Transcript Basic Numerical Procedures - Tian

Basic Numerical Procedures
Chapter 19
資管所 柯婷瑱 2009/07/17
1
Approaches to Derivatives Valuation
• Trees
• Monte Carlo simulation
• Finite difference methods
2
Binomial Trees
• Binomial trees are frequently used to
approximate the movements in the price
of a stock or other asset
• In each small interval of time the stock
price is assumed to move up by a
proportional amount u or to move down
by a proportional amount d
3
Movements in Time Dt
(Figure 19.1, page 408)
Su
S
Sd
4
Generalization (Figure 11.2, page 239)
A derivative lasts for time T and is
dependent on a stock
Su
0
S0
ƒ
ƒu
S0d
ƒd
5
Generalization
(continued)
• Consider the portfolio that is long D shares and short 1
derivative
S0uD – ƒu
S0dD – ƒd
• The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or
ƒu  f d
D
S 0u  S 0 d
6
Generalization
(continued)
• Value of the portfolio at time T is
S0uD – ƒu
• Value of the portfolio today is
(S0uD – ƒu)e–rT
• Another expression for the
portfolio value today is S0D – f
• Hence
ƒ = S0D – (S0uD – ƒu )e–rT
7
Generalization
(continued)
Substituting for D we obtain
ƒ = [ pƒu + (1 – p)ƒd ]e–rT
e d
where p 
ud
rT
8
Tree Parameters for asset
paying a dividend yield of q
Parameters p, u, and d are chosen so that the
tree gives correct values for the mean &
variance of the stock price changes in a riskneutral world
Mean:
Variance:
e(r-q)Dt = pu + (1– p )d
s2Dt = pu2 + (1– p )d 2 – e2(r-q)Dt
A further condition often imposed is u = 1/ d
9
Tree Parameters for asset paying a
dividend yield of q
(continued)
When Dt is small a solution to the equations is
ue
s Dt
d  e s
Dt
ad
p
ud
a  e ( r  q ) Dt
Cox-Ross-Rubinstein binomial tree.
10
The Complete Tree
(Figure 19.2, page 410)
S0u 3
S0u 4
S0u 2
S0u
S0
S0u
S0
S0d
S 0d
S0u 2
S0
S0d 2
S0d 3
S0d 2
S0d 4
11
Backwards Induction
• We know the value of the option at
the final nodes
• We work back through the tree using
risk-neutral valuation to calculate the
value of the option at each node,
testing for early exercise when
appropriate
12
Example: Put Option
(Example 19.1, page 410)
S0 = 50; K = 50; r =10%; s = 40%;
T = 5 months = 0.4167;
Dt = 1 month = 0.0833
The parameters imply
u = 1.1224; d = 0.8909;
a = 1.0084; p = 0.5073
13
Example (continued)
Figure 19.3, page 411
89.07
0.00
79.35
0.00
70.70
0.00
70.70
0.00
62.99
0.00
62.99
0.64
50.00
2.66
50.00
3.77
50.00
4.49
56.12
0.00
56.12
1.30
56.12
2.16
44.55
5.45
44.55
6.38
44.55
6.96
39.69
10.31
39.69
10.36
35.36
14.64
35.36
14.64
31.50
18.50
28.07
21.93
14
Calculation of Delta
Delta is calculated from the nodes at time Dt
2.16  6.96
Delta 
 0.41
5612
.  44.55
15
Calculation of Gamma
Gamma is calculated from the nodes at time
2Dt
0.64  3.77
3.77  10.36
D1 
 0.24; D 2 
 0.64
62.99  50
50  39.69
D1  D 2
Gamma =
 0.03
1165
.
16
Calculation of Theta
Theta is calculated from the central nodes at
times 0 and 2Dt
3.77  4.49
Theta =
  4.3 per year
01667
.
or - 0.012 per calendar day
17
Calculation of Vega
• We can proceed as follows
• Construct a new tree with a volatility of 41%
instead of 40%.
• Value of option is 4.62
• Vega is
4.62  4.49  013
.
per 1% change in volatility
18
Trees for Options on Indices,
Currencies and Futures Contracts
As with Black-Scholes:
– For options on stock indices, q equals the dividend
yield on the index
– For options on a foreign currency, q equals the
foreign risk-free rate
– For options on futures contracts q = r
19
A know dividend yield
s0u
s0
s 0d
ex-dividend date
20
21
A known dollar dividend D
59.47495
70.69895
52.98909
62.98909
56.12
50
56.12
56.11698
49.99731
44.545
50
44.5426
39.68514
35.35549
47.20798
39.99731
44.545
44.89298
35.6336
29.68514
33.3186
26.44649
s
50
p
0.5073
u
1.1224
d
0.8909
22
Binomial Tree for Stock Paying Known
Dividends
• Procedure:
– Draw the tree for the stock price less the
present value of the dividends
– Create a new tree by adding the present
value of the dividends at each node
• This ensures that the tree recombines
and makes assumptions similar to those
when the Black-Scholes model is used
23
Alternative Binomial Tree
• time steps are so large that
s  (r  q Dt )
a -d
p
, a  d  p is negative
u -d
negative probabilities
Alternative Binomial Tree
Instead of setting u = 1/d we can set
each of the 2 probabilities to 0.5 and
ue
( r  q  s 2 / 2 ) Dt  s Dt
d e
( r  q  s 2 / 2 ) Dt  s Dt
it’s not s straightforward to calculate delta,
gamma, and rho because the tree is no longer
centered at the initial stock price
25
Trinomial Tree (Page 424)
Su
u  es
3 Dt
d  1/ u
Dt  s 2  1
 r   
pu 
2 
12s 
2  6
2
pm 
3
Dt  s 2  1
 r   
pd  
2 
12s 
2  6
pu
S
pm
S
pd
Sd
26
Time Dependent Parameters in a
Binomial Tree (page 425)
• We have assumed that r, q, rf ,and are
s
constants.
• In practice, they are usually assumed to be
time dependent.
– r is forward interest rate
27
• Monte Carlo Simulation and Options
28
Sampling Stock Price Movements
(Equations 19.13 and 19.14, page 427)
• In a risk neutral world the process for a
stock price is
 S dt  sS dz
dS  
• We can simulate a path by choosing time
steps of length Dt and using the discrete
version of this
DS  ˆS Dt  sS e Dt
where e is a random sample from f(0,1)
29
A More Accurate Approach
(Equation 19.15, page 428)
Use


d ln S  ˆ  s 2 / 2 dt  s dz
The discrete versionof this is


ln S (t  Dt )  ln S (t )  ˆ  s 2 / 2 Dt  se Dt
or
S (t  Dt )  S (t ) e ˆ s / 2  Dt se
2
Dt
30
Monte Carlo Simulation and Options
When used to value European stock options, Monte
Carlo simulation involves the following steps:
1. Simulate 1 path for the stock price in a risk neutral
world
2. Calculate the payoff from the stock option
3. Repeat steps 1 and 2 many times to get many
sample payoff
4. Calculate mean payoff
5. Discount mean payoff at risk free rate to get an
estimate of the value of the option
31
Table 19.2 Monte Carlo Simulation to
Check Black-Scholes
Extensions
When a derivative depends on several
underlying variables we can simulate
paths for each of them in a risk-neutral
world to calculate the values for the
derivative
33
Number of Trials
• Denote the mean by μ and the standard
deviation by ω. The variable μ is the
simulation’s estimate of the value of the
derivative. The standard error of the
estimate is

M
where M is the number of trails.
• A 95% confidence interval for the price
of the derivative is therefore given by

1.96
M
 f 
1.96
M
Standard Errors in Monte Carlo
Simulation
The standard error of the estimate of the
option price is the standard deviation of
the discounted payoffs given by the
simulation trials divided by the square root
of the number of observations.
To double the accuracy of a simulation, we
must quadruple the number of trials.
35
Example 19.8
• In Table 19.2, the value of the option is
calculated as the average of 1000 numbers.
• The standard deviation of the numbers is 7.68
• In this case, ω=7.68 and M=1000. The
standard error of the estimate is
7.68 1000  0.24
• The spreadsheet therefore gives a 95%
confidence interval for the option value as
(4.98-1.96x0.24) to (4.98+1.96x0.24) , or 4.51
to 5.45.
Application of Monte Carlo Simulation
• Monte Carlo simulation can deal with
path dependent options, options
dependent on several underlying state
variables, and options with complex
payoffs
• It cannot easily deal with American-style
options
37
Determining Greek Letters
For D:
1. Make a small change to asset price
2. Carry out the simulation again using the
same random number streams
3. Estimate D as the change in the option price
divided by the change in the asset price
Proceed in a similar manner for other Greek
letters
38
Sampling Through the Tree
Instead of sampling from the stochastic
process we can sample paths randomly
through a binomial or trinomial tree to value a
derivative
Suppose we have a binomial tree where the probability of an up
movement is 0.6.
At each node, we sample a random number between 0 and 1.
39
Example
89.07
0.00
79.35
0.00
70.70
0.00
62.99
0.64
56.12
2.16
50.00
4.49
70.70
0.00
62.99
0.00
56.12
1.30
50.00
3.77
44.55
6.96
56.12
0.00
50.00
2.66
44.55
6.38
39.69
10.36
44.55
5.45
39.69
10.31
35.36
14.64
35.36
14.64
31.50
18.50
28.07
21.93
40
Example Asian option
Using N-step binomial tree and sampling from the 2Npaths that are
possible.
Trial
Path
Average stock price
Option payoff
1
UUUUD
64.98
14.98
2
UUUDD
59.82
9.82
：
：
：
：
：
：
：
：
：
：
：
：
9
UUUDU
62.25
12.25
10
DDUUD
45.56
0
Average
7.08
discount
6.79
41