Black model

Modelling assumptions

Under a risk-neutral pricing measure the futures or forward price process ${\displaystyle (F_{t})_{0\leq t\leq T}}$ is modelled as a geometric Brownian motion with constant volatility and zero drift. In differential form this can be written as

$${\displaystyle \mathrm {d} F_{t}=\sigma F_{t}\,\mathrm {d} W_{t},\quad 0\leq t\leq T,}$$

where ${\displaystyle \sigma }$ is the volatility parameter and ${\displaystyle (W_{t})}$ is a standard Brownian motion.^[1][4] This implies that ${\displaystyle \ln F_{T}}$ is normally distributed and that the discounted value of the option payoff can be computed by taking expectations with respect to this log-normal distribution.

Derivation

Let ${\displaystyle X_{T}}$ denote a payoff at time ${\displaystyle T}$. Under the risk-neutral measure the time zero price of this payoff is given by the discounted expectation

$${\displaystyle {\text{Price}}_{0}=e^{-rT}\mathbb {E} [X_{T}],}$$

where ${\displaystyle \mathbb {E} }$ denotes expectation.^[3] For a European call option on a futures contract with expiry ${\displaystyle T}$ and strike ${\displaystyle K}$ the payoff is ${\displaystyle (F_{T}-K)^{+}}$. Substituting this payoff into the pricing expression and using the log-normal distribution of ${\displaystyle F_{T}}$ leads, after integration, to the closed form formulas for the call and put prices.^[4]

An alternative derivation views the call on a futures or forward contract as an option to exchange one asset for another. Under this approach, the option holder has the right at expiry to receive the futures payoff and pay the fixed amount ${\displaystyle K}$, which can be represented as an exchange between a position linked to the futures contract and a zero coupon bond.^[11] Under the log-normal assumptions of the Black model and with deterministic interest rates, this exchange option can be valued using Margrabe's formula, and the result reduces to the Black pricing formula for options on forwards and futures.^[3][12]

Futures and commodity options

In commodity and financial futures markets the Black model underlies the pricing of exchange traded and over the counter options on futures. Examples include options on crude oil futures, metal futures and equity index futures.^[1][13] Exchanges and dealers often treat the volatility parameter in the Black formula as the primary quoted quantity and convert between option prices and so called Black implied volatilities when comparing contracts with different strikes and maturities.^[14] Volatility surfaces built from these implied volatilities provide a way to summarise option prices across a range of expiries and strike levels.

Caps, floors and bond options

In fixed income markets the Black framework is used for bond options and for caps and floors. A cap or floor can be decomposed into individual caplets or floorlets. Each component has a payoff that depends on a single forward interest rate over a specified accrual period.^[4][7] Under the assumption that the relevant forward rate is log-normal under an appropriate pricing measure each caplet or floorlet is valued by applying the Black formula to that forward rate and scaling by the notional and accrual factor.^[15] Market participants quote cap and floor prices in terms of Black implied volatilities for standard maturities and strike levels.

Swaptions and forward rate models

European swaptions are treated in a similar spirit. The underlying variable is a forward swap rate and prices for swaptions across different maturities and tenors are commonly quoted in terms of Black implied volatilities.^[6][15] In log-normal forward rate models the Black formula remains the basic building block for valuing individual caplets, floorlets and swaptions. The model specifies joint dynamics for a family of forward rates and is calibrated so that it reproduces observed Black or normal implied volatility surfaces used in swaption and cap markets.^[8][16]

Inflation-linked caps and floors on RPI for pension increases

In the United Kingdom many defined benefit pension schemes provide increases that are linked to the RPI or CPI, subject to an annual floor and cap, a structure known as limited price indexation and often written as LPI(${\displaystyle a}$,${\displaystyle b}$). A typical benefit might promise annual increases in line with RPI subject to a floor of ${\displaystyle 0\%}$ and a cap of ${\displaystyle 5\%}$, denoted LPI(0,5).^[17][18] These guarantees can be valued or hedged using inflation derivatives such as RPI swaps and inflation caps and floors.^[19][20]

For a single year let ${\displaystyle I}$ denote the random annual RPI inflation rate expressed as a proportion. With a floor ${\displaystyle a}$ and cap ${\displaystyle b}$ on the annual increase the limited price indexation rate can be written as

$${\displaystyle L(I)=\min {\bigl (}\max(I,a),b{\bigr )}.}$$

Using the notation ${\displaystyle x^{+}=\max(x,0)}$ this decomposes as

$${\displaystyle L(I)=I-(I-b)^{+}+(a-I)^{+}.}$$

The second term removes outcomes where inflation is above the cap, and the third term replaces very low inflation outcomes with the floor level.^[19][21]

To obtain a tractable formula, assume that under a pricing measure the one year inflation rate ${\displaystyle I}$ is strictly positive and log normal with forward mean ${\displaystyle F}$, volatility ${\displaystyle \sigma }$ and time to maturity ${\displaystyle T}$. This is the same log normal setup used earlier for the Black formula, with ${\displaystyle F}$ playing the role of the forward price and the cap and floor strikes playing the role of the option strike.^[19][20] Under this assumption

$${\displaystyle \mathbb {E} [L(I)]=F-\mathbb {E} {\bigl [}(I-b)^{+}{\bigr ]}+\mathbb {E} {\bigl [}(a-I)^{+}{\bigr ]}.}$$

For a strike ${\displaystyle K}$ define

$${\displaystyle d_{1}^{(K)}={\frac {\ln(F/K)+{\tfrac {1}{2}}\sigma ^{2}T}{\sigma {\sqrt {T}}}},\quad d_{2}^{(K)}=d_{1}^{(K)}-\sigma {\sqrt {T}},}$$

and let ${\displaystyle N}$ denote the cumulative distribution function of a standard normal random variable. Ignoring discounting over a single year, the corresponding call and put expectations are

$${\displaystyle \mathbb {E} {\bigl [}(I-K)^{+}{\bigr ]}=FN{\bigl (}d_{1}^{(K)}{\bigr )}-KN{\bigl (}d_{2}^{(K)}{\bigr )},}$$

$${\displaystyle \mathbb {E} {\bigl [}(K-I)^{+}{\bigr ]}=KN{\bigl (}-d_{2}^{(K)}{\bigr )}-FN{\bigl (}-d_{1}^{(K)}{\bigr )},}$$

which mirror the Black call and put formulas in the formula section, with the exponential discount factor omitted for simplicity.^[3]

For general LPI(${\displaystyle a}$,${\displaystyle b}$) this gives

$${\displaystyle \mathbb {E} [L(I)]=F-{\bigl (}FN{\bigl (}d_{1}^{(b)}{\bigr )}-bN{\bigl (}d_{2}^{(b)}{\bigr )}{\bigr )}+{\bigl (}aN{\bigl (}-d_{2}^{(a)}{\bigr )}-FN{\bigl (}-d_{1}^{(a)}{\bigr )}{\bigr )},}$$

where ${\displaystyle d_{1}^{(a)}}$, ${\displaystyle d_{2}^{(a)}}$ use strike ${\displaystyle a}$ and ${\displaystyle d_{1}^{(b)}}$, ${\displaystyle d_{2}^{(b)}}$ use strike ${\displaystyle b}$.

As an illustration consider a one year LPI(0,5) structure with a forward RPI inflation rate of ${\displaystyle 3.5\%}$, so ${\displaystyle F=0.035}$, a cap ${\displaystyle b=0.05}$, a floor ${\displaystyle a=0}$, volatility parameter ${\displaystyle \sigma =0.02}$ and ${\displaystyle T=1}$. Under the log normal model the floor term is zero, since ${\displaystyle I}$ is strictly positive, and

$${\displaystyle \mathbb {E} [L(I)]=F-{\bigl (}FN{\bigl (}d_{1}^{(0.05)}{\bigr )}-0.05N{\bigl (}d_{2}^{(0.05)}{\bigr )}{\bigr )}.}$$

With ${\displaystyle F}$ substantially below the cap relative to the assumed log normal volatility, the cap rarely bites, the call term is very small, and the expected LPI(0,5) increase is therefore very close to the forward inflation rate of ${\displaystyle 3.5\%}$.

It is common to approximate the one year inflation rate by a normal random variable, in line with the Bachelier model, especially when negative inflation is considered.^[22][23] Under that normal model, with the same central assumption ${\displaystyle F=3.5\%}$, volatility ${\displaystyle \sigma =2\%}$ and LPI(0,5), the expectation ${\displaystyle \mathbb {E} [L(I)]}$ is lower, approximately ${\displaystyle 3.27\%}$, because both the cap at ${\displaystyle 5\%}$ and the floor at ${\displaystyle 0\%}$ are active over a wider range of outcomes.^{[24][25][26][27]}