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max( S T âˆ’ K , 0 )

= discounted to present

(t=0) is

âˆ’ rT

max( S T âˆ’ K , 0 )

e

Call Option Price is..

â€¢ Example: A stock price

â€¢ the expected present

worth $100. I have a call

value (i.e. discounted

option with strike K=$100

to present) of future

maturing in 53 business

returns is days. If stock sells then

âˆ’ rT

max(ST âˆ’ K ,0)} for $120, the present

E{e

value of the payoff

where expectation is

assuming 5% interest

under the Black-

rate and N=252

Scholes model above.

âˆ’.05 ( 53 / 252 )

=e max(120 âˆ’ 100,0)

Simulated value of a call option

for a European call option with value function at maturity

Vo ( x) = max( x âˆ’ K ,0), where K = exercise price and

K = S0 = 10, r = 0.05, Ïƒ = 0.2, T = 0.25

(area under graph below)

3

2 .5

2

V alue of Call Option

1 .5

1

0 .5

0

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1

va lu e o f U

Crude simulation for European

call option

Find by simulation the value of a European call option

= expected payoff from option discounted to present

= E{e max(S 0 e âˆ’ K ,0)} where X is a

- rT X

Normal random variable with mean rT âˆ’ Ïƒ T

2

and

2

variance Ïƒ T and S 0 is the current stock price.

2

Option value= Expectation with

respect to uniform

Suppose X has cumulative distribution function F so

P[ X â‰¤ x] = F ( x).

We may generate X using inverse transform

âˆ’1

X = F (U ) where U is uniform on [0,1].

Then the option price

F âˆ’1 (U )

âˆ’ K ,0)}

- rT

E{e max(S 0 e

1

= âˆ« f (u )du

0

F âˆ’1 ( u )

where f (u ) = e âˆ’ K ,0)

- rT

max(S 0 e

Call option value

is area under this graph

For European call option,

F âˆ’1 ( u )

f(u) = e âˆ’ K ,0) where K = exercise price.

-rT

max(S 0 e

For graph, I chose K = S0 = 10, r = 0.05, Ïƒ = 0.2, T = 0.25

3

2 .5

2

V alue of Call Option

1 .5

f(u)

1

0 .5

0

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1

va lu e o f U

u

Matlab Function for simulated

value call option

â€¢ function v= fn(u)

â€¢ % discounted payoff for call option with

%exercise price K, r=annual interest rate,

%sigma=annual vol, S0=current stock price,

%u=vector of uniform (0,1) input to generate

%normal variate by inverse transform. %T=maturity

(years)

â€¢ %(For Black-Scholes price, integrate over(0,1)).

â€¢ S0=10 ;K=10;r=.05; sigma=.2 ;T=.25 ;

â€¢ ST=S0*exp(norminv(u,T*(r-sigma^2/2),sigma*sqrt(T)));

â€¢ % ST is the stock price at maturity. Discount

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