blksch: Black-Scholes option pricing

In derivmkts: Functions and R Code to Accompany Derivatives Markets

Black-Scholes option pricing

Description

bscall and bsput compute Black-Scholes call and put prices. The functions assetcall, assetput, cashcall, and cashput provide the prices of binary options that pay one share (the asset options) or $1 (the cash options) if at expiration the asset price exceeds the strike (the calls) or is below the strike (the puts). We have the identities

bscall(s, k, v, r, tt, d) = assetcall(s, k, v, r, tt, d) - k*cashcall(s, k, v, r, tt, d)

bsput(s, k, v, r, tt, d) = k*cashput(s, k, v, r, tt, d) - assetput(s, k, v, r, tt, d)

Usage

bscall(s, k, v, r, tt, d)
bsput(s, k, v, r, tt, d)
assetcall(s, k, v, r, tt, d)
cashcall(s, k, v, r, tt, d)
assetput(s, k, v, r, tt, d)
cashput(s, k, v, r, tt, d)

Arguments

s	Price of the underlying asset
k	Strike price
v	Volatility of the asset price, defined as the annualized standard deviation of the continuously-compounded return
r	Annual continuously-compounded risk-free interest rate
tt	Time to maturity in years
d	Dividend yield, annualized, continuously-compounded

Details

Returns a scalar or vector of option prices, depending on the inputs

Value

A Black-Scholes option price. If more than one argument is a vector, the recycling rule determines the handling of the inputs

Note

It is possible to specify the inputs either in terms of an interest rate and a "dividend yield" or an interest rate and a "cost of carry". In this package, the dividend yield should be thought of as the cash dividend received by the owner of the underlying asset, or (equivalently) as the payment received if the owner were to lend the asset.

There are other option pricing packages available for R, and these may use different conventions for specifying inputs. In fOptions, the dividend yield is replaced by the generalized cost of carry, which is the net payment required to fund a position in the underlying asset. If the interest rate is 10% and the dividend yield is 3%, the generalized cost of carry is 7% (the part of the interest payment not funded by the dividend payment). Thus, using the GBS function from fOptions, these two expressions return the same price:

bscall(s, k, v, r, tt, d)

fOptions::GBSOption('c', S=s, K=k, Time=tt, r=r, b=r-d, sigma=v)

Examples

s=40; k=40; v=0.30; r=0.08; tt=0.25; d=0;
bscall(s, k, v, r, tt, d)

## following returns the same price as previous
assetcall(s, k, v, r, tt, d) - k*cashcall(s, k, v, r, tt, d)

## return option prices for different strikes
bsput(s, k=38:42, v, r, tt, d)

derivmkts documentation built on April 11, 2022, 5:10 p.m.