Price option given simulated underlying values
calculates the price of European, American, and Berumdan call/put options based on
risk-neutral simulation of the underlying asset. For American and Bermudan options, the
Longstaff-Schwartz least squares method calculates the early exercise premium.Price
= optpricebysim(RateSpec
,SimulatedPrices
,Times
,OptSpec
,Strike
,ExerciseTimes
)
adds optional name-value pair arguments.Price
= optpricebysim(___,Name,Value
)
Define the option.
S0 = 100; % Initial price of underlying asset Sigma = .2; % Volatility of underlying asset Strike = 110; % Strike OptSpec = 'call'; % Call option Settle = '1-Jan-2013'; % Settlement date of option Maturity = '1-Jan-2014'; % Maturity date of option r = .05; % Risk-free rate (annual, continuous compounding) Compounding = -1; % Continuous compounding Basis = 0; % Act/Act day count convention T = yearfrac(Settle, Maturity, Basis); % Time to expiration in years
Set up the gbm
object and run the Monte Carlo simulation based on Geometric Brownian Motion (GBM) using the simBySolution
method from Financial Toolbox™.
NTRIALS = 1000; NPERIODS = daysact(Settle, Maturity); dt = T/NPERIODS; OptionGBM = gbm(r, Sigma, 'StartState', S0); [Paths, Times, Z] = simBySolution(OptionGBM, NPERIODS, ... 'NTRIALS',NTRIALS, 'DeltaTime',dt,'Antithetic',true);
Create the interest-rate term structure to define RateSpec
.
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ... 'EndDates', Maturity, 'Rate', r, 'Compounding', Compounding, ... 'Basis', Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0500
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1
Price an American option.
SimulatedPrices = squeeze(Paths); OptPrice = optpricebysim(RateSpec, SimulatedPrices, Times, OptSpec, ... Strike, T, 'AmericanOpt', 1)
OptPrice = 6.2028
Define the option.
S0 = 100; % Initial price of underlying asset Sigma = .2; % Volatility of underlying asset Strike = 110; % Strike OptSpec = 'call'; % Call option Settle = '1-Jan-2013'; % Settlement date of option Maturity = '1-Jan-2014'; % Maturity date of option r = .05; % Risk-free rate (annual, continuous compounding) Compounding = -1; % Continuous compounding Basis = 0; % Act/Act day count convention T = yearfrac(Settle, Maturity, Basis); % Time to expiration in years
Set up the gbm
object and run the Monte Carlo simulation based on Geometric Brownian Motion (GBM) using the simBySolution
method from Financial Toolbox™.
NTRIALS = 1000; NPERIODS = daysact(Settle, Maturity); dt = T/NPERIODS; OptionGBM = gbm(r, Sigma, 'StartState', S0); [Paths, Times, Z] = simBySolution(OptionGBM, NPERIODS, ... 'NTRIALS',NTRIALS, 'DeltaTime',dt,'Antithetic',true);
Create the interest-rate term structure to define RateSpec
.
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ... 'EndDates', Maturity, 'Rate', r, 'Compounding', Compounding, ... 'Basis', Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0500
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1
Price an American Asian option (arithmetic mean) by finding the average price over periods.
AvgPrices = zeros(NPERIODS+1, NTRIALS); for i = 1:NPERIODS+1 AvgPrices(i,:) = mean(squeeze(Paths(1:i,:,:))); end AsianPrice = optpricebysim(RateSpec, AvgPrices, Times, OptSpec, ... Strike, T, 'AmericanOpt', 1)
AsianPrice = 1.8540
Define the option.
S0 = 100; % Initial price of underlying asset Sigma = .2; % Volatility of underlying asset Strike = 110; % Strike OptSpec = 'call'; % Call option Settle = '1-Jan-2013'; % Settlement date of option Maturity = '1-Jan-2014'; % Maturity date of option r = .05; % Risk-free rate (annual, continuous compounding) Compounding = -1; % Continuous compounding Basis = 0; % Act/Act day count convention T = yearfrac(Settle, Maturity, Basis); % Time to expiration in years
Set up the gbm
object and run the Monte Carlo simulation based on Geometric Brownian Motion (GBM) using the simBySolution
method from Financial Toolbox™.
NTRIALS = 1000; NPERIODS = daysact(Settle, Maturity); dt = T/NPERIODS; OptionGBM = gbm(r, Sigma, 'StartState', S0); [Paths, Times, Z] = simBySolution(OptionGBM, NPERIODS, ... 'NTRIALS',NTRIALS, 'DeltaTime',dt,'Antithetic',true);
Create the interest-rate term structure to define RateSpec
.
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ... 'EndDates', Maturity, 'Rate', r, 'Compounding', Compounding, ... 'Basis', Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0500
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1
Price an American lookback option by finding the maximum price over periods.
MaxPrices = zeros(NPERIODS+1, NTRIALS); LastPrice = squeeze(Paths(1,:,:))'; for i = 1:NPERIODS+1; MaxPrices(i,:) = max([LastPrice; Paths(i,:)]); LastPrice = MaxPrices(i,:); end LookbackPrice = optpricebysim(RateSpec, MaxPrices, Times, OptSpec, ... Strike, T, 'AmericanOpt', 1)
LookbackPrice = 10.4084
Define the option.
S0 = 80; % Initial price of underlying asset Sigma = .3; % Volatility of underlying asset Strike = 75; % Strike OptSpec = 'put'; % Put option Settle = '1-Jan-2013'; % Settlement date of option Maturity = '1-Jan-2014'; % Maturity date of option ExerciseDates = {'1-Jun-2013', '1-Jan-2014'}; % Exercise dates of option r = .05; % Risk-free rate (annual, continuous compounding) Compounding = -1; % Continuous compounding Basis = 0; % Act/Act day count convention T = yearfrac(Settle, Maturity, Basis); % Time to expiration in years ExerciseTimes = yearfrac(Settle, ExerciseDates, Basis)'; % Exercise times
Set up the gbm
object and run the Monte Carlo simulation based on Geometric Brownian Motion (GBM) using the simBySolution
method from Financial Toolbox™.
NTRIALS = 1000; NPERIODS = daysact(Settle, Maturity); dt = T/NPERIODS; OptionGBM = gbm(r, Sigma, 'StartState', S0); [Paths, Times, Z] = simBySolution(OptionGBM, NPERIODS, ... 'NTRIALS',NTRIALS, 'DeltaTime',dt,'Antithetic',true);
Create the interest-rate term structure to define RateSpec
.
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ... 'EndDates', Maturity, 'Rate', r, 'Compounding', Compounding, ... 'Basis', Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0500
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1
Price the Bermudan option.
SimulatedPrices = squeeze(Paths);
BermudanPrice = optpricebysim(RateSpec, SimulatedPrices, Times, ...
OptSpec, Strike, ExerciseTimes)
BermudanPrice = 5.3950
Define the option.
S1 = 110; % Price of first underlying asset S2 = 100; % Price of second underlying asset Sigma1 = .1; % Volatility of first underlying asset Sigma2 = .15; % Volatility of second underlying asset Strike = 15; % Strike Rho = .3; % Correlation between underlyings OptSpec = 'put'; % Put option Settle = '1-Jan-2013'; % Settlement date of option Maturity = '1-Jan-2014'; % Maturity date of option r = .05; % Risk-free rate (annual, continuous compounding) Compounding = -1; % Continuous compounding Basis = 0; % Act/Act day count convention T = yearfrac(Settle, Maturity, Basis); % Time to expiration in years
Set up the gbm
object and run the Monte Carlo simulation based on Geometric Brownian Motion (GBM) using the simBySolution
method from Financial Toolbox™.
NTRIALS = 1000; NPERIODS = daysact(Settle, Maturity); dt = T/NPERIODS; SpreadGBM = gbm(r*eye(2), diag([Sigma1;Sigma2]),'Correlation',... [1 Rho;Rho 1],'StartState',[S1;S2]); [Paths, Times, Z] = simBySolution(SpreadGBM, NPERIODS,'NTRIALS',NTRIALS,... 'DeltaTime',dt,'Antithetic',true);
Create the interest-rate term structure to define RateSpec
.
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ... 'EndDates', Maturity, 'Rate', r, 'Compounding', Compounding, ... 'Basis', Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0500
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1
Price the American spread option.
Spread = squeeze(Paths(:,1,:) - Paths(:,2,:)); SpreadPrice = optpricebysim(RateSpec, Spread, Times, OptSpec, Strike, ... T, 'AmericanOpt', 1)
SpreadPrice = 9.0007
RateSpec
— Interest-rate term structure of risk-free ratesInterest-rate term structure of risk-free rates (annualized and continuously
compounded), specified by the RateSpec
obtained from intenvset
. The valuation date must be at the settlement date of the
option, and the day-count basis and end-of-month rule must be the same as those used to
calculate the Times
input. For information on the interest-rate
specification, see intenvset
.
Data Types: struct
SimulatedPrices
— Simulated pricesSimulated prices, specified using a (NumPeriods
+
1
)-by-NumTrials
matrix of risk-neutral simulated
prices. The first element of SimulatedPrices
is the initial value
at time 0.
Data Types: single
| double
Times
— Annual time factors associated with simulated pricesAnnual time factors associated with simulated prices, specified using a
(NumPeriods
+ 1
)-by-1
column
vector. Each element of Times
is associated with the corresponding
row of SimulatedPrices
. The first element of
Times
must be 0 (current time).
Data Types: single
| double
OptSpec
— Definition of option 'call'
or
'put'
Definition of option as 'call'
or 'put'
,
specified as a character vector.
Data Types: char
Strike
— Option strike price valuesOption strike price values, specified as a scalar value Strike
price. Strike
for Bermudan options can be specified as a
1
-by-NSTRIKES
vector or a function handle that
returns the value of the strike given the time of the strike.
Data Types: single
| double
| function_handle
ExerciseTimes
— Exercise time for optionExercise time for the option, specified as a vector of exercise times as follows:
For a European or Bermudan option, ExerciseTimes
is a
1
-by-1
(European) or
1
-by-NSTRIKES
(Bermudan) vector of
exercise times. For a European option, there is only one
ExerciseTimes
on the option expiry date.
For an American option, ExerciseTimes
is a
1
-by-2
vector of exercise time boundaries.
The option exercises on any date between, or including, the pair of times on that
row. If ExerciseTimes
is
1
-by-1
, the option exercises between time
0
and the single listed
ExerciseTimes
.
Data Types: double
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
Price =
optpricebysim(RateSpec,Prices,Times,OptSpec,Settle,Strike,ExerciseTimes,'AmericanOpt',1)
'AmericanOpt'
— Option type0
European or Bermudan (default) | scalar flag with value [0,1]
Option type, specified as the comma-separated pair consisting of
'AmericanOpt'
and an integer scalar flag with values:
0
— European or Bermudan
1
— American
For American options, the Longstaff-Schwartz least squares method calculates the early exercise premium.
Data Types: single
| double
Price
— Price of optionPrice of the option, returned as a scalar value.
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