Binomial Model
Pricing And Hedging
Abdelaziz Baihi
MATH-UA 250 Mathematics of Finance
Spring 2024
Abdelaziz Baihi Binomial Model
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Outline
1 Binomial Model
Single period Binomial model
Multi-period Binomial
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Table of Contents
1 Binomial Model
Single period Binomial model
Multi-period Binomial
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Single period Binomial model
In a single period binomial model, we assume that there are only two possible scenarii between
t0 = 0 and T the maturity, where the asset can only go up or down
1.
The initial value of the asset is S0 = S, and its value ST (wu) = Su and ST (wd) = Sd (e.g. u = 1.1
and d = 0.92). We Further assume that P (ST = Su) = p and P (ST = Sd) = 1− p where 0 < p < 1
represents either a subjective probability (i.e. what some analyst may believe from its market
study of what the probability of the asset goes up) or a statistical estimation that we believe to be
good enough.
Let’s consider we sold a European option with payoff g(ST ) that we need to hedge.
The two questions we want to answer are:
1 Can we build a (hedging) portfolio such that PT = g(ST ) ?
2 How much should we charge, i.e. what should be p0?
1Strictly speaking, even the ”down” scenario could be an upward move, but it makes sense to assume that market may either go up or down.
For this model to seem a bit more realistic, we can have in mind that T is short dated, e.g. T = 1/52 for a week (or less) and that Su is the
”most likely” value when the asset goes up, and Sd when it goes down
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Single period Binomial model
Assuming we sold the option at price p0 and bought ∆0 stock, our hedging portfolio at time T is:
PT = ∆0ST + (p0 −∆0S)erT
PT = p0e
rT +∆0(ST − SerT )
If we want to perfectly hedge, we need to have PT = g(ST ) in both scenarii, meaning
PT (wu) = p0 +∆0(Su− S) + (p0 −∆0S)(erT − 1) = g(Su)
PT (wd) = p0 +∆0(Sd− S) + (p0 −∆0S)(erT − 1) = g(Sd)
This is linear system in (p0,∆0); subtracting the second from the first line yields:
∆0 =
g(Su)− g(Sd)
Su− Sd
and
p0 = g(Su)e
−rT e
rT − d
u− d + g(Sd)e
−rT u− erT
u− d
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Single period Binomial model
Recall:
p0 = g(Su)e
−rT e
rT − d
u− d + g(Sd)e
−rT u− erT
u− d
Already we see that neither p0 nor ∆0 depends on the historical probability p. It only depends on
the possible outcomes and funding assumption.
Denoting q = e
rT−d
u−d , we can see that:
p0 = g(Su)e
−rT q + g(Sd)e−rT (1− q)
The initial price p0 appears as the expectation under a probability measure Q of the discounted
terminal payoff, namely:
p0 = E
Q(g(ST )e
−rT )
The probability Q is called the Risk Neutral measure.
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Multi-period Binomial Model
We consider an N -step recombinant tree which represents the variation over (small) periods of time
of some asset price. As before, we consider that the asset can only move up or down at each step
and if Sti is the asset price after the i
th step, then Sti+1 = Stiu with probability p and Sti+1 = Stid
with probability 1− p.2
For k = 0, ..., i, the underlying can go up k times, and go down i− k times (as it has to move i
times), in which case we have Sti = Su
kdi−k.
If after i steps, if the asset has gone k times up, there are
(
i
k
)
= i!k!(i−k)! possibilities of where these
upward moves have occurred, showing that the asset follows a binomial distribution3:
P (Sti = Su
kdi−k) =
(
i
k
)
pk(1− p)i−k
2Given the discrete nature of the model, I could have written Si instead of Sti
but I find it better to see the model as a discretized version
at dates t1 < ... < tN of some continuous time model
3Hence the name of binomial model
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Multi-period Binomial Model
Besides, the model is built such that we have the following conditional probability equality
P (Sti+1 = Stiu|Sti) = p = 1− P (Sti+1 = Stid|Sti)
We see that at each node of the tree (except the last one) including the next up and down leaves is
a single-perdio binomial model. This suggests to work the hedging and pricing problem backward.
As before then, we note q = e
rδt−d
u−d where δt represents the amount of time between each step that
we assume constant.4
Starting from the second to last row in the tree, which represents the time tN−1, there are N nodes
as the underlying can move upward k = 0, ..., N − 1 times and downward N − 1− k times.
Repeating the single-period calculation, we have that from each possible node to the next either up
or down node, we would have a (self-financing) portfolio value PtN which would start with a price
ptN−1 and a quantity of asset ∆tN−1 such that PtN = ptN−1e
rδt +∆tN−1(StN − StN−1erδt) and
∆tN−1 =
g(StN−1u)− g(StN−1d)
StN−1(u− d)
ptN−1 = g(StN−1u)e
−rδtq + g(StN−1d)e
−rδt(1− q)
4We could have written erδt as er or 1 +R as it is usually done in textbooks. I think it is better to introduce time so as to get familiar with
the intuition of dynamic hedging
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Multi-period Binomial Model
We see that ptN−1 = p(StN−1) is a function of StN−1 and g. From time tN−2, this can be seen as a
new payoff, and the (self-financing) portfolio at time tN−2 starting from the value ptN−2 and
quantity of asset ∆tN−2 would yield PtN−1 = ptN−2e
rδt +∆tN−2(StN−1 − StN−2erδt). Equating
PtN−1 = p(StN−1) yields:
∆tN−2 =
p(StN−2u)− p(StN−2d)
StN−2(u− d)
ptN−2 = p(StN−2u)e
−rδtq + p(StN−2d)e
−rδt(1− q)
Some remarks are in order:
We see that again, ptN−2 is a function of StN−2 exactly as ptN−1 is a function of StN−1 but with
g replaced by ptN−1 .
Repeating this procedure gives an iterative way to compute backward all the pti and ∆ti .
We see that again, the risk-neutral expectation can be used to write pti = E
Q(pti+1e
−rδt|Sti).
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Multi-period Binomial Model
In particular we see that pti will be a linear combination of terms
e−r(N−i)δtg(Stiu
kdN−i−k)qk(1− q)N−i−k for k = 0, ..., N − i since at time ti, there remains N − i
steps and the asset can go upward at most N − i times. By induction, and conditionally on which
node Sti is, we have
pti = e
−r(N−i)δt
N−i∑
k=0
(
N − i
k
)
g(SukdN−i−k)qk(1− q)N−i−k = e−r(N−i)δtEQ(g(StN )|Sti)
By induction, we see that the initial price can be expressed as (with tN = T = Nδt):
p0 = E
Q(g(StN e
−rNδt) = EQ(g(S(N)T e
−rT ) = e−rT
N∑
k=0
(
N
k
)
g(SukdN−k)qk(1− q)N−k
Like before the probability p becomes irrelevant: what ever distribition is statistically driving the
underlying up or down will be captured in our hedging portfolio so that at maturity T , the value of
the portfolio will exactly match the payoff function g(ST ) ensuring that we can fullfil our legal
liability toward our clients.
Abdelaziz Baihi Binomial Model
MATH-UA 250 Mathematics of Finance Spring 2024