FIXED-INCOME SECURITIES
Chapter 15
Options on Futures, Caps, Floors and Swaptions
Outline • Options on Futures – Definition and Terminology – Pricing – Uses
• Caps, Floors and Collars – Definition and Terminology – Pricing – Uses
• Swaptions – Definition and Terminology – Pricing – Uses
Options on Futures Definition and Terminology • An option on a futures contract gives the buyer the right to buy from or sell to the seller… – … one unit of a designated futures, called the trading unit … – … a determined price, called the strike price (expressed in the same basis as the futures price) … – … on the maturity date of the option (European option) or at any time during the option life (American option)
• Position – A call enables the buyer to acquire a long position in the underlying futures if he exercises the option – Conversely and if exercised, the seller has a short position in the futures – A put enables the buyer to acquire a short position in the underlying futures if he exercises the option. – Conversely and if exercised, the seller has a long position in the futures
Options on Futures Definition and Terminology CME
CBOT
LIFFE
Eurodollar Options
30-Year US T-Bond Options
Long Gilt Options
5-Year Eurodollar Bundle Options
10-Year US T-Note Options
German Government Bond Options
13 Week T-Bill Options
5-Year US T-Note Options
3-Month Sterling Options
1-Month Libor Options
2-Year US T-Note Options
3-Month Euribor Options
Euro Yen Options
10-Year Agency Note Options
3-Month Euro Swiss Options
5-Year Agency Note Options
2-Year Euro Swapnote Options
Long Term Municipal Bond Index Options
5-Year Euro Swapnote Options
Mortgage Options
10-Year Euro Swapnote Options
CME, CBOT and LIFFE Options on Interest Rate Futures
Options on Futures Uses • One can use futures to hedge away interest-rate risk (see Chapter 11) • One drawback of such hedging strategies is that they do not allow firms to profit from favorable movement in prices or rates • One alternative to using futures is to use options on futures • Because T-bond futures prices are highly correlated with T-bond prices, options on T-bond futures are usually used to hedge T-bonds
Options on Futures Pricing • Black (1976) model – Does not rely on an interest rate model (futures options markets and Black formula were developed prior to good interest rate models) – Price of forward options is same as futures option interest rates are deterministic in the Black (1976) model (F is futures price, E is strike price, is volatility of futures contract) – Pricing formula for a call option (use put-call parity for put price))
[
(
Ct = N × B ( t , T ) × Ft Φ( d f ) − EΦ d f −σ T −t with :
Ft ln +σ 2 (T −t ) 2 E df = σ T −t Φ( d ) =
1 2π
d
∫e
−∞
−
x2 2
dx
)]
Options on Futures The Greeks • Compute sensitivity of option price to small changes in parameter value: the “Greeks” ∂Ct ∆t = = N × B ( t , T ) ×Φ( d f ) ∂Ft B(t , T ) ∂∆t γt = = N × Φ' ( d f ∂Ft σ T −t Ft
)
∂Ct = N × B ( t , T ) × T −t × Ft ×Φ' ( d f ∂σ ∂Ct ρt = = −(T −t ) ×Ct ∂R ( t , T −t )
υt =
)
∂Ct B ( t , T )σFt ( ) θt = = R ( t , T −t )Ct − N × Φ' d f ∂t 2 T −t
Options on Futures Time for an Example! • Example: call option on T-bond futures at t=05/04/02 – – – –
Nominal amount: N=$10,000,000 Maturity date: T=06/04/02 Strike rate: E=103.50% T-bond futures price is 102.99%, its volatility 5% and R(t,T-t) is 5%
• Black model: Ct = $37,773 • Greeks:
∆t = 3,684,851
γ t = 250,537,596 υt =1,128,501 ρt = −3,208 θt = −330,291
Options on Futures Time for an Example! • What is the approximate change in the call value if the futures price goes up to 103.09%? • Call price change is approximately the delta value multiplied by underlying price change: 3,684,851×0.1% = $3,685 • Account for convexity (gamma) to get better estimate: 3,684,851×0.1% + .5×250,537,596×(0.1%)2 = $3,810 (true price variation is $3,811) • What is the approximate change in the call value if the futures volatility goes up to 6%? • Call price change is approximately the vega multiplied by change in volatility: 1,128,501×1% = $11,285
Caps, Floors and Collars Definition • A cap is an over-the-counter contract by which the seller agrees to payoff a positive amount to the buyer of the contract if the reference rate (e.g., LIBOR) exceeds a pre-specified level called the exercise rate of the cap on given future dates • Conversely, the seller of a floor agrees to pay a positive amount to the buyer of the contract if the reference rate falls below the exercise rate on some future dates • A collar is a combination of a cap and a floor – Combination 1: buying a cap and selling a floor at the same time – Combination 2: buying a floor and selling a cap at the same time
Caps, Floors and Collars Terminology • Terminology – The notional or nominal amount is fixed in general – The reference rate is an interest rate index based for example on Libor, T-bill and T-bond yield to maturity, swap rates, ect. – The settlement frequency refers to the frequency with which the reference rate is compared to the exercise rate – The time between two payments is known as the tenor (typically monthly, quarterly, semi-annually and annually) – The maturity of caps, floors and collars can range from several months to 30 years – The premium of caps, floors and collars is expressed as a percentage of the notional amount.
Caps, Floors and Collars Pricing • On each payment date, Tj, the cap holder receives a cash-flow
[
]
C j = N × (T j −1 , T j ) ×max ( R (T j −1 , T j ) − E ,0 )
• Each of these terms can be priced using Black formula; it is typically quoted in terms of implied volatility obtained by inverting Black formula • For floors, the payoff is
[
]
F j = N × (T j −1 , T j ) ×max ( E − R (T j −1 , T j ),0 )
• The price of a collar is simply the difference between the cap price and the floor price
Caps, Floors and Collars Uses: Interest Rate Risk Hedging • Caps – A cap enables the buyer to cap the reference rate associated to a liability – Buyer of a cap is hedged against an increase in interest rates (e.g., for hedging a floating rate loan)
• Floors – A floor enables the buyer to protect the total return of assets – Buyer of a floor is hedged against a decrease in interest rates (e.g., for a floating rate asset)
• A collar (buy a floor and sell a cap) protects the rate of return of a floating rate asset while reducing the cost of the hedge • Drawback is that firm will not benefit if the reference rate goes up above the strike rate of the cap
Swaptions Definition and Terminology • Swaptions are OTC contracts allowing holder to enter a pre-specified swap contract on a pre-specified date • There are two kinds of European swaptions – Receiver option on swap gives the buyer the right to receive the fixed leg of the swap – Payer option on swap gives the buyer the right to pay the fixed leg of the swap
• Terminology – Exercise or strike rate is the specified fixed rate at which buyer can enter into the swap – Maturity or expiry date is the date when the option can be exercised (from several months to ten years) – Premium expressed as a percentage of the principal amount of the swap (computed using a suitable extension of Black’s model, and quoted in terms of implied volatility) – Underlying asset is most commonly a plain vanilla swap whose maturity can range from one year to 30 years
Swaptions Uses • Like caps, floors and collars, swaptions are interest rate derivatives designed to hedge the interest rate risk • A payer swaption can be used in two ways – It enables a firm to fix a maximum limit to its floating rate debt – It enables an investor to transform its fixed-rate assets into floating-rate assets to benefit from a rise in interest rates
• A receiver swaption can also be used in two ways – It enables a firm to transform its fixed rate debt into a floating rate debt in a context of a decrease in interest rates – It enables an investor to protect its floating rate investment