CHAPTER 7Brief Technical Background BLACK-SCHOLES MODEL The basic BSM is summarized as follows: whereΦ is the cumulative standard-normal distribution function S is the value of the forec
Trang 1PART Two
Technical Background
of the Binomial Lattice and Black-Scholes Models
Trang 3CHAPTER 7
Brief Technical Background
BLACK-SCHOLES MODEL
The basic BSM is summarized as follows:
whereΦ is the cumulative standard-normal distribution function
S is the value of the forecast stock price at grant date
X is the option’s contractual strike price
rf is the nominal risk-free rate
σ is the annualized volatility
T is the time to expiration of the option
To illustrate its use, let us assume that an option exists such that both the
stock price (S) and the strike price (X) are $100, the time to expiration (T)
is one year with a 5 percent annualized risk-free rate (rf) for the same
dura-tion, while the annualized volatility (σ) of the underlying asset is 25 per-cent The BSM calculation yields $12.3360:
T
T
T
rf T
rf T
−
−
( )
( )
σ σ
σ σ σ
σ
σ
σσ T
77
Trang 4The cumulative standard-normal distribution function can be solved in Ex-cel by using its “NORMSDIST( )” function
In addition, you can create calculation codes within Excel’s Visual Basic for Applications (VBA) environment Following are the VBA codes for the BSM for estimating call and put options The equations for the BSM are simplified to functions in Excel named “BlackScholesCall” and
“BlackScholesPut.”
Public Function BlackScholesCall(Stock As Double, Strike As Double, Time As Double, Riskfree _
As Double, Volatility As Double) As Double Dim D1 As Double, D2 As Double
D1 = (Log(Stock / Strike) + (Riskfree + 0.5 * Volatility ^ 2 ) * Time) / (Volatility * Sqr(Time))
D2 = D1 – Volatility * Sqr(Time) BlackScholesCall = Stock * Application.NormSDist(D1) – Strike * Exp(–Time * Riskfree) * _
Application.NormSDist(D2) End Function
Public Function BlackScholesPut(Stock As Double, Strike As Double, Time As Double, Riskfree _
As Double, Volatility As Double) As Double Dim D1 As Double, D2 As Double
D1 = (Log(Stock / Strike) + (Riskfree + 0.5 * Volatility ^ 2 ) * Time) / (Volatility * Sqr(Time))
D2 = D1 – Volatility * Sqr(Time) BlackScholesPut = Strike * Exp(–Time * Riskfree) * Application.NormSDist(–D2) – Stock * _ Application.NormSDist(–D1)
End Function
78 BACKGROUND OF THE BINOMIAL LATTICE AND BLACK-SCHOLES MODELS
Call S S X rf T
T
Xe S X rf T
T
rf T
=
−
−
−
Φ
$
ln $
.
$
( )
σ σ
σ σ
2
0
100
100
100 0 05
1
20 25 1
0 25 1
100 ( )
ln $
.
$ ( ) $ ( ) ( ) $ ( ) $ ( ) $
05 1
2
100
100 0 05
1
20 25 1
0 25 1
100 0 3250 100 0 9512 0 0750 100 0 6274 95 12 0 5299 12 3360
Φ
Call
Trang 5As an example, entering the function in Excel:
“=BlackScholesCall(100,100,1,5%,25%)”
results in 12.3360 and entering the function in Excel:
“=BlackScholesPut(100,100,1,5%,25%)”
results in 7.4589
Note that Log is a natural logarithm function in VBA and Sqr is
square root, and make sure there is a space before the underscore in the code
MONTE CARLO SIMULATION MODEL
In the simulation approach for estimating European options, a series of forecast stock prices are created using the Geometric Brownian Motion stochastic process, and the options maximization calculation is applied to the end point of the series, and discounted back to time zero, at the risk-free rate That is, starting with an initial seed value of the underlying stock price, simulate out multiple future pathways using a Geometric Brownian Motion, where That is, the change in asset value
δS t at time t is the value of the asset in the previous period S t–1multiplied
by the Brownian Motion The term rf is the risk-free rate,
δt is the time-steps, σ is the volatility, and ε is the simulated value from a
standard-normal distribution with mean of zero and a variance of one Other variations of Brownian Motions exist but for illustration purposes, this simple version will be used
The first step in Monte Carlo simulation is to decide on the number of time-steps to simulate In the example, 10 steps were chosen for simplicity
Starting with the initial stock price of $100 (S0), the change in value from this initial value to the first period is seen as . Hence, the stock price at the first time-step is equivalent to
The stock price at the second
the way until the terminal tenth time-step Notice that because ε changes
on each simulation trial, each simulation trial will produce an entirely
S2 =S1+δ ( )S2 =S1+S rf t1( δ +σε δt)
S1=S0+δ ( )S1=S0+S rf t0( δ +σε δt)
δS1=S rf t0( ( )δ +σε δt)
rf t( )δ +σε δt
( )
δS t =S t−1(rf t( )δ +σε δt)
Trang 6different asset-evolution pathway At the end of the tenth time-step, the maximization process is then applied That is, for a simple European
op-tion with a $100 implementaop-tion cost, the funcop-tion is simply C 10,i =
Max[S 10,i – X, 0] This is the call value C 10,i at time 10 for the i thsimulation trial This value is then discounted at the risk-free rate to obtain the call
value at time zero, that is, C 0,i = C 10,i e –rf(T) This is a single-point estimate for
a single simulated pathway A forecast distribution of the thousands of simulated pathways is collected and the mean of the distribution is the ex-pected value of the option On the one hand, it must be stressed that Monte Carlo simulation can be applied only to calculate European op-tions, and not American opop-tions, making it less suitable for use in valuing ESOs On the other hand, Monte Carlo can be used to simulate the uncer-tain input variables that go into the customized binomial lattice model as seen in Chapter 5
BINOMIAL LATTICES
Binomial lattices, in contrast to the other methods, are easy to implement and easy to explain They are also highly flexible but require significant com-puting power and time-steps to obtain good approximations, as will be seen later It is important to note, however, that at the limit, results obtained through the use of binomial lattices tend to approach those derived from closed-form solutions, and hence, it is always recommended that both ap-proaches be used to benchmark the results The results from closed-form so-lutions may be used in conjunction with the binomial lattice approach when presenting a complete financial options valuation solution in the most basic European options analysis However, when real-life cases are added into the analysis (forfeitures, vesting, suboptimal exercise behavior multiples, and blackout dates), the results diverge because the closed-form models such as the BSM or GBM cannot account for these added variables
Following is an example to illustrate the point of binomial lattices ap-proaching the results of a closed-form model Let us look again at the Eu-ropean call option presented previously, but this time, calculated using the GBM:
Using the previous example where both the stock price (S) and the strike price (X) are $100, the time to expiration (T) is one year with a 5
per-80 BACKGROUND OF THE BINOMIAL LATTICE AND BLACK-SCHOLES MODELS
T
T
σ
σ σ
Trang 7cent risk-free rate (rf) for the same duration, while the volatility (σ) of
the underlying asset is 25 percent with no dividends (q) The GBM
cal-culation yields $12.3360, similar to the BSM calcal-culations, while using a binomial lattice we obtain the following results:
N = 10 steps $12.0923
N = 20 steps $12.2132
N = 50 steps $12.2867
N = 100 steps $12.3113
N = 1,000 steps $12.3335
N = 10,000 steps $12.3358
N = 50,000 steps $12.3360 Notice that even in this simplified example, as the number of time-steps
(N) gets larger, the value calculated using the binomial lattice approaches
the GBM closed-form solution Suffice it to say, many steps are required for a good estimate using binomial lattices It has been shown in past re-search that 1,000 time-steps are usually sufficient for a good approxima-tion for up to 2 decimals Chapter 10 provides a case example of how to find the optimal number of lattice steps and to test for results convergence
in a binomial lattice
SUMMARY AND KEY POINTS
■ The three mainstream approaches used to solve simple options are the GBM and BSM closed-form models, path-dependent simulation, and binomial lattices
■ BSM is highly inflexible and can be applied to solve only European options
■ Path-dependent simulations are also applicable for solving only Euro-pean options
■ Binomial lattices are more flexible and can be used to solve both Amer-ican and European options and are capable of handling other exotic input variables that exist in real-life ESOs