L o g o

L o g oTrinomial Tree vs. MC

Gopher 6 Consulting: Di(Emma) Wu and Zheng Rong

L o g oAgenda

AAPL stock and Option Price 1

Trinomial Tree & MC from Class2

Functions from Financial Toolbox 3

Comparison of Different Methods 4

L o g oAAPL and Option Price

❖Compare results by using different computational finance methods with the real option price of AAPL

▪All data come from real market data▪Researches include: Yahoo Finance &

S&P Capital IQ’s Stock Report & Thomson Reuters StockReports+

L o g oDiagram (Yahoo Finance)

Data from http://finance.yahoo.com/q?s=AAPL

Implied Volatility 25.05%

L o g oResearch Report

S&P Capital IQ

TargetPrice

Thomson StockReports+

L o g oRisk Free Interest Rate & Option

Risk Free Interest Rate: 13 WEEK TREASURY BILL (^IRX) 0.265 March 4

Choose 0.003Those tests are recommended if time allowed:● Sensitivity Test● Scenario Test ● Statistics Test

L o g oAssumptions we use

Implied σ=25.05%

S0=103 r=0.003

Sk=125

T=319 days

Call Option

Div & Yield=2.02%

L o g oAssumptions we use

Implied σ=27.86%

S0=103 r=0.003

Sk=125

T=319 days

Put Option

Div & Yield=2.02%

L o g o

Monte Carlo Simulationn

Strike price Bid price Ask price Implied Volatility Expire date

call 125 2.90 3.05 25.05% 20-Jan-2017

put 125 25.50 25.90 27.86% 20-Jan-2017

Results:

Call option: We get 3.0323.Put option: We get 25.4100.

L o g oImplied Trinomial Tree

Strike price Bid price Ask price Implied Volatility Expire date

call 125 2.90 3.05 25.05% 20-Jan-2017

put 125 25.50 25.90 27.86% 20-Jan-2017

Results:

Call option: We get 3.2297.Put option: We get 32.4783.

L o g oFunctions from Financial Toolbox

Generality

Robustness

Readability

stttree

StockSepcRateSpecSTTTimeSpec

More Flexibility

•Div•Rf•Structure •Track•Easy

Treeviewer

Standard Trinomial Tree

Usability

L o g oMy personal experience

Goole!Mathworks

Begin with Mimic

Algorithm

Data Structure

Search

Help

Edit

Toolbox

L o g oMatlab Code

Adjusted to Today’s Market Price Create a StockSpec:% Create a StockSpecAssetPrice=102.11;Sigma=0.2525;Div=0.0202;StockSpec=stockspec(Sigma,AssetPrice,'continuous',Div)

L o g oMatlab CodeAdjusted to Today’s Market Price% Create a RateSpecStartDates = 'Mar-7-2016';EndDates='Jan-20-2017';Rates=0.003;Basis=1; % Day count basis; 1 = 30/360 (SIA)Compounding=-1; % ?1 = Continuous compounding% Scalar value representing the rate at which the input% zero rates were compounded when annualized. Default is 2.RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates,

'EndDates', EndDates, 'Rates', Rates,'Compounding', Compounding, 'Basis', Basis)

L o g oMatlab Code% Create a

TimeSpecNumPeriods = 12;TimeSpec =

stttimespec(StartDates, EndDates, NumPeriods);

% Create a Standard

Trinomial TreeSTTTree=stttree(Stoc

kSpec, RateSpec, TimeSpec)

L o g oWait, this looks like black box!Not Really! Use edit functionFor example:

Readable Code and Adjustable!

% Calculate standard probs:Common = sqrt(dT/(12 *

Sigma0^2)) .* (BRates - (Sigma0^2)/2);

for iLevel=1:NumLevels-1 PTree{iLevel}(1,:) = 1/6 +

Common(iLevel); PTree{iLevel}(3,:) = 1/6 -

Common(iLevel); PTree{iLevel}(2,:) = 2/3;end

% Build Tree StructureSTTTree =

classfin('STStockTree');STTTree.StockSpec = StockSpec;STTTree.TimeSpec =

STTTimeSpec;STTTree.RateSpec =

RateSpecOri;STTTree.tObs = TreeTimes';STTTree.dObs = TreeDates';STTTree.STree = STree;STTTree.Probs = PTree;

L o g oOption Prices Vanilla OptionsSettle = '3/7/16';ExerciseDates = [datenum('1/20/17');datenum('1/20/17')];OptSpec = {'call';'put'};Strike =[125;75]; Price = optstockbystt(STTTree, OptSpec, Strike, Settle, ExerciseDates)P=0.30013Asian Options

Price = asianbystt(STTTree, OptSpec, Strike, Settle, ExerciseDates)

P=0.48820

LookBack OPtions Price= lookbackbystt(STTTree, OptSpec, Strike, Settle, ExerciseDates)

P=4.2269

Most Efficient : Use load deri.mat

L o g oComparison with Real Data Real Market Price Now:

Results From Matlab: P=3.0013Why Different? Change Too Quick! Highly Depend On Assumptions!

(Div & Yield, Risk-Free Interest, Volatility)

L o g oMonte Carlo Simulation_Antithetic

AssetPrice = 102.08;Sigma = 0.2512;StockSpec = stockspec(Sigma, AssetPrice)

StartDates = 'Mar-7-2016';EndDates = 'Jan-20-2017';Rates = 0.003-0.0202;RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates, 'EndDates', EndDates, 'Rates', Rates)

OptSpec = 'call';Settle = 'Mar-7-2016';ExerciseDates = 'Jan-20-2017';Strike = 125;

RateSpec

Antithetic = true; Price = optstockbyls(RateSpec, StockSpec, OptSpec, Strike, Settle,ExerciseDates, 'Antithetic', Antithetic)

Price =2.8310

StockSpec

Vanilla Option

L o g oTable

Basic MC

Antithetic MC

Implied Trinomial Tree

Standard Trinomial Tree

Real price

Call 3.0323

2.8310 3.2297 3.0031 2.98

Put 25.4100

32.4763 25.50

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Gopher 6 Consulting