Binomski Model English

8/6/2019 Binomski Model English

1/39

BINOMSKI MODEL

Ale Ah an


2/39

Literatura

St even Shreve-s toc has tic calcu lus f or f inan ce1,2

Paul Wilmott -quan tit ative f inan ceTomas Bjork-arb itrage the ory in continou stimeblogi: wilmott.co mPoljud ne knj igeMy life as a q uan tTraders, g uns an d money


3/39

In troductio n

Tod ay we w ill take a loo k at how to determ inethe pr ice of the optio n

Althou gh it may seem a t f irst glan ce tha t theprices can be choo sen arb itrar ily this is not the caseNamely optio ns are under suit ableass ump tio ns re du ndan t asse ts; can berepl icated/co ns truct ed by mixingstoc ks/f utu res an d bonds/money mk t accou nt


4/39

In troductio n

We are g oing to price optio ns in discre tetime /state (much simpler) using a b inom ial

mod elLet us thus assume tha t the s toc k can g o upor do wn, acco rding to the outco me of thecoin to ss (H,T). In case we ge t hea d the s toc kgoes up to S*u , and if tails S*d , theprobab ility is p an d 1-p


5/39


6/39

Li nk discrete-co ntio nos

If S,u,d=1/u , than s tandard deviatio n equal to (stran 225); useprobab ility p in such a way to repl icate the drif t


7/39

E -ma n

Binom ial mod el is a poo r s man method f orunders tanding an d pricing optio ns

Sou rce: My life as a q uan t ; Eman uel DermanRead the b oo k if u have time, an ot her oneworth the try is Das: Traders, Guns an d money


8/39

Ex ample

Let us price a s imple one per iod call optio n ona stoc k XYZ

Assume a s toc k XYZ worth 50$, tha t caneither r ise f or 100% or fall to 50%. Assumerisk free ra te f or 1 period is 25%. Theprobab ility of stoc k rise is 70%, ( do u nee d it?)How do we pr ice a call optio n with strikeK=50$


9/39

S imple e x ample

Stoc k price dynam ics

$50

$50x(1+1)= $100

$50x(1-0.5) = $25

up state

do wn state

t = now t = now + 1 month


10/39

Call optio n

A call optio n on this stoc k has a s trike pr ice of $50

t=0 t=1

Stoc k Price=$50;Call Value=$c

Stoc k Price=$100;

Call Value=$50

Stoc k Price=$25;

Call Value=$0


11/39

A replicati ng portfolio

C ons ider a p ortf o lio containing ( shares of the s toc k and $B inves ted in risk-free b onds.

The presen t value (price) of this port f olio is ( S + B= $50 ( + B


12/39

P ortfolio value

t=0 t=1

$100 ( + (1+r)B

$25 ( + (1+r)B

$50 ( + B

up state

down state


13/39

A replicati ng portfolio

This portf o lio will replicate the optio n if wecan f ind a ( and a B such tha t

$100 ( + (1+r) B = $50

$25 ( + (1+r) B = $0

and

Portfolio payoff = Option payoff

Up state

Down state


14/39

Th e replicati ng portfolio

So lutio n:( = 2/3

(1+r)B = -50/3.Eg, if r = 25%, then the p ortf o lio contains 2 /3 of a stoc k and short -13.33 $ of money marke t or bond


15/39


Payoffs at ma tu rity

Up state Down state

stoc k

bond

portf olio


16/39


Since the the repl icating p ortf olio has thesame pay off in all states as the call, the two

must also have the same pr ice .The presen t value (pr ice) of the repl icatingportf olio is 2/3*50$ - $13.33 = $20 .Theref ore, c = $20


17/39

A general (1-period) formula

Short summary

( !C

uC

d

S u S d B !

S uC

d S

d C

u

1 r S u S d

p !r d u d

c ! ( S B !pC u 1 p C d

1 r


18/39

An observatio n about (

As the time interval shr inks to ward zero ,delta be comes the der ivative.

( !C u C d S

uS

d

px C x S


19/39

Questio n

Do u notic e any thing par ticu lar on theprev iou s slide

Wha t do es effe ct t he pr ice?Probab ilityVolatility

Drif t


20/39

S toc k dy namics 3 periods


21/39

Replicati ng portfolio-

repetito n More pre ciselly inves ting V0 $ and rebalan cingin time can repl icate any optio n

Since there are n o cashflows with theexcep tio n of V0 $ the pr ice of optio n sh ou ldequal V0 $And if not?


22/39

Ex ample c ntued

As men tio ned there are 2 var iables an d 2equatio ns; repl icatio n possibleOf cou rse the repl icating p ortf o lio is choo sen insuch a way tha t under bot h scenar ios werepl icate optio ns pay off


23/39

Reciepe for replicati ng

1. Sell an optio n2. Buy 0 stoc ks3. Invest the differen ce (surplus) in the mny mk t accou nt

4. The pay off ma tc hes the pay off from the optio n

5. C hoo se 0 and V0 such tha t regar dless of u and d the pay off isma tc he d


24/39

recap

The las t conditio n can be rewr itt en

Delta is than eq ual to


25/39

Bi nomski model- naprej

Let u be s toc k value rising (H)-an d d fall in stoc kvalue even t (T) than we ge t


26/39

Remi nder- ris k neutrale x pectatio n

Loo king a t eq uatio n on prev iou s slide we can see

tha t optio n s value a t time 0 V0 equals thediscou nted value of weigthe d values underdifferen t scenar ios. Instea d of probab ility we usewha t is called a risk neut ral probab ility tha t

depen des on the r isk free ra te, u and dV0 =e-rtEQ(V(t))


27/39

Contio nus time

U sing s im ilar arg umen ts as bef ore one cander ive a f ormula in contio nou s time

Here the parame ters are


28/39


29/39

Multi period model

Wha t if we are deal ing w ith m ulti-per iod mod el

Is the idea of repl icating p ortf olio still validLet us use the same ass ump tio ns as bef oreSell short o r buy long

No lim it on borrowingNo transa ctio n co stTrading n o effe ct o n pr ice


30/39

2 period model

Let us analyze a two per iod mod elAssume a s imple e uropean call optio n tha t

pays the differen ce be tween pr ice S andstrike K at the en d o f 2 per iod s

U sing s im ilar arg umen t as bef ore try to determ ine the pr ice of a optio n V0


31/39

Replicati ng portfolio 2

modelAs bef ore ass ume tha t we b ou gh t 0 stoc kand V0 - 0 S bonds (or short). Af ter 1 per iod

the val ue of this portf olio is

Or


32/39

In termezzo

Af ter 1 per iod the pr ice of stoc k is either u ord ((H) or (T)). The value of the p ortf olio haschange d and we g ot 1 period to go. Wha t to do?Rebalan ce aga in. We sh ou ld change the m ixof stoc ks an d bonds aga in so tha t t he val ue a t the en d o f per iod 2 ma thces optio n s payoff Assume 1 is the n umber of stoc ks af terper iod 1 (this a f unctio n of H,T so 1(T) and

1(H)


33/39


periodsNow it ho lds

Or


34/39


periodsIn this case we have 6 eq uatio ns an d 6uknowns 1(T), 1(H), X1(H), X1(T),V0 , 0

Last two give us the e xpres ion f or 1(T)


35/39

Repli kat - naprej

Iz dele a do bimo vrednost portfelja v primer u padca deln ice X1(T)

Podo bno do bimo tudi


36/39

Repli kat -c ntued

The only thing lef t to do is to determ ine thevalue of the p ortf o lio in per iod 1 and t he ra tio

of stoc ks to bonds (delta)As bef ore


37/39

Multi period model

This can be general ized to


38/39

Assi gment

C alcu late the val ue of a optio n


39/39

Case

Implied vo latility stut gar t boerse

Binomski Model English

Documents

Transcript of Binomski Model English