Finite Difference Methods for Option...

Finite Difference Methods for Option Pricing

Muhammad Usman, Ph.D.

University of Dayton

CASM Workshop - Black Scholes and Beyond: Pricing EquityDerivatives

LUMS, Lahore, Pakistan, May 16–18, 2014

Outline

Introduction

Finite Difference Methods for Option Valuation

Computer Arithmetic

Pricing American Options

Basic Definitions

Derivative.A financial quantity that is derived based upon the value from thebehavior of the underlying asset is called a Derivative.Examples: Options, swaps, futures and forward contracts.Option pricing. Determining the future market value of these sorts ofcontracts is a problem in option pricing.Option. Option is a derivative financial security whose value dependson the value of underlying asset. Options are financial contracts thatgive the holder certain rights. As a holder you buy the rightsstipulated in the contract. It can be the right to buy, sell or exchangeone thing for another. It can be converted to cash at the expense ofcounterparty that issued the option, called the writer of the option.Who charges a fee for the risk of incurring possible loss?

Origin of Option: Ancient Greece

As recorded by Aristotle in Politics the fifth century BC philosopher Thalesof Miletus took part in a sophisticated trading strategy. Reason of this tradewas to confirm that philosophers could become rich if they so chose. This isperhaps the first rejoinder to the famous question If you are so smart, whyarent you rich? which has dogged academics throughout the ages. Thalesobserved that the weather was very favorable to a good olive crop, whichwould result in a bumper harvest of olives. Thales put a deposit on all theolive presses surrounding Miletus. When the olive crop was harvesteddemand for olive presses reached enormous proportions. Thales then subletthe presses for a profit. Note that by placing a deposit on the presses, Thaleswas actually manufacturing an option on the olive crop that is the most hecould lose was his deposit. If he had sold short olive future, he would havebeen liable to an unlimited loss, in the event that the olive crop turned outbad and the price of the olive went up (a surplus of olives would cause theprice of olive to go down, so there were risks involved). In other words, hehad the option on a future of a non storable commodity.

Basic Definitions

I Options is not an obligation

I Options are exercised only when their value is greater thanzero-then it is said that the option is in-the-money.

I All options have expiration time after that they becomeworthless.

I Payoff function describes the value of the option as a function ofthe underlying asset at the time of expiry. A payoff function neednot to be differentiable nor even continuous.

Basic Definitions

I Options is not an obligationI Options are exercised only when their value is greater than

zero-then it is said that the option is in-the-money.

I All options have expiration time after that they becomeworthless.


Basic Definitions


zero-then it is said that the option is in-the-money.I All options have expiration time after that they become

worthless.


Basic Definitions


zero-then it is said that the option is in-the-money.I All options have expiration time after that they become

worthless.I Payoff function describes the value of the option as a function of

the underlying asset at the time of expiry. A payoff function neednot to be differentiable nor even continuous.

Basic Definitions

Types of optionsI Some times there are limitations on options when they can be

exercised.

European option: can be exercised only at expiry.American option: can be exercised at any timebefore the expiry.Bermudan option: can be exercised on only atdates specified in advanced.

I The most basic options are the call option and the put option.

A call option is an option to buy an asset at aprescribed price K (the exercise or strike price)A Put option is an option to sell an asset at aprescribed price K (the exercise or strike price)

Basic Definitions


exercised.European option: can be exercised only at expiry.

American option: can be exercised at any timebefore the expiry.Bermudan option: can be exercised on only atdates specified in advanced.



Basic Definitions


exercised.European option: can be exercised only at expiry.American option: can be exercised at any timebefore the expiry.

Bermudan option: can be exercised on only atdates specified in advanced.



Basic Definitions


exercised.European option: can be exercised only at expiry.American option: can be exercised at any timebefore the expiry.Bermudan option: can be exercised on only atdates specified in advanced.



Basic Definitions





Basic Definitions



I The most basic options are the call option and the put option.A call option is an option to buy an asset at aprescribed price K (the exercise or strike price)

A Put option is an option to sell an asset at aprescribed price K (the exercise or strike price)

Basic Definitions



I The most basic options are the call option and the put option.A call option is an option to buy an asset at aprescribed price K (the exercise or strike price)A Put option is an option to sell an asset at aprescribed price K (the exercise or strike price)

Figure: An Example of European Call Option with Strike price $21. Stockprice is higher than the strike price, so one can buy at the strike price andearn profit.

Exchanges Trading Options

Chicago Board Options Exchange

International Securities Exchange

NYSE Euronext

Eurex (Europe)

and many more

Stochastic Process: A variable whose value is changing randomly issaid to follow a stochastic process.Geometric Brownian Motion:A non-dividend paying asset, S, following GBM is govererned bySDE

dS = µSdt + σSdz,

where µ and σ are constants and dS is the change in the level of theasset price over a small time interval dt.

dSS

= µdt + σdz

M. S. Scholes and R. C. Merton were awarded by thePrize of the Swedish Bank for Economics in thememory of A. Nobel in 1997. Fisher Black died in1995, was mentioned as a contributor by SwedishAcademy.

Fischer Black(1938–1995)

Myron Scholes(1941–)

R. C. Merton(1944)

The Black-Scholes Partial Differential Equation for Valuation ofOptions

∂V∂t

+12σ2S2∂

2V∂S2 + rS

∂V∂S− rV = 0, S > 0, t ∈ [0,T]

The Black-Scholes Model for Pricing Financial Derivative

∂V∂t

+12σ2S2∂

2V∂S2 + rS

∂V∂S− rV = 0.

where σ is the volatility, r risk free interest rate and V(S, t) is optionvalue at time t and stock price S. Initial condition is the terminalpayoff value

V(S,T) =

maxS− X, 0, for call option;maxX − S, 0, for put option.

Where T is the time of maturity and X is strike price.

The Transformed Black-Scholes Equation

I S = E exp(x), t = T − τ12σ, C = Ev(x, τ)

I k =r

12σ

I∂v∂τ

=∂2u∂x2 + (k − 1)

∂v∂τ− kv

I v = e−12 (k−1)x− 1

4 (k+1)2τu(x, τ)

I∂u∂τ

=∂2u∂x2 , −∞ < x <∞, τ > 0



I k =r

12σ

I∂v∂τ

=∂2u∂x2 + (k − 1)

∂v∂τ− kv

I v = e−12 (k−1)x− 1

4 (k+1)2τu(x, τ)

I∂u∂τ

=∂2u∂x2 , −∞ < x <∞, τ > 0



I k =r

12σ

I∂v∂τ

=∂2u∂x2 + (k − 1)

∂v∂τ− kv

I v = e−12 (k−1)x− 1

4 (k+1)2τu(x, τ)

I∂u∂τ

=∂2u∂x2 , −∞ < x <∞, τ > 0



I k =r

12σ

I∂v∂τ

=∂2u∂x2 + (k − 1)

∂v∂τ− kv

I v = e−12 (k−1)x− 1

4 (k+1)2τu(x, τ)

I∂u∂τ

=∂2u∂x2 , −∞ < x <∞, τ > 0



I k =r

12σ

I∂v∂τ

=∂2u∂x2 + (k − 1)

∂v∂τ− kv

I v = e−12 (k−1)x− 1

4 (k+1)2τu(x, τ)

I∂u∂τ

=∂2u∂x2 , −∞ < x <∞, τ > 0


∂u∂τ

=∂2u∂x2 , −∞ < x <∞, τ > 0

I Initial condition for call: u(x, 0) = max(e12 (k+1)x − e

12 (k−1)x, 0)

I Initial condition for put: u(x, 0) = max(e12 (k−1)x − e

12 (k+1)x, 0)


∂u∂τ

=∂2u∂x2 , −∞ < x <∞, τ > 0

I Initial condition for call: u(x, 0) = max(e12 (k+1)x − e

12 (k−1)x, 0)

I Initial condition for put: u(x, 0) = max(e12 (k−1)x − e

12 (k+1)x, 0)

Although this may seem a paradox, all exactscience is dominated by the idea ofapproximation.

Bertrand Russell

Methods to solve ut + Lu = 0

I Finite difference methods

I Method of lines

I Collocation methods

I Finite element methods

I Monte Carlo methods



I Method of lines






I Method of lines






I Method of lines






I Method of lines




Finite Difference Method

I Numerical solution is an approximation of solution

I It is discreteI Discretize the domainI Discretize the PDE/ODEI Solve the linear/nonlinear system


I Numerical solution is an approximation of solutionI It is discrete

I Discretize the domainI Discretize the PDE/ODEI Solve the linear/nonlinear system


I Numerical solution is an approximation of solutionI It is discreteI Discretize the domain

I Discretize the PDE/ODEI Solve the linear/nonlinear system


I Numerical solution is an approximation of solutionI It is discreteI Discretize the domainI Discretize the PDE/ODE

I Solve the linear/nonlinear system


I Numerical solution is an approximation of solutionI It is discreteI Discretize the domainI Discretize the PDE/ODEI Solve the linear/nonlinear system

Synergy of FD methods

Convergence of FD methods

I Accuracy

I ConsistencyI Stability


I AccuracyI Consistency

I Stability


I AccuracyI ConsistencyI Stability

Types of Errors and their Sources

Most of the numerical methods approximate the analytical solution.Often numerical value of the solution is guessed and then usingiterative process that guess is refined. To perform these iterativeprocess we use computers. As computers are faster than humans andare not susceptible to human errors, such as dropping a decimal pointor miscopying a number. Computerized numerical methods providesus convenient techniques that are needed. But convenience comeswith the price i.e. the introduction of error into calculations!



I Modelling ErrorI Discretization and truncation error

One of the important step in numerical computation is converting thecontinuous system into discrete one. This conversion processintroduces the error known as discretization error. Other techniqueinvolve the truncation of the infinite series giving rise to truncationerror.

I Round off and data error Unlike the discretization andtruncation error which arise due to the formulation of thenumerical method, round off error and the data errors are due tothe limitations of the hardware. As soon as we use the computerthere are roundoff errors even we haven’t done any computationat all.

I Human error


Taylor’s Theorem:

Suppose f (x) has (n + 1) derivatives in an interval containing thepoints x0 and x0 + h. Then

f (x0 +h) = f (x0)+hf ′(x0)+h2

2!f ′′(x0)+ . . .+

hn

n!f (n)(x0)+

hn+1

(n + 1)!f (n+1)(ξ)

where ξ is some point between x0 and x0 + h.OR

f (x) = f (x0) + f ′(x0)(x− x0) +f ′′(x0)

2!(x− x0)2 + . . .

+f (n)(x0)

n!(x− x0)n +

f (n+1)(ξ)

(n + 1)!(x− x0)n+1

where ξ is some point between x0 and x.

Taylor’s Theorem:

f (x) = Pn(x) + Rn(x), where

Pn(x) =

n∑k=0

f (k)(x0)

k!(x− x0)k,

is called Taylor’s polynomial of degree n and

Rn(x) =f (k+1)(ξx)

(k + 1)!(x− x0)k+1

is the Remainder (truncation error).

Example:

f (x) = sin(x) =

∞∑k=0

(−1)kx2k+1

(2k + 1)!

sin(x) = x− x3

3!︸︷︷︸P3(x)

+x5

5!

︸︷︷︸P5(x)

−x7

7!+

x9

9!− x11

11!+ . . .

Approximation of derivative of a function at a point x0

I f ′(x0) = limh→0

f (x0 + h)− f (x0)

h

Quick Notes Page 1

Example 1: ∣∣∣∣∣∣∣∣f (x + h)− f (x)

h︸︷︷︸F(h)

− f ′(x)︸︷︷︸L

∣∣∣∣∣∣∣∣ =

∣∣∣∣h2 f ′′(ξ)∣∣∣∣

∣∣∣∣ f (x + h)− f (x)

h− f ′(x)

∣∣∣∣ ≤ M2

h = ch

∴ f ′(x) =f (x + h)− f (x)

h+O(h)

Example 2:Using the central difference (CD) formula.∣∣∣∣ f (x + h)− f (x− h)

2h− f ′(x)

∣∣∣∣ ≤ ch2.

f ′(x) =f (x + h)− f (x− h)

2h+O(h2).

f (x + h)− f (x− h)

2h→ f ′(x),

with the rate of convergence O(h2).

Consistent: A method is consistent if its local truncation errorTi,j → 0 as ∆x→ 0 and ∆t→ 0 . Local truncation error is the errorthat occurs when the exact solution U(xi, tj) is substituted into the FDapproximation at each point of interest.


I Approximate the derivative using a difference formula, instead oftaking h zero, take “small” values of h.

I f ′(x0) ≈ f (x0 + h)− f (x0)

hI f (x) = sin x, x0 = 1.2, fp = cos(1.2)

I cos(x0) ≈ sin(x0 + h)− f (x0)

hI Lets make a table of values of difference quotient with

decreasing h valuesI We hope that with decreasing h the error will become smaller

and smaller



I f ′(x0) ≈ f (x0 + h)− f (x0)

h

I f (x) = sin x, x0 = 1.2, fp = cos(1.2)

I cos(x0) ≈ sin(x0 + h)− f (x0)



and smaller



I f ′(x0) ≈ f (x0 + h)− f (x0)

hI f (x) = sin x, x0 = 1.2, fp = cos(1.2)

I cos(x0) ≈ sin(x0 + h)− f (x0)



and smaller



I f ′(x0) ≈ f (x0 + h)− f (x0)

hI f (x) = sin x, x0 = 1.2, fp = cos(1.2)

I cos(x0) ≈ sin(x0 + h)− f (x0)

h

I Lets make a table of values of difference quotient withdecreasing h values

I We hope that with decreasing h the error will become smallerand smaller



I f ′(x0) ≈ f (x0 + h)− f (x0)

hI f (x) = sin x, x0 = 1.2, fp = cos(1.2)

I cos(x0) ≈ sin(x0 + h)− f (x0)


decreasing h values

I We hope that with decreasing h the error will become smallerand smaller



I f ′(x0) ≈ f (x0 + h)− f (x0)

hI f (x) = sin x, x0 = 1.2, fp = cos(1.2)

I cos(x0) ≈ sin(x0 + h)− f (x0)



and smaller

Example

h Absolute error1e− 8 4.361050e− 10

1e− 9 5.594726e− 81e− 10 1.669696e− 71e− 11 7.398531e− 61e− 13 4.250484e− 41e− 15 8.173146e− 21e− 16 3.623578e− 1

Surprise!!! error gets bigger and bigger. Can you explain?

Example

h Absolute error1e− 8 4.361050e− 101e− 9 5.594726e− 8

1e− 10 1.669696e− 71e− 11 7.398531e− 61e− 13 4.250484e− 41e− 15 8.173146e− 21e− 16 3.623578e− 1


Example

h Absolute error1e− 8 4.361050e− 101e− 9 5.594726e− 81e− 10 1.669696e− 7

1e− 11 7.398531e− 61e− 13 4.250484e− 41e− 15 8.173146e− 21e− 16 3.623578e− 1


Example

h Absolute error1e− 8 4.361050e− 101e− 9 5.594726e− 81e− 10 1.669696e− 71e− 11 7.398531e− 6

1e− 13 4.250484e− 41e− 15 8.173146e− 21e− 16 3.623578e− 1


Example

h Absolute error1e− 8 4.361050e− 101e− 9 5.594726e− 81e− 10 1.669696e− 71e− 11 7.398531e− 61e− 13 4.250484e− 4

1e− 15 8.173146e− 21e− 16 3.623578e− 1


Example

h Absolute error1e− 8 4.361050e− 101e− 9 5.594726e− 81e− 10 1.669696e− 71e− 11 7.398531e− 61e− 13 4.250484e− 41e− 15 8.173146e− 2

1e− 16 3.623578e− 1


Example

h Absolute error1e− 8 4.361050e− 101e− 9 5.594726e− 81e− 10 1.669696e− 71e− 11 7.398531e− 61e− 13 4.250484e− 41e− 15 8.173146e− 21e− 16 3.623578e− 1


Example

h Absolute error1e− 8 4.361050e− 101e− 9 5.594726e− 81e− 10 1.669696e− 71e− 11 7.398531e− 61e− 13 4.250484e− 41e− 15 8.173146e− 21e− 16 3.623578e− 1


Approximating the derivative using a difference formulaI f ′(x0) = lim

h→0

f (x0 + h)− f (x0 − h)

2h

Quick Notes Page 1

Approximating the derivative using a difference formula

I

Error ≤ ε

h+

M6

h2

where M = max[x0−h,x0+h]

|f ′′′(x)|

I Optimal value of h is given by h =

(3εM

)1/3

I Corresponding error is O(ε2/3)


I

Error ≤ ε

h+

M6

h2


|f ′′′(x)|


(3εM

)1/3



I

Error ≤ ε

h+

M6

h2


|f ′′′(x)|


(3εM

)1/3



Quick Notes Page 1

For a successful and acceptable approximationthe approximation error dominates the roundofferror in magnitude.

First Derivative Formulas

I f (1)(a) =

−f (a− h) + f (a + h)

2h+O(h2

)

I f (1)(a) =

−3f (a) + 4f (a + h)− f (a + 2h)

2h+O(h2

)

I f (1)(a) =

−2f (a− h)− 3f (a) + 6f (a + h)− f (a + 2h)

6h+O(h3

)

I f (1)(a) =

−11f (a) + 18f (a + h)− 9f (a + 2h) + 2f (a + 3h)

6h+O(h3

)

I f (1)(a) =

f (a− 2h)− 8f (a− h) + 8f (a + h)− f (a + 2h)

12h+O(h4

)

I f (1)(a) =

−3f (a− h)− 10f (a) + 18f (a + h)− 6f (a + 2h) + f (a + 3h)

12h+O(h4

)

I f (1)(a) =

−25f (a) + 48f (a + h)− 36f (a + 2h) + 16f (a + 3h)− 3f (a + 4h)

12h+O(h4

)


I f (1)(a) =

−f (a− h) + f (a + h)

2h+O(h2

)

I f (1)(a) =

−3f (a) + 4f (a + h)− f (a + 2h)

2h+O(h2

)

I f (1)(a) =

−2f (a− h)− 3f (a) + 6f (a + h)− f (a + 2h)

6h+O(h3

)

I f (1)(a) =

−11f (a) + 18f (a + h)− 9f (a + 2h) + 2f (a + 3h)

6h+O(h3

)

I f (1)(a) =

f (a− 2h)− 8f (a− h) + 8f (a + h)− f (a + 2h)

12h+O(h4

)

I f (1)(a) =

−3f (a− h)− 10f (a) + 18f (a + h)− 6f (a + 2h) + f (a + 3h)

12h+O(h4

)

I f (1)(a) =

−25f (a) + 48f (a + h)− 36f (a + 2h) + 16f (a + 3h)− 3f (a + 4h)

12h+O(h4

)


I f (1)(a) =

−f (a− h) + f (a + h)

2h+O(h2

)

I f (1)(a) =

−3f (a) + 4f (a + h)− f (a + 2h)

2h+O(h2

)

I f (1)(a) =

−2f (a− h)− 3f (a) + 6f (a + h)− f (a + 2h)

6h+O(h3

)

I f (1)(a) =

−11f (a) + 18f (a + h)− 9f (a + 2h) + 2f (a + 3h)

6h+O(h3

)

I f (1)(a) =

f (a− 2h)− 8f (a− h) + 8f (a + h)− f (a + 2h)

12h+O(h4

)

I f (1)(a) =

−3f (a− h)− 10f (a) + 18f (a + h)− 6f (a + 2h) + f (a + 3h)

12h+O(h4

)

I f (1)(a) =

−25f (a) + 48f (a + h)− 36f (a + 2h) + 16f (a + 3h)− 3f (a + 4h)

12h+O(h4

)


I f (1)(a) =

−f (a− h) + f (a + h)

2h+O(h2

)

I f (1)(a) =

−3f (a) + 4f (a + h)− f (a + 2h)

2h+O(h2

)

I f (1)(a) =

−2f (a− h)− 3f (a) + 6f (a + h)− f (a + 2h)

6h+O(h3

)

I f (1)(a) =

−11f (a) + 18f (a + h)− 9f (a + 2h) + 2f (a + 3h)

6h+O(h3

)

I f (1)(a) =

f (a− 2h)− 8f (a− h) + 8f (a + h)− f (a + 2h)

12h+O(h4

)

I f (1)(a) =

−3f (a− h)− 10f (a) + 18f (a + h)− 6f (a + 2h) + f (a + 3h)

12h+O(h4

)

I f (1)(a) =

−25f (a) + 48f (a + h)− 36f (a + 2h) + 16f (a + 3h)− 3f (a + 4h)

12h+O(h4

)


I f (1)(a) =

−f (a− h) + f (a + h)

2h+O(h2

)

I f (1)(a) =

−3f (a) + 4f (a + h)− f (a + 2h)

2h+O(h2

)

I f (1)(a) =

−2f (a− h)− 3f (a) + 6f (a + h)− f (a + 2h)

6h+O(h3

)

I f (1)(a) =

−11f (a) + 18f (a + h)− 9f (a + 2h) + 2f (a + 3h)

6h+O(h3

)

I f (1)(a) =

f (a− 2h)− 8f (a− h) + 8f (a + h)− f (a + 2h)

12h+O(h4

)

I f (1)(a) =

−3f (a− h)− 10f (a) + 18f (a + h)− 6f (a + 2h) + f (a + 3h)

12h+O(h4

)

I f (1)(a) =

−25f (a) + 48f (a + h)− 36f (a + 2h) + 16f (a + 3h)− 3f (a + 4h)

12h+O(h4

)


I f (1)(a) =

−f (a− h) + f (a + h)

2h+O(h2

)

I f (1)(a) =

−3f (a) + 4f (a + h)− f (a + 2h)

2h+O(h2

)

I f (1)(a) =

−2f (a− h)− 3f (a) + 6f (a + h)− f (a + 2h)

6h+O(h3

)

I f (1)(a) =

−11f (a) + 18f (a + h)− 9f (a + 2h) + 2f (a + 3h)

6h+O(h3

)

I f (1)(a) =

f (a− 2h)− 8f (a− h) + 8f (a + h)− f (a + 2h)

12h+O(h4

)

I f (1)(a) =

−3f (a− h)− 10f (a) + 18f (a + h)− 6f (a + 2h) + f (a + 3h)

12h+O(h4

)

I f (1)(a) =

−25f (a) + 48f (a + h)− 36f (a + 2h) + 16f (a + 3h)− 3f (a + 4h)

12h+O(h4

)


I f (1)(a) =

−f (a− h) + f (a + h)

2h+O(h2

)

I f (1)(a) =

−3f (a) + 4f (a + h)− f (a + 2h)

2h+O(h2

)

I f (1)(a) =

−2f (a− h)− 3f (a) + 6f (a + h)− f (a + 2h)

6h+O(h3

)

I f (1)(a) =

−11f (a) + 18f (a + h)− 9f (a + 2h) + 2f (a + 3h)

6h+O(h3

)

I f (1)(a) =

f (a− 2h)− 8f (a− h) + 8f (a + h)− f (a + 2h)

12h+O(h4

)

I f (1)(a) =

−3f (a− h)− 10f (a) + 18f (a + h)− 6f (a + 2h) + f (a + 3h)

12h+O(h4

)

I f (1)(a) =

−25f (a) + 48f (a + h)− 36f (a + 2h) + 16f (a + 3h)− 3f (a + 4h)

12h+O(h4

)

Second Derivative Formulas

I f (2)(a) =

f (a− h)− 2f (a) + f (a + h)

h2+O(h2

)

I f (2)(a) =

2f (a)− 5f (a + h) + 4f (a + 2h)− f (a + 3h)

h2+O(h2

)

I f (2)(a) =

−f (a− 2h) + 16f (a− h)− 30f (a) + 16f (a + h)− f (a + 2h)

12h2+O(h4

)

I f (2)(a) =

11f (a− h)− 20f (a) + 6f (a + h) + 4f (a + 2h)− f (a + 3h)

12h2+O(h3

)

I f (2)(a) =

35f (a)− 104f (a + h) + 114f (a + 2h)− 56f (a + 3h) + 11f (a + 4h)

h2+O(h3

)


I f (2)(a) =

f (a− h)− 2f (a) + f (a + h)

h2+O(h2

)

I f (2)(a) =

2f (a)− 5f (a + h) + 4f (a + 2h)− f (a + 3h)

h2+O(h2

)

I f (2)(a) =

−f (a− 2h) + 16f (a− h)− 30f (a) + 16f (a + h)− f (a + 2h)

12h2+O(h4

)

I f (2)(a) =

11f (a− h)− 20f (a) + 6f (a + h) + 4f (a + 2h)− f (a + 3h)

12h2+O(h3

)

I f (2)(a) =

35f (a)− 104f (a + h) + 114f (a + 2h)− 56f (a + 3h) + 11f (a + 4h)

h2+O(h3

)


I f (2)(a) =

f (a− h)− 2f (a) + f (a + h)

h2+O(h2

)

I f (2)(a) =

2f (a)− 5f (a + h) + 4f (a + 2h)− f (a + 3h)

h2+O(h2

)

I f (2)(a) =

−f (a− 2h) + 16f (a− h)− 30f (a) + 16f (a + h)− f (a + 2h)

12h2+O(h4

)

I f (2)(a) =

11f (a− h)− 20f (a) + 6f (a + h) + 4f (a + 2h)− f (a + 3h)

12h2+O(h3

)

I f (2)(a) =

35f (a)− 104f (a + h) + 114f (a + 2h)− 56f (a + 3h) + 11f (a + 4h)

h2+O(h3

)


I f (2)(a) =

f (a− h)− 2f (a) + f (a + h)

h2+O(h2

)

I f (2)(a) =

2f (a)− 5f (a + h) + 4f (a + 2h)− f (a + 3h)

h2+O(h2

)

I f (2)(a) =

−f (a− 2h) + 16f (a− h)− 30f (a) + 16f (a + h)− f (a + 2h)

12h2+O(h4

)

I f (2)(a) =

11f (a− h)− 20f (a) + 6f (a + h) + 4f (a + 2h)− f (a + 3h)

12h2+O(h3

)

I f (2)(a) =

35f (a)− 104f (a + h) + 114f (a + 2h)− 56f (a + 3h) + 11f (a + 4h)

h2+O(h3

)


I f (2)(a) =

f (a− h)− 2f (a) + f (a + h)

h2+O(h2

)

I f (2)(a) =

2f (a)− 5f (a + h) + 4f (a + 2h)− f (a + 3h)

h2+O(h2

)

I f (2)(a) =

−f (a− 2h) + 16f (a− h)− 30f (a) + 16f (a + h)− f (a + 2h)

12h2+O(h4

)

I f (2)(a) =

11f (a− h)− 20f (a) + 6f (a + h) + 4f (a + 2h)− f (a + 3h)

12h2+O(h3

)

I f (2)(a) =

35f (a)− 104f (a + h) + 114f (a + 2h)− 56f (a + 3h) + 11f (a + 4h)

h2+O(h3

)

Weights and Coefficients of First Derivatives

Weights and Coefficients of Second Derivatives

Order Weights

2 1 −2 1

4 − 112

43 − 5

243 − 1

12

6 190 − 3

2032 − 49

1832 − 3

201

90

8 − 1560

8315 − 1

585 − 205

7285 − 1

58

315 − 1560

.

.

. . . .− 242

232 − 2

222

12 −π23

212 − 2

222

32 − 242 . . .

Third and Fourth Derivative Formulas

I f (3)(a) =

−f (a− 2h) + 2f (a− h)− 2f (a + h) + f (a + 2h)

2h3+O(h2

)

I f (3)(a) =

−3f (a− h) + 10f (a)− 12f (a + h) + 6f (a + 2h)− f (a + 3h)

2h3+O(h2

)

I f (3)(a) =

−5f (a) + 18f (a + h)− 24f (a + 2h) + 14f (a + 3h)− 3f (a + 4h)

2h3+O(h2

)

I f (4)(a) =

f (a− 2h)− 4f (a− h) + 6f (a)− 4f (a + h) + f (a + 2h)

h4+O(h2

)


I f (3)(a) =

−f (a− 2h) + 2f (a− h)− 2f (a + h) + f (a + 2h)

2h3+O(h2

)

I f (3)(a) =

−3f (a− h) + 10f (a)− 12f (a + h) + 6f (a + 2h)− f (a + 3h)

2h3+O(h2

)

I f (3)(a) =

−5f (a) + 18f (a + h)− 24f (a + 2h) + 14f (a + 3h)− 3f (a + 4h)

2h3+O(h2

)

I f (4)(a) =

f (a− 2h)− 4f (a− h) + 6f (a)− 4f (a + h) + f (a + 2h)

h4+O(h2

)


I f (3)(a) =

−f (a− 2h) + 2f (a− h)− 2f (a + h) + f (a + 2h)

2h3+O(h2

)

I f (3)(a) =

−3f (a− h) + 10f (a)− 12f (a + h) + 6f (a + 2h)− f (a + 3h)

2h3+O(h2

)

I f (3)(a) =

−5f (a) + 18f (a + h)− 24f (a + 2h) + 14f (a + 3h)− 3f (a + 4h)

2h3+O(h2

)

I f (4)(a) =

f (a− 2h)− 4f (a− h) + 6f (a)− 4f (a + h) + f (a + 2h)

h4+O(h2

)


I f (3)(a) =

−f (a− 2h) + 2f (a− h)− 2f (a + h) + f (a + 2h)

2h3+O(h2

)

I f (3)(a) =

−3f (a− h) + 10f (a)− 12f (a + h) + 6f (a + 2h)− f (a + 3h)

2h3+O(h2

)

I f (3)(a) =

−5f (a) + 18f (a + h)− 24f (a + 2h) + 14f (a + 3h)− 3f (a + 4h)

2h3+O(h2

)

I f (4)(a) =

f (a− 2h)− 4f (a− h) + 6f (a)− 4f (a + h) + f (a + 2h)

h4+O(h2

)

Convergence of finite difference methods

Convergence: An approximation is said to be convergent if theapproximate values converge to exact values as ∆t→ 0 and ∆x→ 0,mathematically

ukh → u(xi, tk), as ∆t→ 0 and ∆x→ 0


I The Lax Equivalence Theorem

For a well-posed linear IVP a consistent FDscheme is convergent iff it is stable

I Consistency: A FD scheme is consistent if the local truncationerror τ j

i → 0 as ∆t→ 0 and ∆x→ 0 (in other words as the meshsize approaches to zero). Truncation error is the amount bywhich a finite difference scheme fails to satisfy the PDE.

I Stability: A method is stable if error at the initial step does notgrow with iteration.


I The Lax Equivalence TheoremFor a well-posed linear IVP a consistent FDscheme is convergent iff it is stable














Example: FTCS Scheme

Forut = kuxx

the Forward-Time-Center-Space (FTCS) scheme has the truncationerror of

τ =∆t2∂2u∂t2 + O(∆x)2

τ → 0 as ∆t→ 0, ∆x→ 0.

Hence the FTCS scheme is consistent with PDE

ut = kuxx.

Example: Dufort-Frankel Scheme

ut = kuxx

the Dufort-Frankel Scheme

uk+1i − uk−1

i2∆t

= k(uk

i+1 + uki−1)− (uk+1

i + uk−1i )

∆x2

has the truncation error:

τ = k(∆x)2

12∂4u∂x4 − k

(∆t∆x

)2 ∂2u∂t2 −

(∆t)2

6∂3u∂t3 .

If lim∆x→0∆t→0

∆t∆x

= 0 then scheme is consistent.

Example: Dufort-Frankel Scheme

ut = kuxx

the Dufort-Frankel Scheme

uk+1i − uk−1

i2∆t

= k(uk

i+1 + uki−1)− (uk+1

i + uk−1i )

∆x2

If lim∆x→0∆t→0

∆t∆x

= β 6= 0, then lim∆x→0∆t→0

τ 6= 0

Stability of finite difference methods

I Matrix stability analysis

I von Neumann Stability Analysis:

Based upon Fourier analysis.A crude way is to use Un

m = gneimθ in FD scheme.A numerical scheme for an evolution equation isstable if and only if the associated largestamplification factor satisfies

|g| = 1 +O(∆t)


I Matrix stability analysisI von Neumann Stability Analysis:



|g| = 1 +O(∆t)



Based upon Fourier analysis.

A crude way is to use Unm = gneimθ in FD scheme.

A numerical scheme for an evolution equation isstable if and only if the associated largestamplification factor satisfies

|g| = 1 +O(∆t)




m = gneimθ in FD scheme.

A numerical scheme for an evolution equation isstable if and only if the associated largestamplification factor satisfies

|g| = 1 +O(∆t)





|g| = 1 +O(∆t)


I Explicit Euler, Implicit Euler, and the Crank-Nicolson method.

I Explicit method (also called explicit Euler) is the easiest methodbut unstable for certain choices of domain discretization.

I Implicit Euler and Crank-Nicolson are implicit methods, whichgenerally require a system of linear equations to be solved ateach time step, which can be computationally intensive on a finemesh.

I Crank-Nicolson exhibits the greatest accuracy of the three for agiven domain discretization.

I Finite Difference methods can be applied to American (earlyexercise)


I Explicit Euler, Implicit Euler, and the Crank-Nicolson method.I Explicit method (also called explicit Euler) is the easiest method

but unstable for certain choices of domain discretization.

I Implicit Euler and Crank-Nicolson are implicit methods, whichgenerally require a system of linear equations to be solved ateach time step, which can be computationally intensive on a finemesh.





but unstable for certain choices of domain discretization.I Implicit Euler and Crank-Nicolson are implicit methods, which

generally require a system of linear equations to be solved ateach time step, which can be computationally intensive on a finemesh.















Stencil for Explicit Finite Difference Scheme

Discretization of BS-PDE using the Explicit Euler Method.

V ji − V j−1

i∆t

+12σ2(i∆S)2 V j

i+1 − 2V ji + V j

i−1

∆S2

+ r(i∆S)V j

i+1 − V ji−1

2∆S− rV j

i = 0

V j−1i = AiV

ji−1 + BiV

ji + CiV

ji+1

where

Ai =12

∆t(σ2i2−ri), Bi = 1−(σ2i2 +r)∆t, Ci =12

∆t(ri+σ2i2)

Value of European Call Option using the Explicit Euler Method.

Figure: Solution of the Black-Scholes equation using Explicit Euler Methodfor European Call option, for K = 10, r = 0.2, σ = 0.25 and T = 1

Value of European Call Option using the Explicit Euler Method.

Figure: Solution of the Black-Scholes equation using Explicit Euler Methodfor European Call option, for K = 10, r = 0.2, σ = 0.25 at T = 0, T/2 andat expiry

Stencil for Implicit Finite Difference Scheme

Value of European Put using the Implicit Euler Method.I Mesh: 0,∆S, 2∆S, . . . ,M∆S where ∆S = Smax/M

0,∆t, 2∆t, . . . ,N∆t where ∆t = T/Nwhere Smax is the maximum value of S chosen sufficiently largeand and V j

i = V(i∆S, j∆t), i = 0, 1, ...,M, j = 0, 1, ...,N

I The initial and boundary conditions for the European Put are:

V(S,T) = max(K − S, 0), V(0, t) = Ke−r(T−t), V(Smax, t) = 0

Discretized BCs are:

VNi = max(K − (i∆S), 0), i = 0, 1, ...,M

V j0 = Ke−r(N−j)∆t, j = 0, 1, ...,N

V jM = 0, j = 0, 1, ...,N

I Since we are given the payoff at expiry, our problem is to solvethe Black-Scholes PDE backwards in time from expiry to thepresent time (t = 0).



i = V(i∆S, j∆t), i = 0, 1, ...,M, j = 0, 1, ...,NI The initial and boundary conditions for the European Put are:



VNi = max(K − (i∆S), 0), i = 0, 1, ...,M

V j0 = Ke−r(N−j)∆t, j = 0, 1, ...,N

V jM = 0, j = 0, 1, ...,N




i = V(i∆S, j∆t), i = 0, 1, ...,M, j = 0, 1, ...,NI The initial and boundary conditions for the European Put are:



VNi = max(K − (i∆S), 0), i = 0, 1, ...,M

V j0 = Ke−r(N−j)∆t, j = 0, 1, ...,N

V jM = 0, j = 0, 1, ...,N


Discretization of PDE using the Implicit Euler Method.

V ji − V j−1

i∆t

+12σ2(i∆S)2 V j−1

i+1 − 2V j−1i + V j−1

i−1

∆S2

+ r(i∆S)V j−1

i+1 − V j−1i−1

2∆S− rV j−1

i = 0

V ji = AiV

j−1i−1 + BiV

j−1i + CiV

j−1i+1

where

Ai =12

∆t(ri−σ2i2), Bi = 1+(σ2i2+r)∆t, Ci = −12

∆t(ri+σ2i2)

We will use the Backslash operator to invert the tridiagonal matrix ateach time step. Results for the values

K = 50, r = 0.05, σ = 0.2,T = 3.

are given in the figure.

Figure: Solution of the Black-Scholes equation using Implicit Euler Methodfor European Put option, for K = 50, r = 0.05, σ = 0.2 and T = 3

Boundary Conditions for Options

The boundary conditions for a European call are given by

C(S,T) = max(S− E, 0); S > 0

C(0, t) = 0; t > 0

C(S, t) ∼ Ee−r(T−t) as S→∞; t > 0

Boundary Conditions for Options

The boundary conditions for the European put are

P(S,T) = max(E − S, 0); S > 0

P(0, t) = Ee−r(T−t); t > 0

P(S, t) → 0 as S→ 1; t > 0

Finite Difference Methods for the Black-Scholes Eq.

I Let Ω denote the interior of the grid and ∂Ω the boundary points

I We apply θ weighted method to discretize the PDE, whereθ ∈ [0, 1]. This is a generalization of three methods, namely,explicit, implicit and Crank-Nicolson method.

θ(uj+1Ω −uj

Ω+AujΩ+Buj

∂Ω)+(1−θ)(uj+1Ω −uj

Ω+Auj+1Ω +Buj+1

∂Ω ) = 0

(I − θA)ujΩ = (I + (1− θ)A)uj+1

Ω + θBuj∂Ω + (1− θ)Buj+1

∂Ω


I Let Ω denote the interior of the grid and ∂Ω the boundary pointsI We apply θ weighted method to discretize the PDE, whereθ ∈ [0, 1]. This is a generalization of three methods, namely,explicit, implicit and Crank-Nicolson method.

θ(uj+1Ω −uj

Ω+AujΩ+Buj

∂Ω)+(1−θ)(uj+1Ω −uj

Ω+Auj+1Ω +Buj+1

∂Ω ) = 0

(I − θA)ujΩ = (I + (1− θ)A)uj+1

Ω + θBuj∂Ω + (1− θ)Buj+1

∂Ω


θ Stability Convergence Linear system needs to be solved0 Conditional O(h2 + k) No

1/2 Unconditional O(h2 + k2) Yes1 Unconditional O(h2 + k) Yes

I θ = 0, Explicit method,

I θ = 1, Implicit method,

I θ = 1/2, Crank-Nicolson method.













Comparison of three methods

Figure: Solution of untransformed BS equation for Put Option withparametersE = 10; r = 0.05; T = 6/12;σ = .2; D = 0; Smin = 0; Smax = 100;

American Options

I American option allows the holder to exercise the option at anypoint in time up to and including expiry.

I Will consider the finite difference method for American Put.I When should the holder of option exercise instead of waiting for

expiry?I At expiry, the payoff of a (European or American) Put is the

same, hence the boundary condition at t = T is:

P(S,T) = max(K − S, 0)

I At S = 0, as in the European case, we expect that the payoff willagain be K, discounted in time at the risk free rate, so thatP(0, t) = Ke−r(T−t).

American Options


I Will consider the finite difference method for American Put.

I When should the holder of option exercise instead of waiting forexpiry?

I At expiry, the payoff of a (European or American) Put is thesame, hence the boundary condition at t = T is:

P(S,T) = max(K − S, 0)


American Options



expiry?

I At expiry, the payoff of a (European or American) Put is thesame, hence the boundary condition at t = T is:

P(S,T) = max(K − S, 0)


American Options





P(S,T) = max(K − S, 0)


American Options





P(S,T) = max(K − S, 0)


American Options

I For the boundary as S→∞, we expect that the payoff to bezero, i.e. P(S→∞, t) = 0.

I When is optimal to exercise?I Strategy for American Put:

PAm(S, t) = max(K − S,PEu(S, t))

I We solve for American Put using the parameter values:

K = 50, r = 0.05, σ = 0.25 and T = 3.

American Options


I When is optimal to exercise?

I Strategy for American Put:



K = 50, r = 0.05, σ = 0.25 and T = 3.

American Options





K = 50, r = 0.05, σ = 0.25 and T = 3.

American Options





K = 50, r = 0.05, σ = 0.25 and T = 3.

American Options

I At each time step we need to check V ji = max(K − iδS,V j

i ).

I For explicit method it is easy, as we have just computed V ji .

I For implicit method, this cannot be done because at each timestep we need to solve the linear system. We don’t know V j

i untilwe get to next step.

I Use iterative solver to solve the linear systemI Examples: Jacobi iteration, Gauss-Siedel method or Successive

Over Relaxation or SOR iteration.

American Options


i ).






American Options


i ).






American Options


i ).




I Use iterative solver to solve the linear system

I Examples: Jacobi iteration, Gauss-Siedel method or SuccessiveOver Relaxation or SOR iteration.

American Options


i ).






SOR method to solve Ax = b

for k = 1, 2, · · · , kmax dofor i = 1, 2, · · · , n do

yk+1i =

1aii

bi −i−1∑j=1

aijxk+1j −

n∑j=i+1

aijxkj

xk+1

i = ωyk+1i + (1− ω)yk

i

end forend forwhere ω is called the relaxation parameter.

Example: American Put withK = 50, r = 0.05, σ = 0.25, ω = 1.2

S Value with implicit Euler+SOR50 5.854755 4.295560 3.154165 2.320270 1.711975 1.266880 0.9391

References

Paul Wilmott, Sam Howison, and Jeff Dewynne. The Mathematics of Financial Derivatives: A student introductionCambridge University Press, Cambridge, UK, First edition, 1995.

John C. Hull. Options, Futures, and Other Derivatives. Prentice-Hall, New Jersey, international edition, 1997.

Fisher Black & Myron Scholes The Pricing of Options and Corporate Liabilities. The Journal of Political Economy. Vol81, Issue 3, 1973.

Morton & Mayers Numerical Solution of Partial Differential Equations: An Introduction Cambridge University Press,Second edition, 2005.

Markus Leippold An option pricing tool, Semester Thesis in Quantitative Finance (2004) Swiss Banking Institut,University of Zurich.

Cox, Ross & Rubinstein Option pricing: A simplified approach The Journal of Financial Economics, Vol. 7, (July1979), pp. 229 - 263.

References







References







References







References







References







Desmond J. Higham Nine Ways to Implement the Binomial Method for Option Valuation in MATLAB SIAMREVIEW, Vol. 44, No. 4, pp. 661-677.

P.A. Forsyth An Introduction to Computational Finance Without Agonizing Pain, 2013.

Alonso Pena, Option Pricing with Radial Basis Functions: A Tutorial

Haug, E. Complete Guide to Option Pricing Formulas, 2nd ed, McGraw Hill, New York, 1998.

Higham, D. J. An Introduction to Financial Option Valuation: Mathematics, Stochas- tics and Computation, CambridgeUniversity Press, New York, 2004.

Horfelt, P. Extension of the corrected barrier approximation by Broadie, Glasserman, and Kou, Finance and Stochastic7(1): 231243, 2003.

Hull, J. Options, futures and other derivatives, 4th ed, Pearson Education, Upper saddle river, NJ, 2000.











































Brandimarte, P. Numerical Methods in Finance and Economics: A MATLAB-Based Introduction, 2nd ed, John WileySons, Hoboken, NJ, 2006

Cheney, W. and Kincaid, D. Numerical Mathematics and Computing, 5th ed, Thomson, Brooks/Cole, USA, 2004.

Brandimarte, P. Numerical Methods in Finance and Economics: A MATLAB-Based Introduction, 2nd ed, John WileySons, Hoboken, NJ, 2006

Cheney, W. and Kincaid, D. Numerical Mathematics and Computing, 5th ed, Thomson, Brooks/Cole, USA, 2004.

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