COMP6212讲解、辅导MATLABCOMP6212 Computational Finance

COMP6212 Computational Finance 2017/18
Assignment (Part I (2 & 3 (merged) of 3 from MN): 30%)
Issue 11 March 2019
Due 26 March 2019 (15:00)
Please work on Section 1 [20 marks] and either one of sections 2 or 3 [10 marks each].
1 Derivatives Pricing [20 Marks]
Some prices of call and put options written on the FTSE100 Index are available in the course notes page
(file FTSEOPtionsData). These are daily data during the period January 2018 to January 2019, all of
which are European style exercise and maturing in January 2019. Select parts of this data for the exercises
below.
a Study how the Black-Scholes model of pricing options was derived before attempting this task. The
notation is : K, the strike price; S, the value of the underlying asset; r the risk-free interest rate;
T, the time of maturity; t, the current time and N (x), the cummulative normal distribution.
1. Write the expression for N
0
(x) (the derivative of N (x)).
2. Show that S N0
(d1) = K exp(r(T t)) 0
(d2) where d1 and d2 were defined as:
d1 =log(S/K) + (r + σ2/2)(T t)σp(T t)
d2 = d1 σq(T t)
3. Calculate the derivaties d1/S and d2/S
4. With the solution for the call option price given by
c = SN (d1) K exp(r(T t)) N (d2)
show that its derivative with respect to time is c
t =rK exp(r(T t)) N (d2) SN0(d1)σ2p(T t)
Show that c/S = N (d1)
5. Differentiating again to get,
and substituting in the relevant expression, show that the expression for the call option price
indeed the solution to the Black-Scholes differential equation.
b Evaluate how well the option prices in the given data satisfy the Black-Scholes model. For each option,
evaluate the price given by Black-Scholes from T /4 + 1 to T (the length of the time series), using
volatility estimated using a sliding window of range t ? T /4 to t. Compare the price obtained by
using the formula with the true price of the option. Are there any systematic differences? For
estimating volatility from historic data, see for example Hull [1] Section 13.4 or a similar source.
1c On a random set of 30 days in the range T /4 + 1 : T, compute the implied volatilities and plot them
as scatter plot against the corresponding volatilities you estimated from data. For prices on any
particular day, is there any systematic variation of implied volatilities computed from options with
different strike prices (Hint: Look up the term volatility smile)?
d Compare pricing a call option with European style exercise using the Black-Scholes and Binomial lattice
methods. Using one random set of values for the parameters (strike price, time to maturity, interest
rate and volatility) taken from the data given to you, evaluate how the binomial lattice method
approximates Black-Scholes as the step time δt is decreased. Plot a graph of the absolute difference
between the two methods as a function of δt.
Consider pricing an option using a binomial lattice. The code for pricing a put option with American
style exercise included the following lines[2]:
[...]
for tau=1:N
for i= (tau+1):2:(2*N+1-tau)
hold = p_u*PVals(i+1) + p_d*PVals(i-1);
PVals(i) = max(hold, K-SVals(i));
end
end
[...]
Explain in your own words the steps involved in this part of the code. How will this change if you
were pricing a call option (with American style exercise)?
2 Data-driven Modelling of Option Prices [10 Marks]
The relationship between the price of an option contract and the underlying parameters relating to the
exercise of the option is a complex one. Hutchinson et al. [3] show that this relationship can be reasonably
approximated by nonparametric neural network type models. In particular, they train a Radial Basis
Functions (RBF) model on data simulated from a Black-Scholes model and show that not only good
approximations to the options price can be obtained, but also the sensitivity Delta (= C/S) can be
relaibly extracted from it. In this assignment we will explore if some of the claims made in their paper
are true.
1. Study the paper by Hutchinson et al. [3]. You need to focus on Figures 4 and 5, and how these
were arrived at. Skip their evaluation part beyond this point.
2. Construct a dataset by generating call option prices from the Black-Scholes formula. You may take
input parameters (strike prices, interest rates, volatilities and the underlying asset price) from the
data used in Assignment 2.
With this data, reproduce Fig. 4 and Fig. 5 in [3] by training an RBF model (Eqn. 9 in [3]).
With the data vector x = [ S/X (T t) ]T
, the RBF model used is
λj φj (x) + wTx + w0
where the nonlinear function φj (x) = h(x mj )
T Σj (x mj ) + bji1/2
is a local Mahalanobis
distance with a bias term and J = 4. In this task, we will ignore the bias terms within
the nonlinearity (i.e. bj = 0).
Use the MATLAB function fitgmdist for fitting a Gaussian mixture model and extract the
parameters mj and Σj of the nonlinear terms in the model.
Now construct the design matrix of inputs – this matrix will have as many rows as you have
data points and seven columns (four for mapping the data to each of the nonlinear distances,
two for the two data dimensions of the linear part and the last column will have ones which
multiply w0).
2 Estimate the weights of the model (λj , j = 1, ..., 4; w and w0) by solving the resulting least
squares problem.
3. How well does such a non-parametric model learn the data used in Section 1?
4. Compute the “hedge ratio” (i.e. the derivative of option price with respoect to the underlying asset
price) by differentiating the network and from Black-Scholes formula. How well do they compare?
3 Time Series Analysis [10 Marks]
In a recent paper, Mahler [4] shows how the Kalman filter for sequential estimation of a state space model
can be combined with the Lasso approach [5] for l1 penalized regression. The result is to select lags
(past values) of explanatory variables that are relevant for the prediction of future values of a time series.
The work shows how residual signal arising from a time-series model might be explained by exogenous
information in the environment.
Implement this model and critically evaluate the claims in the paper:
First implement a simple second / third order autoregressive model on the index data and calculate the
residual. The variance of this residual can be an estimate you can use as observation noise variance
for the Kalman filter.
Implement a Kalman filter to estimate AR model parameters recursively. The process noise variance
has to be tuned; observe if there is anything systematic about the role of this hyper-parameter.
Implementing a Kalman filter is tricky, so you should first construct a synthetic time series (in
which you know the correct answer to the AR model coefficients), gain some experience in working
with this before applying the model to real data.
Model the error signal as a linear function of the variables considered in Mahler’s paper ([4]) with a
sparsity inducing regulariser (lasso) and explore their relative relevance. The residual computed
from the batch solution to the AR model could be used here, if your Kalman filter did not produce
stable results.
Use the convex programming package cvx from http://cvxr.com/ to implement sparse regression.
You are familiar with this from the index tracking exercise.
S&P500 time series can be obtained at http://finance.yahoo.com
Other data used in [4] can be obtained from http://research.stlouisfed.org/fred2/
Report
Write a report of no more than eight pages describing the work you have done.
References
[1] J. C. Hull, Options, Futures and Other Derivatives. Prentice Hall, 2009.
[2] P. Brandimarte, Numerical Methods in Finance and Economics. Wiley, 2006.
[3] J. Hutchinson, A. Lo, and T. Poggio, “A nonparametric approach to pricing and hedging derivative
securities via learning networks,” The Journal of Finance, vol. 49, no. 3, pp. 851–889, 1994.
[4] N. Mahler, “Modeling the S & P 500 index using the Kalman filter and the LagLasso,” in Machine
Learning for Signal Processing, 2009. MLSP 2009. IEEE International Workshop on, Sept 2009, pp.
1–6.
[5] R. Tibshirani, “Regression srinkage and selection via the lasso,” Journal of the Royal Statistical Society,
Series B, vol. 58, no. 1, pp. 267–288, 1996.