iii) pay F P 0,T = S 0 e δt when stock has dividend yield δ.

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1 Fnal s Wed May 7, 12:50-2:50 You are allowed 15 sheets of notes and a calculator The fnal s cumulatve, so you should know everythng on the frst 4 revews Ths materal not on those revews 184) Suppose S t s the prce of the underlyng asset for 0 t T Then the theoretcal prce of the long forward at tme t s P L (t) = S t S 0 e rt = S t S 0 (1 + ) t The theoretcal prce of the short forward at tme t s P S (t) = S 0 e rt S t = S 0 (1 + ) t S t Note that the two prces sum to 0 The theoretcal prces are farly accurate estmates of the market prces at tme t 185) There are 4 common ways to buy stock If prces below are for one share of stock and J shares of stock are purchased, then multply the payments by J Recall that r s the rsk free nterest rate compounded contnuously = rsk free force of nterest A share of stock has (contnuous) dvdend yeld (rate) δ f the number of shares grows contnuously: so 1 share wll grow to e δt shares at tme T Contnuous dvdend yeld s common for shares of ndex funds or mutual funds You may need to substtute 1 + = e r, et cetera, where s the rsk free nterest rate The payment prce for the stock assumes that the dvdend payment tmes and amounts are known (More advanced prcng technques have dvdend uncertanty reflected n the calculated stock prce) a) Outrght purchase: receve stock at tme t = 0 and pay S 0 at tme t = 0 b) Forward contract: receve stock at tme T and pay at tme T (buyer does not receve dvdends pad n [0, T)): ) pay F 0,T = S 0 e rt = S 0 (1 + ) T for nondvdend payng stock, ) pay F 0,T = S 0 e rt FV(dvdends) = S 0 (1 + ) T FV(dvdends) for stock whch pays dscrete dvdends (eg quarterly), ) pay F 0,T = S 0 e (r δ)t when stock has dvdend yeld δ c) Prepad forward = prepay: receve stock at tme T and pay at tme 0 (buyer does not receve any dvdends pad n [0, T)): ) pay F P 0,T = S 0 for nondvdend payng stock, ) pay F P 0,T = S 0 PV(dvdends) for stock whch pays dscrete dvdends, ) pay F P 0,T = S 0 e δt when stock has dvdend yeld δ d) Fully leveraged poston: Receve stock at tme 0, pay S 0 e rt = S 0 (1 + ) T at tme T (Borrow S 0 to pay for the stock at tme 0, then repay the loan at tme T) The forward prce = (prepad forward prce) (1 + ) T = (prepad forward prce) e rt 186) For the buyer, the forward payoff s S T F 0,T and the prepad forward payoff s S T F P 0,T The broker s (trader s or amrket maker s) payoffs are F 0,T S T or F P 0,T S T If the payoffs (= profts) are for 1 share of stock and the contract s for J shares, multply the payoffs by J 187) An n the money opton s an opton, f exercsed now, would have a postve payoff An at the money opton s an opton, f exercsed now, would have approxmately 0 payoff An out of the money opton s an opton, f exercsed now, would have a negatve payoff By mmedately, assume the current tme s t where 0 t T In determnng moneyness act af a European opton could be executed mmedately (or sold for the strke prce at tme 0) 1

2 A purchased call n the money f the current spot prce s greater than the strke prce: S t > K The long call s out of the money f S t < K A purchased put n the money f the current spot prce s less than the strke prce: S t < K The long put s out of the money s S t > K 188) An asset prce contngency s the condton for the underlyng asset to be bought or sold The asset s exercsed f t s bought or sold for an opton The asset s always bought or sold for a forward 189) The followng table gves the maxmum loss (negatve proft) and maxmum gan for some dervatves The entres that are not unlmted occur f S T = 0 or f the payoff = 0 dervatve maxmum loss maxmum gan long forward F 0,T unlmted short forward unlmted F 0,T long call F V (C) unlmted short call unlmted F V (C) long put FV (P) K FV (P) short put FV (P) K FV (P) 190) A futures contract s smlar to a forward but has several dfferences Futures are wdely bought and sold daly n markets durng the lfe [0, T] of the contract Forwards are typcally sold by brokers The amount needed to open a futures account s called an ntal margn To mantan order n futures markets, there are daly lmts on the movement of future prces At the end of each tradng day, a futures account s markedto-market whch means that any proft or loss resultng from a change n the future s prce from the prevous tradng day s close s added or deducted from the account balance The holder of the long futures contract s oblgated to buy and the holder of the short futures contract s oblgated to sell the asset at expry tme T day futures prce prce change credt or debt margn account balance S t = S t 1 + C t C t = S t S t 1 JC t B t = B t 1 e r/365 + JC t 0 S 0 B 0 1 S 1 C 1 JC 1 B 1 2 S 2 C 2 JC 2 B 2 T 1 S T 1 C T 1 JC T 1 B T 1 T S T C T JC T B T 191) The above table s used to compute the balance B t = B t 1 e r/365 + JC t n the margn account where 1 day = (1/365)th of a year Replace 365 by 52 f weeks are used and by 12 f months are used Let S t be the prce of 1 unt of the future on day t (at the close of the market) Let C t = S t S t 1 be the prce change of 1 unt of the future Let J be the multpler = number of untn the future contract so the sze of the contract s JS 0 Then B 0 = D% (sze) = D%JS 0, eg D% = 10% Note that S T = S 0 +C 1 +C 2 + +C T Then long future proft = B T B 0 e rt/365 = B T AV (B 0 ) 192) If a long forward has F 0,T = S 0, then the proft on the long forward s J(S T S 0 ) whch s the proft of the long future when there s no nterest: δ = 0 (Wth nterest, F 0,T = S 0 e rt, assumng no ncome, such as dvdends, from the asset durng [0, T]) 2

3 193) The purchaser of a futures contract may be requred to add funds to the margn account f B t falls below a predefned level called the mantenance margn 194) A swap s a contract that covers a stream of payments over a perod of tme (Optons and forwards are sngle payment swaps) A swap s an agreement to exchange (swap) one set of payments for another set of payments over tme 195) Let P(0, t ) = (1 + ) t = v t be the PV of a $1 zero coupon bond (treasury bll) maturng at tme t wth (rsk free) spot rate 196) For a commodty swap, suppose a commodty s needed at tmes t 1,, t n and a swap contract s avalable wth guaranteed prces F t1,, F tn (eg from n forward contracts wth expraton dates T = t ) A prepad swap has prce = PV of the guaranteed prces at the rsk free nterest spot rates s t Thus prepad prce = n =1 F t v t 197) The customary way to handle payments of a swap s to make level payments R at tmes t such that the PV of the payments = prepad prce Thus n =1 Rv t = n =1 F t v t F t v t s where the level payment s the swap prce R = F = P(0, t )F t = v t P(0, t ) Be able to fnd R gven the F t and a table of the t and s t Often t j = j 198) Let r 0 (t 1, t ) be the forward rate that s effectve from tme t 1 to tme t Then r 0 (j, j + 1) = (j, j + 1) = f j, the j year forward rate for year j + 1 startng at tme j See 133) - 135) Hence r 0 (j, j + 1) = f j = (1 + s j+1) j+1 1 Suppose nonlevel (1 + s j ) j nterest payments r 0 (t 1, t ) are made at tmes t for = 1,, n where t 0 = 0 If level nterest payments R are made at tmes t such that the level payment stream and nonlevel payment stream have the same PV, then n =1 Rv t = n =1 r 0 (t 1, t )v t where swap rate r 0 (t 1, t )v t s R = P(0, t )r 0 (t 1, t ) = v t = 1 P(0, t n) P(0, t ) P(0, t ) 199) To create a synthetc long forward, at tme 0 buy a purchased (long) call and sell a wrtten (short) put for the underlyng asset where both the call and the put expre at tme T wth strke prce K The synthetc long forward prce = C P = C(K, T) P(K, T) where C s the call premum and P s the put premum Then the synthetc long forward payoff = S T K = long forward payoff The synthetc long forward proft = S T K FV (C P) = S T K (C P)(1+) T = S T K (C P)e rt The purchaser of the synthetc long forward buys the asset at tme T for the strke prce K Here s the effectve annual nterest rate and r s the annual nterest rate compounded contnuously = force of nterest As usual, other formulas may be needed to compute F V (C P) The proft and payoff formulas assume that the asset earns no ncome (eg dvdends) durng [0, T] 200) To create a synthetc short forward, at tme 0 buy a purchased (long) put and sell a wrtten (short) call for the underlyng asset where both the call and the put expre at tme T wth strke prce K The synthetc short forward prce = P C = P(K, T) C(K, T) Then the synthetc short forward payoff = K S T = short forward payoff The synthetc short forward proft = K S T FV (P C) = K S T (P C)(1 + ) T = K S T (P C)e rt The purchaser of a synthetc short forward sells the asset at tme T for the strke prce K 201) The put call party: C P = PV (F 0,T K) The put call party relatonshp 3

4 arses snce the PV of the cash outflows to buy an asset should be the same, regardless of the method A long forward wth cash outflow of F 0,T at tme T has a PV of PV (F 0,T ) = F 0,T (1 + ) T = F 0,T v T A long synthetc forward has a cash outflow of C P at tme 0 and a cash outflow of K at tme T The PV of these outflows C P + PV (K) Usng S 0 = C P +PV (K) = PV (F 0,T ) can gve several relatonshps, ncludng the put call party and C P = S 0 PV (K), S 0 = PV (F 0,T ), and F 0,T = FV (S 0 ) = S 0 e rt = S 0 (1 + ) T 202) The put call party: C P = PV (F 0,T K) holds wth ) F 0,T = S 0 e rt = S 0 (1 + ) T for nondvdend payng stock, ) F 0,T = S 0 e rt FV(dvdends) = S 0 (1 + ) T FV(dvdends) for stock whch pays dscrete dvdends (eg quarterly), ) F 0,T = S 0 e (r δ)t when stock has dvdend yeld δ 203) The opportunty to make a sure proft wth no rsk s called arbtrage Such opportuntes can t survve long n the marketplace A fundamental assumpton made n prcng dervatves that arbtrage s not possble Hence ths prcng s called no-arbtrage prcng Ths prcng s used to derve the put call party 204) An off-market (long) forward has a premum C P and forward prce (for the asset) of F 0,T = K, the same cash flows as a synthetc long forward 205) Let L = long (purchased) and S = short (wrtten) so LCall(K) s a long Call wth stke prce K and and SPut(K) s a short put wth strke prce K Memorze the payoff graphs for the LCall, SCall, LPut and SPut For combnatons of optons, ) want to know the strategy, ) know the shape of the proft graph, and ) know the combnaton of optons that make up the dervatve The (short) wrtten proft graph wll be the mrror mage (about the x-axs) of the (long) purchased proft graph, and the proft s the combnaton of profts that make up the dervatve Wrtten dervatve proft = purchased dervatve proft 206) A (long) straddle ) bets prce wll ether go up or down from K, ) s V shaped, ) = LPut(K) + LCall(K) So straddle proft = max(0, K S T ) FV (P) + max(0, S T K) FV (C) A wrtten straddle ) bets prce wll be near K, ) shape s an upsde down V, ) = SPut(K) + SCall(K) 207) A strangle ) bets prce wll ether go up or down from K, ) decreases lnearly, then flat from K D to K+D, then ncreases lnearly, ) = LPut(K D) + LCall(K+D) So strangle proft = max(0, K D S T ) FV (P(K D)) + max(0, S T K D) FV (C(K + D)) where P(K D) s the premum of a put wth strke prce K D and C(K + D) s the premum of a call wth strke prce K + D 208) Bull spread = LCall(K 1 )+ SCall(K 2 ), K 2 > K 1 Bear spread = SCall(K 1 )+ LCall(K 2 ), K 2 > K 1 Collar = LPut(K 1 )+ SCall(K 2 ), K 2 > K 1 4

5 K1 K2 K3 Kn t0 t1 t2 t3 tn t_(n+1)=t 209) Consder the above tme dagram where deposts of K are made at tmes t 1, t 2,, t n Often t j = j for j = 0, 1,, n, n + 1 and often K j = K for j = 1,, n Sometmes t 0 = t 1 and T = t n+1 = t n The table below gves PV (t 0 ) and AV (T) for some mportant methods PV (t 0 ) AV (T) compound K j (1 + ) (t j t 0 ) K j (1 + ) (T t j) smple K j [1 + (t j t 0 )] 1 n K j [1 + (T t j )] a(t 0 ) a(t) a(t) not smple K j K j a(t j ) a(t j ) spot nterest K j (1 + s j ) (t j t 0 ) K j (1 + s j ) (T t j) mmedate, K j = K, t j = j + d, T = t n K a n K s n due K j = K, t j = j + d, t 0 = t 1 K ä n K S n 5