Package 'amsSim'

AMS Adaptive Multilevel Splitting estimator for rare-event option payoffs.

Description

Pipeline per iteration:

• Simulate $n$ paths under the chosen model (BS/Heston-family).

• Compute continuation scores $a_{i,j}$ via function_AMS_Cpp.

• Set level $L =$ K-th order statistic of $\max_j a_{i,j}$.

• Identify survivors (top $n-K$) and parents ($K$ indices that cleared the level).

• For each parent, cut at first index that exceeds $L$ and resimulate the suffix.

• Repeat until $L \ge L_{\max}$. Then compute discounted payoff on the final population.

Usage

AMS(
  model,
  type,
  funz,
  n,
  t,
  p,
  r,
  sigma,
  S0,
  rho = NULL,
  rim = 0L,
  v0 = 0.04,
  Lmax = 0,
  strike = 1,
  K = 1L
)

Arguments

model	1 = Black–Scholes; 2,3,4 = Heston variants (as in simulate_AMS).
type	Payoff type passed to payoff() and function_AMS_Cpp (1..6).
funz	1 = BS digital proxy in continuation; 2 = raw feature (signed).
n	Population size (> K).
t	Maturity in years (>0).
p	Total time steps (>0).
r	Risk–free rate.
sigma	BS volatility (used by continuation; >0 if funz == 1).
S0	Initial spot.
rho	Correlation for Heston models (required for model >= 2, in $[-1,1]$).
rim	Left-trim for simulation (keep last p - rim steps; 0 <= rim < p).
v0	Initial variance for Heston models (>=0).
Lmax	Stopping level: iterate while $L < L_{\max}$.
strike	Strike $K$ used by continuation and final payoff.
K	Number of resampled offspring per iteration (1..n-1).

Value

List with price and std.

Examples

out <- AMS(model = 2, type = 3, funz = 1, n = 500, t = 1, p = 252, r = 0.03,
             sigma = 0.2, rho = -0.5, S0 = 1, rim = 0, Lmax = 0.5, strike = 1.3, K = 200)
  str(out)

---

simulate_AMS Monte Carlo simulation of price paths under: 1 = Black–Scholes (exact solution) 2 = Heston (Euler discretisation) 3 = Heston (Milstein discretisation) 4 = Heston (Quadratic–Exponential scheme, Andersen 2008)

Description

simulate_AMS Monte Carlo simulation of price paths under: 1 = Black–Scholes (exact solution) 2 = Heston (Euler discretisation) 3 = Heston (Milstein discretisation) 4 = Heston (Quadratic–Exponential scheme, Andersen 2008)

Usage

simulate_AMS(model, n, t, p, r, sigma, S0, rho = NULL, rim = 0L, v0 = 0.04)

Arguments

model	Integer in $\{1,2,3,4\}$ selecting the model.
n	Number of simulated paths (>0).
t	Maturity in years (>0).
p	Total time steps (>0).
r	Risk–free rate.
sigma	Black–Scholes volatility (>=0, used only when model == 1).
S0	Initial spot price (>0).
rho	Correlation between asset and variance Brownian motions (required for Heston models, finite in $[-1,1]$).
rim	Left–trim: discard the first rim time steps (0 <= rim < p). Returned matrices keep p - rim + 1 columns including the initial time.
v0	Initial variance for Heston models (>=0).

Value

List: for model 1 returns S ($n \times (p-rim+1)$); for Heston models returns S and V.

Examples

b <- simulate_AMS(1, n = 50, t = 1, p = 10, r = 0.01, sigma = 0.2, S0 = 100, rho = NULL)
  str(b)

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