The Term Structure of Interest Rates Chapter 3 報告者 張富昇 陳郁婷 指導教授 戴天時 博士 Modeling Fixed-Income Securities and Interest Rate Option, 2nd Edition, Copyright © Robert A. Jarrow 2002

Outline The economy The traded securities Interest rates Forward contracts Futures contracts Option contracts

The Economy Frictionless ： -no transaction costs, no bid/ask spreads, no restrictions on trade, no taxes -If these traders determine prices, then this model approximates actual pricing and hedging well -frictionless markets v.s friction ‑ filled markets

Competitive ： -perfectly (infinitely) liquid -organized exchanges v.s over ‑ the ‑ counter markets discrete trading ： {0, 1, 2,..., τ} -Continuous trading The Economy

The Traded Securities Money Market Account- shortest term zero-coupon bond Zero-coupon bond price - default free, strictly positive prices 0T B(0) =$1 P(t,T) tT $1

Term Structure of Interest Rates The interest rates vary with maturity. Concerned with how interest rates change with maturity. The set of yields to maturity for bonds forms the term structure. -The bonds must be of equal quality. -They differ solely in their terms to maturity.

Yield

Forward rate

(3.3) Drive an expression for the bond’s price in terms of the various maturity forward rates ： (3.4)

Derivation of Expression (3.4)

Spot rate

Interest rates MarkNameMeaning Zero-coupon bond price 到期日 T 的零息債券在時間 t 的價格 Money market account 時間 t 到 T ，以利率 r(t) 投資 1 元至到 期時的金額。在此表示，將 1 元投 入極短期 zero-coupon bond Yield Internal rate of return ；時間 t 到 T 的 平均利率 Forward rate 在時間點 t 下，將來時間點 T 的瞬間 利率 Spot rate ； Zero rate 時間 t 的瞬時利率

Forward Contracts Forward contract – forward price a prespecified price that determined at the time the contract is written) – delivery or expiration date a prespecified date. – The contract has zero value at initiation.

Forward Contracts forward contracts on zero ‑ coupon bonds: – the date the contract is written (t) – the date the zero ‑ coupon bond is purchased or delivered (T 1 ) – the maturity date of the zero ‑ coupon bond (T 2 ) – The dates must necessarily line up as

Forward Contracts – We denote the time t forward price of a contract with expiration date T 1 on the T 2 ‑ maturity zero ‑ coupon bond as F(t,T 1 :T 2 ) – – The boundary condition or payoff to the forward contract on the delivery date is

Figure 3.1: Payoff Diagram for a Forward Contract with Delivery Date T 1 on a T 2 -maturity Zero-coupon Bond P(T 1, T 2 ) P(T 1, T 2 ) - F(t, T 1 : T 2 ) 0 F(t, T 1 : T 2 )

Futures Contracts Futures contract – futures price A given price at the time the contract is written. The futures price is paid via a sequence of random and unequal installments over the contract's life. – delivery or expiration date a prespecified date. – The contract has zero value at initiation.

Futures Contracts futures contracts on zero ‑ coupon bonds: – the date the contract is written (t) – the date the zero ‑ coupon bond is purchased or delivered (T 1 ) – the maturity date of the zero ‑ coupon bond (T 2 ) – The dates must necessarily line up as

Futures Contracts – We denote the time t futures price of a contract with expiration date T 1 on the T 2 ‑ maturity zero ‑ coupon bond as – – The cash flow to the futures contract at time t+1 is the change in the value of the futures contract over the preceding period [t,t+1], i.e

Futures Contracts – This payment occurs at the end of every period over the futures contract’s life. – This cash payment to the futures contract is called marking to the market.

Let us decide whether a long position in a forward contract is preferred to a long position in a futures contract with delivery date on the same -maturity bond. If the forward contract is preferred, then the forward price should be greater than the futures price. i.e.

IF spot rate zero-coupon bond price the current futures price the change in the futures price is negative we need to borrow cash to cover the loss, and spot rates are high.

This is a negative compared to the forward contract that has no cash flow and an implicit borrowing rate set before rates increased.

IF spot rate zero-coupon bond price the current futures price the change in the futures price is positive after getting this cash profit, we need to invest it and spot rates are low.

This is a negative compared to the forward contract that has no cash flow and an implicit investment rate set before rates decreased.

Option Contracts A call option of the European a financial security that gives its owner the right to purchase a commodity at a prespecified price (strike price or exercise price) and at a predetermined date(maturity date or expiration date). A call option of the American it allows the purchase decision to be made at any time from the date the contract is written until the maturity date.

Option Contracts A put option of the European a financial security that gives its owner the right to sell a commodity at a prespecified price (strike price or exercise price) and at a predetermined date(maturity date or expiration date). A put option of the American it allows the sell decision to be made at any time from the date the contract is written until the maturity date.

Option Contracts a European call option with strike price K and maturity date written on this zero- coupon bond. Its time t price is denoted At maturity its payoff is: C(T 1, T 1, K: T 2 ) = max [P(T 1, T 2 ) - K, 0]

35 Figure 3.2: Payoff Diagram for a European Call Option on the T 2 -maturity Zero-coupon Bond with Strike K and Expiration Date T 1 K P(T 1, T 2 ) In-the-money Out-of-the- money

Option Contracts a European put option with strike price K and maturity date written on this zero- coupon bond. Its time t price is denoted At maturity its payoff is:

37 Figure 3.3: Payoff Diagram for a European Put Option on the T 2 -maturity Zero-coupon Bond with Strike K and Expiration Date T 1 K P(T 1, T 2 ) K Out-of-the-money In-the-money

Option Contracts Put-call parity

Protfolio A : Protfolio B: ABAB