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%% Pricing European Call Options Using Different Equity Models
%
% This example illustrates how the Financial Instruments Toolbox(TM) is used
% to price European vanilla call options using different equity models.
%
% The example compares call option prices using the Cox-Ross-Rubinstein model,
% the Leisen-Reimer model and the Black-Scholes closed formula.
% Copyright 1995-2014 The MathWorks, Inc.
%% Define the Call Instrument
% Consider a European call option, with an exercise price of $30 on January 1, 2010.
% The option expires on Sep 1, 2010. Assume that the underlying stock provides
% no dividends. The stock is trading at $25 and has a volatility of 35% per annum.
% The annualized continuously compounded risk-free rate is 1.11% per annum.
% Option
Settle = 'Jan-01-2010';
Maturity = 'Sep-01-2010';
Strike = 30;
OptSpec = 'call';
% Stock
AssetPrice = 25;
Sigma = .35;
%% Create the Interest Rate Term Structure
StartDates = '01 Jan 2010';
EndDates = '01 Jan 2013';
Rates = 0.0111;
ValuationDate = '01 Jan 2010';
Compounding = -1;
RateSpec = intenvset('Compounding',Compounding,'StartDates', StartDates,...
'EndDates', EndDates, 'Rates', Rates,'ValuationDate', ValuationDate);
%% Create the Stock Structure
% Suppose we wish to create two scenarios. The first one assumes that
% AssetPrice is currently $25, the option is out of the money (OTM).
% The second scenario assumes that the option is at the money (ATM), and
% therefore AssetPriceATM = 30.
AssetPriceATM = 30;
StockSpec = stockspec(Sigma, AssetPrice);
StockSpecATM = stockspec(Sigma, AssetPriceATM);
%% Price the Options Using the Black-Scholes Closed Formula
% Use the function 'optstockbybls' in the Financial Instruments Toolbox to
% compute the price of the European call options.
% Price the option with AssetPrice = 25
PriceBLS = optstockbybls(RateSpec, StockSpec, Settle, Maturity, OptSpec, Strike);
% Price the option with AssetPrice = 30
PriceBLSATM = optstockbybls(RateSpec, StockSpecATM, Settle, Maturity, OptSpec, Strike);
%% Build the Cox-Ross-Rubinstein Tree
% Create the time specification of the tree
NumPeriods = 15;
CRRTimeSpec = crrtimespec(ValuationDate, Maturity, NumPeriods);
% Build the tree
CRRTree = crrtree(StockSpec, RateSpec, CRRTimeSpec);
CRRTreeATM = crrtree(StockSpecATM, RateSpec, CRRTimeSpec);
%% Build the Leisen-Reimer Tree
% Create the time specification of the tree
LRTimeSpec = lrtimespec(ValuationDate, Maturity, NumPeriods);
% Use the default method 'PP1' (Peizer-Pratt method 1 inversion)to build
% the tree
LRTree = lrtree(StockSpec, RateSpec, LRTimeSpec, Strike);
LRTreeATM = lrtree(StockSpecATM, RateSpec, LRTimeSpec, Strike);
%% Price the Options Using the Cox-Ross-Rubinstein (CRR) Model
PriceCRR = optstockbycrr(CRRTree, OptSpec, Strike, Settle, Maturity);
PriceCRRATM = optstockbycrr(CRRTreeATM, OptSpec, Strike, Settle, Maturity);
%% Price the Options Using the Leisen-Reimer (LR) Model
PriceLR = optstockbylr(LRTree, OptSpec, Strike, Settle, Maturity);
PriceLRATM = optstockbylr(LRTreeATM, OptSpec, Strike, Settle, Maturity);
%% Compare BLS, CRR and LR Results
sprintf('PriceBLS: \t%f\nPriceCRR: \t%f\nPriceLR:\t%f\n', PriceBLS, ...
PriceCRR, PriceLR)
sprintf('\t== ATM ==\nPriceBLS ATM: \t%f\nPriceCRR ATM: \t%f\nPriceLR ATM:\t%f\n', PriceBLSATM, ...
PriceCRRATM, PriceLRATM)
%% Convergence of CRR and LR Models to a BLS Solution
% The following tables compare call option prices using the CRR and LR models
% against the results obtained with the Black-Scholes formula.
%
% While the CRR binomial model and the Black-Scholes model converge as the
% number of time steps gets large and the length of each step gets small, this
% convergence, except for at the money options, is anything but smooth or
% uniform.
%
% The tables below show that the Leisen-Reimer model reduces the size of the
% error with even as few steps of 45.
%
%
% Strike = 30, Asset Price = 30
% -------------------------------------
% #Steps LR CRR
% 15 3.4986 3.5539
% 25 3.4981 3.5314
% 45 3.4980 3.5165
% 65 3.4979 3.5108
% 85 3.4979 3.5077
% 105 3.4979 3.5058
% 201 3.4979 3.5020
% 501 3.4979 3.4996
% 999 3.4979 3.4987
%
%
%
%
% Strike = 30, Asset Price = 25
% -------------------------------------
% #Steps LR CRR
% 15 1.2758 1.2950
% 25 1.2754 1.2627
% 45 1.2751 1.2851
% 65 1.2751 1.2692
% 85 1.2751 1.2812
% 105 1.2751 1.2766
% 201 1.2751 1.2723
% 501 1.2751 1.2759
% 999 1.2751 1.2756
%
%% Analyze the Effect of the Number of Periods on the Price of the Options
% The following graphs show how convergence changes as the number
% of steps in the binomial calculation increases, as well as the impact on
% convergence to changes to the stock price.
% Observe that the Leisen-Reimer model removes the oscillation and
% produces estimates close to the Black-Scholes model using only a small
% number of steps.
NPoints = 300;
% Cox-Ross-Rubinstein
NumPeriodCRR = 5 : 1 : NPoints;
NbStepCRR = length(NumPeriodCRR);
PriceCRR = nan(NbStepCRR, 1);
PriceCRRATM = PriceCRR;
for i = 1 : NbStepCRR
CRRTimeSpec = crrtimespec(ValuationDate, Maturity, NumPeriodCRR(i));
CRRT = crrtree(StockSpec, RateSpec, CRRTimeSpec);
PriceCRR(i) = optstockbycrr(CRRT, OptSpec, Strike,ValuationDate, Maturity) ;
CRRTATM = crrtree(StockSpecATM, RateSpec, CRRTimeSpec);
PriceCRRATM(i) = optstockbycrr(CRRTATM, OptSpec, Strike,ValuationDate, Maturity) ;
end;
% Now with Leisen-Reimer
NumPeriodLR = 5 : 2 : NPoints;
NbStepLR = length(NumPeriodLR);
PriceLR = nan(NbStepLR, 1);
PriceLRATM = PriceLR;
for i = 1 : NbStepLR
LRTimeSpec = lrtimespec(ValuationDate, Maturity, NumPeriodLR(i));
LRT = lrtree(StockSpec, RateSpec, LRTimeSpec, Strike);
PriceLR(i) = optstockbylr(LRT, OptSpec, Strike,ValuationDate, Maturity) ;
LRTATM = lrtree(StockSpecATM, RateSpec, LRTimeSpec, Strike);
PriceLRATM(i) = optstockbylr(LRTATM, OptSpec, Strike,ValuationDate, Maturity) ;
end;
%%
% First scenario: Out of the Money call option
% For Cox-Ross-Rubinstein
plot(NumPeriodCRR, PriceCRR);
hold on;
plot(NumPeriodCRR, PriceBLS*ones(NbStepCRR,1),'Color',[0 0.9 0], 'linewidth', 1.5);
% For Leisen-Reimer
plot(NumPeriodLR, PriceLR, 'Color',[0.9 0 0], 'linewidth', 1.5);
% Concentrate in the area of interest by clipping on the Y axis at 5x the
% LR Price:
YLimDelta = 5*abs(PriceLR(1) - PriceBLS);
ax = gca;
ax.YLim = [PriceBLS-YLimDelta PriceBLS+YLimDelta];
% Annotate Plot
titleString = sprintf('\nConvergence of CRR and LR models to a BLS Solution (OTM)\nStrike = %d, Asset Price = %d', Strike , AssetPrice);
title(titleString)
ylabel('Option Price')
xlabel('Number of Steps')
legend('CRR', 'BLS', 'LR', 'Location', 'NorthEast')
%%
% Second scenario: At the Money call option
% For Cox-Ross-Rubinstein
figure;
plot(NumPeriodCRR, PriceCRRATM);
hold on;
plot(NumPeriodCRR, PriceBLSATM*ones(NbStepCRR,1),'Color',[0 0.9 0], 'linewidth', 1.5);
% For Leisen-Reimer
plot(NumPeriodLR, PriceLRATM, 'Color',[0.9 0 0], 'linewidth', 1.5);
% Concentrate in the area of interest by clipping on the Y axis at 5x the
% LR Price:
YLimDelta = 5*abs(PriceLRATM(1) - PriceBLSATM);
ax = gca;
ax.YLim = [PriceBLSATM-YLimDelta PriceBLSATM+YLimDelta];
% Annotate Plot
titleString = sprintf('\nConvergence of CRR and LR models to a BLS Solution (ATM)\nStrike = %d, Asset Price = %d', Strike , AssetPriceATM);
title(titleString)
ylabel('Option Price')
xlabel('Number of Steps')
legend('CRR', 'BLS', 'LR', 'Location', 'NorthEast')