Option Pricing Fundamentals

A European call price, strike, time to maturity, current stock price, and risk-free rate are given, and you must find the corresponding put price.

Use put-call parity: C - P = S - K e^{-rT} (for non-dividend stock) or C - P = S e^{-qT} - K e^{-rT} (with continuous dividend yield q). Solve for the unknown. Check that the put price is non-negative; if not, recheck the inputs and parity sign.

C - P = S e^{-qT} - K e^{-rT} (European, continuous dividend q).

A stock can move up by u or down by d in each period with given probabilities, and you must price a European call or put using the binomial model.

Compute the risk-neutral probability p* = (e^{rh} - d) / (u - d) per period. Walk back from terminal payoffs using p* and discounting by e^{-rh} per step. For American options, take the max of the discounted expected continuation value and the exercise value at each node. Match h, u, d, and r to a consistent time unit.

p* = (e^{rh} - d) / (u - d); option price = e^{-rh} (p* V_u + (1 - p*) V_d).

Stock price, strike, time, volatility, risk-free rate, and dividend yield are given and you must price a European call or put.

Compute d_1 = [ln(S/K) + (r - q + σ^2/2) T] / (σ √T) and d_2 = d_1 - σ √T. Look up N(d_1) and N(d_2) from a standard normal table. Call price = S e^{-qT} N(d_1) - K e^{-rT} N(d_2). Use put-call parity to get the put price if needed.

C = S e^{-qT} N(d_1) - K e^{-rT} N(d_2); d_1, d_2 as above.