FRM: Risk neutral valuation in option pricing model

A difficult idea, but maybe the key idea in option pricing: we can price the option under the riskless assumption and yet it will be valid it the real (risky) world! For more financial risk videos, visit our website! http://www.bionicturtle.com

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hi this is David Harper a banach turtle
I like to illustrate an idea that is
difficult but central to option pricing
models and that's called risk-neutral
valuation I'd like to do it with a
single step binomial tree right here and
I've adapted an example from John holes
book chapter 11 on binomial trees
we're doing option pricing so as usual
we need some input assumptions I made
these up stock price of 10 strike price
of ten dollars that's also the exercise
price we call that you is the size of an
up jump
d is the size of a Down jump
I just made up 1.24 up point for down a
riskless rate here to notify our of four
percent and we need the length of time
between the nodes in our binomial tree
i'm going to use three months or
one-fourth of a year we're only going to
do a single step so that just means
today is x 0 and we'll go forward one
step in the tree to three months so here
is the price of that European call
option under the risk free world
assumption and just to show you how that
works the stock price here starts at ten
dollars and the binomial idea is that it
can go either up to 12 or down to 8 12 i
got by multiplying 10 by you my
assumption for the size of an up jump or
up step 1.2 times 10 is 12 d is point
eight so if i don't go up i'm going to
go down with the stock and green 28
that's the stock price we're talking
about the call option so what's the
value of the call option if the stock
goes up to 12 it's going to be two
dollars
it's in the money by 12-10 or two
dollars if the stock goes down 28 the
option has a strike of 10 so it's
underwater the option can't have a
negative value and so it's going to have
a value of zero so what we have here
under this elegant binomial approach
two possible about future values for the
stock option and then we just apply time
value of money concepts to get the price
of the option we don't have to do two
things
there's two future outcomes so we just
take a weighted average it's not quite
the average which would be one dollar
would be a simple average its average by
the probability that each happen and it
happens to be here that the probability
under this risk free world of an up step
here is a little bit about fifty percent
or 50 2.5% i'm not going to go to math
of that it's a function of und if we
don't go up we're going to go down with
probability of 1 minus P or a little bit
less than forty-eight percent so we take
a weighted average of those two outcomes
and that will give us the expected
future value of the option about the
option value in three months so we know
we need to discount that to the present
that will get us a dollar and four and
just to show you here's the weighted
average part and then here's the
discounting exponential function of
negative a rate that's that's continuous
compounding or really continuous
discounting here and I'm discounting at
our or four percent the riskless rate so
that just confirms that i am indeed here
in the risk free world where discounting
these future expected values at a
riskless rate to compute the price of
the stock option here it's a little bit
above a dollar and so in addition to
being elegant the first protect second
time you see this you might think
wait a second you just competed the
price in an assumption of the risk free
world that's not realistic this is a
stock you expected to return more than
four percent and you're right the
counter intuitive idea of the
risk-neutral valuation is that even
though you're right we can still use
this option in the real world which has
risked so to compare on the Left want to
keep that risk for your world and here
on the right is the risky were
the real world so to speak and this
should be time no 2.25 ok so we still
have the stock going that starting at
ten today and going up to 12 are down to
eight so we still have a future
intrinsic price on the stock under the
binomial of either 2 or 0 let me just
change the discount rate to four percent
so it's consistent with the risk free
world now in the real world
let's try to make a more realistic
assumption in the risk free world we set
the probability of an up jump is about
50 2.5% and let's just to sit say I'm
just making this up that in the real
world weather is risk we think the
probability of the stock going up is
sixty percent that means we expect the
future stock price to be ten dollars and
forty cents and then if I just compute
the implied return on our stock here i
can just take the natural log of that
ten dollars and 40 / the ten dollars
that i started with and by saying the
probability of an up jump is sixty
percent I'm implying that in the real
world weather is risk i expect the stock
to return sixteen percent so that's
quite a bit more than our riskless rate
of four percent and now what happens is
I going to have more weight the sixty
percent apply to the to sew on a
weighted average i'm going to get more
than I get on the risk free world and
then I'll discount that back still at
the four percent and I do in fact get an
option price that's higher so you can
see here because i increased the
probability the chance that we go up and
then discount and then i do the same
discounted value here
I do in fact get an option price that's
higher the thing is though I committed a
fallacy i discounted at the risk Lee
risk-free rate and I can't really do
that because if i'm going to go over
here to the risky world and treat the
asset
as having a higher expected return I
then need to acknowledge that it's a
risky asset and I can't use a discount
rate that's the riskless rate I can't
take a stock and discounted to the
present value at a rate that implies
it's more like a Treasury what rate
should i use well I can actually figure
that out here with goal seek by setting
this option value equal to the option
value on the left a dollar for this
looks circular but i'm only using it to
figure out what the correct discount
rate is and i'm going to change it here
solver tells me that the discount rate
here of if i put in fifty-seven percent
that would give me an option value of a
dollar for equal to what I got on the
left and so okay to find the discount
rate was circular but the point remains
now what I've done on the right is I
still have a higher expected return on
the stock probability of an up jump of
sixty percent implies an expected return
of the stock of sixteen percent so I do
have a I do have a higher expected
future value of the stock option
this is true that's a difference here on
the real world in the right
however the key thing is if i'm going to
do that I need to discount use it use a
higher discount rate to get back to the
present in this case it should be 57%
and i will get here an option price on
the right of a dollar and for which is
equal the option price on the left and
again the key thing is if i'm going to
shift over here to the real world and
assume that the stock has an expected
return that's higher than the riskless
rate i need to be consistent and use a
discount rate that's also higher than
the risk this rate and when i do that i
will find that the price of the option
and the risky world is the same as what
i got in the risk free world
put another way i'm ok / the idea of
risk evaluation to compute the option
price over here
and then assume it's going to be valid
in the risky world and so that's the
idea of risk neutral valuation this is
david harper demonic turtle thanks for
your time