Cap

Um Cap é geralmente um limite (tecto). Por exemplo, muitas hipotecas incluem um cap (limite) na taxa de juro, acima da qual esta não poderá variar. Ou seja, se a taxa de juro indexante fixar acima do cap, a taxa de juro aplicada ao empréstimo será o cap.

Também acontece produtos estruturados, nomeadamente de capital garantido, estabelecerem um cap para a rendibilidade, acima do qual o investidor não beneficia de valorização adicional. Se o cap for estabelecido por exemplo a 15%, atingidos os 15% o produto não valoriza mais mesmo que o subjacente continue a valorizar.

Isto permite produzir produtos estruturados mais baratos, pois na prática o investidor está a vender uma opção que dará a quem monta o produto a rendibilidade acima do cap, opção essa que tem valor.

Cap de taxa de juro

Existe um outro sentido. Um Cap de taxa de juro também pode ser um derivado que paga quando a taxa de juro ultrapassa o cap (strike). Um exemplo seria receber juros para cada mês que a Euribor excede 5%. Esta é a opção que faz valer o sentido anterior, pois estabelece um limite superior para a taxa de juro para quem a compra.

Um cap de taxa de juro pode ser analisado como uma série de calls europeias, ou "caplets", que existe para cada período em que o acordo de cap está em vigor.

A fórmula do payoff de um caplet com taxa e strike é:

Onde é o montante do contrato e é a contagem de dias (como fracção de um ano) à qual L se aplica. Por exemplo, suponhamos que detemos um caplet na USD Libor 6 meses, com expiração em 1 de Fevereiro de 2007 e strike a 2.5% para um montante de 1 milhão de dólares. Se a USD Libor faz 3% em 1 de Fevereiro, receberia 1m*0.5*max(0.03-0.025,0) = $2500. Por costume o pagamento é feito no final do período da taxa, neste caso 1 de Agosto.

Floor de taxa de juro

Um floor de taxa de juro é um inverso do cap, e corresponde a uma série de puts europeias ou floorlets com uma determinada taxa de referência, usualmente a LIBOR. O comprador de um floot recebe dinheiro se na maturidade de qualquer dos floorlets, a taxa de referência fixar abaixo do strike do floor.

Valuation of interest rate caps

Black

The simplest and most common valuation of interest rate caplets is via the Black model. Under this model we assume that the underlying rate is distributed log-normally with volatility <math>\sigma</math>. Under this model, a caplet on a LIBOR expiring at t and paying at T has present value

<math> V = P(0,T)(FN(d_1) - KN(d_2))</math>

where

P(0,T) is today's discount factor for T

F is the forward price of the rate. For LIBOR rates this is equal to <math> {1\over \alpha }\left(\frac{P(0,t)}{P(0,T)} - 1\right)</math>

K is the strike

<math>d_1 = \frac{\ln(F/K) + 0.5 \sigma^2t}{\sigma\sqrt{t}}</math>

and

<math>d_2 = d_1 - \sigma\sqrt{t}</math>

Notice that there is a one-to-one mapping between the volatility and the present value of the option. Because all the other terms arising in the equation are indisputable, there is no ambiguity in quoting the price of a caplet simply by quoting its volatility. This is what happens in the market. The volatility is known as the "Black vol" or implied vol.

As a bond put

It can be shown that a cap on a LIBOR from t to T is equivalent to a multiple of a t-maturity put on a T-maturity bond. Thus if we have an interest rate model in which we are able to value bond puts, we can value interest rate caps. Similarly a floor is equivalent to a certain bond call. Several popular short rate models, such as the Hull-White model have this degree of tractability. Thus we can value caps and floors in those models..

What about Collars?

Interest rate collar

…the simultaneous purchase of an interest rate cap and sale of an interest rate floor on the same index for the same maturity and notional principal amount.

• • The cap rate is set above the floor rate.
• The objective of the buyer of a collar is to protect against rising interest rates.
  • The purchase of the cap protects against rising rates while the sale of the floor generates premium income.
• A collar creates a band within which the buyer’s effective interest rate fluctuates

And Reverse Collars?

…buying an interest rate floor and simultaneously selling an interest rate cap.

• The objective is to protect the bank from falling interest rates.
• The buyer selects the index rate and matches the maturity and notional principal amounts for the floor and cap.
• Buyers can construct zero cost reverse collars when it is possible to find floor and cap rates with the same premiums that provide an acceptable band.

The size of cap and floor premiums are determined by a wide range of factors

• The relationship between the strike rate and the prevailing 3-month LIBOR
  • premiums are highest for in the money options and lower for at the money and out of the money options
• Premiums increase with maturity.
  • The option seller must be compensated more for committing to a fixed-rate for a longer period of time.
• Prevailing economic conditions, the shape of the yield curve, and the volatility of interest rates.
  • upsloping yield curve -- caps will be more expensive than floors.
  • the steeper is the slope of the yield curve, ceteris paribus, the greater are the cap premiums.
  • floor premiums reveal the opposite relationship.

Volatilidades implícitas

• An important consideration is cap and floor volatilities. Caps consist of caplets with volatilities dependent on the corresponding forward LIBOR rate. But caps can be represented by a "flat volatility", so the net of the caplets still comes out to be the same. (15%,20%,....,12%) ---> (16.5%,16.5%,....,16.5%)

• • So one cap can be priced at one vol.

• • Another important intuition is that caps and floors are duals. Cap-Floor = Swap.

• • Caps and floors have the same implied vol too for a given strike.

Imagine a cap with 20% and floor with 30%. Long cap, short floor gives a swap with no vol. Now, interchange the vols. Cap price goes up, floor price goes down. But the net price of the swap is unchanged. So, if a cap has x vol, floor is forced to have x vol else you have arbitrage.

Ver também

• Swap de taxa de juro

Links relevantes

• Hedging básico de renda fixa - Artigo no Financial-edu.com.