Forward Start Options [Loxx]

2022년 10월 5일

A forward start option with time to maturity T starts at-the-money or proportionally in- or out-of-the-money after a known elapsed time t in the future. The strike is set equal to a positive constant a times the asset price S after the known time t. If a is less than unity, the call (put) will start 1 - a percent in-the-money (out-of-the- money); if a is unity, the option will start at-the-money; and if a is larger than unity, the call (put) will start a - 1 percentage out-of-the- money (in-the-money).A forward start option can be priced using the Rubinstein (1990) formula: (via "The Complete Guide to Option Pricing Formulas")

c = S*e^(b-r)t * (e^(b-r)(T-t) * N(d1)) - alpha * e^-r(T-t) * N(d2))

p = S*e^(b-r)t * (alpha*e^r(T-t) * N(-d2)) - e^-(b-r)(T-t) * N(-d1))

where

d1 = (log(1/alpha) + (b + v^2/2)(T-1))/v*(T-t)^0.5

d2 = d1 - v*(T-t)^0.5

Application
Employee options are often of the forward starting type. Ratchet options (aka cliquet options) consist of a series of forward starting options.

b=r options on non-dividend paying stock
b=r-q options on stock or index paying a dividend yield of q
b=0 options on futures
b=r-rf currency options (where rf is the rate in the second currency)

Inputs
S = Stock price.
a = Alpha
T1 = Time to forward start
T = Time to expiration in years.
r = Risk-free rate
c = Cost of Carry
v = volatility of the underlying asset price

Numerical Greeks or Greeks by Finite Difference
Analytical Greeks are the standard approach to estimating Delta, Gamma etc... That is what we typically use when we can derive from closed form solutions. Normally, these are well-defined and available in text books. Previously, we relied on closed form solutions for the call or put formulae differentiated with respect to the Black Scholes parameters. When Greeks formulae are difficult to develop or tease out, we can alternatively employ numerical Greeks - sometimes referred to finite difference approximations. A key advantage of numerical Greeks relates to their estimation independent of deriving mathematical Greeks. This could be important when we examine American options where there may not technically exist an exact closed form solution that is straightforward to work with. (via VinegarHill FinanceLabs)

Things to know
Only works on the daily timeframe and for the current source price.
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