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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

`gbm`

object and run the Monte Carlo simulation based on Geometric Brownian Motion (GBM) using the `simBySolution`

method from Financial Toolbox™.

Create the interest-rate term structure to define `RateSpec`

.

`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

`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 = `*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 ratesstructure

Interest-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 pricesmatrix

Simulated 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 pricesvector

Annual 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 character vector with values

`'call'`

or
`'put'`

Definition of option as `'call'`

or `'put'`

,
specified as a character vector.

**Data Types: **`char`

`Strike`

— Option strike price valuesscalar | function handle

Option 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 optionvector of exercise times

Exercise 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 type`0`

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 optionscalar

Price of the option, returned as a scalar value.

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