how do we valuate a call option?

(post from old site)

1)by establishing a hedged portfolio (ie a portfolio value will not be affected by the underlying price) and work backward!!
2) risk-neutral valuation

1)
for example, we have 
an underlying with price @t0 = 100
a call option that can be exercised @t1 at 100
what is the call option worth at t0?

hedged portfolio: buy a fraction of the underlying and sell the call
(hedge ratio)*Underlying - c
h*u-c

let's assume @t1, underlying price will either go up by 15 to 115 (u=1.15) or down by 5 to 95 (d=0.95)
if up
underlying is worth 115
call option is worth 15
if down
underlying is worth 95
call option is worth 0

h=? if we want the portfolio to have the same value no matter the underlying goes up or down?

h*100*1.15-15 = h*100*.95-0
h = (15-0)/[100(1.15-0.95)]
h= 15/20 = 0.75

=> it means that if we establish our portfolio at t0 by buying 0.75 share of the underlying and selling the call option @t0, then
@t1, our portfolio will be worth 71.25 regardless of the underlying price going up or down
up = 0.75*100*1.15 -15 = 71.25
down = 0.75*100 *0.95 = 71.25

ok, so what does the above have anything to do with the call price at t0?
since the hedged portfolio will be worth $71.25 @t1 without RISK (ie, no matter the price going up or down), the portfolio has to earn risk less rate due to no arb pricing

say the risk-free rate is 5% and t1-t0 = 1 year, we discount the hedged portfolio value at t1 with the risk-free rate
hedged portfolio value @t0 = 71.25/1.05 ~= 67.857
solve for c
67.857 = h*u-c
= 0.75*100 -c
c = 7.143

in general, h = (cu - cd)/S(u-d)
= (call value if up - call value if down)/Share price@t0(underlying price up change - underlying price down change)

in the example h= (15-0)/[100(1.15-0.95)]

note that the call price is independent of the likelihood of the underlying going up or down!!!!!!!! (that's quite counter intuitive if u think about it!)

2) risk-neutral approach
continuing with the above example, we assumed that risk-free rate=5% and the underlying/Share we use will either go up by 15% or down by 5% from t0 to t1

so what is the implied probability of it going up and going down?
=> ALL ASSETS should be expected make risk-free return in a risk neutral world. if not, arb opportunities exist.
so the Share should make 5% too =>
5% = 15%(prob of going up) + -5%(prob of going down)
= 0.15(p)+ -0.05(1-p)
p = 50% <- implied prob of going up
1-50% = 50% <- implied prob of going down

going back to the call option, with 50% going up (worth $15) and 50% going down (worth 0), the call option should be worth @t1:
0.5*$15+0.5*0= 7.5
discounting w/ rfr to get call value @t0 => 7.5/1.05 = 7.143

guess what? c is worth the same as with approach 1!! (whew...)


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wait a second! we assumed that the S price will only go from 100 up to 115 or down to 95. that doesn't sound realistic!

u're right. the above simplifies everything with a simple one-step binomial tree. to model the real world, we will have to create multiple-step binomial trees. with the above one step binomial tree, we've got 2 expected values of the S price. As we increase the number of steps (n-> infinity), the binomial distribution would approx to a continuous normal distribution.