# Εισαγωγή στην τιμολόγηση παραγωγών

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2 2.1. - P , (.. , ..). , P , , . .. P () r , , , P + Pr = P(1 + r). , .. , (LIBOR: London Interbank Offered Rate EURIBOR: European Interbank Offered Rate ..), , .. , ( ), . , ( .. ). 2.1.1. k () P r, k , k , r/k. , P r/k,

r P 1 + . k () r/k

r r r P1 + 1 + = P1 + , k k k ... k- ( ) P r P 1 + . k , m , P(1 + r / k ) mk .Boutsikas M.V. (2005-7), & , k

2

20

, P = 100000 (.. .. ) 6% (. k = 4). r 0.06 P1 + = 1000001 + = 106136 . 4 k 6.136% 6% . .2.1.2. k 4

() P r, , k .

r lim P (1 + ) k = Pe r , k k , t (t(0,)) P Pe rt . , P = 100000 6% 100000e0.06 = 106184 ( 6.184%). (t = 1/12) 100000e0.06/12 = 100501. r . .2.1.3.

: P r (, , ). P(t) t (P(0) = P). (t, t+t] P(t) P(t)rt ( rt), ,

P(t + t ) P(t ) + P(t )rt

t 0 ( t)1

P (t + dt ) = P (t ) + P(t )rdt dP (t ) = P (t )rdt P(t ) = dP (t ) = P (t )r dt ( t).

P(t) ( P(t) P(t)). P(t) :

1

, P(t + t ) = P(t ) + P(t )rt + o(t ) (t) (t)/t 0.

Boutsikas M.V. (2005-7), & ,

21

P(t ) = P(t )r

P(t ) = r (ln P (t )) = r ( t), P(t )

t [0, x]

x 0

(ln P(t ))dt = rdt ln P( x) ln P(0) = rx P( x) = P(0)e rx .0

x

r. r . . , r(t) r, (t, t+dt] P(t) P(t )r (t )dt , ,

P(t + dt ) = P(t ) + P(t )r (t )dt dP(t ) = P(t )r (t )dt ( t), :

x 0

(ln P(t ))dt = r (t )dt ln P ( x) ln P (0) = r (t )dt P( x) = P (0)e0 0

x

x

0 r (t ) dt

x

.

2.1.4. .

r ( ) ( ). r (risk-free interest rate) . , r ( ). () P, t ( ) Pe rt ( P r, t Pe rt). t A, Ae rt ( Ae rt r, t Ae rt e rt = A ).. 0, t, 2t, , (n1)t ( r). H A + Ae rt + Ae r 2t + ... + + Ae r ( n 1) t = A (e rt )i = Ai =0 n 1

1 (e rt ) n 1 e rt

..

Boutsikas M.V. (2005-7), & ,

22

A

1 (e rt ) n r e . 1 e rt

.. P n 0, t, 2t, , (n1)t P=A 1 (e rt ) n 1 e rt A=P 1 e rt . 1 e rnt

.. P = 100000, r = 4%, n = 1230, t = 1/12, = 476.21. , , (.. t). ( ).

2.2. Forwards Futures (Forwards) (, Futures) (delivery price). ( ). , , C . . . . , . . , , . (transaction costs), ( ). (, ) r (risk free interest rate) . , , . , arbitrage (. , ), ( ). Boutsikas M.V. (2005-7), 23 & ,

(. arbitrage). , .. C , arbitrage, arbitrage. (delivery price) (Forwards) (, Futures). (.. long position) (.. 100 ) T(0,) . T ( short position) K. K; ; St (.. 100 ) t. , , S0 T ST. S0 ST . K , K = E(ST). , , . K; . K arbitrage. K arbitrage, , , arbitrage ( .. K , long position E K). (.. ).

2.2.1. (delivery price) T , arbitrage

K = S 0 e rT , S0 r .

.

K > S 0 e rT . short position ( ) S0 ( r) (. 100 ). , ( 0) K S0erT. Boutsikas M.V. (2005-7), 24 & ,

S0erT > 0, arbitrage . ,

K < S 0 e rT , long position ( ) (short sell) S0 ( r). , ( ) S0e rT. S0erT > 0, arbitrage . , = S0erT arbitrage. .

2.2.1. 100 ( ). 20 (. S0 = 2000) 6%. , arbitrage (delivery price)

K = S 0 e rT = (20 100)e

0.06

1 4

2030.23 .

delivery price ( arbitrage) . f (t) t[0,T], T ( long short f (t)). :

2.2.2. ( T ) t < , arbitrage

f (t ) = S t Ke r (T t ) , St t r .. . :- long forward ( ) erT 0. - . T , . , T erTe rT = K Boutsikas M.V. (2005-7), & ,

25

long forward . , . no-arbitrage . .. t < T (short sell) . (arbitrage). t f (t ) + Ke rT e rt = f (t ) + Ke r (T t ) St. , arbitrage

f (t ) + Ke r (T t ) = S t , t < T .

2.2.1. f (0) = S 0 Ke r (T 0 ) . (t = 0),

f (0) = 0 S 0 Ke r (T 0 ) = 0 K = S 0e rT 2.2.1. 2.2.2. ( 2.2.1) 100 ( ) = 2030 ( noarbitrage 2030.23 arbitrage no-arbitrage , .. ). AAA 22 ( 20 ). (no-arbitrage value);

2.2.2, arbitrage , (t = 1/12) (long forward)

f (t ) = S t Ke

r (T t )

= 22 100 2030 e

1 1 0.06 ( ) 4 12

190.2 .

2.3.

. . 2.3.1. .. , , , ... 1, 2, , . , Si(t) i t ( ). (.. ). 2.3.2. (portfolio) x = (x1, x2, ,x). xi Boutsikas M.V. (2005-7), 26 & ,

i ( xi < 0 short xi i). , , x1, x2, , x . 2.3.3. A (x1, x2, ,x) t Vt = x S(t ) = ( x1 , x 2 ,..., x ) ( S1 (t ), S 2 (t ),...., S (t )) = xi S i (t ) ,i =1

( ). S(t) = (S1(t), S2(t), , S(t)) t.

2.3.1. , 2.2.2. 3 :(1) 1 ( 0), r ( t ert), (2) 2 , (3) 3 long forward 1 K T. 2.2.2 , : (Ke-rT, 0, 1)

B: (0, 1, 0)

t < T (ert, St, f (t)) t = T (erT, S, ST K). t : - t:

( Ke rT ,0,1) (e rt , St , f (t )) = Ke rT e rt + 0 St + 1 f (t ) = Ke r (T t ) + f (t ), t < T Vt = ( Ke rT ,0,1) (e rT , ST , ST K ) = Ke rT e rT + 0 ST + 1 ( ST K ) = ST , t = T A

- B t: (0,1,0) (e rt , St , f (t )), t < T Vt = (0,1,0) (e rT , ST , ST K ), t = T B

= St , t T .

T ST t < T (, 2.2.2, arbitrage),

Ke r (T t ) + f (t ) = S t f (t ) = S t Ke r (T t ) .

Boutsikas M.V. (2005-7), & ,

27

2.3.2. 2.2.2 2.3.1 :- : (0, 0, 1), long forward AAA, - B: (Ke-rT, 1, 0), (short) KerT 0 ( KerT 0, .. ) AAA. T ST K no-arbitrage ,

f (t ) = S t Ke r (T t ) , t < T. . . long forward . replicating portfolio hedging portfolio ( ) (. long forward ) B ( : (Ke-rT, 1, 0) ( ) . .

2.3.3. ( ) , . K = ( S 0 I )e rT f (t ) = S t I Ke r (T t ) , t = 0 ( ).

2.4. - C (option price option premium) ( , ) . C arbitrage. ( ) . . (.. ) . no-arbitrage . .. long callBoutsikas M.V. (2005-7), & ,

28

, (.. T) (. replicating hedging portfolio). ( .. ). : 0 S0 S1, S1 S0a S0b (0 < a < b). ( 0) S1 (S0b, S0a) , p 1 p.p

S1 = S0b

S01p

S1 = S0a T t

0

0 < a < e r T < b , a < b < e rT (arbitrage) short . , arbitrage e rT < a < b ( ). . . C (call option), (holder) (writer) T ( ) writer K ( T 0). .

2.4.1. no-arbitrage ( ) C= (e r T b 1)( S 0 a K ) + + (1 e r T a )( S 0 b K ) + . ba

. :(1) 1 ( 0), r ( t ert), (2) 2 (.. ), (3) 3 long call 1 exercise price K, option price C exercise date T. :Boutsikas M.V. (2005-7), & ,

29

- : (0, 0, 1) ( long call) - B: (0, 0, 0) ( 0 0 ). U0 U1 0 . long call , U1 = (S1 K)+. U0 long call 0, long call (option price option premium). 0, 0 . T:

( S1 K ) + = 0 e rT + 0 S1 . S1 , S0a . p S0b . 1 p. T S1, ,

(S 0 b K ) + (S 0 a K ) + ( S 0 a K ) + = 0 e r T + 0 S 0 a 0 = S 0 (b a) ( S b K ) = e r T + S b = e

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