Derivatives Introduction

Finance 70523 Spring 2000Assistant Professor Steven C. MannThe Neeley School of Business at TCU

Forward contractsFutures contractsCall and put option contracts

NotationDefinitionsGraphical representations

Bond prices and interest ratesnotation and definitions

Agreement to trade at a future date (T), at price set today (t): delivery price: K (t).

Forward Contracts

cash flow only at contract maturity (T)contracted buyer is Longcontracted seller is Short

t T time

Define forward price: current market price for future delivery forward price at t for delivery at T = f (t,T)

contract value is zero if : K(t) = f (t,T)

Forward Contract value at maturity

Define spot price: current market price for immediate deliverySpot price at time t = S(t)

Value of long forward contract at maturity

Value at T = spot price - delivery price= S(T) - K(t,T)

Value

0

S(T)K(t)

Value of longforward contract

Example: forward value at maturity

Forward mark price: f (0,60 days) = $0.5507 / DMenter into forward contract to buy DM 10,000,000 in 60 days

at delivery price K(0) =$0.5507.K(0) = $0.5507 = f(0,60 days) ; cost today is zero.

S(60 days)3/12/98 spot rate

.53 .54 .55 .56 .57

Value

0

If S(60) = $0.5657then value = [(S(60) - K(0) ) $/DM] x DM contract size

= [($0.5657 - 0.5507) $/DM] x DM 10,000,000= $0.0150 x 10,000,000 = $150,000

Futures Contracts

Define futures price: current futures market price for future delivery futures price at t for delivery at T = F (t,T)

contract value is zero if : K(t) = F (t,T)

Futures versus Forwards:

Futures Forwards

regulated (CFTC) unregulated *daily cash flows cash flow only at maturitystandardized customclearinghouse minimizes credit risk is important

credit riskF( t,T) f (t,T)

“Futures-spot basis”

Note that F(T,T) = S(T) (at T, delivery at T is same as spot delivery)prior to T, prices usually diverge.

Definebasis = F(t,T) - S(t)

Ttime time

T0 0

positive basis negative basis

basis

Futures price F(t,T)

Spot price S(t)Futures price F(t,T)

Spot price S(t)

basis

Options

Right, but not the obligation, to either buy or sell at a fixed price over a time period (t,T)

Call option - right to buy at fixed pricePut option - right to sell at fixed price

fixed price (K) : strike price, exercise priceselling an option: write the option

Notation: call value (stock price, time remaining, strike price)= c ( S(t) , T-t, K)

at expiration (T):c (S(T),0,K) = 0 if S(T) < K

S(T) - K if S(T) K

or: c(S(T),0,K) = max (0,S(T) - K)

Call value at maturity

Value

5

0

K (K+5) S(T)

Call value =max (0, S(T) - K)

c (S(T),0,K) = 0 ; S(T) < K S(T) - K ; S(T) K

Short position in Call: value at maturity

Value

0

-5

K (K+5) S(T)

Short call value =min (0, K -S(T))

c (S(T),0,K) = 0 ; S(T) < K S(T) - K ; S(T) K

short is opposite:-c(S(T),0,K) = 0 ; S(T) < K

-[S(T)-K] ; S(T) K

Call profit at maturity

Value

0

K S(T)

Profit = c(S(T),0,K) - c(S(t),T-t,K)

Call value at T: c(S(T),0,K) = max(0,S(T)-K)

Profit is value at maturity less initial price paid.

Breakeven point

Callprofit

$80.00 Asset price (S) $71.00

date and time of valuation: Option Strike (X) $80.00

expiration date: 6/19/98 Years to maturity (T) 0.44

Call Fair Value $3.92 riskless rate 5.40%volatility () 35%

Calculations are below:

1/10/98 13:03Intel April 98 Call; strike =

Call Option Value at maturity ( CT) (Black-Scholes)

-10

-5

0

5

10

15

20

25

30

35

Underlying asset value

Val

ue

at

Exp

irat

ion

value 0 0.0 0.0 0.0 0.0 0.0 0.0 5.0 10.0 15.0 20.0 25.0

Price 55.0 60.0 65.0 70.0 75.0 80.0 85.0 90.0 95.0 100.0 105.0

Put value at maturity

Value

5

0

(K-5) K S(T)

Put value =max (0, K - S(T))

p(S(T),0,K) = K - S(T) ; S(T) K 0 ; S(T) > K

Short put position: value at maturity

Value

0

-5

(K-5) K S(T)

Short put value =min (0, S(T)-K)

p(S(T),0,K) = K - S(T) ; S(T) K 0 ; S(T) > K

short is opposite:-p(S(T),0,K) = S(T) - K ; S(T) K

0 ; S(T) > K

Put profit at maturity

Value

0

K S(T)

Put value at T: p(S(T),0,K) = max(0,K-S(T))

put profit

Profit =p(S(T),0,K) - p(S(t),T-t,K)

Breakeven point

Profit is value at maturity less initial price paid.

Option values at maturity (payoffs)

0

K

KK

K

00

0

long call

short put

long put

short call

Bond prices and interest rate definitions

Default free bonds (Treasuries)zero coupon bond price, stated as price per dollar:

B(t,T) = price, at time t, for dollar to be received at T

Interest ratesdiscount rate (T-bill market)simple interestdiscrete compoundingcontinuous compounding

Rate differences due to:compoundingday-count conventions

actual/actual; 30/360; actual/360; etc.

Discount rate: id (T)

B(0,T) = 1 - id (T)

T 360

T (days) B(0,T) id (T)30 0.9967 3.9660 0.9931 4.1490 0.9894 4.24

180 0.9784 4.32

T(days) id(T) B(0,T)30 3.96 0.996760 4.14 0.993190 4.24 0.9894

180 4.32 0.9784

Example:30-day discount rate id = 3.96%B(0,30) = 1 - (0.0396)(30/360)

= 0.9967

Current quotes: www.bloomberg.comid = 100 (1 - B(0,T)) 360

T

Example:90-day bill price B(0,90) = 0.9894 id (90) = 100 (1- 0.9894)(360/90)

= 4.24%

Simple interest rate: is (T)

B(0,T) =

T (days) B(0,T) is (T)

30 0.9967 4.0360 0.9931 4.2390 0.9894 4.34

180 0.9784 4.48

T(days) is(T) B(0,T)

30 4.03 0.996760 4.28 0.993090 4.34 0.9894

180 4.48 0.9784

Example:30-day simple rate is = 4.03%B(0,30) = 1/ [1+ (0.0403)(30/365)]

= 0.9967

Current quotes: www.bloomberg.comis= 100 [ - 1] 365

T

Example:90-day bill price B(0,90) = 0.9894 is (90) = 100 [(1/0.9894) -1](365/90)

= 4.34%

1

1 + is (T)(T/365)

1 B(0,T)

Discretely compounded rate: ic(h)

compounding for h periods

B(t,t+h) =

ic(h) = h [ (1/B)(1/h) - 1 ]

Example:1 year zero-coupon bond price = 0.9560

semiannually compounded rate

ic(2) = 2 [ (1/0.9560) (1/2)

- 1 ]

= 4.551%

1

[1 + ic(h)/h] h 0.9560

periods rate1 4.603%2 4.551%3 4.534%4 4.525%5 4.520%6 4.517%

12 4.508%52 4.502%

365 4.500%8760 4.500%

1-year zero price =

Continuously compounded rates: r(T) note T must be defined as year

B(0,T) = = exp(-r(T)T)Example:1 year zero-coupon bond price = 0.9560continuously compounded rater(1) = - ln (.9560) /1 = 4.50%6-month zero:T=0.4932, B(0,0.4932) = 0.9784r(0.4932) = -ln(.9874)/.4932 = 4.43%

1exp(r(T)T)

Note: ln [exp(a)] = a = exp[ln(a)]

thus ln[exp(-r(T)T)] = -r(T)Tand ln(B(0,T)) = -r(T)T

r(T) = -ln[B(0,T)]/T

(days) T B(0,T) r(T)30 0.0822 0.9967 4.02%60 0.1644 0.9931 4.21%90 0.2466 0.9894 4.32%

180 0.4932 0.9784 4.43%365 1.0000 0.956 4.50%