently change the structure of hedge portfolioduring all the hedging horizon. It is obvious that the dynamic hedging expectmaking many deals. Prices of all our options have to equally change during aminimum period. We need dynamic hedging if other instruments or too expensiveor not possible from different reason.

We will consider the Static hedging in this chapter. We will use it for nar-rowing of the envelopes of the option price. Over the last time interesting andmany significant results about the Static hedging have appeared. One of the first

15

16 Chapter 3. The Static Hedging

theoretical work in this field was published by Bowie & Carr (1994), later areDerman, Ergener & Kani (1995) and Avellaneda, Levy & Paras (1995). There isstill a very active research area. Some of the latest paper are from Andersen, An-dreasen & Eliezer (2002), Carr & Wu (2002) and Meyer (2004). We will proceedclose to Meyer’s paper [1] here.

3.1 The narrowing procedure for the envelopes

with the static hedging

The envelopes for the option price V (S, t) may lie very wide from each otheralthough the bounds for the values the volatility σ and the interest rate r aretight. We can narrow the envelopes with help of the static hedging. We assumethat the related instruments are traded.

Meyer assumed only the validity of the Black-Scholes equation, and relyingon the maximum principle, described the static hedging in the context of PDEswith the uncertain volatility.

Let us define the static hedging for an European option with an other Eu-ropean option having the same maturity date. We price in this chapter theEuropean option with the strike price K and the expiration T today. We againset t = T − τ where τ is the calendar time. The option value V (S, t) satisfies theBlack-Scholes equation

L(σ, r)V (S, t) = 0, (3.1)

the initial and boundary conditions characterize the option. As in previous chap-ter the value of σ and r are uncertain but bounded above and below by

σ0(S, t) ≤ σ(S, t) ≤ σ1(S, t),

r0(S, t) ≤ r(S, t) ≤ r1(S, t),

and ”today” we have following bounds for V (S(T ), T ), which cannot be improved

V0(S(T ), T ) ≤ V (S(T ), T ) ≤ V1(S(T ), T ),

where V0 and V1 satisfy conditions (or equations)(2.17),(2.18).We shall assume that N other European options on the same asset are freely

traded. They have the same expiration and various pay-offs at t = 0. Weknow their values at the expiration. Let us denote their quoted price today byWi(S(T ), T ), where i = 1, ..., N .

We suppose like in [1] the option prices Wi(S, t), where i = 1, ..., N satisfythe Black-Scholes equation (3.1) for the same uncertain volatility and the sameinterest rate functions as the option V (S, t) which we wish to bound.


Let us constitute the following portfolio:

π(S, t,−→c ) = V (S, t)−M∑i=1

ciWi(S, t), (3.2)

where −→c is the set c1, .., cM of real numbers. We choose −→c once and it remainunchanged(i.e. static) for the all portfolio life time. The sum

∑Mi=1 ciWi(S, t) is

called a static hedge of V (S, t).

Now we can apply Black-Scholes operator for the portfolio π(S, t,−→c ). Weobtain the following expression

L(σ, r)π(S, t,−→c ) = 0,

with the initial condition

π(S, 0,−→c ) = V (S, 0)−M∑i=1

ciWi(S, 0).

Furthermore the values for π(S, t,−→c ) at S0 and S1 are known.

Similar to the previous chapter we can bound portfolio π(S, t,−→c ) from belowwith the BSB equation (2.17) for all values of the constants ci.

For all σ and r in their specified ranges the lower bound π0 for π(S(T ), T,−→c )gives the lower bound on V (S(T ), T )

V (S(T ), T ) = π(S(T ), T,−→c ) +M∑i=1

ciWi(S(T ), T ).

In the same way we determine an upper bound π1 for a portfolio π

π1(S, t,−→d ) = V (S, t)−

N∑j=1

djWj(S, t), (3.3)

which gives us an upper bound on V (S, t).

For simplicity, we assume that the lower and upper bounds are independenton each other. The portfolios (3.2) and (3.3) may have different numbers andtypes of options.

Our goal is to define the coefficients ci and di. If we look on the seller positionthen to determine the lowest upper bound we have to find ci which minimizesthe function

Eu(−→c ) = π1(S(T ), T,−→c ) +N∑

i=1

ciWi(S(T ), T ). (3.4)


Similarly, if we look on the buyer position then to determine the biggest lowerbound from V (S(T ), T ) we have to find dj which maximizes

El−→d ) = π0(S(T ), T,

−→d ) +

N∑j=1

djWj(S(T ), T ). (3.5)

Let us define

V1(S(T ), T ) = minc1,...,cMEu(−→c ),

V0(S(T ), T ) = mind1,...,dNEl(

−→d ).

It is easy to see, that

V0(S(T ), T ) ≤ V1(S(T ), t) ≤ Eu(−→0 ) = V1(S(T ), T ),

V0(S(T ), T ) = El(−→0 ) ≤ V0(S(T ), t) ≤= V1(S(T ), T ),

where V0 and V1 satisfy (2.7) and (2.17) at t = T .The arguments remain unchanged if the options Wi expire at different times.

3.2 Numerical examples for the static hedging

Now we represent some examples of the narrowing procedure for the envelopeswith the static hedging. We repeat partly calculations given in [1]. Improve ofthe previous results. We values the European Call option with T = 0.5, K = S =100, r = 0.05 and 0.2 ≤ σ(S, t) ≤ 0.3.

Example 3.2.1 We consider a European Put W (S, t) with K = 100, T = 0.5and r = 0.05.We know W (100, 0.5) = 5.791, which corresponds to an impliedvolatility of σ = 0.25. For the static hedging we construct the portfolio

π(S, t, c) = V (S, t)− cW (S, t).

The Black-Scholes equation hold true

L(σ, r)π = 0 (3.6)

where the initial conditions are

π(S, 0, c) = (S −K)+ − c(K − S)+ (3.7)

and the boundary conditions defined by

limS→∞

π(S, t, c) = limS→∞

(S −Ke−rt), (3.8)

π(0, t, c) = −cKe−rt. (3.9)


Figure 3.2.1 Upper- and lower-bounds

for the European Put (3.6)-(3.9) depen-

dence on the parameter c.

Figure 3.2.2 The surface Eu(c1, c2) for

the portfolio (3.10).

Now we prove that the static hedging give us a very good improvement. If weapply the BSB equation to the above Call we obtain the following result

6.89 = V0(100; 0.5) < V (100; 0.5) < V1(100; 0.5) = 9.64.

There is too large spread between upper and lower envelopes.The static hedging give us zero spread in this case. We presented this result

in Figure 3.2.1 for c = 1.Let us consider more complicated example.

Example 3.2.2 Let us consider two European Call options W1 and W2 withstrikes K1 = 90 and K2 = 110. They are freely traded with option pricesW1(100, 0.5) = 14.42 and W2(100, 0.5) = 4.22, respectively (an implied volatil-ity for both of them is equal to σ = 0.25).

We construct the portfolio

π(S, t, c) = V (S, t)− c1W (S, t)− c2W (S, t). (3.10)

Now we solveL(σ, r)π = 0,

with the initial conditions

π(S, 0, c) = (S − 100)+ − c1(S − 90)+ − c2(S − 110)+,

and the boundary conditionsπ(0, t, c) = 0,

π(X, t, c) = (X − 100e−rt)− c1(X − 90e−rt)− c2(X − 110e−rt).


The example gave us following results

V0(S(T ), T ) = 7.76012 at (c1, c2) = (0.3, 1),

V1(S(T ), T ) = 8.5237 at (c1, c2) = (1, 0.5).

In [1] Meyer obtain following results

V0(S(T ), T ) = 7.760 at (c1, c2) = (0.6, 0.6),

V1(S(T ), T ) = 8.667 at (c1, c2) = (0.7, 0.5).

In his paper the spread for the same case was equal to 1.038, in our work thespread is equal to 0.9225. We have some improvement about 11, 13%.

Chapter 4

The numerical approach

4.1 The Finite Difference Method

Let us represent some citation from [5] to explain idea of the finite differencemethod.

”The idea underlying finite-difference methods is to replace the partial deriva-tives occurring in partial differential equation by approximations based on Taylorseries expansions of functions near the point or points of interest.

For example, the partial derivative ∂V/∂t may be defined to be the limitingdifference

∂V

∂t(S, t) = lim

∆t

V (S, t + ∆t)− V (S, t)

∆t.

If, instead of taking the limit ∆t → 0, we regard ∆t as nonzero but small, weobtain the approximation

∂V

∂t(S, t) ≈ V (S, t + ∆t)− V (S, t)

∆t+ O(∆t).

This is called a finite-difference approximation or finite difference of ∂V/∂t be-cause it involves small, but not infinitesimal, differences of the dependent variableV . This particular finite-difference approximation is called a forward difference,since the differencing is in the forward t direction; only the values of V at t andt+∆t are used. As the O(∆t) term suggests, the smaller ∆t is, the more accuratethe approximation.

We also have

∂V

∂t(S, t) = lim

∆t→0

V (S, t)− V (S, t−∆t)

∆t,

so that the approximation

∂V

∂t(S, t) ≈ V (S, t)− V (S, t−∆t)

∆t+ O(∆t).

21

22 Chapter 4. The numerical approach

is likewise a finite-difference approximation for ∂V/∂t.We call this finite-difference approximation a backward difference.We can also define central differences by noting that

∂V

∂t(S, t) = lim

∆t→0

V (S, t + ∆t)− V (S, t−∆t)

2∆t.

This gives rise to the approximation

∂V

∂t(S, t) ≈ lim

∆t→0

V (S, t + ∆t)− V (S, t−∆t)

2∆t+ O((∆t)2).

Note that central differences are more accurate (for small ∆t) than eitherforward or backward differences.

We can define finite-difference approximations for the x-partial derivative of Vin exactly the same way. For example, the central finite-difference approximationis easily seen to be

∂V

∂t≈ V (S, t + ∆t/2)− V (S, t−∆t/2)

∆t+ O((∆t)2).

For second partial derivatives, such as ∂2V/∂S2, we can define a symmetricfinite-difference approximation as the forward difference of backward-differenceapproximations to the first derivative or as the backward-difference of forward-difference approximations to the first derivative. In either case we obtain thesymmetric central-difference approximation

∂2V

∂S2(S, t) ≈ V (S + ∆S, t)− 2V (S, t) + V (S −∆S, t)

(∆t)2+ O((∆t)2).

Although there are other approximation, this approximation to ∂2V/∂V 2 ispreferred, as its symmetry preserves the reflectional symmetry of the secondpartial derivative. It is left invariant by reflections of the form x 7→ −x. It is alsomore accurate than other similar approximations.”

4.1.1 Application of the explicit finite difference methodto the values of the option V (S, t)

In paper of Ioffe & Ioffe [2] the finite difference method is applied to pricing barrieroptions. Authors considered the finite difference method for the usual Black-Scholes equation (2.1). We apply this method to the nonlinear BSB equation.

To implement a finite difference method we take a grid where the horizontaldirection represents discrete values of the variable S with constant step ∆S and


the vertical direction represents discrete points in time with constant step ∆t. Inany node (i, j) we calculate the values of time t and variable S as

ti = i ∆T, i = 0, ..., Nt − 1,

Sj = S1 + i ∆S, j = 0, ..., NS − 1,

where

∆S =S2 − S1

NS

,

∆t =T

Nt

,

where NS is a number of nodes in horizontal direction, and Nt in vertical direction.Partial derivatives we approximate in the following way:

∂V

∂t=

V (j, i + 1)− V (j, i)

∆t,

∂V

∂S=

V (j + 1, i)− V (j − 1, i)

2∆S,

∂2V

∂S2=

V (j + 1, i) + V (j − 1, i)− 2V (j, i)

∆S2,

For ∂V/∂t we use the backward approximation, because we have to do witha backward differential equation (2.7).

For ∂V/∂S we use central approximation, because this approximation is themost precesses one.

The expressions (2.8),(2.9) will take a form:

g0(r) =

{r0(S, t), Sj(V (j + 2, i)− V (j, i)) ≥ 2V (j + 1, i),r1(S, t), Sj(V (j + 2, i)− V (j, i)) < 2V (j + 1, i),

and, correspondingly

f0(σ) =

{σ0(S, t), V (j + 2, i)− V (j, i) ≥ V (j + 1, i),σ1(S, t), V (j + 2, i)− V (j, i) < V (j + 1, i),

The PDEs (2.7) is now reduced to an algebraic system

1

2f0(σ)2S2

j

V (j + 1, i) + V (j − 1, i)− 2V (j, i)

∆S2+

g0(r)

(V (j + 1, i)− V (j − 1, i)

2∆SSj − V (j, i)

)− V (j, i + 1)− V (j, i)

∆t= 0.


Otherwise

V (j + 1, i)

(1

2

f0(σ)2

∆S2S2

j +g0(r)

2∆SSj

)+ V (j, i)

(1

∆t− f0(σ)2

∆S2S2

j − g0(r)

)+V (j − 1, i)

(1

2

f0(σ)2

∆S2S2

j −g0(r)

2∆SSj

)− V (j, i + 1)

∆t= 0.

We obtain the following finite difference equation to calculate values of the func-tion V (S, t) in the grid nodes

V (j, i + 1) = ∆t

(12

(f0(σ)∆S

)2

S2j −

g0(r)2∆S

Sj

)V (j − 1, i)

+

(1−

(f0(σ)∆S

)2

S2j ∆t− g0(r)∆t

)V (j, i)

+

(12

(f0(σ)∆S

)2

S2j + g0(r)

2∆SSj

)∆tV (j + 1, i).

(4.1)

The boundary conditions in Example 2.0.1 are

V (0, i) = V (NS − 1, i) = 0, (4.2)

and initial conditions are

V (j, 0) = (Sj −K)+ + (K − Sj)+. (4.3)

The initial and boundary conditions can be similarly obtain for another examples.

4.1.2 Application of the explicit finite difference methodto the delta function

The same transformations as in part 4.1.1 for equations (2.27),(2.29), (2.28),(2.25)give us:

δt =δ(j, i + 1)− δ(j, i)

∆t,

δS =δ(j + 1, i)− δ(j − 1, i)

2∆S,

δSS =δ(j + 1, i) + δ(j − 1, i)− 2δ(j, i)

∆S2,

g0(r) =

{r0(S, t), δ(j + 1, i) ≥ δ(j − 1, i),r1(S, t), δ(j + 1, i) < δ(j − 1, i),

k0(r) =



and

f0(σ) =

{σ0(S, t), δ(j + 1, i) + δ(j − 1, i) ≥ 2δ(j, i),σ1(S, t), δ(j + 1, i) + δ(j − 1, i) < 2δ(j, i).

The PDEs (2.23) is now reduced to an algebraic system of equations

1

2f0(σ)2S2

j

δ(j + 1, i) + δ(j − 1, i)− 2δ(j, i)

∆S2

+(k0(σ)2 + g0(r)

) δ(j + 1, i)− δ(j − 1, i)

2∆SSj −

δ(j, i + 1)− δ(j, i)

∆t= 0.

Otherwise

δ(j + 1, i)

(1

2f0(σ)2

S2j

∆S2+(k0(σ)2 + g0(r)

) Sj

2∆S

)+δ(j − 1, i)

(1

2f0(σ)2

S2j

∆S2−(k0(σ)2 + g0(r)

) Sj

2∆S

)−δ(j, i)

(f0(σ)2

S2j

∆S2− 1

∆t

)− δ(j, i + 1)

1

∆t= 0.

We obtain the following finite difference equation for the function δ(j, i + 1)

δ(j, i + 1) = ∆t2

Sj

∆S

(f0(σ)2 Sj

∆S+ k0(σ)2 + g0(r)

)δ(j + 1, i)

+∆t2

Sj

∆S

(f0(σ)2 Sj

∆S− k0(σ)2 − g0(r)

)δ(j − 1, i)

−δ(j, i)(f0(σ)2∆t

S2j

∆S− 1)

.

(4.4)

The initial conditions in Example 2.2.1 can be rewritten in terms of the finitedifferences as

δ(j, 0) =

{1, S > K,0, S ≤ K,

and boundary conditions are

δ0(0, i) = δ0(NS − 1, i) = 0.

This conditions done for Example 2.2.1 and can be easy repeated for anothercase.

Remark 4.1.1 The program for this calculations is presented in the Appendix.

4.2 Stability of the finite difference method

4.2.1 Stability of the finite difference method by valuatingof the option V (S, t)

Now we consider stability of the scheme (4.1). In [3] M.Avellaneda and R.Buffstudied the stability problem. The introduce the special inequalities to solve this


problem in case of BS equation. Now we recalculate the inequalities in our case.We have following conditions on A0, B0 and C0

A0 = ∆t

(1

2

(f0(σ)

∆S

)2

S2 − g0(r)

2∆SS

)> 0,

B0 = 1−(

f0(σ)

∆S

)2

S∆t− g0(r)∆t > 0,

C0 = ∆t

(1

2

(f0(σ)

∆S

)2

S2 +g0(r)

2∆SS

)> 0,

or the in other words A0 > 0, C0 > 0, A0 + C0 < 12.

It is easy to see, that C0 is positive for all S because it is the sum of positiveterms. Let us rewrite expression for A0. We know that ∆t > 0, S > 0. In thiscase we can divide the expression by ∆t and S

1

2

(f0(σ)

∆S

)2

S − g0(r)

2∆S> 0,

We also know, that ∆S > 0. Let us multiple this equation by 2 and ∆S

1

2

f0(σ)2

∆SS > g0(r),

finally, for ∆S we obtain the expression:

∆S <f0(σ)2S

g0(r). (4.5)

For A0 + C0 we have the following inequality:

∆tS2f 20 (σ)

2∆S2<

1

2,

Otherwise

∆t

∆S2<

1

2f 20 (σ)S2

(4.6)

We have to update (4.5)to prove it for all S and t, where S1 < S < S2, andthe expressions for f0 (2.9),g0 (2.8). We have the following expression:

∆S < minS,t

f0(σ)2S

g0(r). (4.7)

For (4.6) we should minimize the function 12f2

0 (σ)S2 . Finally, we obtain

∆t

∆S2< min

S,t

1

2f 20 (σ)S2

(4.8)


4.2.2 Stability of the finite difference method by the deltafunction valuation

Similarly, to study the stability of the (4.15), where

A0 = ∆tS

∆S

(1

2f0(σ)

S

∆S− k0

2(σ)− g0(r)

)> 0,

B0 = 1− f0(σ)2S2∆t

∆S2> 0,

C0 = ∆tS

∆S

(1

2f0(σ)2 S

∆S+ k0

2(σ) + g0(r)

)> 0.

It is easy to see, that C0 always positive. Let us rewrite the inequality for B0 as

f0(σ)2S2∆t

∆S2< 1.

After evaluation∆t

∆S2<

1

f0(σ)2S2. (4.9)

Let us divide the inequality for A0 by the positive number ∆t∆S

S and obtain

1

2f0(σ)2 S

∆S> k0(σ)2 + g0(r),

so that

∆S <Sf0(σ)2

2(k0(σ)2 + g0(r)

) . (4.10)

We have to determine the general expression for ∆S for all values of S. We know,that S1 ≤ S ≤ S2 and the expressions for f0(σ), g0(r), k0(σ) are given in (2.27),(2.29), (2.28).

To prove (4.10) we approximate the value ∆S for all S and t

minS,t

Sf0(σ)2

2(k0(σ)2 + g0(r)

) ,

i.e.

∆S < minS,t

Sf0(σ)2

2(k0(σ)2 + g0(r)

) .

The same transformations as before we use to prove the inequality (4.9)

∆t

∆S2< min

S,t

1

Sf0(σ)2.


4.3 Implicit method for the option price V (S, t)

In this section we consider a semi implicit finite difference method. We use socalled ”frozen” procedure in our work. This procedure will be applied to thenonlinear part in BSB equation. Instead to use functions g0, g1, f0, f1 on currenttime step use it from previous time step (”froze” their values). This method ispresented in [7].

The partial derivatives in this case we approximate in the following way

∂V

∂t=

V (j, i)− V (j, i− 1)

∆t,

∂V

∂S=

V (j + 1, i)− V (j − 1, i)

2∆S,

∂2V

∂S2=

V (j + 1, i) + V (j − 1, i)− 2V (j, i)

∆S2,

For ∂V/∂t we use the forward approximation, because we have to do with a back-ward differential equation (2.7). In the case of implicit finite difference methodwe should do our calculations in following direction from unknown data to knowndata.

The expressions (2.8),(2.9) will take a form

g0(r) =

{r0(S, t), Sj(V (j + 2, i)− V (j, i)) ≥ 2V (j + 1, i),r1(S, t), Sj(V (j + 2, i)− V (j, i)) < 2V (j + 1, i),

and, correspondingly,

f0(σ) =

{σ0(S, t), V (j + 2, i)− V (j, i) ≥ V (j + 1, i),σ1(S, t), V (j + 2, i)− V (j, i) < V (j + 1, i).


1

2f0(σ)2S2

j

V (j + 1, i) + V (j − 1, i)− 2V (j, i)

∆S2+

g0(r)

(V (j + 1, i)− V (j − 1, i)

2∆SSj − V (j, i)

)− V (j, i)− V (j, i− 1)

∆t= 0,

i = 1, ..., Nt − 1,

j = 1, ..., NS − 1.

Otherwise

V (j + 1, i)

(1

2

f0(σ)2

∆S2S2

j +g0(r)

2∆SSj

)+ V (j, i)

(− 1

∆t− f0(σ)2

∆S2S2

j − g0(r)

)+V (j − 1, i)

(1

2

f0(σ)2

∆S2S2

j −g0(r)

2∆SSj

)+

V (j, i− 1)

∆t= 0.


We obtain the following finite difference equation to calculate values of thefunction V (S, t) in the grid nodes

V (j, i− 1) = ∆t

(12

(f0(σ)∆S

)2

S2j −

g0(r)2∆S

Sj

)V (j − 1, i)

+

(1 +

(f0(σ)∆S

)2

S2j ∆t + g0(r)∆t

)V (j, i)

+(−f0(σ)2Sj

∆S− g0(r)

)Sj∆t

2∆S∆tV (j + 1, i).

(4.11)

Let us denote

Ai = 12

(f0(σ)∆S

)2

S2j −

g0(r)2∆S

Sj,

Bi = 1 +(

f0(σ)∆S

)2

S2j ∆t + g0(r)∆t,

Ci =(−f0(σ)2Sj

∆S− g0(r)

)Sj∆t

2∆S∆t.

We can rewrite (4.11) as a linear systemB1 C1 0 · · · 0A2 B2 C2 · · · 00 A3 B3 C3 0...

......

. . ....

0 · · · 0 AM−1 BM−1

V (1, i + 1)V (2, i + 1)V (3, i + 1)

...V (M − 1, i + 1)

=

V (1, i)− A1V (0, i)

V (2, i)V (3, i)

...V (M, i)− CM−1V (M, i)

.

Finally we obtain the following expression for V (j, i− 1)

V (1, i + 1)...

V (M − 1, i + 1)

=

B1 C1 0 · · · 0A2 B2 C2 · · · 00 A3 B3 C3 0...

......

. . ....

0 · · · 0 AM−1 BM−1

V (1, i)− A1V (0, i)

V (2, i)V (3, i)

...V (M, i)− CM−1V (M, i)

.

The boundary conditions in Example 2.0.1 are

V (0, i) = V (NS − 1, i) = 0, (4.12)

and initial conditions are

V (j, 0) = (Sj −K)+ + (K − Sj)+. (4.13)

The initial and boundary conditions can be similarly obtain for another examples.


4.4 Implicit method for the function δ

The same transformations as described in part 4.3 we calculate for equation

1

2f0(σ)2S2δ0SS +

(k0(σ)2) + g0(r)

)Sδ0S − δ0t = 0. (4.14)

The partial derivatives in this case we approximate in the following way

δt =δ(j, i)− δ(j − 1, i)

∆t,

δS =δ(j + 1, i)− δ(j − 1, i)

2∆S,

δSS =δ(j + 1, i) + δ(j − 1, i)− 2δ(j, i)

∆S2,

the functions (2.27), (2.28), (2.29) are rewritten as

g0(r) =


k0(r) =


and

f0(σ) =

{σ0(S, t), δ(j + 1, i) + δ(j − 1, i) ≥ 2δ(j, i),σ1(S, t), δ(j + 1, i) + δ(j − 1, i) < 2δ(j, i).


1

2f0(σ)2S2

j

δ(j + 1, i) + δ(j − 1, i)− 2δ(j, i)

∆S2

+(k0(σ)2 + g0(r)

) δ(j + 1, i)− δ(j − 1, i)

2∆SSj −

δ(j, i)− δ(j, i− 1)

∆t= 0,

i = 1, ..., Nt − 1,

j = 1, ..., NS − 1.

Otherwise

δ(j + 1, i)

(1

2f0(σ)2

S2j

∆S2+(k0(σ)2 + g0(r)

) Sj

2∆S

)+δ(j − 1, i)

(1

2f0(σ)2

S2j

∆S2−(k0(σ)2 + g0(r)

) Sj

2∆S

)+δ(j, i)

(−f0(σ)2

S2j

∆S2− 1

∆t

)+ δ(j, i− 1)

1

∆t= 0,


We obtain the following finite difference equation for the function δ(j, i− 1):

δ(j, i− 1) = ∆t2

Sj

∆S

(f0(σ)2 Sj

∆S+ k0(σ)2 + g0(r)

)δ(j + 1, i)

+∆t2

Sj

∆S

(f0(σ)2 Sj

∆S− k0(σ)2 − g0(r)

)δ(j − 1, i)

+δ(j, i)(−f0(σ)2∆t

S2j

∆S− 1)

.

(4.15)

Let us denote

Ai = ∆t2

Sj

∆S

(f0(σ)2 Sj

∆S− k0(σ)2 − g0(r)

),

Bi = −f0(σ)2∆tS2

j

∆S− 1,

Ci = ∆t2

Sj

∆S

(f0(σ)2 Sj

∆S+ k0(σ)2 + g0(r)

).

Following we can represent the system of difference equation correspondingto (4.14) in the form

B1 C1 0 · · · 0A2 B2 C2 · · · 00 A3 B3 C3 0...

......

. . ....

0 · · · 0 AM−1 BM−1

δ(1, i + 1)δ(2, i + 1)δ(3, i + 1)

...V (M − 1, i + 1)

=

δ(1, i)− A1δ(0, i)

δ(2, i)δ(3, i)

...δ(M, i)− CM−1δ(M, i)

Finally we reduce this system to the form

δ(1, i + 1)δ(2, i + 1)δ(3, i + 1)

...V (M − 1, i + 1)

=

B1 C1 0 · · · 0A2 B2 C2 · · · 00 A3 B3 C3 0...

......

. . ....

0 · · · 0 AM−1 BM−1

δ(1, i)− A1δ(0, i)

δ(2, i)δ(3, i)

...δ(M, i)− CM−1δ(M, i)

The initial conditions in Example 2.2.2 can be rewritten in terms of the finite

differences as

δ(j, 0) =

{1, S > K,0, S ≤ K,

and boundary conditions are

δ0(0, i) = δ0(NS − 1, i) = 0.

This conditions done for Example 2.2.2 and can be easy repeated for anothercase.


Chapter 5

Conclusions

Nonlinear models appear often by the modelling of the volatility risk. This modelsdemand methods using of advanced numerical and algorithmic techniques.

In our work we considered uncertain volatility model. To valuate the upperand lower bonds we used numerical methods to solve BSB equation. To reducethe volatility risk we used static hedging. We achieved improvements of theprevious results, using the implicit finite difference method.

In Chapter 2 we considered the uncertain volatility model for the sensitivityparameter delta. We represented examples where the lower and upper bound ofvolatility σ, i.e. functions σ0 and σ1 are the functions dependent on time t.

In Chapter 3 we use the static hedging for narrowing procedure of the envelopof the option price. We improved spread between upper and lower bounds.

In Chapter 4 we represented the explicit and implicit finite difference methodsto valuate the different types of European and American options. We studied thestability problem for the explicit method. Our algorithms can be applied todifferent types of European and American options, for instance, to straddles,butterflies, double barrier and others options.

33

34 Chapter 5. Conclusions

Notation

r The risk-free interest rate,constant over time, in an environmentwith no liquidity constraints.

S A stock’s price.

t The current time.

T The expiration date of a Put optionand a Call option.

K The strike price of the Put optionand Call option.

C(S, t) The price of the Call optionwhen the current stockprice is S and the current time is t.

P (S, t) The price of the Put optionwhen the current stockprice is S and the current time is t.

P (S, t) = C(S, t)− S + Ke−r(T−t) Put-Call parity is therelationship between the price of a Put optionand a Call option on a stock accordingto a standard BS model.

35

36 Chapter 5. Conclusions

Glossary

The interest rate The interest rate is the yearly price charged by a lenderto a borrower in order for the borrower to obtain a loan.This is usually expressed as a percentage of the totalamount loaned. Usually denoted by r.

Expiry The last date an option can be traded or exercised. De-noted by T .

Expiration The time when the option contract ceases toexist. Usually denoted by T .

Underlying The asset that the parties agree to exchange in a deriva-tive contract.

In derivatives, the security that must be delivered if thecontract is exercised.

Stock A Stock (also known as an equity or a share) is a portionof the ownership of a corporation. A share in a corpora-tion gives the owner of the stock a stake in the companyand its profits.

Option A contract which gives the holder of the contract theright to buy or sell a commodity or financial asset for agiven price before a specified date or only at the expira-tion date.

Call Options A Call option is a contract that gives the bearer the rightto buy a share at a given price at the ahead prescribedtime. Usually these options expire after a certain date.

Put Option A Put option is a security which conveys the right to sella specified quantity of an underlying asset at or beforea fixed date.

37

Straddle Straddle is an options trading strategy of buying a Calloption and a Put option on the same stock with thesame strike price and the same expiration date. Such astrategy would result in a profitable position if the stockprice is far enough from the strike price.

Put-Call parity Put-Call parity is the relationship between the price ofa Put option and a Call option on a stock according to astandard BS model. The relationship is derived from thefact that combinations of options can make portfoliosthat are equivalent to holding the stock through timeT , and that must return exactly the same amount or anarbitrage would be available to traders.

Delta Delta in context of the options theory: the rate of changeof a financial derivative’s price with respect to changein the price of the underlying asset. Formally this is apartial derivative of the option price along asset price S.

A derivative is perfectly delta-hedged if it is in a port-folio with a delta equal to zero. Financial companies usedelta-hedged portfolios.

Portfolio The entire collection of financial assets held by aninvestor.

European option An option that may be exercised only at the expirationdate.

American option An option that may be exercised at the expiration dateor before it.

Maximum principle Suppose V (S, t) satisfies the inequality LV > 0 in therectangular region D = (0, X) × (0, T ] then V cannothave (a local) maximum at any interior point.

http://economics.about.comhttp://glossary.itlocus.com

38

Bibliography

[1] G. H. Meyer. (2006) The Black Scholes Barenblatt Equation for Options withUncertain Volatility and its Application to Static Hedging, Int. J. Theoreticaland Appl. Finance 9 (2006), pp. 673-703.

[2] G. Ioffe, M. Ioffe. Application of Finite Difference Method for pricing BarrierOptions, http://www.egartech.com/research financial mathematics.asp.

[3] M. Avellaneda and R. Buff. (1998) Combinatorial implications of nonlinearuncertain volatility models: the case of barrier options, Appl. Math. Finance6, pp. 1-18.

[4] M. Avellaneda, A. Levy and A. Paras. (1995) Pricing and hedging derivativesecurities in markets with uncertain volatilities, Appl. Math. Finance 2, pp.73-88.

[5] P. Wilmott, S. Howison, J. Dewynne. (1995) The Mathematics of FinancialDerivatives, Cambridge University Press, pp. 132-155.

[6] G. H. Meyer and J. van der Hoek. (1997) The valuation of American optionswith the method of lines, Advances in Futures and Options Reseach 9, pp.265-285.

[7] D. Pooley. (2003) Numerical Methods for Nonlinear Equation in Option Pric-ing.Int. J. Theoretical and Applied Finance 4, pp. 467-489.

[8] D. M. Pooey, P. A. Forsyth and K. R. Vetzal. (2003) Numerical convergenceproperties of option pricing PDEs with uncertain volatility. IMA. J. of Nu-merical Analysis 23, pp. 241-267.

39

Chapter 6

Appendix

In this Chapter we provide some examples of programmes used for evaluation ofthe option prices in Chapters 2 and 3.

We will use following variables:

T - maturity date;

S1, S2 - bound of asset price in calculations;

K - strike price;

σ0, σ1 - bound of volatility;

r0, r1 - bound of interest rate;

NS - a number of nodes in horizontal direction;

Nt - a number of nodes in vertical direction;

dt - time step;

dS - asset price step;

V 0, V 1 - envelopes of the option price ;

The sense of variables f0, f1, g0, g1, A0, B0, C0, A1, B1, C1 follows from con-siderations in previous chapters of our work.

41

Program 1 Calculation of envelopes V0, V1 and Black-Scholes Solution V forthe European Double Barrier Straddle (the explicit finite difference method).

Program 2 Application of the implicit finite difference method to the valu-ation of the option price V (S, t).

Program 3 Application of the explicit finite difference method to the valua-tion of the delta function δ for V (S, t).

Program 4 Application of the implicit finite difference method to the valu-ation of the delta function of Up-and-Out European Call (Example 2.2.1).

Program 5 Application of the static hedging procedure to simple EuropeanCall.

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Valuation of portfolios under uncertain volatility: Black-Scholes …hh.diva-portal.org/smash/get/diva2:238852/FULLTEXT01.pdf · 2009-09-26 · The Uncertain Volatility Method (UVM)