Upper picture: 20 discrete asset paths.. ♦ 7.2 Timescale invariance The next computational example reveals a key property of the asset price model.. 68 Asset price model: Part IIfor the
Trang 17.1 Computing asset paths 65
Fig 7.3 Upper picture: 20 discrete asset paths Lower picture: sample mean
of 104discrete asset paths.
Trang 266 Asset price model: Part II
Fig 7.4 Upper picture: 50 discrete asset paths over [0, T ] with S 0= 1, µ =
0.05, σ = 0.5, T = 1 and δt = 10 −2 Lower picture: histogram for S(T ) from
104such paths, with lognormal density function (6.10) superimposed.
it is visually indistinguishable from the exact mean S0e µt that we derived
We next give a test that confirms the lognormal behaviour of the asset model
Computational example Here, we set S0= 1, µ = 0.05 and σ = 0.5, and
com-puted discrete paths over [0, T ], with T = 1 We used a uniform time spacing of
t i+1− t i = δt = 10−2 The upper picture in Figure 7.4 shows 50 such paths In
the lower picture we give a kernel density estimate for the asset price at expiry.This was computed in the manner discussed in Section 4.3, using a histogramwith 45 bins of width 0.05 The corresponding lognormal density function (6.10),
which is superimposed as a dashed line, gives a good match ♦
7.2 Timescale invariance
The next computational example reveals a key property of the asset price model
The jaggedness looks the same over a range of different timescales In other
words, zooming in or out of the picture, we see the same qualitative behaviour
We saw the same effect when we moved from daily to weekly data in Figures 5.1and 5.2
Trang 3Computational example To generate Figure 7.5, we computed a single asset
path for S0= 1, µ = 0.05 and σ = 0.5 at equally spaced time points in [0, 1] a
distance 10−4apart Using this data, we plot three pictures Each picture shows
the path at 100 equally spaced time points
• The upper plot shows the path at 100 equally spaced points in [0, 1].
• The middle plot shows the path at 100 equally spaced points in [0, 0.1].
• The lower plot shows the path at 100 equally spaced points in [0, 0.01]
We see that zooming in on the path in this manner does not reveal any change inthe qualitative features – the path is ‘jagged’ at all time scales ♦
To understand why the pictures have this ‘timescale stability’ we go back to thediscrete model (6.2) and consider
• a small time interval δt,
• very small time interval δt = δt/L, where L is a large integer (In Figure 7.5 we used quite a moderate value, L= 10.)
Using (6.2) to get from time t = 0 to t = δt we have
Trang 468 Asset price model: Part II
for the change in S (t) From time t = 0 to t = δt, increments like this add up:
that the return over the next interval lies between a and b, but, of course, we cannot
predict with any certainty what actual return will be seen
By contrast with the uncertainty of returns, we can show that the sum-of-square
equally spaced subintervals [0, t1 ], [t1, t2], , [t L−1, tL ], with t i = iδt and δt =
This random variable has a variance proportional toδt, and hence is essentially
1 Some justification for this type of approximation can be found in Section 8.2.
Trang 57.4 Notes and references 69
0.9 1 1.1
0 0.1 0.2 0.3 0.4 0.5 0
0.01 0.02 0.03 0.04 0.05
Computational example Figure 7.6 confirms the sum-of-square returns result.
We use S0= 1, µ = 0.05 and σ = 0.3 Ten asset paths over [0, 0.5] are shown
in the upper left plot The paths were computed using equally spaced time points
a distanceδt = 0.5/100 = 5 × 10−3 apart, so L = 100 The lower left pictureplots the running sum-of-square returns
against t k for each path The sum is seen to approximate σ2t k; the height
σ2/2 is shown as a dotted line The right-hand pictures repeat the experiment
with L = 103, so δt = 5 × 10−4 We see that reducing δt has improved the
7.4 Notes and references
Our treatment of timescale invariance in Section 7.2 can be made rigorous, but the
concepts required are beyond the scope of this book (The essence is that if W (t) is
Trang 670 Asset price model: Part II
a Brownian motion then so is W (c2t )/c, for any constant c > 0; see, for example,
(Brze´zniak and Zastawniak, 1999, Exercise 6.28) and (Brze´zniak and Zastawniak,
1999, Exercise 7.20), and their solutions, for details of this result and why it applies
to the asset model.)
There have been numerous attempts to develop generalizations or alternatives tothe lognormal asset price model Many of these are motivated by the observation
that real market data has fat tails – extreme events occur more frequently than a
model based on normal random variables would predict
One approach is to allow the volatility to be stochastic, see (Duffie, 2001; Hull,2000; Hull and White, 1987), for example Another is to allow the asset to undergo
‘jumps’, see (Duffie, 2001; Hull, 2000; Kwok, 1998), for example Jump modelsare especially popular for modelling assets from the utility industries, such as elec-
trical power The article (Cyganowski et al., 2002) discusses some implementation
issues
An alternative is to take a general, parametrized class of random variables andfit the parameters to stock market data, see (Rogers and Zane, 1999), for example
A completely different approach is to abandon any attempt to understand the
processes that drive asset prices (in particular to pay no heed to the efficient ket hypothesis) and instead to test as many models as possible on real market data,and use whatever works best as a predictive tool A group of mathematical physi-cists with expertise in chaos and nonlinear time series, led by Doyne Farmer andNorman Packard, took up this idea They founded The Prediction Company inSanta Fe The company has a website at www.predict.com/html/ introduction.htmlwhich makes the claim that
mar-Our technology allows us to build fully automated trading systems which can handle huge amounts of data, react and make decisions based on that data and execute transactions based on those decisions – all in real time Our science allows us to build accurate and consistent predictive models of markets and the behavior of financial instruments traded in those markets.
The book (Bass, 1999) gives the story behind the foundation and early years of thecompany and has many insights into the practical issues involved in collecting andanalysing vast amounts of financial data
E X E R C I S E S
may define the continuously compounded rate of return for an asset over
that R ∼N(µ − σ2/2, σ2/t).
Trang 77.5 Program of Chapter 7 and walkthrough 71
7.5 Program of Chapter 7 and walkthrough
The program ch07, listed in Figure 7.7, produces a plot of 50 asset paths in the style of the upper ture in Figure 7.4 Having initialized the parameters, we make use of the cumulative product function, cumprod, to produce an array of asset paths Generally, given an M by L array X, cumprod(X) cre-
pic-ates an M by L array whose (i, j) element is the product X(1,j)*X(2,j)*X(3,j)* *X(i,j).
Supplying a second argument set to 2 causes the cumulative product to be taken along the
sec-ond index – across rows rather than down columns, so cumprod(X,2) creates an M by L array
whose(i, j) element is the product X(i,1)*X(i,2)*X(i,3)* *X(i,j) We also supply two
arguments to the randn function: randn(M,L) produces an M by L array with elements from the
randn pseudo-random number generator.
It follows that
Svals = S*cumprod(exp((mu-0.5*sigma^2)*dt + sigma*sqrt(dt)*randn(M,L)),2);
creates an M by L array whose i th row represents a single discrete asset path, as in (6.9) The next
line
Svals = [S*ones(M,1) Svals]; % add initial asset price
adds the initial asset as a first column, so that the i th row Svals(i,1),Svals(i,2), ,
Svals(i,L+1) represents the asset path at times 0,dt,2dt,3dt, ,T.
P R O G R A M M I N G E X E R C I S E S
P7.1 Write a program that illustrates the timescale invariance of the asset model,
in the style of Figure 7.5
P7.2 Usemean and std to verify the approximations (7.4) and (7.5) for (7.3)
%CH07 Program for Chapter 7
Trang 872 Asset price model: Part II
Quotes
But as a warning,
let me note that a trader with a better model might still not be able to transform
this knowledge into money.
Finance is consistent in its ability to build good models
and consistent in its inability to make easy money.
The purpose of the model is to understand the factors
that influence and move option prices
but in the absence of an ability to forecast these factors
the transformation into money remains non-trivial.
D I L I P B M A D A N (Madan, 2001) Evidence countering the efficient market hypothesis
comes in the form of stock market anomalies.
These are events that violate the assumption that stock returns
are randomly distributed.
They include the size effect
(big-company stocks out-perform small-company stocks or vice versa);
the January effect
(stock returns are abnormally high during the first few days of January);
the week-of-the-month effect
(the market goes up at the beginning and down at the end of the month);
and the hour-of-the-day effect
(prices drop during the first hour of trading on Monday and rise on other days).
Prices fall faster than they rise;
the market suffers from ‘roundaphobia’
(the Dow breaking ten thousand is a big deal);
and the market tends to overreact
(aggressive buying after good news is followed by nervous selling,
no matter what the news).
Finally, the efficient market hypothesis is incapable of explaining
stock market bubbles and crashes, insider trading, monopolies,
and all the other messy stuff that happens outside its perfect models.
T H O M A S A B A S S (Bass, 1999) Prices reflect intelligent behavior of rational investors and traders,
but they also reflect screaming mass hysteria.
A L E X A N D E R E L D E R (Elder, 2002)
Trang 9At this stage we have defined what we mean by a European call or put option on
an underlying asset and we have developed a model for the asset price movement
We are ready to address the key question: what is an option worth? More precisely,
can we systematically determine a fair value of the option at t = 0?
The answer, of course, is yes, if we agree upon various assumptions Although
our basic aim is to value an option at time t = 0 with asset price S(0) = S0, we
will look for a function V (S, t) that gives the option value for any asset price S ≥ 0
at any time 0≤ t ≤ T Moreover, we assume that the option may be bought and
sold at this value in the market at any time 0≤ t ≤ T In this setting, V (S0, 0) is
the required time-zero option value We are going to assume that such a function
respect to these variables exist It was mentioned in Section 7.1 that S (t) is not a
smooth function of t – it is jagged, without a well-defined first derivative However,
it is still perfectly possible for the option value V (S, t) to be smooth in S and t.
Looking ahead, Figures 11.3 and 11.4 illustrate this fundamental disparity.Our analysis will lead us to the celebrated Black–Scholes partial differential
equation (PDE) for the function V The approach is quite general and the PDE
is valid in particular for the cases where V (S, t) corresponds to the value of a
European call or put
73
Trang 1074 Black–Scholes PDE and formulas
The key idea in this chapter is hedging to eliminate risk To reinforce the idea,
and emphasize that it is a concrete tool as well as a theoretical device, the nextchapter is devoted to computational experiments that illustrate hedging in practice.Before launching into a description of hedging, we first introduce one of themain ingredients that goes into the analysis
8.2 Sum-of-square increments for asset price
To make progress, we need to work on two timescales For the rest of the chapter
we use
• a small timescale, determined by a time increment t, and
• a very small timescale, determined by a time increment δt = t/L, where L is a large
integer.
We consider some general time t ∈ [0, T ] and general asset price S(t) ≥ 0, and cus on the small time interval [t , t + t] This is broken down into equally spaced,
fo-very small, subintervals of length δt, giving [t0, t1 ], [t1, t2], , [t L−1, tL] with
t0 = t, t L = t + t and, generally, t i = t + iδt.
We will let
δS i := S(ti+1) − S(t i )
denote the change in asset price over a very small time increment Before ing to derive the Black–Scholes PDE, we need to establish a preliminary resultabout the sum-of-square increments, L−1
attempt-i=0 δS2
i A similar analysis was done inSection 7.3 for the sum-of-square returns,L−1
i=0(δS i /S(t i ))2.Returning to the discrete model (6.2) we have
We now make this summation amenable to the Central Limit Theorem by replacing
each S (t i ) by S(t) This approximation, which is discussed further in the next
Trang 118.2 Sum-of-square increments for asset price 75Working out the mean and variance of the random variables inside the summationand appealing to the Central Limit Theorem suggests the approximate relation
see Exercise 8.1 Becauseδt is very small, the variance of that final expression is
tiny, leading us to conclude that the sum-of-square increments is approximately a
The step of replacing each S (t i ) in (8.1) by S(t) can be loosely justified as
follows Our model (6.9) shows that
Using e x ≈ 1 + x for small x, we have
rigorous argument – Z is a random variable, not simply a real number – but it can
be shown that the overall conclusion is valid
Computational example Although we are not in a position to prove (8.4)
rigo-rously, we can certainly illustrate the result via a computational experiment Wemay copy the way that Figure 7.6 was produced, but now computing the sum-
of-square increments, instead of the sum-of-square returns We set S0= 1, µ =
0.05 and σ = 0.3 The upper left plot in Figure 8.1 shows ten discrete asset
paths over [0, t] with t = 0.5, using equally spaced points a distance δt =
t/100 = 5 × 10−3 apart So L = 100 and t = 0 The lower left picture plots
the running sum-of-square increments
Trang 1276 Black–Scholes PDE and formulas
Fig 8.1 Upper pictures: asset paths Lower pictures: running sum-of-square increments (8.5).
against t k for each path We see that the sum typically approximatesσ2t =
0.045 as k approaches L The right-hand pictures give the same information for
an example witht = 0.1 and L = 1000, so δt = 10−4 We see that the quality
8.3 Hedging
Now, to find a fair option value, we set up a replicating portfolio of asset and cash, that is, a combination of asset and cash that has precisely the same risk as the option at all time The portfolio will consist of a cash deposit D and a number A
of units of asset We allow D and A to be functions of asset price S and time t.
The portfolio value, denoted by, thus satisfies
We must specify how the asset holding A (S, t) and cash deposit D(S, t) are
going to vary with S and t Before delving into the details it is perhaps useful to
remind ourselves of some basic assumptions that are being made, all of which havebeen introduced earlier:
• there are no transaction costs,
• the asset can be bought/sold in arbitrary units,
Trang 138.3 Hedging 77
• short selling is permitted,
• no dividends are paid,
• the interest rate r is constant,
• trading of the asset (and option) can take place in continuous time.
To avoid unreadably long equations we will also introduce some shorthand
nota-tion A subscript i denotes evaluation of a function at (S(t i ), t i ), so
V i means V (S(t i ), t i ), i means(S(t i ), t i ), etc.
No subscript denotes evaluation at(S(t), t), so
V means V (S(t), t), means (S(t), t), etc.
The symbolδ denotes the difference over a timestep of length δt, so
portfolio has two sources
(1) The asset price fluctuation The changeδS i produces a change A i δS i in the portfolio value.
(2) Interest accrued on the cash deposit Using the discrete version for convenience (see
(2.7) in Exercise 2.2), we may write this contribution to the portfolio change as r D i δt.
We have kept theδS2
i term in (8.8) because experience from the previous two ters suggests that it will make a contribution of size proportional toδt Subtracting
chap-(8.7) from (8.8) in order to compare the change in the portfolio with that in theoption value, we find