Valuation of Interest Rate Derivatives › Worked Example: Hedging Floating-Rate Exposure

# Corporate hedge parameters
notional_corporate = 100_000_000  # $100 million
strike_cap = 0.06  # 6% cap strike
cap_tenor = 3  # 3 years
current_rate = 0.05  # 5% current rate
cap_vol = 0.22  # 22% implied volatility

# Quarterly resets
payment_dates_corp = np.arange(0.25, cap_tenor + 0.25, 0.25)
n_caplets = len(payment_dates_corp)
times_to_expiry_corp = payment_dates_corp - 0.25

# Assume a gently upward-sloping rate curve
forward_rates_corp = current_rate + 0.001 * np.arange(n_caplets)

# Discount factors (using current rate for simplicity)
discount_factors_corp = np.exp(-current_rate * payment_dates_corp)
day_count_fractions_corp = np.full(n_caplets, 0.25)

# Price each caplet
caplet_details = []
for i in range(n_caplets):
    if times_to_expiry_corp[i] <= 0:
        price = 0
    else:
        price = black_caplet_price(
            forward_rates_corp[i],
            strike_cap,
            cap_vol,
            times_to_expiry_corp[i],
            discount_factors_corp[i],
            day_count_fractions_corp[i],
            notional_corporate,
        )
    caplet_details.append(
        {
            "period": i + 1,
            "fixing_time": times_to_expiry_corp[i],
            "forward_rate": forward_rates_corp[i],
            "price": price,
        }
    )

total_cap_cost = sum(c["price"] for c in caplet_details)

# Breakeven analysis: at what average rate above strike does cap pay for itself?
# Assume rates jump immediately to a constant level above strike


def cap_payoff_at_rate(
    constant_rate, strike, notional, n_periods, df_array, dcf_array
):
    """Calculate total cap payoff if rates stay constant above strike."""
    if constant_rate <= strike:
        return 0
    payoffs = [
        (constant_rate - strike) * notional * dcf * df
        for dcf, df in zip(dcf_array, df_array)
    ]
    return sum(payoffs)


# Find breakeven rate
from scipy.optimize import brentq


def net_value(rate):
    payoff = cap_payoff_at_rate(
        rate,
        strike_cap,
        notional_corporate,
        n_caplets,
        discount_factors_corp,
        day_count_fractions_corp,
    )
    return payoff - total_cap_cost


# Breakeven is where net value = 0
breakeven_rate = brentq(net_value, strike_cap, 0.15)