Documentation

# optstockbyblk

Price options on futures and forwards using Black option pricing model

## Description

example

Price = optstockbyblk(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike) computes option prices on futures or forward using the Black option pricing model.

### Note

optstockbyblk calculates option prices on futures and forwards. If ForwardMaturity is not passed, the function calculates prices of future options. If ForwardMaturity is passed, the function computes prices of forward options. This function handles several types of underlying assets, for example, stocks and commodities. For more information on the underlying asset specification, see stockspec.

example

Price = optstockbyblk(___,Name,Value) adds an optional name-value pair argument for ForwardMaturity to compute option prices on forwards using the Black option pricing model.

## Examples

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This example shows how to compute option prices on futures using the Black option pricing model. Consider two European call options on a futures contract with exercise prices of \$20 and \$25 that expire on September 1, 2008. Assume that on May 1, 2008 the contract is trading at \$20, and has a volatility of 35% per annum. The risk-free rate is 4% per annum. Using this data, calculate the price of the call futures options using the Black model.

Strike = [20; 25];
AssetPrice = 20;
Sigma = .35;
Rates = 0.04;
Settle = 'May-01-08';
Maturity = 'Sep-01-08';

% define the RateSpec and StockSpec
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle,...
'EndDates', Maturity, 'Rates', Rates, 'Compounding', -1);

StockSpec = stockspec(Sigma, AssetPrice);

% define the call options
OptSpec = {'call'};

Price = optstockbyblk(RateSpec, StockSpec, Settle, Maturity,...
OptSpec, Strike)
Price = 2×1

1.5903
0.3037

This example shows how to compute option prices on forwards using the Black pricing model. Consider two European options, a call and put on the Brent Blend forward contract that expires on January 1, 2015. The options expire on October 1, 2014 with an exercise price of \$200 and \$90 respectively. Assume that on January 1, 2014 the forward price is at \$107, the annualized continuously compounded risk-free rate is 3% per annum and volatility is 28% per annum. Using this data, compute the price of the options.

Define the RateSpec.

ValuationDate = 'Jan-1-2014';
EndDates = 'Jan-1-2015';
Rates = 0.03;
Compounding = -1;
Basis = 1;
RateSpec  = intenvset('ValuationDate', ValuationDate, ...
'StartDates', ValuationDate, 'EndDates', EndDates, 'Rates', Rates,....
'Compounding', Compounding, 'Basis', Basis')
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9704
Rates: 0.0300
EndTimes: 1
StartTimes: 0
EndDates: 735965
StartDates: 735600
ValuationDate: 735600
Basis: 1
EndMonthRule: 1

Define the StockSpec.

AssetPrice = 107;
Sigma = 0.28;
StockSpec  = stockspec(Sigma, AssetPrice);

Define the options.

Settle = 'Jan-1-2014';
Maturity = 'Oct-1-2014';  %Options maturity
Strike = [200;90];
OptSpec = {'call'; 'put'};

Price the forward call and put options.

ForwardMaturity = 'Jan-1-2015';  % Forward contract maturity
Price = optstockbyblk(RateSpec, StockSpec, Settle, Maturity, OptSpec, Strike,...
'ForwardMaturity', ForwardMaturity)
Price = 2×1

0.0535
3.2111

Consider a call European option on the Crude Oil Brent futures. The option expires on December 1, 2014 with an exercise price of \$120. Assume that on April 1, 2014 futures price is at \$105, the annualized continuously compounded risk-free rate is 3.5% per annum and volatility is 22% per annum. Using this data, compute the price of the option.

Define the RateSpec.

ValuationDate = 'January-1-2014';
EndDates = 'January-1-2015';
Rates = 0.035;
Compounding = -1;
Basis = 1;
RateSpec  = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate,...
'EndDates', EndDates, 'Rates', Rates, 'Compounding', Compounding, 'Basis', Basis')
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9656
Rates: 0.0350
EndTimes: 1
StartTimes: 0
EndDates: 735965
StartDates: 735600
ValuationDate: 735600
Basis: 1
EndMonthRule: 1

Define the StockSpec.

AssetPrice = 105;
Sigma = 0.22;
StockSpec  = stockspec(Sigma, AssetPrice)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.2200
AssetPrice: 105
DividendType: []
DividendAmounts: 0
ExDividendDates: []

Define the option.

Settle = 'April-1-2014';
Maturity = 'Dec-1-2014';
Strike = 120;
OptSpec = {'call'};

Price the futures call option.

Price = optstockbyblk(RateSpec, StockSpec, Settle, Maturity, OptSpec, Strike)
Price = 2.5847

## Input Arguments

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Interest-rate term structure (annualized and continuously compounded), specified by the RateSpec obtained from intenvset. For information on the interest-rate specification, see intenvset.

Data Types: struct

Stock specification for the underlying asset. For information on the stock specification, see stockspec.

stockspec handles several types of underlying assets. For example, for physical commodities the price is StockSpec.Asset, the volatility is StockSpec.Sigma, and the convenience yield is StockSpec.DividendAmounts.

Data Types: struct

Settlement or trade date, specified as serial date number or date character vector using a NINST-by-1 vector.

Data Types: double | cell

Maturity date for option, specified as serial date number or date character vector using a NINST-by-1 vector.

Data Types: double | cell

Definition of the option as 'call' or 'put', specified as a NINST-by-1 cell array of character vectors with values 'call' or 'put'.

Data Types: cell

Option strike price value, specified as a nonnegative NINST-by-1 vector.

Data Types: double

### Name-Value Pair Arguments

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.

Example: Price = optstockbyblk(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,'ForwardMaturity',ForwardMaturity)

Maturity date or delivery date of forward contract, specified as the comma-separated pair consisting of 'ForwardMaturity' and a NINST-by-1 vector using serial date numbers or date character vectors.

Data Types: double | cell

## Output Arguments

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Expected option prices, returned as a NINST-by-1 vector.

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### Futures Option

A futures option is a standardized contract between two parties to buy or sell a specified asset of standardized quantity and quality for a price agreed upon today (the futures price) with delivery and payment occurring at a specified future date, the delivery date.

The futures contracts are negotiated at a futures exchange, which acts as an intermediary between the two parties. The party agreeing to buy the underlying asset in the future, the "buyer" of the contract, is said to be "long," and the party agreeing to sell the asset in the future, the "seller" of the contract, is said to be "short."

A futures contract is the delivery of item J at time T and:

• There exists in the market a quoted price $F\left(t,T\right)$, which is known as the futures price at time t for delivery of J at time T.

• The price of entering a futures contract is equal to zero.

• During any time interval [t,s], the holder receives the amount $F\left(s,T\right)-F\left(t,T\right)$ (this reflects instantaneous marking to market).

• At time T, the holder pays $F\left(T,T\right)$ and is entitled to receive J. Note that $F\left(T,T\right)$ should be the spot price of J at time T.

### Forwards Option

A forwards option is a non-standardized contract between two parties to buy or to sell an asset at a specified future time at a price agreed upon today.

The buyer of a forwards option contract has the right to hold a particular forward position at a specific price any time before the option expires. The forwards option seller holds the opposite forward position when the buyer exercises the option. A call option is the right to enter into a long forward position and a put option is the right to enter into a short forward position. A closely related contract is a futures contract. A forward is like a futures in that it specifies the exchange of goods for a specified price at a specified future date.

The payoff for a forwards option, where the value of a forward position at maturity depends on the relationship between the delivery price (K) and the underlying price (ST) at that time, is:

• For a long position: ${f}_{T}={S}_{T}-K$

• For a short position: ${f}_{T}=K-{S}_{T}$