Type:	Package
Title:	Asian Option Pricing with Price Impact
Version:	0.1.0
Date:	2025-12-04
Maintainer:	Priyanshu Tiwari <tiwari.priyanshu.iitk@gmail.com>
Description:	Implements binomial tree pricing for geometric and arithmetic Asian options incorporating market price impact from hedging activities. Uses the Cox-Ross-Rubinstein (CRR) model with the replicating portfolio method. Provides exact pricing for geometric Asian options and bounds for arithmetic Asian options based on Jensen's inequality. The price impact mechanism models how hedging volumes affect stock prices, leading to modified risk-neutral probabilities. Based on the methodology described in Tiwari and Majumdar (2025) <doi:10.48550/arXiv.2512.07154>.
License:	GPL (≥ 3)
URL:	https://github.com/plato-12/AsianOption
BugReports:	https://github.com/plato-12/AsianOption/issues
Encoding:	UTF-8
Depends:	R (≥ 4.0.0)
Imports:	Rcpp (≥ 1.0.0)
LinkingTo:	Rcpp
Suggests:	testthat (≥ 3.0.0), covr
RoxygenNote:	7.3.3
NeedsCompilation:	yes
Packaged:	2025-12-17 20:11:49 UTC; priyanshutiwari
Author:	Priyanshu Tiwari [aut, cre], Sourav Majumdar [ctb]
Repository:	CRAN
Date/Publication:	2025-12-22 18:20:02 UTC

AsianOption: Asian Option Pricing with Price Impact

Description

Implements binomial tree pricing for geometric and arithmetic Asian options incorporating market price impact from hedging activities. Uses the Cox-Ross-Rubinstein (CRR) model with the replicating portfolio method.

Main Functions

• price_geometric_asian: Exact pricing for geometric Asian calls

• arithmetic_asian_bounds: Bounds for arithmetic Asian calls

• compute_p_adj: Compute adjusted risk-neutral probability

• compute_adjusted_factors: Compute modified up/down factors

• check_no_arbitrage: Validate no-arbitrage condition

Price Impact Mechanism

When market makers hedge options, their trading volume causes price movements following the Kyle (1985) linear impact model. This modifies the binomial tree dynamics through adjusted up/down factors and risk-neutral probability.

Mathematical Framework

The package uses the replicating portfolio approach where the option value equals the cost of constructing a portfolio that replicates its payoff, leading to risk-neutral pricing.

For geometric Asian options, the geometric average of stock prices is used. For arithmetic Asian options, rigorous upper and lower bounds are computed using Jensen's inequality. A tighter path-specific bound is available via exact path enumeration.

No-Arbitrage Condition

For valid pricing, the model requires that the adjusted down factor is less than the risk-free rate, which is less than the adjusted up factor. This ensures a valid risk-neutral probability. All pricing functions validate this condition automatically.

For detailed mathematical formulations, see the package vignettes and the reference paper.

Computational Complexity

The implementation enumerates all 2^n possible price paths:

• n \leq 15: Fast (< 1 second)

• n = 20: ~1 million paths (~10 seconds)

• n > 20: Warning issued automatically

Author(s)

Maintainer: Priyanshu Tiwari tiwari.priyanshu.iitk@gmail.com (ORCID)

Other contributors:

• Sourav Majumdar souravm@iitk.ac.in [contributor]

References

Tiwari, P., & Majumdar, S. (2025). Asian option valuation under price impact. arXiv preprint. doi:10.48550/arXiv.2512.07154

See Also

Useful links:

• https://github.com/plato-12/AsianOption

• Report bugs at https://github.com/plato-12/AsianOption/issues

Examples

# Price geometric Asian option with price impact
price_geometric_asian(
  S0 = 100, K = 100, r = 1.05,
  u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1,
  n = 3
)

# Compute bounds for arithmetic Asian option
bounds <- arithmetic_asian_bounds(
  S0 = 100, K = 100, r = 1.05,
  u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1,
  n = 3
)
print(bounds)

# Check no-arbitrage condition
check_no_arbitrage(r = 1.05, u = 1.2, d = 0.8,
                   lambda = 0.1, v_u = 1, v_d = 1)

# Compute adjusted factors
factors <- compute_adjusted_factors(u = 1.2, d = 0.8,
                                    lambda = 0.1, v_u = 1, v_d = 1)
print(factors)

Bounds for Arithmetic Asian Option with Price Impact

Description

Computes lower and upper bounds for the arithmetic Asian option (call or put) using the relationship between arithmetic and geometric means (Jensen's inequality).

Usage

arithmetic_asian_bounds(
  S0,
  K,
  r,
  u,
  d,
  lambda,
  v_u,
  v_d,
  n,
  option_type = "call",
  compute_path_specific = FALSE,
  validate = TRUE
)

Arguments

Details

Computes rigorous upper and lower bounds for arithmetic Asian options using Jensen's inequality. The lower bound is the geometric Asian option price (from AM-GM inequality). Two types of upper bounds are available:

Global upper bound: Uses a worst-case spread parameter applicable to all paths.

Path-specific upper bound: Computes tighter bounds by using path-specific spread parameters. This requires exact enumeration of all 2^n paths in the binomial tree (no sampling or approximation). The path-specific bound is typically much tighter than the global bound.

For detailed mathematical formulations, see the package vignettes and the reference paper.

Value

List containing:

lower_bound

Lower bound for arithmetic option (= geometric option price)

upper_bound

Upper bound for arithmetic option (global bound, for backward compatibility)

upper_bound_global

Global upper bound using \rho^*

upper_bound_path_specific

Path-specific upper bound (only if compute_path_specific=TRUE, otherwise NA)

rho_star

Spread parameter \rho^*

EQ_G

Expected geometric average under risk-neutral measure

V0_G

Geometric Asian option price (same as lower_bound)

n_paths_used

Number of paths used for path-specific bound (2^n if computed, 0 otherwise)

References

Tiwari, P., & Majumdar, S. (2025). Asian option valuation under price impact. arXiv preprint. doi:10.48550/arXiv.2512.07154

See Also

price_geometric_asian

Examples

# Compute basic bounds (global bound only) for call option
bounds <- arithmetic_asian_bounds(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1, n = 3, option_type = "call"
)

print(bounds)

# Compute bounds for put option
bounds_put <- arithmetic_asian_bounds(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1, n = 3, option_type = "put"
)

# Compute with path-specific bound (uses exact enumeration of all 2^n paths)
bounds_ps <- arithmetic_asian_bounds(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1, n = 5,
  compute_path_specific = TRUE
)

print(bounds_ps)

# Estimate arithmetic option price as midpoint of path-specific bounds
if (!is.na(bounds_ps$upper_bound_path_specific)) {
  estimated_price <- mean(c(bounds_ps$lower_bound,
                            bounds_ps$upper_bound_path_specific))
  cat("Estimated price:", estimated_price, "\n")
}

Check No-Arbitrage Condition

Description

Verifies that the no-arbitrage condition \tilde{d} < r < \tilde{u} holds.

Usage

check_no_arbitrage(r, u, d, lambda, v_u, v_d)

Arguments

Value

Logical: TRUE if condition holds, FALSE otherwise

Examples

check_no_arbitrage(r = 1.05, u = 1.2, d = 0.8, lambda = 0.1, v_u = 1, v_d = 1)

Compute Adjusted Up and Down Factors

Description

Calculates the modified up and down factors after incorporating price impact from hedging.

Usage

compute_adjusted_factors(u, d, lambda, v_u, v_d)

Arguments

Value

List with elements u_tilde and d_tilde

Examples

compute_adjusted_factors(u = 1.2, d = 0.8, lambda = 0.1, v_u = 1, v_d = 1)

Compute Adjusted Risk-Neutral Probability

Description

Calculates the adjusted risk-neutral probability incorporating price impact from hedging activities.

Usage

compute_p_adj(r, u, d, lambda, v_u, v_d)

Arguments

Value

Adjusted risk-neutral probability (numeric)

Examples

compute_p_adj(r = 1.05, u = 1.2, d = 0.8, lambda = 0.1, v_u = 1, v_d = 1)

Black-Scholes European Call Option Price

Description

Computes the exact price of a European call option using the classical Black-Scholes (1973) analytical formula. This is the continuous-time benchmark for comparison with discrete binomial models.

Usage

price_black_scholes_call(S0, K, r, sigma, time_to_maturity)

Arguments

Details

The Black-Scholes formula for a European call option is:

C = S_0 N(d_1) - K e^{-rT} N(d_2)

where:

d_1 = \frac{\log(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

d_2 = d_1 - \sigma\sqrt{T}

and N(\cdot) is the cumulative standard normal distribution function.

This formula assumes:

• Stock price follows geometric Brownian motion: dS_t = rS_t dt + \sigma S_t dW_t

• No dividends

• Constant risk-free rate and volatility

• Continuous trading with no transaction costs or price impact

Value

European call option price (numeric)

References

Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654. doi:10.1086/260062

Examples

price_black_scholes_call(S0 = 100, K = 100, r = 0.05, sigma = 0.2,
                         time_to_maturity = 1)

Black-Scholes European Put Option Price

Description

Computes the exact price of a European put option using the classical Black-Scholes (1973) analytical formula.

Usage

price_black_scholes_put(S0, K, r, sigma, time_to_maturity)

Arguments

Details

The Black-Scholes formula for a European put option is:

P = K e^{-rT} N(-d_2) - S_0 N(-d_1)

where:

d_1 = \frac{\log(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

d_2 = d_1 - \sigma\sqrt{T}

and N(\cdot) is the cumulative standard normal distribution function.

Alternatively, the put price can be derived from put-call parity:

P = C - S_0 + K e^{-rT}

Value

European put option price (numeric)

Put-Call Parity

The Black-Scholes put and call prices satisfy:

C - P = S_0 - K e^{-rT}

This relationship holds exactly for European options without dividends.

References

Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654. doi:10.1086/260062

See Also

price_black_scholes_call

Examples

price_black_scholes_put(S0 = 100, K = 100, r = 0.05, sigma = 0.2,
                        time_to_maturity = 1)

Price European Option with Price Impact

Description

Computes the exact price of a European option (call or put) using the Cox-Ross-Rubinstein (CRR) binomial model with price impact from hedging activities.

Usage

price_european(
  S0,
  K,
  r,
  u,
  d,
  lambda,
  v_u,
  v_d,
  n,
  option_type = "call",
  validate = TRUE
)

Arguments

Details

Computes exact prices for European options (call or put) using the binomial model with price impact. Price impact from hedging activities modifies the stock dynamics through adjusted up/down factors and risk-neutral probability.

Unlike path-dependent Asian options, European options only depend on the terminal stock price, allowing for efficient O(n) computation instead of O(2^n). See the package vignettes and reference paper for detailed mathematical formulations.

Value

European option price (numeric)

References

Tiwari, P., & Majumdar, S. (2025). Asian option valuation under price impact. arXiv preprint. doi:10.48550/arXiv.2512.07154

See Also

price_geometric_asian, compute_p_adj

Examples

# Call option with no price impact
price_european(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0, v_u = 0, v_d = 0, n = 10, option_type = "call"
)

# Put option with price impact
price_european(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1, n = 10, option_type = "put"
)

# Verify put-call parity
call <- price_european(100, 100, 1.05, 1.2, 0.8, 0.1, 1, 1, 10, "call")
put <- price_european(100, 100, 1.05, 1.2, 0.8, 0.1, 1, 1, 10, "put")

Price Geometric Asian Option with Price Impact

Description

Computes the exact price of a geometric Asian option (call or put) using the Cox-Ross-Rubinstein (CRR) binomial model with price impact from hedging activities. Uses exact enumeration of all 2^n paths.

Usage

price_geometric_asian(
  S0,
  K,
  r,
  u,
  d,
  lambda,
  v_u,
  v_d,
  n,
  option_type = "call",
  validate = TRUE
)

Arguments

Details

Computes exact prices for geometric Asian options using complete path enumeration in a binomial tree. Price impact from hedging activities modifies the stock dynamics through adjusted up/down factors and risk-neutral probability.

This function enumerates all 2^n possible paths in the binomial tree for exact pricing (no approximation or sampling). For large n (> 20), this requires significant computation time and memory. See the package vignettes and reference paper for detailed mathematical formulations.

Value

Geometric Asian option price (numeric).

References

Tiwari, P., & Majumdar, S. (2025). Asian option valuation under price impact. arXiv preprint. doi:10.48550/arXiv.2512.07154

See Also

arithmetic_asian_bounds, compute_p_adj

Examples

# Basic example
price_geometric_asian(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0, v_u = 0, v_d = 0, n = 10
)

# With price impact
price_geometric_asian(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1, n = 15
)

# Put option
price_geometric_asian(
  S0 = 100, K = 100, r = 1.05, u = 1.2, d = 0.8,
  lambda = 0.1, v_u = 1, v_d = 1, n = 10,
  option_type = "put"
)

Kemna-Vorst Arithmetic Average Asian Option

Description

Calculates the price of an arithmetic average Asian option using Monte Carlo simulation with variance reduction via the geometric average control variate. This implements the Kemna & Vorst (1990) method WITHOUT price impact.

Usage

price_kemna_vorst_arithmetic(
  S0,
  K,
  r,
  sigma,
  T0,
  T_mat,
  n,
  M = 10000,
  option_type = "call",
  use_control_variate = TRUE,
  seed = NULL,
  return_diagnostics = FALSE
)

Arguments

Value

If return_diagnostics = FALSE, returns a numeric value (the estimated option price). If return_diagnostics = TRUE, returns a list with components:

price

Estimated option price

std_error

Standard error of the estimate

lower_ci

Lower 95% confidence interval

upper_ci

Upper 95% confidence interval

geometric_price

Analytical geometric average price (control variate)

correlation

Correlation between arithmetic and geometric payoffs

variance_reduction_factor

Ratio of variances (with/without control)

n_simulations

Number of Monte Carlo simulations used

n_steps

Number of time steps in each simulation

References

Kemna, A.G.Z. and Vorst, A.C.F. (1990). "A Pricing Method for Options Based on Average Asset Values." Journal of Banking and Finance, 14, 113-129.

Examples

price_kemna_vorst_arithmetic(
  S0 = 100, K = 100, r = 0.05, sigma = 0.2,
  T0 = 0, T_mat = 1, n = 50, M = 10000
)

Kemna-Vorst Geometric Average Asian Option

Description

Calculates the price of a geometric average Asian call option using the closed-form analytical solution from Kemna & Vorst (1990). This is the standard benchmark implementation WITHOUT price impact.

Usage

price_kemna_vorst_geometric(S0, K, r, sigma, T0, T_mat, option_type = "call")

Arguments

Details

The geometric average at maturity is defined as:

G_T = \exp\left(\frac{1}{T-T_0} \int_{T_0}^{T} \log(S(\tau)) d\tau\right)

For the discrete case with n+1 observations:

G_T = \left(\prod_{i=0}^{n} S(T_i)\right)^{1/(n+1)}

The closed-form solution for a call option is:

C = S_0 e^{d^*} N(d) - K N(d - \sigma_G\sqrt{T-T_0})

where:

d^* = \frac{1}{2}(r - \frac{\sigma^2}{6})(T - T_0)

d = \frac{\log(S_0/K) + \frac{1}{2}(r + \frac{\sigma^2}{6})(T-T_0)}{\sigma\sqrt{(T-T_0)/3}}

and N(\cdot) is the cumulative standard normal distribution function.

Value

Numeric. The analytical price of the geometric average Asian option.

References

Kemna, A.G.Z. and Vorst, A.C.F. (1990). "A Pricing Method for Options Based on Average Asset Values." Journal of Banking and Finance, 14, 113-129.

Examples

price_kemna_vorst_geometric(
  S0 = 100, K = 100, r = 0.05, sigma = 0.2,
  T0 = 0, T_mat = 1, option_type = "call"
)

Print Method for Arithmetic Asian Bounds

Description

Print Method for Arithmetic Asian Bounds

Usage

## S3 method for class 'arithmetic_bounds'
print(x, ...)

Arguments

Value

Invisible x

Print Method for Kemna-Vorst Arithmetic Results

Description

Print Method for Kemna-Vorst Arithmetic Results

Usage

## S3 method for class 'kemna_vorst_arithmetic'
print(x, ...)

Arguments

Value

Invisibly returns the input object x. Called for side effects (printing).

Summary Method for Kemna-Vorst Arithmetic Results

Description

Summary Method for Kemna-Vorst Arithmetic Results

Usage

## S3 method for class 'kemna_vorst_arithmetic'
summary(object, ...)

Arguments

Value

Invisibly returns the input object object. Called for side effects (printing).

Validate Input Parameters for Asian Option Pricing

Description

Validate Input Parameters for Asian Option Pricing

Usage

validate_inputs(S0, K, r, u, d, lambda, v_u, v_d, n)

Arguments

Value

NULL (throws error if validation fails)