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Use Black-Scholes Model to Price Asian Options with Several Equity Pricers

This example shows how to compare arithmetic and geometric Asian option prices using the BlackScholes model and various pricing methods. The pricing methods are: the Kemna-Vorst, Levy, Turnbull-Wakeman, and Cox-Ross-Rubinstein methods and Monte Carlo simulation. This example also demonstrates how variations in spot prices affect option and delta sensitivity values on European vanilla and Asian options.

Create ratecurve Object

Create a ratecurve object using ratecurve.

Settle = datetime(2019,01,01);
Maturity = datetime(2025,01,01);
Rate = 0.035;
Compounding = -1;
Basis = 1;
ZeroCurve = ratecurve('zero',Settle,Maturity,Rate,'Compounding',Compounding,'Basis', Basis)
ZeroCurve = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 1
                Dates: 01-Jan-2025
                Rates: 0.0350
               Settle: 01-Jan-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create BlackScholes Model Object

Use finmodel to create a BlackScholes model object.

Volatility = .20;
BSModel = finmodel("BlackScholes",'Volatility',Volatility)
BSModel = 
  BlackScholes with properties:

     Volatility: 0.2000
    Correlation: 1

Create Asian Instrument Objects

Use fininstrument to create two Asian instrument objects, one using an arithmetic average and the other using a geometric average.

ExerciseDates = datetime(2020,01,01);
Strike = 90;
OptionType = 'call';
AverageType = 'geometric';

AsianOptArith = fininstrument("Asian",'ExerciseDate',ExerciseDates,'Strike',Strike,...
                              'OptionType',OptionType,'Name',"CallAsianArith")
AsianOptArith = 
  Asian with properties:

          OptionType: "call"
              Strike: 90
         AverageType: "arithmetic"
        AveragePrice: 0
    AverageStartDate: NaT
       ExerciseStyle: "european"
        ExerciseDate: 01-Jan-2020
                Name: "CallAsianArith"

AsianOptGeo = fininstrument("Asian",'ExerciseDate',ExerciseDates,'Strike',Strike,...
                            'OptionType',OptionType,'AverageType', AverageType,'Name',"CallAsianGeo")
AsianOptGeo = 
  Asian with properties:

          OptionType: "call"
              Strike: 90
         AverageType: "geometric"
        AveragePrice: 0
    AverageStartDate: NaT
       ExerciseStyle: "european"
        ExerciseDate: 01-Jan-2020
                Name: "CallAsianGeo"

Create Analytic, AssetTree, and AssetMonteCarlo Pricer Objects

Use finpricer to create BlackScholes, AssetTree, and AssetMonteCarlo pricer objects and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

SpotPrice = 100;
InpSensitivity = "delta";

% Analytic Pricers
LevyPricer = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', SpotPrice, ...
                       'DiscountCurve', ZeroCurve, 'PricingMethod', "Levy");

TWPricer = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', SpotPrice, ...
                     'DiscountCurve', ZeroCurve, 'PricingMethod', "TurnbullWakeman");

KVPricer = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', SpotPrice, ...
                       'DiscountCurve', ZeroCurve, 'PricingMethod', "KemnaVorst");

% AssetTree Pricer
% Define the number of levels of the tree
NumPeriods = 50;
CRRPricer = finpricer("AssetTree",'DiscountCurve',ZeroCurve,'Model',BSModel, 'SpotPrice',SpotPrice, ...
                      'PricingMethod',"CoxRossRubinstein",'NumPeriods', NumPeriods,...
                      'Maturity', ExerciseDates);

% AssetMonteCarlo Pricer
% Define the number of simulation trials
NumTrials = 2000;
SimDates =[Settle:days(2):ExerciseDates ExerciseDates];

MCPricer = finpricer("AssetMonteCarlo", 'Model', BSModel, 'SpotPrice', SpotPrice, 'DiscountCurve', ZeroCurve,...
                     'SimulationDates',  SimDates, 'NumTrials', NumTrials);

Calculate the Price of the Arithmetic and Geometric Asian Options Using Different Pricers

Calculate the Asian option prices using the price function for the Analytic, AssetTree, and AssetMonetCarlo pricing methods.

% Analytic
[LevyPrice,LevyoutPR] = price(LevyPricer, AsianOptArith,InpSensitivity);

[TWPrice, TWoutPR] = price(TWPricer, AsianOptArith, InpSensitivity);

[KVPrice, KVoutPR] = price(KVPricer, AsianOptGeo, InpSensitivity);

% Cox-Ross-Rubinstein
[CRRArithPrice, CRRArithoutPR] = price(CRRPricer, AsianOptArith, InpSensitivity);

[CRRGeoPrice, CRRGeooutPR] = price(CRRPricer, AsianOptGeo, InpSensitivity);

% Monte Carlo
[MCArithPrice, MCArithoutPR] = price(MCPricer, AsianOptArith, InpSensitivity);

[MCGeoPrice, MCGeooutPR] = price(MCPricer, AsianOptGeo, InpSensitivity);

Compare Asian Option Prices

Compare the Asian option call prices using the displayPricesAsianCallOption function defined in Local Functions.

displayPricesAsianCallOption(KVPrice,LevyPrice,TWPrice,CRRArithPrice,CRRGeoPrice,MCArithPrice,MCGeoPrice)
Comparison of Asian prices:

Arithmetic Asian
Levy:                     12.164734
Turnbull-Wakeman:         12.164734
Cox-Ross-Rubinstein:      12.126509
Monte Carlo:              12.082420

Geometric Asian
Kemna-Vorst:              11.862580
Cox-Ross-Rubinstein:      11.852462
Monte Carlo:              12.039533

The table contrasts the results from closed approximation models against price simulations implemented using the Cox-Ross-Rubinstein binomial tree and Monte Carlo pricing methods. Observe that arithmetic average Asian options are more expensive than their geometric average counterparts.

Compare Asian and Vanilla 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. You can compare the price and delta sensitivity values of Asian options against their vanilla counterparts.

Create Vanilla Instrument Object

Use fininstrument to create a Vanilla instrument object with same Maturity and Strike as the two Asian options.

EuropeanCallOption = fininstrument("Vanilla",'ExerciseDate',ExerciseDates,'Strike',Strike,...
                                   'OptionType',OptionType,'Name',"CallVanilla")
EuropeanCallOption = 
  Vanilla with properties:

       OptionType: "call"
    ExerciseStyle: "european"
     ExerciseDate: 01-Jan-2020
           Strike: 90
             Name: "CallVanilla"

Create Analytic Pricer Object

Use finpricer to create a BlackScholes pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

BLSPricer = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', SpotPrice, 'DiscountCurve', ZeroCurve)
BLSPricer = 
  BlackScholes with properties:

    DiscountCurve: [1x1 ratecurve]
            Model: [1x1 finmodel.BlackScholes]
        SpotPrice: 100
    DividendValue: 0
     DividendType: "continuous"

Compute price and delta sensitivity.

[BLSPrice, BLSoutPR] = price(BLSPricer, EuropeanCallOption, InpSensitivity);

Compare Prices for Asian and Vanilla Options

Compare option prices using the displayVanillaAsianComparison function defined in Local Functions.

displayVanillaAsianComparison('Prices', BLSPrice, KVPrice,LevyPrice,TWPrice)
Comparison of Vanilla and Asian Option Prices:

Vanilla BLS:              15.743809
Asian Kemna-Vorst:        11.862580
Asian Levy:               12.164734
Asian Turnbull-Wakeman:   12.164734

Observe that both the geometric and arithmetic Asian option prices are lower than their vanilla counterparts.

Compare Delta Sensitivity for Asian and Vanilla Options

The delata value measures the option price sensitivity to changes in the price of the underlying asset. As the underlier changes, the proportions of the instruments forming the portfolio might need to be adjusted to keep the sensitivities within the desired range.

displayVanillaAsianComparison('Delta', BLSoutPR.Results.Delta, KVoutPR.Results.Delta, LevyoutPR.Results.Delta,TWoutPR.Results.Delta)
Comparison of Vanilla and Asian Option Delta:

Vanilla BLS:              0.788666
Asian Kemna-Vorst:        0.844986
Asian Levy:               0.852806
Asian Turnbull-Wakeman:   0.852806

The table shows the delta values for both the vanilla and arithmetic and geometric Asian options. Observe that the geometric Asian delta value is lower than the delta value for the arithmetic Asian option.

Analyze Effect of Variations of Underlying Asset on Option Prices

Examine the effect of changes of underlying asset prices. Create a plot to show the effect of variations in the price of the underlying asset on the vanilla and Asian option prices.

StockPrices = (50:5:120)';
PriceBLS = nan(size(StockPrices));
PriceKV = PriceBLS;
PriceLevy = PriceBLS;
PriceTW = PriceBLS;
DeltaBLS = PriceBLS;
DeltaLevy = PriceBLS;
DeltaTW = PriceBLS;
DeltaKV = PriceBLS;
InpSensitivity = "delta";
idx = 1;
for AssetPrice = StockPrices'

    PricerBLS = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', AssetPrice,...
                          'DiscountCurve', ZeroCurve);
    [PriceBLS(idx), outPRBLS] = price(PricerBLS, EuropeanCallOption, InpSensitivity);
    DeltaBLS(idx) = outPRBLS.Results.Delta;

    PricerLevy = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', AssetPrice, ...
                           'DiscountCurve', ZeroCurve, 'PricingMethod', "Levy");
    [PriceLevy(idx), outPRLevy] = price(PricerLevy, AsianOptArith, InpSensitivity);
    DeltaLevy(idx) = outPRLevy.Results.Delta;

    PricerTW = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', AssetPrice, ...
        'DiscountCurve', ZeroCurve, 'PricingMethod', "TurnbullWakeman");
    [PriceTW(idx), outPRBTW] = price(PricerTW, AsianOptArith, InpSensitivity);
    DeltaTW(idx) = outPRBTW.Results.Delta;

    PricerKV = finpricer('Analytic', 'Model', BSModel, 'SpotPrice', AssetPrice, ...
        'DiscountCurve', ZeroCurve, 'PricingMethod', "KemnaVorst");
    [PriceKV(idx), outPRKV] = price(PricerKV, AsianOptGeo, InpSensitivity);
    DeltaKV(idx) = outPRKV.Results.Delta;

    idx = idx+1;
end

figure('menubar', 'none', 'numbertitle', 'off')
plot(StockPrices, [PriceBLS PriceKV PriceLevy PriceTW]);
xlabel 'Spot Price ($)'
ylabel 'Option Price ($)'
title 'Asian and Vanilla Option Price Comparison'
legend('Vanilla', 'KV Geometric Asian', 'Levy Arithmetic Asian', 'TW Arithmetic Asian', 'location','northwest');

Figure contains an axes. The axes with title Asian and Vanilla Option Price Comparison contains 4 objects of type line. These objects represent Vanilla, KV Geometric Asian, Levy Arithmetic Asian, TW Arithmetic Asian.

The plot displays vanilla and Asian option prices with respect to the underlying asset price. Observe that the price of the Asian options is cheaper than the price of the vanilla option.

Analyze Effect of Variations of the Underlying Asset on the Options Delta

Examine the effect of changes of underlying asset prices on delta sensitivity. The following plot demonstrates the behavior of the delta value for the Vanilla and Asian options as a function of the underlying price.

figure('menubar', 'none', 'numbertitle', 'off')
plot(StockPrices, [DeltaBLS DeltaKV DeltaLevy DeltaTW]);
xlabel 'Spot Price ($)'
ylabel 'Call Delta'
title 'Asian and Vanilla Option Delta Comparison (Strike Price = $90)'
legend('Vanilla', 'KV Geometric Asian', 'Levy Arithmetic Asian', 'TW Arithmetic Asian', 'location', 'northwest');

Figure contains an axes. The axes with title Asian and Vanilla Option Delta Comparison (Strike Price = $90) contains 4 objects of type line. These objects represent Vanilla, KV Geometric Asian, Levy Arithmetic Asian, TW Arithmetic Asian.

The plot displays the vanilla and Asian delta sensitivity values with respect to the underlying asset price. 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 is the case for an out-of-the-money option. Observe that the Asian delta value is lower for out-of-the-money options and higher for in-the-money options compared with the vanilla European counterpart.

Local Functions

function displayPricesAsianCallOption(KVPrice,LevyPrice,TWPrice,CRRArithPrice,CRRGeoPrice,MCArithPrice,MCGeoPrice)
fprintf('Comparison of Asian prices:\n');
fprintf('\n');
fprintf('Arithmetic Asian\n');
fprintf('Levy:                     %f\n', LevyPrice);
fprintf('Turnbull-Wakeman:         %f\n', TWPrice);
fprintf('Cox-Ross-Rubinstein:      %f\n', CRRArithPrice);
fprintf('Monte Carlo:              %f\n', MCArithPrice);
fprintf('\n');
fprintf('Geometric Asian\n');
fprintf('Kemna-Vorst:              %f\n', KVPrice);
fprintf('Cox-Ross-Rubinstein:      %f\n', CRRGeoPrice);
fprintf('Monte Carlo:              %f\n', MCGeoPrice);
end

function displayVanillaAsianComparison(type, BLS, KV, Levy, TW)
fprintf('Comparison of Vanilla and Asian Option %s:\n', type);
fprintf('\n');
fprintf('Vanilla BLS:              %f\n', BLS);
fprintf('Asian Kemna-Vorst:        %f\n', KV);
fprintf('Asian Levy:               %f\n', Levy);
fprintf('Asian Turnbull-Wakeman:   %f\n', TW);
end

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