Malliavin_Geometric_Asian_Greeks {greeks}

Computes the Greeks of a geometric Asian option with the Malliavin Monte Carlo Method in the Black Scholes- or Jump diffusion model

Description

In contrast to Asian options (see Malliavin_Asian_Greeks), geometric Asian options evaluate the geometric average \exp \left( \frac{1}{T} \int_0^T \ln S_t dt \right), where S_t is the price of the underlying asset at time t and T is the time-to-maturity of the option (see

en.wikipedia.org/wiki/Asian_option#European_Asian_call_and_put_options_with_geometric_averaging). For more details on the definition of Greeks see Greeks, and for a description of the Malliavin Monte Carlo Method for Greeks see for example (Hudde & Rüschendorf, 2023).

Usage

Malliavin_Geometric_Asian_Greeks(
  initial_price = 100,
  exercise_price = 100,
  r = 0,
  time_to_maturity = 1,
  volatility = 0.3,
  dividend_yield = 0,
  payoff = "call",
  greek = c("fair_value", "delta", "rho", "vega", "theta", "gamma"),
  model = "black_scholes",
  lambda = 0.2,
  alpha = 0.3,
  jump_distribution = function(n) stats::rt(n, df = 3),
  steps = round(time_to_maturity * 252),
  paths = 10000,
  seed = 1,
  antithetic = FALSE
)

Arguments

initial_price

• initial price of the underlying asset, can also be a vector

exercise_price

• strike price of the option, can also be a vector

r

• risk-free interest rate

time_to_maturity

• time to maturity in years

volatility

• volatility of the underlying asset

dividend_yield

• dividend yield

payoff

• the payoff function, either a string in ("call", "put", "digital_call", "digital_put"), or a function

greek

• the Greek to be calculated

model

• the model to be chosen in ("black_scholes", "jump_diffusion")

lambda

• the lambda of the Poisson process in the jump-diffusion model

alpha

• the alpha in the jump-diffusion model influences the jump size

jump_distribution

• the distribution of the jumps, choose a function which generates random numbers with the desired distribution

steps

• the number of integration steps

paths

• the number of simulated paths

seed

• the seed of the random number generator

antithetic

• if TRUE, antithetic random numbers will be chosen to decrease variance

Value

Named vector containing the values of the Greeks specified in the parameter greek.

References

Hudde, A., & Rüschendorf, L. (2023). European and Asian Greeks for Exponential Lévy Processes. Methodol Comput Appl Probab, 25 (39). doi:10.1007/s11009-023-10014-5

See Also

BS_Geometric_Asian_Greeks for exact and fast computation in the Black Scholes model and for put- and call payoff functions

Examples

Malliavin_Asian_Greeks(initial_price = 110, exercise_price = 100,
r = 0.02, time_to_maturity = 4.5, dividend_yield = 0.015, volatility = 0.22,
greek = c("fair_value", "delta", "rho"), payoff = "put")