Computational Finance

1
Computational Finance

• Lecture 6
• Numerical Procedure Finite Difference

2
Review of Black-Scholes Formula

• must satisfy the following PDE
• and

3
Review of Black-Scholes Formula

• Black-Scholes formula
• European call
• European put
• where

4
Some Extensions to Black-Scholes Formula

• But we assumed non-dividend stocks as the
  underlying assets when deriving the B-S formula.
• How about other kinds of assets?

5
Options on Dividend-Paying Stocks

• Options on dividend-paying stocks
• Usually stock prices will go down by an amount at
  the dividend payable date reflecting the dividend
  paid per share.
• So we should take this change on the stock price
  into account when pricing options based upon such
  securities.

6
Options on Dividend-Paying Stocks

• We can view the whole stock prices as the sum of
  two parts
• Riskless component that corresponds to the known
  dividend during the life of the option
• Risky component.
• Reset to be the current stock price minus the
  present value of dividends. Then we can use the
  B-S formula.

7
Options on Dividend-Paying Stocks

• Consider a 6-month European call option on a
  stock when there are two dividend payments
  expected in two months and five months.
• The dividend of each payment is expected to be
  0.5.
• Current stock price 40, volatility 30 per
  annum, risk free interest 9 per annum
• Strike price 40

8
Options on Dividend-Paying Stocks

• The present value of the dividends is
• The stock price after deduction
• Plugging in B-S formula,

9
Currency Options

• Recall that in the binomial tree model to
  duplicate options on currencies, we use a
  portfolio of foreign currency and risk free
  saving account.
• Foreign currency generates interests according to
  foreign risk-free interest rates, which is
  different from the domestic risk-free interest
  rate.

10
Currency Options

• So, such options can be viewed as options on an
  underlying asset with continuous-time-paying
  dividends.

11
Currency Options

• Consider a four-month European call option traded
  in the US market on the British pound.
• Current exchange rate US1.3/pound
• Strike price US1.4/pound
• Risk free interest rates 2 in US, 3 in UK
• Exchange rate volatility 20

12
Some Extensions of Black-ScholesCurrency Options

• The duplication argument will lead to
• The B-S formula then is transformed to

13
Some Extensions of Black-ScholesCurrency Options

• In the previous example, the interest rate in US
  is considered as domestic and the one in UK as
  foreign

14
Some Extensions Black-Scholes Can Not Handle

• How to price American options is still open so
  far. That is why we introduce numerical methods
  to overcome the difficulty.

15
Grids

• associates a pair of t and S with the
  value of an option when time is t and the
  underlying asset price is S.
• S
• T t

16
Grids

• Impossible to calculate all values for each pair
  of t and S
• Finite difference method is to calculate the
  values at the nodes of a grid.
• S
• 
  t

17
Node Labels

• Introduce the following labels to simplify our
  notations
• for node (t, S), we denote it by (i, j) if
• We intend to calculate

18
Boundary Conditions

• The values at some nodes are clear

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Black-Scholes PDE

• But we do not know the values for most of nodes.
  
• We would like to derive some relationship among
  nodes via Black-Scholes PDE

20
Differentiation on Grids

• Approximating Theta
• If the distance of time grid is , then

21
Differentiation on Grids

• Approximating Delta
• We can use the following Taylor expansion
• Then,

22
Differentiation on Grids

• Approximating Gamma
• By Taylor expansion,
• Then,

23
Explicit Finite Difference Method

• Plugging all of the approximations back to PDE,
  we have

24
Explicit Finite Difference Method

• After rearrangement,
• where

25
Explicit Finite Difference Method

• Under our labeling system, the recursion becomes
• where

26
Explicit Finite Difference Method

• Relationship in the explicit finite difference
  method
• S
• 
  t

is calculated from there
Option Value
27
Pseudo Codes

• Pseudo codes
• Suppose that
• Set values for all V(M, j)
• For i M to 0
• Calculate
• end

28
One Tip Truncation of Stock Price Range

• In theory, stock price could be any number from 0
  to infinity. But it is impossible to calculate
  infinite numbers in a computer.
• We truncate the range of stock price to be finite
  by choosing a sufficiently high value, .
  

29
One Tip Truncation of Stock Price Range

• The recursion relation does not work for nodes at
  the boundary.
• One convention is as follows
• Call option
• Put option

30
Numerical Example

• Pricing a European put with strike price 50 and
  maturity in 5 months. The underlying stock is
  non-dividend-paying and the current price is 40
  and its volatility is 40 per year. The risk free
  interest rate is 10 per year.

31
American Options Pricing

• American options entitle the right of early
  exercising.
• When implementing finite difference for American
  options, we need to evaluate every node taking
  such right into account.