finite- difference approach

Finite-difference approach contains two methods such as explicit finite-difference and implicit finite-difference. How can we to distinguish them? We can distinguish them through the solving process. The explicit finite-difference method can compute the solution straightforwardly. Compared with the explicit method, the implicit finite-difference method is more complicated. The implicit method must compute the solution by solving a group of algebraic equations. What are the advantages and disadvantages of these two kinds of method? For the explicit, the advantage is that easy to solving, however, it not stable sometimes. The shortcoming for the implicit is that big amount of calculation but can get a constringency result. In practical these two numerical approaches have a high application value.

Definition of calculus of continuous function 𝜇(𝑥), then we get first derivation

𝑑𝜇𝑑𝑥

=lim

𝜇(𝑥+𝛿)−𝜇(𝑥)(𝑥+𝛿)−𝑥

𝛿→0

=lim

𝜇(𝑥+𝛿)−𝜇(𝑥)

𝛿

𝑑𝜇𝑑𝑥

𝛿→0

. Omit limitation

𝜇(𝑥+𝛿)−𝜇(𝑥)

𝛿

calculation we get difference approximation

≈

and

this particular finite-difference approximation is called a forward difference. We also have

𝑑𝜇𝑑𝑥

≈

𝜇(𝑥)−𝜇(𝑥−𝛿)

𝛿

which is called

backward difference. We also define central difference by noting that

𝑑𝜇𝑑𝑥

≈

𝜇(𝑥+𝛿)−𝜇(𝑥−𝛿)

2𝛿

.

Let say at time t stock price is S𝑡 option value is 𝑉𝑡 , so we get a

two variables function such as 𝑉𝑡=V(S𝑡,𝑡) .So we can express the max(𝑆𝑇−𝐸,0),(𝑐𝑎𝑙𝑙 𝑜𝑝𝑡𝑖𝑜𝑛);

option profit like this: V(S,t)={

()()max𝐸−𝑆𝑇,0,𝑝𝑢𝑡 𝑜𝑝𝑡𝑖𝑜𝑛.Where E is strike price or exercise price, T is stock expiration time. Let ∆t=𝑇⁄𝑁 where T is expiration, T is divided to N+1 the same small interval time such as 0, ∆t,2∆t,⋯,T.

Let ∆S=𝑆𝑚𝑎𝑥⁄𝑀 where 𝑆𝑚𝑎𝑥 is assumed as the maximum stock price, so we can get 𝑀+1 stock prices such as 0, ∆S,2∆S,⋯,𝑆𝑚𝑎𝑥.

The implicit finite-difference approach: This

𝜕𝑉𝜕𝑡

2

+

1

𝜕𝑉

𝜎2𝑆222𝜕𝑆

+𝑟𝑆

𝜕𝑉𝜕𝑆

−𝑟𝑉=0 is Black-Scholes partial

differential equation, where 𝑟 is free risk interest and 𝜎 is volatility.

Now we can use difference approximation instead of derivation.

𝜕𝑉𝜕𝑡

=

𝑉𝑖+1,𝑗−𝑉𝑖,𝑗

∆𝑡

,

𝜕𝑉𝜕𝑆

=

𝑉𝑖,𝑗+1−𝑉𝑖,𝑗−1

2∆𝑆

,

𝜕2𝑉𝜕𝑆2=

𝑉𝑖,𝑗+1−2𝑉𝑖,𝑗+𝑉𝑖,𝑗−1

∆𝑆2 . Now put

above difference approximation into Black-Scholes partial differential equation, then we get

12

𝑉𝑖+1,𝑗−𝑉𝑖,𝑗

∆𝑡

+𝑟𝑗 ∆𝑆×

𝑉𝑖,𝑗+1−𝑉𝑖,𝑗−1

2∆𝑆

+

𝜎𝑗∆𝑆×

222

𝑉𝑖,𝑗+1−2𝑉𝑖,𝑗+𝑉𝑖,𝑗−1

∆𝑆2=𝑟𝑉𝑖,𝑗 .Then collection of like terms

12

12

get

12

12

(1+∆𝑡𝜎2𝑗2+∆𝑡𝑟)𝑉𝑖,𝑗+(−∆𝑡𝑟𝑗−∆𝑡𝜎2𝑗2)𝑉𝑖,𝑗+1+

(∆𝑡𝑟𝑗−∆𝑡𝜎2𝑗2)𝑉𝑖,𝑗−1=𝑉𝑖+1,𝑗. The explicit finite-difference approach:

𝜕𝑉𝜕𝑡

𝜕2𝑉𝜕𝑆2=

𝑉𝑖+1,𝑗−𝑉𝑖,𝑗

∆𝑡

,

𝜕𝑉𝜕𝑆

=

𝑉𝑖+1,𝑗+1−𝑉𝑖,𝑗−1

2∆𝑆

,

=

𝑉𝑖+1,𝑗+1−2𝑉𝑖+1,𝑗+𝑉𝑖+1,𝑗−1

∆𝑆2 .

Substitute the derivation, and

𝜕𝑡

𝜕𝑉𝜕𝑉𝜕𝑆1

𝜕2𝑉𝜕𝑆21212

in Black-Scholes partial

12

differential equation, we get

1

1+𝑟∆𝑡1

∆𝑡𝜎2𝑗2)𝑉𝑖+1,𝑗+12

×(1−∆𝑡𝜎2𝑗2)𝑉𝑖+1,𝑗+

=𝑉𝑖,𝑗 .

1+𝑟∆𝑡11+𝑟∆𝑡

×(∆𝑡𝑟𝑗+∆𝑡𝜎2𝑗2)𝑉𝑖+1,𝑗−1+×(∆𝑡𝑟𝑗+