The Black-Scholes-Merton Formula and
Risk-Neutral Pricing
An Introduction to Mathematical Finance
Phillip Monin
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712
pmonin@math.utexas.edu
March 31, 2010
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P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
“Derivatives are financial weapons of mass destruction.”
(Warren Buffett, 2003)
“Although the benefits and costs of derivatives remain the
subject of spirited debate, the performance of the economy
and the financial system in recent years suggests that those
benefits have materially exceeded the costs.”
(Alan Greenspan, 2003)
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Outline
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
• Overview of Math Finance
• Binomial Asset Pricing Model
• Geometric Brownian Motion
• The Black-Scholes-Merton Formula
B-S-M and
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Outline
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
1 Overview of Math Finance
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
2 The Binomial Asset Pricing Model
3 Geometric Brownian Motion
4 The Black-Scholes-Merton Formula
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History
P. Monin
Overview of
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Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
• 1900: Bachelier - Brownian motion & stock prices
• 1950-60s: Markowitz and Sharpe - portfolio
optimization - risk/return tradeoff
• 1970s: Black-Scholes, Merton - pricing options
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P. Monin
Mathematical Finance
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Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
• risk-neutral pricing/ APT
• utility maximization, duality methods
• general equilibrium pricing
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The Math in Math Finance
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
• Real analysis and PDEs
• Probability
• Functional analysis
• Stochastic analysis, SDEs
• Convex duality
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Outline
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
1 Overview of Math Finance
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
2 The Binomial Asset Pricing Model
3 Geometric Brownian Motion
4 The Black-Scholes-Merton Formula
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The Binomial Asset Pricing
Model: An Illustration of
Risk-Neutral Pricing
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
• One period, no transactions costs, borrow/lend at same
rate r > 0
• One riskless bond with price at time t denoted by Bt
• One risky stock with price at time t denoted by St .
S1 (H) = uS0
r
rrr
r
r
rr
rrr
S0 LL
LLL
LLL
q=1−p LLL
p
S1 (T ) = dS0
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Arbitrage
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
Arbitrage:
Geometric
Brownian
Motion
• “something for nothing:” chance to make money with
The BlackScholesMerton
Formula
• real markets sometimes exhibit arbitrage, but this is
no possible loss
necessarily fleeting; as soon as someone discovers it,
trading takes place that removes it.
In the binomial model, to preclude arbitrage, we must
assume
0<d<1+r <u
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P. Monin
Overview of
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Bin. Asset
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Why?
1 If d ≥ 1 + r
• Start with nothing.
• Borrow from bond to buy stock.
• Money earned from stock at time one enough to pay
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
debt; positive probability of profit.
2
If u ≤ 1 + r
• Start with nothing.
• Short stock and buy bond.
• Money earned from bond at time one enough to sell
stock; positive probability of profit.
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P. Monin
Financial Derivatives
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
A derivative is a financial contract whose payout depends
on the value of some other (underlying) security.
A European call option: right, but not the obligation, to
buy one share of stock at time one for strike price K.
• We shall assume that S1 (T ) < K < S1 (H).
• If we get a tail on the toss, the option expires worthless.
• If we get a head on the coin toss, the option is
exercised and yields a profit of S1 (H) − K.
Thus the option’s payout is (S1 − K)+ .
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Let’s play a game
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
1
d= ,
2
u = 2,
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
r=
1
4
S1 (H) = 8
qq
qqq
q
q
qqq
p
S0 = 4M
MMM
MMM
q=1−p MMM
S1 (T ) = 2
B0 = 1
B1 = 1.25
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P. Monin
Overview of
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Geometric
Brownian
Motion
The BlackScholesMerton
Formula
You will buy the European call from me. The call has strike
K = 5 and so its payout looks like:
C1 (H) = 3
q
qqq
q
q
q
qqq
p
C0 =?M
MMM
MMM
q=1−p MMM
C1 (T ) = 0
B-S-M and
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P. Monin
Overview of
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Finance
You will buy the European call from me. The call has strike
K = 5 and so its payout looks like:
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
C1 (H) = 3
q
qqq
q
q
q
qqq
p
C0 =?M
MMM
MMM
q=1−p MMM
C1 (T ) = 0
How much will you pay me for this game?
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Replication
P. Monin
Overview of
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Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
•
•
•
•
•
•
Suppose my initial wealth is X0 = 1.20.
Buy ∆0 = 21 shares stock, which costs 2.
Only have 1.20, so borrow .80 at an interest rate of 25%.
If H, then stock position worth 12 · 8 = 4.
If T , then stock position worth 12 · 2 = 1.
At time one, owe 1 with certainty.
Therefore, total wealth at time one looks like
X1 (H) = 3
o
p oooo
ooo
ooo
X0 = 1.20
O
OOO
OOO
O
q=1−p OOO
X1 (T ) = 0
regardless of what p and q are!
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P. Monin
Overview of
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Bin. Asset
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Geometric
Brownian
Motion
The BlackScholesMerton
Formula
We have replicated the call option by investing in the stock
and the money market. This is the risk-neutral pricing or
arbitrage pricing theory approach. Why the names?
Since we have replicated the call with initial wealth
X0 = 1.20, then the price of the call must be C0 = 1.20 or
else there will be arbitrage.
1
If C0 = 1.21, then I use 1.20 to replicate the call and
take 0.01 and put in the bond.
2
If C0 = 1.19, then I buy the option and reverse the
replicating strategy: sell 21 shares stock short, receive
income of 2. Use 1.19 to buy the option, put .80 in
money market, and put 0.01 in another money market.
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Why is it called risk-neutral
pricing?
P. Monin
Overview of
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Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
Let
pe =
(1 + r) − d
> 0,
u−d
qe =
u − (1 + r)
= 1 − pe > 0
u−d
• pe + qe = 1, and so we can regard them as probabilities.
• Investors are neutral about risk:
S0 =
1
[e
pS1 (H) + qeS1 (T )]
1+r
Investors must be neutral about risk since they do not
require extra compensation to assume it, and are not
willing to pay for it.
• Under these weights, we price by expectation:
C0 =
1
[e
pC1 (H) + qeC1 (T )]
1+r
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Hedging
P. Monin
Overview of
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Bin. Asset
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Geometric
Brownian
Motion
The BlackScholesMerton
Formula
So, how much do I put in the stock at time zero to replicate
the call?
One can show
∆0 =
C1 (H) − C1 (T )
S1 (H) − S1 (T )
the so called delta-hedging formula.
This tells me how much to invest in the stock to replicate
the call, that is, to hedge a short position in the call.
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P. Monin
Multi-period Binomial Model
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
• Used in practice because with a sufficient number of
periods, it provides a reasonably good, computationally
tractable approximation to continuous-time models.
• Use backward induction.
• Price by expectation using risk-neutral probabilities,
and hedge using delta-hedging formula.
In multi-period or continuous time setting, all discounted
portfolios are martingales under the risk-neutral measure,
which is sometimes called equivalent martingale
measure.
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Fundamental Theorem of Asset
Pricing (First part)
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Overview of
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Finance
Bin. Asset
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Geometric
Brownian
Motion
The BlackScholesMerton
Formula
There is not always a unique risk-neutral measure. For a
given market, the set of EMMs is similar to the set of
solutions to linear systems.
Either
• There is no risk-neutral measure.
• There is a unique risk-neutral measure.
• There are infinitely many risk-neutral measures.
Theorem
No arbitrage ⇐⇒ there exists a risk-neutral measure
B-S-M and
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Outline
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
1 Overview of Math Finance
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
2 The Binomial Asset Pricing Model
3 Geometric Brownian Motion
4 The Black-Scholes-Merton Formula
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P. Monin
Real data
Overview of
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Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
Figure: Google price history–3/26/10
Figure: Google price history–4/09-3/10
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P. Monin
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Bin. Asset
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Geometric
Brownian
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The BlackScholesMerton
Formula
Paths of Geometric Brownian
Motion
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P. Monin
Geometric Brownian Motion
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
We say that {St }t≥0 follows a geometric Brownian
motion with drift µ and volatility σ, if for all t, s ≥ 0 the
random variable
St+s
St
is independent of all prices up to time t and if
St+s
log
St
is a normal random variable with mean µt and variance tσ 2 .
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P. Monin
Overview of
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Geometric
Brownian
Motion
The BlackScholesMerton
Formula
Geometric Brownian motion as a
limit of binomial models
Let ∆ > 0 be a small increment in time, and suppose that
every ∆ time units, the price of a security either goes up by
a factor u with probability p or goes down by a factor d with
probability 1 − p where
√
√
1
µ√ u = eσ ∆ ,
d = e−σ ∆ ,
p=
∆
1+
2
σ
Proposition
As ∆ → 0, the collection of prices becomes a geometric
Brownian motion.
Theorem (Central Limit Theorem)
{Xn }n∈N
Pn sequence of iid random variables, and
Sn = i=1 Xi . Then for large n, Sn will approximately be
N (nµ, nσ 2 ).
B-S-M and
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P. Monin
Overview of
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Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
B-S-M and
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P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
B-S-M and
R-N
Outline
P. Monin
Overview of
Math
Finance
Bin. Asset
Pricing
1 Overview of Math Finance
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
2 The Binomial Asset Pricing Model
3 Geometric Brownian Motion
4 The Black-Scholes-Merton Formula
B-S-M and
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The Black-Scholes-Merton
Formula
P. Monin
Overview of
Math
Finance
• Price a call with strike K and expiration T .
Bin. Asset
Pricing
• Interest rate r, compounded continuously.
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
• Price of stock- GBM with volatility σ 2
Idea: find risk-neutral price in n-periods, send n to infinity
Approximate GBM (note no µ)
u = eσ
√
d = e−σ
T /n
√
T /n
≈1+σ
p
σ2T
T /n +
2n
≈1−σ
p
σ2T
T /n +
2n
Let Y ∼ Binomial(n, p)
p=
p
p
1 + rT /n − d
1 r T /n σ T /n
≈ +
−
u−d
2
2σ
4
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P. Monin
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Geometric
Brownian
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The unique risk-neutral price for the n-period model is
C = (1 + rT /n)−n E[(S0 uY dn−Y − K)+ ]
"
+ #
u Y
−n
n
= (1 + rT /n) E S0
d −K
d
√
√
= (1 + rT /n)−n E[(S0 e2σ T /n Y e−σ nT − K)+ ]
The BlackScholesMerton
Formula
= (1 + rT /n)−n E[(S0 eW − K)+ ]
where
W = 2σ
p
√
T /nY − σ nT .
• Y ∼ Binomial(n, p) implies Y → N (np, np(1 − p))
• Linear combos of normals are normal, so W converges
to a normal too.
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Geometric
Brownian
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The BlackScholesMerton
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Since E[Y ] = np,
p
√
T /nE[Y ] − σ nT
p
√
= 2σ T /nnp − σ nT
√
= 2σ nT (p − 1/2)
!
p
p
√
r T /n σ T /n
≈ 2σ nT
−
2σ
4
E[W ] = 2σ
Moreover, Var(Y ) = np(1 − p) and p ≈ 1/2 for large n, so we
have that
p
Var(W ) = (2σ T /n)2 Var(Y )
= 4σ 2 T p(1 − p)
≈ σ2t
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P. Monin
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Geometric
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The BlackScholesMerton
Formula
Thus as n → ∞, the unique cost of the option that does not
result in an arbitrage when the underlying security’s price
follows a geo BM with volatility parameter σ is
C0 = e−rT E[(S0 eL − K)+ ]
where L ∼ N ((r − σ 2 /2)T, σ 2 T ). Thus, we can derive the
Black-Scholes-Merton option pricing formula:
√
C0 = S0 Φ(ω) − Ke−rt Φ(ω − σ T )
where
ω=
rT − σ 2 T /2 − log(K/S0 )
√
σ T
and
Z
x
Φ(x) =
−∞
1
2
√ e−z /2 dz
2π
is the cumulative distribution function of the standard
normal random variable.
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Overview of
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Using Girsanov’s theorem, it can be shown that the
risk-neutral measure in this situation corresponds to the
stock following a GBM with µ and σ where
µ = r − σ 2 /2.
Bin. Asset
Pricing
Geometric
Brownian
Motion
The BlackScholesMerton
Formula
Thus, the BSM formula is an example of risk-neutral pricing.
If C(S0 , T, K) is the BSM valuation of the option then
∆0 =
∂
C(S0 , T, K) = Φ(ω) > 0
∂S0
We can use this to construct hedges.
• One can see that C is increasing in S0 , T, σ, r and
decreasing in K.
• σ is estimated from historical data.
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Overview of
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Geometric
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The BlackScholesMerton
Formula
In practice...
What if the real cost differs from the BSM price?
Probably not.
• BSM assumes continuous trading in the stock, which is
impossible and expensive because of transactions costs.
• Even if we assume our estimate of σ is very accurate, it
can change over the option’s life.
• Our assumption that the stock follows a geometric BM
is only an approximation to reality. Many studies show
this not to be the case.
If you believe that GBM is a reasonable approximate model,
then BSM formula gives a reasonable option price. If the
price is significantly above/below the market price, then a
strategy of buying/selling options and selling/buying stock
can be devised. Such a strategy, although not yielding a
certain win, can often yield a gain that has a positive
expected value along with a small variance.
B-S-M and
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P. Monin
Overview of
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Bin. Asset
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Geometric
Brownian
Motion
The BlackScholesMerton
Formula
BSM used in practice, usually as a reference point.
Other limitations:
• the underestimation of extreme moves
• stationarity of the process, risk-free nonconstant &
must be estimated
• continuous and frictionless trading
Thank you for your attention!
Feel free to email me at
pmonin@math.utexas.edu
for a set of references.