0 0.1 0.2 0.3 0.4 0.5 0.6
0.8 1 1.2 1.4 1.6
dt = 5 × 10−3 dt = 5 × 10−4
Asset paths
0 0.1 0.2 0.3 0.4 0.5 0
0.01 0.02 0.03 0.04 0.05
Sum-of-square returns
σ2/2 σσ22/2/2
0 0.1 0.2 0.3 0.4 0.5 0.8
0.9 1 1.1
Asset paths
0 0.1 0.2 0.3 0.4 0.5 0
0.01 0.02 0.03 0.04 0.05
Sum-of-square returns
σ2/2
Fig. 7.6. Upper pictures: asset paths. Lower pictures: running sum-of-square returns (7.6).
constant. Thus, although the individual returns are unpredictable, the sum of the squared returns taken over a large number of small intervals is approximately equal toσ2t.
Computational example Figure 7.6 confirms the sum-of-square returns result.
We use S0=1,µ=0.05 andσ =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 picture plots the running sum-of-square returns
k i=1
S(ti+1)−S(ti) S(ti)
2
(7.6) against tk for each path. The sum is seen to approximate σ2tk; 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
match. ♦
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 ifW(t)is
a Brownian motion then so isW(c2t)/c, for any constantc >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 to the lognormal asset price model. Many of these are motivated by the observation that real market data hasfat 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 models are especially popular for modelling assets from the utility industries, such as elec- trical power. The article (Cyganowskiet al., 2002) discusses some implementation issues.
An alternative is to take a general, parametrized class of random variables and fit the parameters to stock market data, see (Rogers and Zane, 1999), for example.
A completely different approach is to abandon any attempt tounderstand the processes that drive asset prices (in particular to pay no heed to the efficient mar- 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 and Norman Packard, took up this idea. They founded The Prediction Company in Santa Fe. The company has a website at www.predict.com/html/ introduction.html which makes the claim that
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 the company and has many insights into the practical issues involved in collecting and analysing vast amounts of financial data.
E X E R C I S E S
7.1. Confirm the results (7.4) and (7.5).
7.2. By analogy with the continuously compounded interest rate model, we may define the continuously compounded rate of return for an asset over [0,t] to be the random variableRsatisfyingS(t)=S0eRt. Using (6.8), show thatR ∼N(µ−σ2/2, σ2/t).
7.5 Program of Chapter 7 and walkthrough 71 7.5 Program of Chapter 7 and walkthrough
The programch07, listed in Figure 7.7, produces a plot of 50 asset paths in the style of the upper pic- 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 anMbyLarrayX,cumprod(X)cre- ates anMbyL array whose(i,j)element is the productX(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, socumprod(X,2)creates an M by L array whose(i,j)element is the productX(i,1)*X(i,2)*X(i,3)*...*X(i,j). We also supply two arguments to therandnfunction:randn(M,L)produces an Mby Larray with elements from the randnpseudo-random number generator.
It follows that
Svals = S*cumprod(exp((mu-0.5*sigma^2)*dt + sigma*sqrt(dt)*randn(M,L)),2);
creates anMbyL array whoseith 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 theith rowSvals(i,1),Svals(i,2), . . . , Svals(i,L+1)represents the asset path at times0,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. Usemeanandstdto verify the approximations (7.4) and (7.5) for (7.3).
%CH07 Program for Chapter 7
%
% Plot discrete sample paths randn(’state’,100)
clf
%%%%%%%%% Problem parameters %%%%%%%%%%%
S = 1; mu = 0.05; sigma = 0.5; L = 1e2; T = 1; dt = T/L; M = 50;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tvals = [0:dt:T];
Svals = S*cumprod(exp((mu-0.5*sigmaˆ2)*dt + sigma*sqrt(dt)*randn(M,L)),2);
Svals = [S*ones(M,1) Svals]; % add initial asset price plot(tvals,Svals)
title(’50 asset paths’) xlabel(’t’), ylabel(’S(t)’)
Fig. 7.7. Program of Chapter 7:ch07.m.
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)
8
Black–Scholes PDE and formulas
O U T L I N E
• sum-of-squares for asset price
• replicating portfolio
• hedging
• Black–Scholes PDE
• Black–Scholes formulas for a European call and put