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%% Pricing Asian Options
%
% This example shows how to price a European Asian option using four methods
% in the Financial Instruments Toolbox(TM). This example demonstrates two closed form approximations
% (Levy and Kemna-Vorst), a lattice model (Cox-Ross-Rubinstein), and Monte
% Carlo simulation. All these methods involve some tradeoffs between
% numerical accuracy and computational efficiency. This example also
% demonstrates how variations in spot prices, volatility, and strike prices
% affect option prices on European Vanilla and Asian options.
%
%% Overview of Asian Options
% Asian options are securities with payoffs that depend on the average value
% of an underlying asset over a specific period of time. Underlying assets
% can be stocks, commodities, or financial indices.
%
% Two types of Asian options are found in the market: average price options
% and average strike options. Average price options have a fixed strike
% value and the average used is the asset price. Average strike options
% have a strike equal to the average value of the underlying asset.
%
% The payoff at maturity of an average price European Asian option is:
%
% $max (0, Savg - K)$ for a call
%
% $max (0, K - Savg)$ for a put
%
% The payoff at maturity of an average strike European Asian option is:
%
% $max (0, St - Savg)$ for a call
%
% $max (0, Savg - St)$ for a put
%
% where _Savg_ is the average price of underlying asset, _St_ is the
% price at maturity of underlying asset, and _K_ is the strike price.
%
% The average can be arithmetic or geometric.
%
% Copyright 2014 The MathWorks, Inc.
%% Pricing Asian Options Using Closed Form Approximations
% The Financial Instruments Toolbox(TM) supports two closed form
% approximations for European Average Price options. The Levy model is based
% on the arithmetic mean of the price of the underlying during the life of
% the option [1]. The Kemna-Vorst method provides a closed form pricing
% solution to geometric averaging options [2].
%%
% The pricing functions |asianbylevy| and |asianbykv| take an interest rate
% term structure and stock structure as inputs.
%
% Consider the following example:
% Create RateSpec from the interest rate term structure
StartDates = '12-March-2014';
EndDates = '12-March-2020';
Rates = 0.035;
Compounding = -1;
Basis = 1;
RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates, ...
'EndDates', EndDates, 'Rates', Rates, 'Compounding', ...
Compounding, 'Basis', Basis);
% Define StockSpec with the underlying asset information
Sigma = 0.20;
AssetPrice = 100;
StockSpec = stockspec(Sigma, AssetPrice);
% Define the Asian option
Settle = '12-March-2014';
ExerciseDates = '12-March-2015';
Strike = 90;
OptSpec = 'call';
% Levy model approximation
PriceLevy = asianbylevy(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates);
% Kemna-Vorst closed form model
PriceKV = asianbykv(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates);
% Comparison of calculated prices for the geometric and arithmetic options
% using different closed form algorithms.
fprintf('Comparison of Asian Arithmetic and Geometric Prices:\n\n');
fprintf('Levy: %f\n', PriceLevy);
fprintf('Kemna-Vorst: %f\n', PriceKV);
%% Computing Asian Options Prices Using the Cox-Ross-Rubinstein Model
% In addition to closed form approximations, the Financial Instruments
% Toolbox(TM) supports pricing European Average Price options using CRR
% trees via the function |asianbycrr|.
%
% The lattice pricing function |asianbycrr| takes an interest rate tree
% ( |CRRTree| ) and stock structure as inputs. We can price the previous
% options by building a |CRRTree| using the interest rate term structure and
% stock specification from the example above.
% Create the time specification of the tree
NPeriods = 20;
TreeValuationDate = '12-March-2014';
TreeMaturity = '12-March-2024';
TimeSpec = crrtimespec(TreeValuationDate, TreeMaturity, NPeriods);
% Build the tree
CRRTree = crrtree(StockSpec, RateSpec, TimeSpec);
% Price the European Asian option using the CRR lattice model.
% The function 'asianbycrr' computes prices of arithmetic and geometric
% Asian options.
AvgType = {'arithmetic';'geometric'};
AmericanOpt = 0;
PriceCRR20 = asianbycrr(CRRTree, OptSpec, Strike, Settle, ExerciseDates,...
AmericanOpt, AvgType);
% Increase the numbers of periods in the tree and compare results
NPeriods = 40;
TimeSpec = crrtimespec(TreeValuationDate, TreeMaturity, NPeriods);
CRRTree = crrtree(StockSpec, RateSpec, TimeSpec);
PriceCRR40 = asianbycrr(CRRTree, OptSpec, Strike, Settle, ExerciseDates,...
AmericanOpt, AvgType);
% Display prices
fprintf('Asian Prices using the CRR lattice model:\n\n');
fprintf('PriceArithmetic(CRR20): %f\n', PriceCRR20(1));
fprintf('PriceArithmetic(CRR40): %f\n', PriceCRR40(1));
fprintf('PriceGeometric (CRR20): %f\n', PriceCRR20(2));
fprintf('PriceGeometric (CRR40): %f\n', PriceCRR40(2));
%%
% The results above compare the findings from calculating both
% geometric and arithmetic Asian options, using CRR trees with 20 and 40
% levels. It can be seen that as the number of levels increases, the
% results approach the closed form solutions.
%% Calculating Prices of Asian Options Using Monte Carlo Simulation
% Another method to price European Average Price options with the Financial
% Instruments Toolbox(TM) is via Monte Carlo simulations.
%
% The pricing function |asianbyls| takes an interest rate term structure
% and stock structure as inputs. The output and execution time of the Monte
% Carlo simulation depends on the number of paths ( |NumTrials| ) and the
% number of time periods per path ( |NumPeriods| ).
%
% We can price the same options of previous examples using Monte Carlo.
% Simulation Parameters
NumTrials = 500;
NumPeriods = 200;
% Price the arithmetic option
PriceAMC = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates,'NumTrials', NumTrials, ...
'NumPeriods', NumPeriods);
% Price the geometric option
PriceGMC = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates,'NumTrials', NumTrials, ...
'NumPeriods', NumPeriods, 'AvgType', AvgType(2));
% Use the antithetic variates method to value the options
Antithetic = true;
PriceAMCAntithetic = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates,'NumTrials', NumTrials, 'NumPeriods',...
NumPeriods, 'Antithetic', Antithetic);
PriceGMCAntithetic = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates,'NumTrials', NumTrials, 'NumPeriods',...
NumPeriods, 'Antithetic', Antithetic,'AvgType', AvgType(2));
% Display prices
fprintf('Asian Prices using Monte Carlo Method:\n\n');
fprintf('Arithmetic Asian\n');
fprintf('Standard Monte Carlo: %f\n', PriceAMC);
fprintf('Variate Antithetic Monte Carlo: %f\n\n', PriceAMCAntithetic);
fprintf('Geometric Asian\n');
fprintf('Standard Monte Carlo: %f\n', PriceGMC);
fprintf('Variate Antithetic Monte Carlo: %f\n', PriceGMCAntithetic);
%%
% The use of variate antithetic accelerates the conversion process by
% reducing the variance.
%%
% We can create a plot to display the difference between the geometric Asian
% price using the Kemna-Vorst model, standard Monte Carlo and antithetic
% Monte Carlo.
nTrials = [50:5:100 110:10:250 300:50:500 600:100:2500]';
PriceKVVector = PriceKV * ones(size(nTrials));
PriceGMCVector = nan(size(nTrials));
PriceGMCAntitheticVector = nan(size(nTrials));
TimeGMCAntitheticVector = nan(length(nTrials),1);
TimeGMCVector = nan(length(nTrials),1);
idx = 1;
for iNumTrials = nTrials'
PriceGMCVector(idx) = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates,'NumTrials', iNumTrials, 'NumPeriods',...
NumPeriods,'AvgType', AvgType(2));
PriceGMCAntitheticVector(idx) = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates,'NumTrials', iNumTrials, 'NumPeriods',...
NumPeriods, 'Antithetic', Antithetic,'AvgType', AvgType(2));
idx = idx+1;
end
figure('menubar', 'none', 'numbertitle', 'off')
plot(nTrials, [PriceKVVector PriceGMCVector PriceGMCAntitheticVector]);
title 'Variance Reduction by Antithetic'
xlabel 'Number of Simulations'
ylabel 'Asian Option Price'
legend('Kemna-Vorst', 'Standard Monte Carlo', 'Variate Antithetic Monte Carlo ', 'location', 'northeast');
%%
% The graph above shows how oscillation in simulated price is reduced
% through the use of variate antithetic.
%% Compare Pricing Model Results
% Prices calculated by the Monte Carlo method will vary depending on the
% outcome of the simulations. Increase |NumTrials| and analyze the results.
NumTrials = 2000;
PriceAMCAntithetic2000 = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle, ExerciseDates,...
'NumTrials', NumTrials, 'NumPeriods', NumPeriods, 'Antithetic', Antithetic);
PriceGMCAntithetic2000 = asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates,'NumTrials', NumTrials, 'NumPeriods',...
NumPeriods, 'Antithetic', Antithetic,'AvgType', AvgType(2));
% Comparison of calculated Asian call prices
fprintf('Comparison of Asian call prices:\n\n');
fprintf('Arithmetic Asian\n');
fprintf('Levy: %f\n', PriceLevy);
fprintf('Cox-Ross-Rubinstein: %f\n', PriceCRR40(1));
fprintf('Monte Carlo(500 trials): %f\n', PriceAMCAntithetic);
fprintf('Monte Carlo(2000 trials): %f\n\n', PriceAMCAntithetic2000);
fprintf('Geometric Asian\n');
fprintf('Kemna-Vorst: %f\n', PriceKV);
fprintf('Cox-Ross-Rubinstein: %f\n', PriceCRR40(2));
fprintf('Monte Carlo(500 trials): %f\n', PriceGMCAntithetic);
fprintf('Monte Carlo(2000 trials): %f\n', PriceGMCAntithetic2000);
%%
% The table above contrasts the results from closed approximation models
% against price simulations implemented via CRR trees and Monte Carlo.
%% Asian and Vanilla Call Options
% Asian options are popular instruments since they tend to be less expensive
% than comparable Vanilla calls and puts. This is because the volatility in
% the average value of an underlier tends to be lower than the volatility
% of the value of the underlier itself.
%
% The Financial Instruments Toolbox(TM) supports several algorithms for
% pricing vanilla options. Let's compare the price of Asian options against
% their Vanilla counterpart.
%
% First, we compute the price of a European Vanilla Option using the Black
% Scholes model.
PriceBLS = optstockbybls(RateSpec, StockSpec, Settle, ExerciseDates,...
OptSpec, Strike);
% Comparison of calculated call prices.
fprintf('Comparison of Vanilla and Asian Prices:\n\n');
fprintf('Vanilla BLS: %f\n', PriceBLS);
fprintf('Asian Levy: %f\n', PriceLevy);
fprintf('Asian Kemna-Vorst: %f\n', PriceKV);
%%
% Both geometric and arithmetic Asians price lower than their Vanilla
% counterpart.
%%
% We can analyze options prices at different levels of the underlying asset.
% Using the Financial Instruments Toolbox(TM), it is possible to observe
% the effect of different parameters on the price of the options. Consider
% for example, the effect of variations in the price of the underlying
% asset.
StockPrices = (50:5:150)';
PriceBLS = nan(size(StockPrices));
PriceLevy = nan(size(StockPrices));
PriceKV = nan(size(StockPrices));
idx = 1;
for So = StockPrices';
SP = stockspec(Sigma, So);
PriceBLS(idx) = optstockbybls(RateSpec, SP, Settle, ExerciseDates,...
OptSpec, Strike);
PriceLevy(idx) = asianbylevy(RateSpec, SP, OptSpec, Strike, Settle,...
ExerciseDates);
PriceKV(idx) = asianbykv(RateSpec, SP, OptSpec, Strike, Settle,...
ExerciseDates);
idx = idx+1;
end
figure('menubar', 'none', 'numbertitle', 'off')
plot(StockPrices, [PriceBLS PriceLevy PriceKV]);
xlabel 'Spot Price ($)'
ylabel 'Option Price ($)'
title 'Call Price Comparison'
legend('Vanilla', 'Arithmetic Asian', 'Geometric Asian', 'location', 'northwest');
%%
% It can be observed that the price of the Asian option is cheaper than the
% price of the Vanilla option.
%%
% Additionally, it is possible to observe the effect of changes in the
% volatility of the underlying asset.
% The table below shows what happens to Asian and Vanilla option prices when
% the constant volatility changes.
%
% Call Option (ITM)
% Strike = 90 AssetPrice = 100
% ------------------------------------------------
% Volatility Levy Kemna-Vorst BLS
% 10% 11.3987 11.3121 13.4343
% 20% 12.1647 11.8626 15.7438
% 30% 13.6512 13.0338 18.8770
% 40% 15.4464 14.4086 22.2507
%
%
% A comparison of the calculated prices show that Asian options are less
% sensitive to volatility changes, since averaging reduces the volatility
% of the value of the underlying asset. Also, Asian options that use
% arithmetic average are more expensive than those that use geometric average.
%%
% Now, examine the effect of strike on option prices.
Strikes = (90:5:120)';
NStrike = length(Strikes);
PriceBLS = nan(size(Strikes));
PriceLevy = nan(size(Strikes));
PriceKV = nan(size(Strikes));
idx = 1;
for ST = Strikes';
SP = stockspec(Sigma, AssetPrice);
PriceBLS(idx) = optstockbybls(RateSpec, SP, Settle, ExerciseDates,...
OptSpec, ST);
PriceLevy(idx) = asianbylevy(RateSpec, SP, OptSpec, ST, Settle,...
ExerciseDates);
PriceKV(idx) = asianbykv(RateSpec, SP, OptSpec, ST, Settle,...
ExerciseDates);
idx = idx+1;
end
figure('menubar', 'none', 'numbertitle', 'off')
plot(Strikes, [PriceBLS PriceLevy PriceKV]);
xlabel 'Strike Price ($)'
ylabel 'Option Price ($)'
title 'Effect of Strike on Option Prices'
legend('Vanilla', 'Arithmetic Asian', 'Geometric Asian', 'location', 'northeast');
%%
% The figure above displays the option price with respect to strike price.
% Since call option value decreases as strike price increases, the Asian
% call curve is under the Vanilla call curve. It can be observed that the
% Asian call option is less expensive than the Vanilla call.
%% Hedging
% Hedging is an insurance to minimize exposure to market movements on the
% value of a position or portfolio. As the underlying changes, the proportions
% of the instruments forming the portfolio may need to be adjusted to keep
% the sensitivities within the desired range. Delta measures the option
% price sensitivity to changes in the price of the underlying.
%
% Assume that we have a portfolio of two options with the same strike and
% maturity. We can use the Financial Instruments Toolbox(TM) to compute
% Delta for the Vanilla and Average Price options.
OutSpec = 'Delta';
% Vanilla option using Black Scholes
DeltaBLS = optstocksensbybls(RateSpec, StockSpec, Settle, ExerciseDates,...
OptSpec, Strike, 'OutSpec', OutSpec);
% Asian option using Levy model
DeltaLevy = asiansensbylevy(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates, 'OutSpec', OutSpec);
% Asian option using Kemna-Vorst method
DeltaLKV= asiansensbykv(RateSpec, StockSpec, OptSpec, Strike, Settle,...
ExerciseDates, 'OutSpec', OutSpec);
% Delta Comparison
fprintf('Comparison of Vanilla and Asian Delta:\n\n');
fprintf('Vanilla BLS: %f\n', DeltaBLS);
fprintf('Asian Levy: %f\n', DeltaLevy);
fprintf('Asian Kemna-Vorst: %f\n', DeltaLKV);
%%
% The following graph demonstrates the behavior of Delta for the Vanilla
% and Asian options as a function of the underlying price.
StockPrices = (40:5:120)';
NStockPrices = length(StockPrices);
DeltaBLS = nan(size(StockPrices));
DeltaLevy = nan(size(StockPrices));
DeltaKV = nan(size(StockPrices));
idx = 1;
for SPrices = StockPrices'
SP = stockspec(Sigma, SPrices);
DeltaBLS(idx) = optstocksensbybls(RateSpec, SP, Settle, ...
ExerciseDates, OptSpec, Strike, 'OutSpec', OutSpec);
DeltaLevy(idx) = asiansensbylevy(RateSpec, SP, OptSpec, Strike,...
Settle, ExerciseDates, 'OutSpec', OutSpec);
DeltaKV(idx) = asiansensbykv(RateSpec, SP, OptSpec, Strike, ...
Settle, ExerciseDates,'OutSpec', OutSpec);
idx = idx+1;
end
figure('menubar', 'none', 'numbertitle', 'off')
plot(StockPrices, [DeltaBLS DeltaLevy DeltaKV]);
xlabel 'Spot Price ($)'
ylabel 'Call Delta'
title 'Delta Comparison (Strike Price = $90)'
legend('Vanilla', 'Arithmetic Asian', 'Geometric Asian', 'location', 'northwest');
%%
% A Vanilla, or Asian, in the money (ITM) call option is more sensitive
% to price movements than an out of the money (OTM) option. If the asset
% price is deep in the money, then it is more likely to be exercised. The
% opposite occurs for an out of the money option. Asian delta is lower for
% out of the money options and is higher for in the money options than its
% Vanilla European counterpart. The geometric Asian delta is lower than
% the arithmetic Asian delta.
%% References
%
% [1] Levy, E.,"Pricing European Average Rate Currency Options", Journal of
% International Money and Finance,14,474-491,(1992).
%
% [2] Kemna, A. & Vorst, A.,"A Pricing Method for Options Based on Average
% Asset Values", Journal of Banking and Finance,14,113-129,(1990).