Derivatives Introduction Finance Spring 2000 Assistant Professor Steven C. Mann The Neeley School of Business at TCU Forward contracts Futures contracts Call and put option contracts Notation Definitions Graphical representations Bond prices and interest rates notation 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 Long contracted seller is Short tTtime 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 delivery Spot 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 long forward contract

Example: forward value at maturity Forward mark price: f (0,60 days) = $ / DM enter into forward contract to buy DM 10,000,000 in 60 days at delivery price K(0) =$ K(0) = $ = f(0,60 days) ; cost today is zero. S(60 days) 3/12/98 spot rate Value 0 If S(60) = $ then value = [(S(60) - K(0) ) $/DM] x DM contract size = [($ ) $/DM] x DM 10,000,000 = $ 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: FuturesForwards regulated (CFTC)unregulated * daily cash flowscash flow only at maturity standardizedcustom clearinghouse minimizescredit risk is important credit risk F( 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. Define basis = F(t,T) - S(t) T time T 0 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 price Put option - right to sell at fixed price fixed price (K) : strike price, exercise price selling 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) = 0if 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 KS(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 Call profit

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 K K 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 rates discount rate (T-bill market) simple interest discrete compounding continuous compounding Rate differences due to: compounding day-count conventions actual/actual; 30/360; actual/360; etc.

Discount rate: i d (T) B(0,T) = 1 - i d (T) T 360 Example: 30-day discount rate i d = 3.96% B(0,30) = 1 - (0.0396)(30/360) = Current quotes: i d = 100 (1 - B(0,T)) 360 T Example: 90-day bill price B(0,90) = i d (90) = 100 ( )(360/90) = 4.24%

Simple interest rate: i s (T) B(0,T) = Example: 30-day simple rate i s = 4.03% B(0,30) = 1/ [1+ (0.0403)(30/365)] = Current quotes: i s = 100 [ - 1] 365 T Example: 90-day bill price B(0,90) = i s (90) = 100 [(1/0.9894) -1](365/90) = 4.34% i s (T)(T/365) 1 B(0,T)

Discretely compounded rate: i c (h) compounding for h periods B(t,t+h) = i c (h) = h [ (1/B) (1/h) - 1 ] Example: 1 year zero-coupon bond price = semiannually compounded rate i c (2) = 2 [ (1/0.9560) (1/2) - 1 ] = 4.551% 1 [1 + i c (h)/h] h

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 = continuously compounded rate r(1) = - ln (.9560) /1 = 4.50% 6-month zero: T=0.4932, B(0,0.4932) = r(0.4932) = -ln(.9874)/.4932 = 4.43% 1 exp(r(T)T) Note: ln [exp(a)] = a = exp[ln(a)] thus ln[exp(-r(T)T)] = -r(T)T and ln(B(0,T)) = -r(T)T r(T) = -ln[B(0,T)]/T