Binomial-Trinomial-Asian-Option-Pricing

A computational implementation of binomial and trinomial tree models for pricing European and American options, computing Greeks, and simulating dynamic delta hedging of Asian options.

This project implements tree-based methods for derivatives pricing, covering the full lifecycle of option valuation: from discrete-time binomial trees for European and American options, through QuantLib-powered trinomial trees across the moneyness spectrum, to dynamic delta hedging of path-dependent Asian options using Monte Carlo simulation with a geometric control variate. All model inputs spot price, historical volatility, and risk-free rate are sourced live from market data via yfinance, and the Asian option hedging simulation runs on a real historical price path rather than synthetic data.

Model: Cox-Ross-Rubinstein (CRR) binomial tree with n=100 steps.

Parameters: Spot price S0 and strike K (ATM proxy, nearest $5) sourced from SPY via yfinance. Volatility is calibrated as the annualised standard deviation of daily log-returns over the trailing 63 trading days. The risk-free rate is sourced from the 3-month US T-bill yield (^IRX) via yfinance. Expiry is set to T=3 months.

Tree construction:

Up and down factors: u = exp(sigma * sqrt(dt)), d = 1/u

Risk-neutral probability: p = (exp(r * dt) - d) / (u - d)

Backward induction for European options:

V(i,j) = exp(-r*dt) * [p * V(i+1,j+1) + (1-p) * V(i+1,j)]

For American options, at each node the continuation value is compared against immediate exercise:

V(i,j) = max(intrinsic_value, continuation_value)

European Pricing: Call and put prices are computed at n=100 with a convergence plot across step counts 5 to 200, confirming stability.

Delta: First-order finite difference at the root node:

Delta = (V_up - V_down) / (S_up - S_down)

Call delta is positive (0 to 1); put delta is negative (-1 to 0). Delta represents the instantaneous sensitivity of option value to a unit change in the underlying price and is used directly as the hedge ratio.

Volatility Sensitivity (Vega): Prices are recomputed at sigma=25% to measure the impact of a 200 basis point volatility increase. Both calls and puts increase in value with higher volatility; vega is always positive for long option positions.

American Options: The American put carries an early exercise premium over the European put due to the value of exercising before expiry when deeply in-the-money. The American call on a non-dividend-paying stock has no early exercise premium.

Model: JarrowRudd binomial engine via QuantLib, equivalent to trinomial tree pricing, with n=200 steps. Uses QuantLib's BlackScholesMertonProcess with flat yield curve and constant volatility surface. Parameters are sourced from SPY via yfinance, consistent with the binomial tree module.

Strike prices tested: Five strikes spanning a ±10% moneyness range around the live spot price S0:

European Calls: Call prices decrease monotonically as strike increases. Deep ITM calls carry significant intrinsic value while deep OTM calls are dominated by time value.

European Puts: Put prices increase monotonically with strike. Put-call parity is verified across all moneyness levels.

American Options: American puts show an early exercise premium relative to European puts, increasing for deep ITM strikes where discounting costs exceed the optionality value of waiting. American calls on non-dividend-paying stocks are equal to European calls.

Parameters: S0 and sigma are drawn from a real 6-month SPY price path fetched via yfinance. N=25 evenly-spaced observations are sampled from the historical window. Strike K is set to the nearest $5 to S0 (ATM proxy).

Asian Put Pricing:

Arithmetic Asian options have no closed-form solution. Pricing uses Monte Carlo simulation with a geometric Asian control variate to reduce variance:

price_cv = E[payoff_arithmetic] - b* * (E[payoff_geometric] - price_geometric_exact)

where b* = Cov(payoff_arithmetic, payoff_geometric) / Var(payoff_geometric)

The geometric Asian option has a known closed-form solution with modified volatility and drift:

sigma_asian = sigma * sqrt((2N+1) / (6(N+1)))
r_asian = r - sigma^2/2 + sigma_asian^2/2

Hedging Simulation:

Dynamic delta hedging rebalances the hedge portfolio at each of the N=25 real price observations. At each point, the geometric Asian delta formula provides an approximation to the true arithmetic Asian delta:

delta_geometric = exp((r_asian - r)*T) * N(d1)   [for a call]

The hedging portfolio consists of a short option position offset by a dynamic stock holding of (-delta) shares. Cash flows from buying/selling shares are accumulated in a risk-free cash account earning r. At maturity, the stock position is closed, the option payoff is settled, and the residual cash balance represents the hedging error.

Two plots are produced: the delta path along the simulated stock price trajectory (showing how the hedge ratio evolves) and the cumulative portfolio value over time (showing hedging error accumulation). Discrete rebalancing introduces path-dependent hedging error; finer rebalancing reduces but does not eliminate this error.

• CRR binomial tree converges to stable option prices by n=50, validating the choice of n=100.
• European and American calls on non-dividend-paying stocks price identically with no early exercise premium.
• The American put carries an early exercise premium that increases for deep ITM strikes.
• Trinomial tree prices confirm monotone call/put price profiles across the moneyness spectrum.
• Asian option delta hedging with geometric approximation produces small but non-zero hedging errors due to discrete rebalancing and the approximation in the delta formula.

Python 3.x
numpy
scipy
matplotlib
yfinance
QuantLib (trinomial tree via JarrowRudd engine)

git clone https://github.com/QuantSingularity/Binomial-Trinomial-Asian-Option-Pricing.git
cd Binomial-Trinomial-Asian-Option-Pricing
pip install numpy scipy matplotlib yfinance QuantLib
jupyter notebook Binomial_Trinomial_Asian_Option_Pricing.ipynb

• Cox, J.C., Ross, S.A., & Rubinstein, M. (1979). Option Pricing: A Simplified Approach. Journal of Financial Economics, 7(3), 229-263.
• Jarrow, R., & Rudd, A. (1983). Option Pricing. Dow Jones-Irwin.
• Hull, J.C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
• Kemna, A.G.Z., & Vorst, A.C.F. (1990). A Pricing Method for Options Based on Average Asset Values. Journal of Banking and Finance, 14(1), 113-129.
• QuantLib Documentation: https://www.quantlib.org