Transcript Basic Numerical Procedures - E
Basic Numerical Procedures
Chapter 17
Zheng Zhenlong, Dept of Finance,XMU
Tree Approaches to Derivatives Valuation
Trees Monte Carlo simulation Finite difference methods
Zheng Zhenlong, Dept of Finance,XMU
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 Zheng Zhenlong, Dept of Finance,XMU
Movements in Time
D
t
(Figure 17.1, page 392
)
S Su Sd Zheng Zhenlong, Dept of Finance,XMU
1. 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 risk-neutral world Mean: Variance:
e
(
r-q
) D
t
s 2 D
t
=
pu
=
pu
2 + (1– + (1–
p p
)
d
)
d
2 – e 2(
r-q
) D
t
A further condition often imposed is
u
= 1/
d Zheng Zhenlong, Dept of Finance,XMU
2. Tree Parameters for asset paying a dividend yield of q (Equations 17.4 to 17.7)
When D
t
is small a solution to the equations is
u
e
s D
t d
e
s D
t p a
a u
d d e
(
r
q
) D
t Zheng Zhenlong, Dept of Finance,XMU
The Complete Tree (Figure 17.2, page 394)
S
0
S
0
u S
0
d S
0
u 2 S
0
S
0
d 2 S
0
u 3 S
0
u S
0
d S
0
d 3 Zheng Zhenlong, Dept of Finance,XMU S
0
u 4 S
0
u 2 S
0
S
0
d 2 S
0
d 4
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
Zheng Zhenlong, Dept of Finance,XMU
Example: Put Option (Example 17.1, page 394)
S
0 = 50;
K
= 50;
r
=10%; s = 40%;
T
D
t
= 5 months = 0.4167; = 1 month = 0.0833
The parameters imply
u
= 1.1224;
d
= 0.8909;
a
= 1.0084;
p
= 0.5073
Zheng Zhenlong, Dept of Finance,XMU
Example (continued) Figure 17.3, page 395
50.00
4.49
56.12
2.16
44.55
6.96
62.99
0.64
50.00
3.77
39.69
10.36
70.70
0.00
56.12
1.30
44.55
6.38
35.36
14.64
79.35
0.00
62.99
0.00
50.00
2.66
39.69
10.31
31.50
18.50
Zheng Zhenlong, Dept of Finance,XMU
89.07
0.00
70.70
0.00
56.12
0.00
44.55
5.45
35.36
14.64
28.07
21.93
Calculation of Delta
Delta is calculated from the nodes at time D
t
Delta .
Zheng Zhenlong, Dept of Finance,XMU
Calculation of Gamma
Gamma is calculated from the nodes at time 2 D
t
D 1 .
.
. ; .
Gamma = D 1 D 2 D 2 .
.
.
.
Zheng Zhenlong, Dept of Finance,XMU
Calculation of Theta
Theta is calculated from the central nodes at times 0 and 2 D
t
Theta = .
per year per calendar day
Zheng Zhenlong, Dept of Finance,XMU
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 per 1% change in volatility
Zheng Zhenlong, Dept of Finance,XMU
Trees for Options on Indices, Currencies and Futures Contracts
As with Black-Scholes: For options on stock indices,
q
dividend yield on the index equals the For options on a foreign currency,
q
equals the foreign risk-free rate For options on futures contracts
q
=
r Zheng Zhenlong, Dept of Finance,XMU
Binomial Tree for Dividend Paying Stock
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
Zheng Zhenlong, Dept of Finance,XMU
Extensions of Tree Approach
Time dependent interest rates The control variate technique
Zheng Zhenlong, Dept of Finance,XMU
Alternative Binomial Tree (Section 17.4, page 406)
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 ) D
t
s D
t d
e
(
r
q
s 2 / 2 ) D
t
s D
t Zheng Zhenlong, Dept of Finance,XMU
Trinomial Tree (Page 409)
u
e
s
p u
3 D
t d
1 /
u
D
t
12 s 2
r
s 2 2 1 6
p m
2 3
p d
D
t
12 s 2
r
s 2 2 1 6
S Zheng Zhenlong, Dept of Finance,XMU Su p u p m S p d Sd
Time Dependent Parameters in a Binomial Tree (page 409)
Making
r
or
q
a function of time does not affect the geometry of the tree. The probabilities on the tree become functions of time. We can make s a function of time by making the lengths of the time steps inversely proportional to the variance rate.
Zheng Zhenlong, Dept of Finance,XMU
Adaptive Mesh Model
This is a way of grafting a high resolution tree on to a low resolution tree We need high resolution in the region of the tree close to the strike price and option maturity
Zheng zhenlong, Dept. of Finance, XMU
21
Monte Carlo Simulation
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
Zheng Zhenlong, Dept of Finance,XMU
Sampling Stock Price Movements (Equations 17.13 and 17.14, page 411)
In a risk neutral world the process for a stock price is
dS
s We can simulate a path by choosing time steps of length D
t
and using the discrete version of this where D
S
e ˆ
S
D
t
s
S
e D
t
is a random sample from f (0,1)
Zheng Zhenlong, Dept of Finance,XMU
A More Accurate Approach (Equation 17.15, page 412)
Use
d
ln
S
s 2 / 2
dt
s
dz
The discrete version of ln
S
(
t
D
t
) ln
S
(
t
) this is s 2 / 2 D
t
se or
S
(
t
D
t
)
S
(
t
)
e
s 2 / 2 D
t
s e D
t
D
t Zheng Zhenlong, Dept of Finance,XMU
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
Zheng Zhenlong, Dept of Finance,XMU
Sampling from Normal Distribution (Page 414)
One simple way to obtain a sample from f (0,1) is to generate 12 random numbers between 0.0 & 1.0, take the sum, and subtract 6.0
In Excel =NORMSINV(RAND()) gives a random sample from f (0,1)
Zheng Zhenlong, Dept of Finance,XMU
To Obtain 2 Correlated Normal Samples
Obtain independent normal samples
x
1
x
2 and set e e 1 2
x
1
x
1
x
2 1 2 and A procedure known as Cholesky’s decomposition when samples are required from more than two normal variables
Zheng Zhenlong, Dept of Finance,XMU
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.
Zheng Zhenlong, Dept of Finance,XMU
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
Zheng Zhenlong, Dept of Finance,XMU
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
Zheng Zhenlong, Dept of Finance,XMU
Variance Reduction Techniques
Antithetic variable technique Control variate technique Importance sampling Stratified sampling Moment matching Using quasi-random sequences
Zheng Zhenlong, Dept of Finance,XMU
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
Zheng Zhenlong, Dept of Finance,XMU
Finite Difference Methods
Finite difference methods aim to represent the differential equation in the form of a difference equation We form a grid by considering equally spaced time values and stock price values Define
i
D
t
ƒ
i,j
as the value of ƒ at time when the stock price is
j
D
S Zheng Zhenlong, Dept of Finance,XMU
Finite Difference Methods (continued)
In ƒ
t
we set
rS
ƒ
S
ƒ
S
1 2 s 2
S
2 2
S
ƒ 2 ƒ
i
,
j
1 2 D
S
ƒ
i
,
j
1
r
ƒ 2
S
2
S
ƒ ƒ 2 2 ƒ
i
,
j
1 D
S
ƒ
i
,
j
ƒ
i
,
j
D
S
ƒ
i
, ƒ
i
,
j
1 ƒ
i
,
j
1 D
S
2 2 ƒ
i
,
j j
1 D
S
or
Zheng Zhenlong, Dept of Finance,XMU
Implicit Finite Difference Method (Equation 17.25, page 420)
If we also set ƒ
t
ƒ
i
1 ,
j
ƒ D
t
we obtain the implicit finite difference method.
This involves solving simultaneous equations of the form:
a j
ƒ 1
b j
ƒ
c j
ƒ 1 ƒ
i
1 ,
j Zheng Zhenlong, Dept of Finance,XMU
Explicit Finite Difference Method (page 422-428)
If
f
S
and 2
f
S
2 are assumed to be the same at the (
i
1
,j
) point as they are at the (
i,j
) point we obtain the explicit finite difference method This involves solving equations of the form : ƒ
i
,
j
a
*
j
ƒ
i
1 ,
j
1
b
*
j
ƒ
i
1 ,
j
c
*
j
ƒ
i
1 ,
j
1
Zheng Zhenlong, Dept of Finance,XMU
Implicit vs Explicit Finite Difference Method
The explicit finite difference method is equivalent to the trinomial tree approach The implicit finite difference method is equivalent to a multinomial tree approach
Zheng Zhenlong, Dept of Finance,XMU
Implicit vs Explicit Finite Difference Methods (Figure 17.16, page 425)
ƒ
i , j
+1 ƒ
i
+1
, j
+1 ƒ
i , j
ƒ
i
+1
, j
ƒ
i , j
ƒ
i
+1
, j
ƒ
i , j
–1 Implicit Method Explicit Method
Zheng Zhenlong, Dept of Finance,XMU
ƒ
i
+1
, j
–1
Other Points on Finite Difference Methods
It is better to have ln
S
rather than
S
as the underlying variable Improvements over the basic implicit and explicit methods: Hopscotch method Crank-Nicolson method
Zheng Zhenlong, Dept of Finance,XMU