OPEN-SOURCE SCRIPT
Fade-in Options [Loxx]
A fade-in call has the same payoff as a standard call except the size of the payoff is weighted by how many fixings the asset price were inside a predefined range (L, U). If the asset price is inside the range for every fixing, the payoff will be identical to a plain vanilla option. More precisely, for a call option, the payoff will be max(S(T) - X, 0) X 1/n Sum(n(i)), where n is the total number of fixings and n(i) = 1 if at fixing i the asset price is inside the range, and n(i) = 0 otherwise. Similarly, for a put, the payoff is max(X - S(T), 0) X 1/n Sum(n(i)).
Brockhaus, Ferraris, Gallus, Long, Martin, and Overhaus (1999) describe a closed-form formula for fade-in options. For a call the value is given by
max(X - S(T), 0) X 1/n Sum(n(i))
describe a closed-form formula for fade-in options. For a call the value is given by
c = 1/n * Sum(S^((b-r)*T) * (M(-d5, d1; -p) - M(-d3, d1; -p)) - Xe^(-rT) * (M(-d6, d2; -p) - M(-d4, d2; -p))
where n is the number of fixings, p = (t1^0.5/T^0.5), t1 = iT/n
d1 = (log(S/X) + (b + v^2/2)*T) / (v * T^0.5) ... d2 = d1 - v*T^0.5
d3 = (log(S/L) + (b + v^2/2)*t1) / (v * t1^0.5) ... d4 = d3 - v*t1^0.5
d5 = (log(S/U) + (b + v^2/2)*t1) / (v * t1^0.5) ... d6 = d5 - v*t1^0.5
The value of a put is similarly
p = 1/n * Sum(Xe^(-rT) * (M(-d6, -d2; -p) - M(-d4, -d2; -p))) - S^((b-r)*T) * (M(-d5, -d1; -p) - M(-d3, -d1; -p)
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
Asset price ( S )
Strike price ( K )
Lower barrier ( L )
Upper barrier ( U )
Time to maturity ( T )
Risk-free rate ( r )
Cost of carry ( b )
Volatility ( s )
Fixings ( n )
cnd1(x) = Cumulative Normal Distribution
nd(x) = Standard Normal Density Function
cbnd3() = Cumulative Bivariate Distribution
convertingToCCRate(r, cmp ) = Rate compounder
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.
You can adjust the text size to fit the screen
Brockhaus, Ferraris, Gallus, Long, Martin, and Overhaus (1999) describe a closed-form formula for fade-in options. For a call the value is given by
max(X - S(T), 0) X 1/n Sum(n(i))
describe a closed-form formula for fade-in options. For a call the value is given by
c = 1/n * Sum(S^((b-r)*T) * (M(-d5, d1; -p) - M(-d3, d1; -p)) - Xe^(-rT) * (M(-d6, d2; -p) - M(-d4, d2; -p))
where n is the number of fixings, p = (t1^0.5/T^0.5), t1 = iT/n
d1 = (log(S/X) + (b + v^2/2)*T) / (v * T^0.5) ... d2 = d1 - v*T^0.5
d3 = (log(S/L) + (b + v^2/2)*t1) / (v * t1^0.5) ... d4 = d3 - v*t1^0.5
d5 = (log(S/U) + (b + v^2/2)*t1) / (v * t1^0.5) ... d6 = d5 - v*t1^0.5
The value of a put is similarly
p = 1/n * Sum(Xe^(-rT) * (M(-d6, -d2; -p) - M(-d4, -d2; -p))) - S^((b-r)*T) * (M(-d5, -d1; -p) - M(-d3, -d1; -p)
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
Asset price ( S )
Strike price ( K )
Lower barrier ( L )
Upper barrier ( U )
Time to maturity ( T )
Risk-free rate ( r )
Cost of carry ( b )
Volatility ( s )
Fixings ( n )
cnd1(x) = Cumulative Normal Distribution
nd(x) = Standard Normal Density Function
cbnd3() = Cumulative Bivariate Distribution
convertingToCCRate(r, cmp ) = Rate compounder
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.
You can adjust the text size to fit the screen
Open-source script
In true TradingView spirit, the author of this script has published it open-source, so traders can understand and verify it. Cheers to the author! You may use it for free, but reuse of this code in publication is governed by House rules. You can favorite it to use it on a chart.
Public Telegram Group, t.me/algxtrading_public
VIP Membership Info: patreon.com/algxtrading/membership
VIP Membership Info: patreon.com/algxtrading/membership
Disclaimer
The information and publications are not meant to be, and do not constitute, financial, investment, trading, or other types of advice or recommendations supplied or endorsed by TradingView. Read more in the Terms of Use.