jumps {derivmkts}

Option pricing with jumps

Description

The functions cashjump, assetjump, and mertonjump return call and put prices, as vectors named "Call" and "Put", or "Call1", "Call2", etc. in case inputs are vectors. The pricing model is the Merton jump model, in which jumps are lognormally distributed.

Usage

assetjump(s, k, v, r, tt, d, lambda, alphaj, vj, complete)
cashjump(s, k, v, r, tt, d, lambda, alphaj, vj, complete)
mertonjump(s, k, v, r, tt, d, lambda, alphaj, vj, complete)

Arguments

s

Stock price

k

Strike price of the option

v

Volatility of the stock, 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

lambda

Poisson intensity: expected number of jumps per year

alphaj

Mean change in log price conditional on a jump

vj

Standard deviation of change in log price conditional on a jump

complete

Return inputs along with prices, all in a data frame

Details

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

Value

A vector of call and put prices computed using the Merton lognormal jump formula.

See Also

McDonald, Robert L., Derivatives Markets, 3rd Edition (2013) Chapter 24

bscall bsput

Examples

s <- 40; k <- 40; v <- 0.30; r <- 0.08; tt <- 2; d <- 0;
lambda <- 0.75; alphaj <- -0.05; vj <- .35;
bscall(s, k, v, r, tt, d)
bsput(s, k, v, r, tt, d)
mertonjump(s, k, v, r, tt, d, 0, 0, 0)
mertonjump(s, k, v, r, tt, d, lambda, alphaj, vj)

## following returns the same price as previous
c(1, -1)*(assetjump(s, k, v, r, tt, d, lambda, alphaj, vj) -
k*cashjump(s, k, v, r, tt, d, lambda, alphaj, vj))

## return call prices for different strikes
kseq <- 35:45
cp <- mertonjump(s, kseq, v, r, tt, d, lambda, alphaj,
    vj)$Call

## Implied volatilities: Compute Black-Scholes implied volatilities
## for options priced using the Merton jump model
vimp <- sapply(1:length(kseq), function(i) bscallimpvol(s, kseq[i],
    r, tt, d, cp[i]))
plot(kseq, vimp, main='Implied volatilities', xlab='Strike',
    ylab='Implied volatility', ylim=c(0.30, 0.50))