Cap and Floor Pricing¶

Interest rate caps and floors are among the most actively traded OTC derivatives. A cap protects a floating-rate borrower against rising rates by paying the difference whenever the reference rate exceeds a strike, while a floor protects a lender against falling rates. Since a cap is a portfolio of individual caplets, and each caplet in the Hull-White model reduces to a put on a zero-coupon bond, the entire cap can be priced analytically. This section derives the cap and floor pricing formulas and compares the Hull-White result with Black's formula.

Cap and Floor Structure¶

An interest rate cap on a notional $N$ with strike $K$ and payment dates $T_1, T_2, \ldots, T_n$ consists of $n$ individual caplets. Each caplet pays at $T_k$ based on the floating rate observed at $T_{k-1}$.

Definition: Cap and Floor

\[\begin{array}{lllll} \displaystyle \text{Cap}(t_0) &=& \displaystyle\sum_{k=1}^{n} \text{Caplet}_k(t_0) \\[8pt] \displaystyle \text{Floor}(t_0) &=& \displaystyle\sum_{k=1}^{n} \text{Floorlet}_k(t_0) \end{array}\]

where each caplet and floorlet has payoff at $T_k$:

\[\begin{array}{lllll} \displaystyle \text{Caplet}_k(T_k) &=& N\tau_k\max(l_k(T_{k-1}) - K,\; 0) \\[4pt] \displaystyle \text{Floorlet}_k(T_k) &=& N\tau_k\max(K - l_k(T_{k-1}),\; 0) \end{array}\]

Here $\tau_k = T_k - T_{k-1}$ is the accrual fraction and $l_k(T_{k-1}) = l(T_{k-1}; T_{k-1}, T_k)$ is the simply compounded forward rate set at $T_{k-1}$.

Hull-White Caplet as a ZCB Put¶

The key insight for pricing individual caplets in the Hull-White model is that each caplet is equivalent to a put option on a zero-coupon bond. This equivalence allows direct application of the Hull-White ZCB option formula.

Theorem: Caplet-ZCB Put Equivalence

A caplet with reset date $T_{k-1}$, payment date $T_k$, strike $K$, and notional $N$ equals

\[ \text{Caplet}_k(t_0) = \hat{N}\,V_p^{\text{ZCB}}\!\left(t_0,\, T_{k-1},\, T_k;\, \hat{K}\right) \]

where $\hat{N} = N(1 + \tau_k K)$ and $\hat{K} = \frac{1}{1 + \tau_k K}$.

Proof

Starting from the risk-neutral pricing formula:

\[\begin{array}{lllll} \displaystyle \text{Caplet}_k(t_0) &=&\displaystyle \mathbb{E}^{\mathbb{Q}}\!\left[\frac{M(t_0)}{M(T_k)}\,N\tau_k\max(l_k(T_{k-1}) - K, 0)\,\Big|\,\mathcal{F}(t_0)\right] \end{array}\]

Using the tower property and the relation $l_k(T_{k-1}) = \frac{1}{\tau_k}\!\left(\frac{1}{P(T_{k-1}, T_k)} - 1\right)$:

\[\begin{array}{lllll} \displaystyle &=&\displaystyle \mathbb{E}^{\mathbb{Q}}\!\left[\frac{M(t_0)}{M(T_{k-1})}\,N\max\!\left(\frac{1}{P(T_{k-1}, T_k)} - (1 + \tau_k K),\; 0\right)P(T_{k-1}, T_k)\,\Big|\,\mathcal{F}(t_0)\right] \end{array}\]

Rearranging:

\[\begin{array}{lllll} \displaystyle &=&\displaystyle \mathbb{E}^{\mathbb{Q}}\!\left[\frac{M(t_0)}{M(T_{k-1})}\,N(1 + \tau_k K)\max\!\left(\frac{1}{1 + \tau_k K} - P(T_{k-1}, T_k),\; 0\right)\,\Big|\,\mathcal{F}(t_0)\right] \\[6pt] &=&\displaystyle \hat{N}\,V_p^{\text{ZCB}}(t_0,\, T_{k-1},\, T_k;\, \hat{K}) \end{array}\]

$\square$

Similarly, a floorlet equals a scaled ZCB call:

\[ \text{Floorlet}_k(t_0) = \hat{N}\,V_c^{\text{ZCB}}\!\left(t_0,\, T_{k-1},\, T_k;\, \hat{K}\right) \]

Hull-White Cap Formula¶

Combining the caplet decomposition with the Hull-White ZCB option pricing formula, the cap price is:

\[ \text{Cap}(t_0) = \sum_{k=1}^{n} \hat{N}_k\,V_p^{\text{ZCB}}\!\left(t_0,\, T_{k-1},\, T_k;\, \hat{K}_k\right) \]

where for each caplet $k$:

\[\begin{array}{lllll} \hat{N}_k &=& N(1 + \tau_k K) \\[4pt] \hat{K}_k &=& \frac{1}{1 + \tau_k K} \end{array}\]

Each ZCB put $V_p^{\text{ZCB}}$ uses the Hull-White closed-form formula with option maturity $T_{k-1}$, bond maturity $T_k$, and strike $\hat{K}_k$.

Implementation Note

Unlike Jamshidian's trick for coupon bond options, no root-finding for $r^*$ is needed. Each caplet is independently priced as a ZCB put, and the cap price is simply the sum.

Cap-Floor Parity¶

The difference between a cap and a floor with the same strike equals the value of a payer swap:

\[ \text{Cap}(t_0) - \text{Floor}(t_0) = N\!\left(P(t_0, T_0) - P(t_0, T_n) - K\sum_{k=1}^{n}\tau_k\,P(t_0, T_k)\right) = \text{IRS}^{\text{Payer}}(t_0) \]

Proof

For each reset period, the caplet-floorlet parity gives

\[ \text{Caplet}_k - \text{Floorlet}_k = N\tau_k\,P(t_0, T_k)\,(l_k(t_0) - K) \]

Summing over all periods:

\[\begin{array}{lllll} \displaystyle \text{Cap} - \text{Floor} &=&\displaystyle N\sum_{k=1}^{n}\tau_k\,P(t_0, T_k)(l_k(t_0) - K) \;=\; N\!\left(P(t_0, T_0) - P(t_0, T_n)\right) - NK\sum_{k=1}^{n}\tau_k\,P(t_0, T_k) \end{array}\]

This is the payer swap value. $\square$

Comparison with Black's Formula¶

In the market convention, caplets are quoted using Black's formula with an implied volatility $\sigma_k^{\text{Black}}$:

\[ \text{Caplet}_k^{\text{Black}}(t_0) = N\tau_k\,P(t_0, T_k)\!\left[l_k(t_0)\,\Phi(d_1) - K\,\Phi(d_2)\right] \]

where

\[\begin{array}{lllll} \displaystyle d_1 &=& \displaystyle\frac{\log(l_k(t_0)/K) + \frac{1}{2}v_k^2}{v_k} \\[8pt] \displaystyle d_2 &=& d_1 - v_k \\[4pt] v_k &=& \sigma_k^{\text{Black}}\sqrt{T_{k-1} - t_0} \end{array}\]

Black's formula assumes the forward rate $l_k(t)$ is lognormal under $\mathbb{Q}^{T_k}$, while the Hull-White model produces a Gaussian (normal) short rate distribution. The two approaches give different implied volatility structures:

Black's formula: Constant implied volatility across strikes produces a flat volatility smile.
Hull-White: Since bond prices are exponential in the Gaussian $r$, the model generates implied volatilities that vary with strike, producing a volatility skew.

Proposition: Hull-White Implied Volatility

For at-the-money caplets, the Hull-White and Black formulas produce similar prices. Away from ATM, the Hull-White model generates a systematic skew because the lognormal distribution (Black) and the distribution implied by the exponential of a Gaussian (Hull-White) have different tail behavior.

Flat Volatility vs Spot Volatility¶

Cap prices are often quoted using two volatility conventions:

Flat (running) volatility: A single volatility $\sigma^{\text{flat}}$ applied uniformly to all caplets in the cap. The flat vol equates the cap price to $\sum_k \text{Caplet}_k^{\text{Black}}(\sigma^{\text{flat}})$.
Spot volatility: Each caplet $k$ uses its own volatility $\sigma_k^{\text{spot}}$. These are stripped from market cap quotes.

The Hull-White model with parameters $(\lambda, \sigma)$ implies a term structure of spot volatilities that is typically monotonically decreasing with maturity, consistent with the empirical pattern of a downward-sloping cap volatility curve.

Numerical Example¶

Consider a 5-year annual cap with $N = \$1{,}000{,}000$, strike $K = 0.04$, annual resets at $T_0 = 0, T_1 = 1, \ldots, T_5 = 5$, and Hull-White parameters $\lambda = 0.05$, $\sigma = 0.01$.

For each caplet, $\hat{N}_k = 1{,}000{,}000 \times (1 + 0.04) = 1{,}040{,}000$ and $\hat{K}_k = 1/1.04 \approx 0.9615$.

The cap price equals the sum of five ZCB put prices:

\[ \text{Cap}(0) = \sum_{k=1}^{5} 1{,}040{,}000 \times V_p^{\text{ZCB}}(0,\, T_{k-1},\, T_k;\, 0.9615) \]

Each ZCB put is priced using the Hull-White closed-form formula. The total cap price can then be backed out to a flat Black volatility for quoting purposes.

python def price_cap(hw, N, K, T_reset_dates, T_payment_dates): """Price an interest rate cap in the Hull-White model.""" cap_price = 0.0 for T_reset, T_pay in zip(T_reset_dates, T_payment_dates): tau_k = T_pay - T_reset N_hat = N * (1 + tau_k * K) K_hat = 1.0 / (1 + tau_k * K) zbp = hw.compute_ZCB_Option_Price( K_hat, T_reset, T_pay, CP=OptionType.PUT ) cap_price += N_hat * zbp return cap_price

Summary¶

Interest rate caps and floors are portfolios of caplets and floorlets. In the Hull-White model, each caplet reduces to a put on a zero-coupon bond via the equivalence $\text{Caplet}_k = \hat{N}\,V_p^{\text{ZCB}}(t_0, T_{k-1}, T_k; \hat{K})$, enabling fully analytic pricing. The cap-floor parity $\text{Cap} - \text{Floor} = \text{Payer Swap}$ holds model-independently. Compared to Black's formula, the Hull-White model generates a volatility skew due to its Gaussian short rate distribution, providing a richer structure than the flat smile implied by lognormal forward rates.

Exercises¶

Exercise 1. For a 3-year annual cap with $N = \$1{,}000{,}000$, $K = 0.05$, compute $\hat{N}_k$ and $\hat{K}_k$ for each of the three caplets. Describe the ZCB put that needs to be priced for each caplet.

Solution to Exercise 1

For a 3-year annual cap with $N = \$1{,}000{,}000$, $K = 0.05$, the payment dates are $T_1 = 1$, $T_2 = 2$, $T_3 = 3$ with reset dates $T_0 = 0$, $T_1 = 1$, $T_2 = 2$. Each accrual fraction is $\tau_k = 1$.

Caplet 1 (reset at $T_0 = 0$, pay at $T_1 = 1$):

\[ \hat{N}_1 = 1{,}000{,}000 \times (1 + 1 \times 0.05) = 1{,}050{,}000 \]

\[ \hat{K}_1 = \frac{1}{1 + 1 \times 0.05} = \frac{1}{1.05} \approx 0.95238 \]

This caplet requires pricing a ZCB put with option maturity $T_0 = 0$, bond maturity $T_1 = 1$, and strike $0.95238$.

Caplet 2 (reset at $T_1 = 1$, pay at $T_2 = 2$):

\[ \hat{N}_2 = 1{,}050{,}000, \quad \hat{K}_2 = 0.95238 \]

ZCB put with option maturity $T_1 = 1$, bond maturity $T_2 = 2$, strike $0.95238$.

Caplet 3 (reset at $T_2 = 2$, pay at $T_3 = 3$):

\[ \hat{N}_3 = 1{,}050{,}000, \quad \hat{K}_3 = 0.95238 \]

ZCB put with option maturity $T_2 = 2$, bond maturity $T_3 = 3$, strike $0.95238$.

Since $\tau_k$ and $K$ are the same for all three caplets, $\hat{N}_k$ and $\hat{K}_k$ are identical. Each caplet is priced via the Hull-White ZCB put formula with the appropriate option and bond maturities.

Exercise 2. Derive the floorlet-ZCB call equivalence: show that $\text{Floorlet}_k(t_0) = \hat{N}\,V_c^{\text{ZCB}}(t_0, T_{k-1}, T_k; \hat{K})$ by starting from the floorlet payoff and applying the same algebra as the caplet derivation.

Solution to Exercise 2

The floorlet payoff at $T_k$ is $N\tau_k\max(K - l_k(T_{k-1}), 0)$. Starting from the risk-neutral pricing formula:

\[ \text{Floorlet}_k(t_0) = \mathbb{E}^{\mathbb{Q}}\!\left[\frac{M(t_0)}{M(T_k)}\,N\tau_k\max(K - l_k(T_{k-1}), 0)\,\Big|\,\mathcal{F}(t_0)\right] \]

Using the tower property to condition on $\mathcal{F}(T_{k-1})$ and the relation $\mathbb{E}^{\mathbb{Q}}\!\left[\frac{1}{M(T_k)}\Big|\mathcal{F}(T_{k-1})\right] = \frac{P(T_{k-1},T_k)}{M(T_{k-1})}$:

\[ = \mathbb{E}^{\mathbb{Q}}\!\left[\frac{M(t_0)}{M(T_{k-1})}\,N\tau_k\max(K - l_k(T_{k-1}), 0)\,P(T_{k-1},T_k)\,\Big|\,\mathcal{F}(t_0)\right] \]

Substituting $l_k(T_{k-1}) = \frac{1}{\tau_k}\!\left(\frac{1}{P(T_{k-1},T_k)} - 1\right)$:

\[ N\tau_k\max\!\left(K - \frac{1}{\tau_k}\!\left(\frac{1}{P(T_{k-1},T_k)} - 1\right), 0\right)P(T_{k-1},T_k) = N\max\!\left((1+\tau_k K)P(T_{k-1},T_k) - 1, 0\right) \]

Factoring:

\[ = N(1+\tau_k K)\max\!\left(P(T_{k-1},T_k) - \frac{1}{1+\tau_k K}, 0\right) = \hat{N}\max(P(T_{k-1},T_k) - \hat{K}, 0) \]

Therefore:

\[ \text{Floorlet}_k(t_0) = \hat{N}\,\mathbb{E}^{\mathbb{Q}}\!\left[\frac{M(t_0)}{M(T_{k-1})}\max(P(T_{k-1},T_k) - \hat{K}, 0)\,\Big|\,\mathcal{F}(t_0)\right] = \hat{N}\,V_c^{\text{ZCB}}(t_0, T_{k-1}, T_k; \hat{K}) \]

which is $\hat{N}$ times a European call on the ZCB.

Exercise 3. Verify the cap-floor parity $\text{Cap} - \text{Floor} = \text{IRS}^{\text{Payer}}$ by summing the individual caplet-floorlet parities. Explain why this relationship is model-independent (holds for any arbitrage-free model).

Solution to Exercise 3

Verification: For each $k$, the caplet-floorlet parity is:

\[ \text{Caplet}_k - \text{Floorlet}_k = N\tau_k\,P(t_0, T_k)(l_k(t_0) - K) \]

This follows from $\max(x,0) - \max(-x,0) = x$ applied to $x = l_k(T_{k-1}) - K$, plus the forward measure pricing. Summing over $k = 1, \ldots, n$:

\[ \text{Cap} - \text{Floor} = N\sum_{k=1}^n \tau_k\,P(t_0, T_k)(l_k(t_0) - K) \]

Using $\tau_k l_k(t_0) = \frac{P(t_0,T_{k-1})}{P(t_0,T_k)} - 1$, so $\tau_k P(t_0,T_k) l_k(t_0) = P(t_0,T_{k-1}) - P(t_0,T_k)$:

\[ = N\sum_{k=1}^n\!\left(P(t_0,T_{k-1}) - P(t_0,T_k)\right) - NK\sum_{k=1}^n \tau_k P(t_0,T_k) \]

The first sum telescopes: $\sum_{k=1}^n (P(t_0,T_{k-1}) - P(t_0,T_k)) = P(t_0,T_0) - P(t_0,T_n)$.

\[ \text{Cap} - \text{Floor} = N(P(t_0,T_0) - P(t_0,T_n)) - NK\sum_{k=1}^n \tau_k P(t_0,T_k) = \text{IRS}^{\text{Payer}}(t_0) \]

Model independence: The identity $\max(x,0) - \max(-x,0) = x$ holds pathwise for every realization. Therefore the parity involves only the unconditional forward value of $l_k(T_{k-1}) - K$, which equals $l_k(t_0) - K$ under any arbitrage-free model (by the martingale property of $l_k$ under $\mathbb{Q}^{T_k}$). No distributional assumptions are needed.

Exercise 4. The Hull-White model implies a decreasing term structure of cap volatilities. Explain why: as the caplet maturity increases, what happens to the effective volatility of the underlying forward rate in the Hull-White model?

Solution to Exercise 4

In the Hull-White model, the conditional variance of $r(T_{k-1})$ given $r(t_0)$ is:

\[ v_r^2(t_0, T_{k-1}) = \frac{\sigma^2}{2\lambda}(1 - e^{-2\lambda(T_{k-1} - t_0)}) \]

The effective volatility of the bond price $P(T_{k-1}, T_k) = e^{A(\tau_k) + B(\tau_k)r(T_{k-1})}$ depends on $|B(\tau_k)|\,v_r(t_0, T_{k-1})$. While $v_r$ increases with $T_{k-1}$, the mean reversion causes it to saturate at $\sqrt{\sigma^2/(2\lambda)}$ for large maturities.

The key effect is that the forward LIBOR volatility is proportional to $|B(\tau_k)|\,v_r / l_k(t_0)$ in relative terms. As maturity increases:

The variance $v_r^2$ grows but saturates due to mean reversion.
The forward rate $l_k(t_0)$ remains at a similar level (determined by the yield curve).

The result is that the Black implied volatility, which measures relative (percentage) volatility of the forward rate, decreases with maturity. For short maturities, $v_r^2 \approx \sigma^2(T_{k-1} - t_0)$ grows linearly, but for long maturities the saturation effect dominates. This produces the empirically observed decreasing term structure of cap volatilities.

Exercise 5. Explain the difference between flat (running) volatility and spot volatility for cap pricing. Given a flat cap volatility of 20% for a 5-year cap, describe how you would strip the individual spot volatilities $\sigma_k^{\text{spot}}$ for each caplet.

Solution to Exercise 5

Flat (running) volatility $\sigma^{\text{flat}}$ is a single number applied to all caplets in a cap simultaneously. The flat vol is defined implicitly by:

\[ \text{Cap}^{\text{market}}(t_0) = \sum_{k=1}^n \text{Caplet}_k^{\text{Black}}(\sigma^{\text{flat}}) \]

This is a convenient quoting convention but does not reflect the different volatilities of individual forward rates.

Spot volatility $\sigma_k^{\text{spot}}$ is specific to each caplet $k$, so that $\text{Caplet}_k^{\text{market}} = \text{Caplet}_k^{\text{Black}}(\sigma_k^{\text{spot}})$.

Stripping procedure for spot volatilities from a flat vol of 20% for a 5-year cap:

The 1-year cap consists of a single caplet. Its flat vol equals its spot vol: $\sigma_1^{\text{spot}} = \sigma^{\text{flat}}_{1Y}$.
For the 2-year cap, $\text{Cap}_{2Y} = \text{Caplet}_1(\sigma_1^{\text{spot}}) + \text{Caplet}_2(\sigma_2^{\text{spot}})$. Since $\sigma_1^{\text{spot}}$ is known, solve for $\sigma_2^{\text{spot}}$.
Continue bootstrapping: at each step $k$, the first $k-1$ spot vols are known. Solve for $\sigma_k^{\text{spot}}$ from the $k$-year cap price.

In practice, with only the 5-year flat vol of 20%, one needs additional market data (caps at intermediate maturities) to uniquely strip all five spot volatilities. With a single flat vol, an assumption about the term structure shape (e.g., flat or interpolated from other cap quotes) is required.

Exercise 6. Using the Python code provided, modify the price_cap function to also return the individual caplet prices. For Hull-White parameters $\lambda = 0.05$ and $\sigma = 0.01$ with $K = 0.04$, which caplet contributes the most to the total cap value?

Solution to Exercise 6

Modify the function to return both total price and individual prices:

python def price_cap_detailed(hw, N, K, T_reset_dates, T_payment_dates): """Price an interest rate cap, returning total and individual caplet prices.""" caplet_prices = [] for T_reset, T_pay in zip(T_reset_dates, T_payment_dates): tau_k = T_pay - T_reset N_hat = N * (1 + tau_k * K) K_hat = 1.0 / (1 + tau_k * K) zbp = hw.compute_ZCB_Option_Price( K_hat, T_reset, T_pay, CP=OptionType.PUT ) caplet_prices.append(N_hat * zbp) return sum(caplet_prices), caplet_prices

For Hull-White parameters $\lambda = 0.05$, $\sigma = 0.01$, $K = 0.04$:

Short-dated caplets (e.g., $k = 1$) have small variance $v_r^2$ because $T_{k-1} - t_0$ is small, leading to small ZCB put prices.
Intermediate caplets (e.g., $k = 3$ or $k = 4$) have the largest prices because $v_r^2$ has grown substantially but the ZCB option is still well within the range where the Gaussian distribution produces significant probability mass beyond the strike.
Long-dated caplets see diminishing marginal increases in $v_r^2$ due to mean reversion saturation.

Typically, for moderate mean reversion ($\lambda = 0.05$), the caplet contributing the most is an intermediate-maturity one (around year 3-4 for a 5-year cap), because the variance has grown enough to make the option valuable, but saturation has not yet fully kicked in.

Exercise 7. Compare the cap prices produced by the Hull-White model versus Black's formula for at-the-money and deep out-of-the-money strikes. At which strikes do the two models diverge most, and why?

Solution to Exercise 7

At-the-money (ATM) strikes: When $K \approx l_k(t_0)$ (the current forward rate), both models produce very similar prices. At ATM, option prices are dominated by at-the-money volatility, and the differences between the lognormal (Black) and Gaussian-exponential (Hull-White) distributions are minimized. The two distributions agree well near their central values.

Deep out-of-the-money (OTM) strikes: For $K \gg l_k(t_0)$, the caplet price depends on the upper tail of the forward rate distribution. The lognormal distribution (Black) has a heavier right tail than the distribution implied by the Hull-White model. Conversely, for $K \ll l_k(t_0)$ (deep in-the-money caplets or equivalently deep OTM floorlets), the lower tails differ.

The two models diverge most at strikes far from ATM because:

Tail behavior differs: The lognormal distribution has a right-skewed, heavy-tailed shape, while the Hull-White distribution of $l_k$ (being a function of a Gaussian variable) has different tail characteristics.
Symmetry vs asymmetry: The Gaussian $r$ produces a distribution of $1/P$ that is right-skewed but in a different way from the lognormal. For low strikes, the Hull-White model may assign more probability (since $r$ can go negative, making $P > 1$ and $l_k < 0$ possible), while Black's formula confines $l_k > 0$.

In practice, the largest price differences appear at deep OTM strikes where the tail probabilities are most sensitive to the distributional assumption.