def implied_volatility(market_price, S, K, T, r, option_type='call', tol=1e-6, max_iter=100):
    sigma = math.sqrt((2 * math.pi) / T) * (market_price / S)     # Initial Guess (Manaster-Koehler seed for SPX)
    for i in range(max_iter):
        # If sigma leaves bounds, go to Bisection immediately
        if not (0.0001 < sigma < 5.0):
            return implied_volatility_bisection(market_price, S, K, T, r, sigma, option_type, tol, max_iter)
        
        if option_type == 'call':
            price = call(S, K, T, r, sigma)
        else:
            price = put(S, K, T, r, sigma)
            
        vega_value = vega(S, K, T, r, sigma)
        price_diff = price - market_price
        
        if abs(price_diff) < tol:
            return sigma
            
        # If Vega is too flat, go to Bisection
        if abs(vega_value) < 1e-10:
            return implied_volatility_bisection(market_price, S, K, T, r, sigma, option_type, tol, max_iter)
            
        sigma -= price_diff / vega_value
    # If no convergence, use  final guess to warm-start bisection
    return implied_volatility_bisection(market_price, S, K, T, r, sigma, option_type, tol, max_iter)
def implied_volatility_bisection(market_price, S, K, T, r, sigma_guess, option_type='call', tol=1e-6, max_iter=100):
    low = 0.0001
    high = 5.0
    
    # Warm start
    # Shrink search space to speed up bisection if Newton-Raphson guess is reasonable
    if 0.0001 < sigma_guess < 5.0:
        if option_type == 'call':
            price_at_guess = call(S, K, T, r, sigma_guess)
        else:
            price_at_guess = put(S, K, T, r, sigma_guess)
            
        if price_at_guess > market_price:
            high = sigma_guess  # Guess was too high, it becomes our new ceiling
        else:
            low = sigma_guess   # Guess was too low, it becomes our new floor
    # Normal Bisection
    for i in range(max_iter):
        mid = (low + high) / 2
        
        if option_type == 'call':
            price = call(S, K, T, r, mid)
        else:
            price = put(S, K, T, r, mid)
            
        if abs(price - market_price) < tol:
            return mid
        
        if price > market_price:
            high = mid
        else:
            low = mid
            
    return (low + high) / 2