American Approximation Bjerksund & Stensland 2002 [Loxx]

4 באוק׳ 2022

American Approximation Bjerksund & Stensland 2002 [Loxx] is an American Options pricing model. This indicator also includes numerical greeks. You can compare the output of the American Approximation to the Black-Scholes-Merton value on the output of the options panel.

The Bjerksund & Stensland (2002) Approximation
The Bjerksund and Stensland (2002) approximation divides the time to maturity into two parts, each with a separate flat exercise boundary. It is thus a straightforward generalization of the Bjerksund-Stensland 1993 algorithm. The method is fast and efficient and should be more accurate than the Barone-Adesi and Whaley (1987) and the Bjerksund and Stensland (1993b) approximations. The algorithm requires an accurate cumulative bivariate normal approximation. Several approximations that are described in the literature are not sufficiently accurate, but the Genze algorithm works.

C = alpha2*S^B - alpha2*phi(S, t1, B, I2, I2)
+ phi(S, t1, I2, I2) - phi(S, t1, I, I1, I2)
- X*phi(S, t1, 0, I2, I2) + X*phi(S, t1, 0, I1, I2)
+ alpha1*phi(X, t1, B, I1, I2) - alpha1*psi*St, T, B, I1, I2, I1, t1)
+ psi(S, T, 1, I1, I2, I1, t1) - psi(S, T, 1, X, I2, I1, t1)
- X*psi(S, T, 0, I1, I2, I1, t1) + psi(S, T, 0 ,X, I2, I1, t1)

where

alpha1 = (I1 - X)*I1^-B

alpha2 = (I2 - X)*I2^-B

B = (1/2 - b/v^2) + ((b/v^2 - 1/2)^2 + 2*(r/v^2))^0.5

The function psi(S, T, y, H, I) is given by

psi(S, T, gamma, H, I) = e^lambda * S^gamma * (N(-d) - (I/S)^k * N(-d2))

d = (log(S/H) + (b + (gamma - 1/2) * v^2) * T) / (v * T^0.5)

d2 = (log(I^2/(S*H)) + (b + (gamma - 1/2) * v^2) * T) / (v * T^0.5)

lambda = -r + gamma * b + 1/2 * gamma * (gamma - 1) * v^2

k = 2*b/v^2 + (2 * gamma - 1)

and the trigger price I is defined as

I1 = B0 + (B(+infi) - B0) * (1 - e^h1)

I2 = B0 + (B(+infi) - B0) * (1 - e^h2)

h1 = -(b*t1 + 2*v*t1^0.5) * (X^2 / ((B(+infi) - B0))*B0)

h2 = -(b*T + 2*v*T^0.5) * (X^2 / ((B(+infi) - B0))*B0)

t1 = 1/2 * (5^0.5 - 1) * T

B(+infi) = (B / (B - 1)) * X

B0 = max(X, (r / (r - b)) * X)

Moreover, the function psi(S, T, gamma, H, I2, I1, t1) is given by

psi(S, T, gamma, H, I2, I1, t1, r, b, v) = e^(lambda * T) * S^gamma * (M(-e1, -f1, rho) - (I2/S)^k * M(-e2, -f2, rho)
- (I1/S)^k * M(-e3, -f3, -rho) + (I1/I2)^k * M(-e4, -f4, -rho))

where (see screenshot for e and f values)



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.
K = Strike price of option.
T = Time to expiration in years.
r = Risk-free rate
c = Cost of Carry
V = Variance of the underlying asset price
cnd1(x) = Cumulative Normal Distribution
cbnd3(x) = Cumulative Bivariate Normal Distribution
nd(x) = Standard Normal Density Function
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
  

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