Notes I made in June 2013 on the derivation and use of the Black-Scholes equation. If you can forgive the terseness, you can look forward to some nifty stochastic partial differential equation twirling! Any and all corrections are welcome!

1. Black-Scholes
Inon Sharony
§1 Black-Scholes model
http://en.wikipedia.org/wiki/Black%E2%80%93Scholes
http://www.youtube.com/channel/HCfGnqiXcZWX8
1.1 Wiener process (i.e. Brownian motion)
Wt −Ws ∼ N µ,σ2
f (Wt) =
1
√
2πt
exp −
1
2
x2
t
E[Wt] = 0
Var[Wt] = E W2
t = t
1.2 Geometric (i.e. exponential) Brownian motion
• Constant drift, µ
• Constant diffusion (a.k.a. stock volatility), σ
dXt = µXtdt +σXtdWt
Xt = X0 exp µ −
1
2
σ2
t +σWt
1

2. Black-Scholes.lyx Created by Inon Sharony
1.3 It¯o’s lemma
For an It¯o drift-diffusion process, dSt = µtdt +σtdWt, and any well-behaved function f (t,Xt):
d f (t,Xt) =
∂ f
∂t
+ µt
∂ f
∂x
+
1
2
σt
∂2 f
∂x2
dt +σt
∂ f
∂x
dWt
This can be thought of as a Taylor series expansion to second order in x and first order in t.
1.4 Hedge
• A financial investment position intended to offset potential losses/gains that may be incurred by a
companion investment.
• For example, when (long) investing in a given stock (by buying a certain value of shares), hedge out the
industry-related risk by short selling an equal value of shares from the competitors. If the competition
is weaker, the hedge will pay off even when the entire industry takes a hit.
1.5 Black-Scholes equation
• S(t) – the price of a stock at time t
• V (S,t) – the price of some derivative of the stock (e.g. call options or put options)
• σ – the volatility of the stock.
• r – annual interest rate
• t – time (from now, 0, to expiry, T)
• Π – the value of a portfolio
∂tV +
1
2
σ2
S2
∂2
S V = rV −rS∂SV
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3. Black-Scholes.lyx Created by Inon Sharony
1.5.1 Interpretation using “the Greeks”
• ∆ ≡ ∂SV
• Θ ≡ ∂tV < 0 – Time decay, reflecting the loss in value due to having less time for exercising the option.
For a European call on an underlying without dividends, it is always negative.
• Γ ≡ ∂2
S V > 0
Also,
• rV – risk-less return from long position
• −rS∆ – risk-less return from short position of ∆ many shares.
1.5.2 Derivation
¶1.5.2.1 Stock price
Stock price follows a Geometric Brownian Motion
dSt
St
= µdt +σdWt
Where dWt is a Weiner process with mean 0 and variance t. Therefore, the infinitesimal rate of return on
the stock, dSt
St
, has E dSt
St
= µdt and Var dSt
St
= σ2dt.
¶1.5.2.2 Option price
V (S,T) is known (terminal condition). We get V (S,t) from It¯o’s lemma:
dV (t,St) = ∂tV + µS∂SV +
1
2
σS2
∂2
S V dt +σS∂SVdWt
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4. Black-Scholes.lyx Created by Inon Sharony
¶1.5.2.3 Value of the Delta-Hedge Portfolio (Short 1, long ∆)
(∆ ≡ ∂SV)
Π = −V +S∆
= −V +S∂SV
dΠ = dV +(∂SV)dS
= −∂tV −
1
2
σ2
S2
∂2
S V dt
No dWt – no uncertainty, and therefore no risk!
¶1.5.2.4 No arbitrage
All risk-free instruments must yield the same gains, if no arbitrage is to be upheld.
Risk free interest:
dΠ = rΠdt = r(−V +S∂SV)dt
Equating both expression for dΠ:
−∂tV −
1
2
σ2
S2
∂2
S V dt = dΠ = r(−V +S∂SV)dt
−∂tV −
1
2
σ2
S2
∂2
S V = −rV +rS∂SV
This is the Black-Scholes equation.
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5. Black-Scholes.lyx Created by Inon Sharony
1.6 Transformation to the diffusion equation
e.g. Black-Scholes equation for a call option:
−∂tC −
1
2
σ2
S2
∂2
S C = −rC +rS∂SC
Use the following boundary conditions:
• C(0,t) = 0 ∀t
• C(S,t) −→
S→∞
S
• C(S,T) = max{S−K,0}
1. Solving for C(S,t):
∂tC +
1
2
σ2
(S∂S)2
C + r −
1
2
σ2
S∂SC −rC = 0
Which is a Cauchy-Euler equation (i.e. mixed first and second order PDE with variable coefficients such
that all terms are of the same order in S).
2. We use the following change of variables:
y ≡ lnS −→ ∂y = S∂S
τ ≡ T −t −→ ∂τ = −∂t
∂τC +
1
2
σ2
∂2
y C + r −
1
2
σ2
∂2
y C −rC = 0
now with constant coefficients.
3. We now make another substitution, u ≡ Cerτ
∂τu−
1
2
σ2
∂2
y u− r −
1
2
σ2
∂yu = 0
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6. Black-Scholes.lyx Created by Inon Sharony
4. Finally, we reduce the term which is first order in ∂y, using the substitution x ≡ y+ r − 1
2σ2 τ
∂τu =
1
2
σ2
∂2
x u
which is the parabolic PDE called Fick’s equation (a.k.a. the diffusion equation or the heat equation).
5. The terminal condition (C(S,T) = max{S−K,0}) is now an initial condition:
u(x,0) = u0 (x) = max exp
1
2
[a+1]x −exp
1
2
[a−1]x ,0
a ≡ 2r/σ2
6. Use the Green’s function solution for the heat equation
u(x,τ) =
ˆ ∞
−∞
u0 (s)G(x−s;τ)ds =
1
σ
√
2πτ
ˆ ∞
−∞
u0 (s)exp −
(x−s)2
2σ2τ
ds
7. Return to the old variables, {x,τ,u} −→ {S,t,C}, to get the Black-Scholes formula.
http://quant.stackexchange.com/questions/84/transformation-from-the-black-scholes-different
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7. Black-Scholes.lyx Created by Inon Sharony
§2 Black-Scholes formula
2.1 Stock options
• The owner of an option contract has the right to exercise it, and thus require that the financial transac-
tion specified by the contract is to be carried out immediately between the two parties, whereupon the
option contract is terminated.
• European options: May only be exercised on expiration.
• call option: The owner of the option buys the underlying stock at the strike price from the option seller
• put option: The owner of the option sells the underlying stock at the strike price to the option seller.
2.1.1 Moneyness
• In-the-money (ITM) – A stock option that would would make money if it expired now.
• Out-the-money (OTM) – A stock option that would would loose money if it expired now.
• At-the-money (ATM) – A stock option that would would break even if it expired now.
• Spot – when discussing the current moneyness.
• Forward – when discussing some future moneyness.
• Simple moneyness – S/K
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8. Black-Scholes.lyx Created by Inon Sharony
2.2 Calculation of the prices of put and call options, consistent with the Black-Scholes
model.
• C(S,t) – the price of a European call option
• P(S,t) – the price of a European put option
• K – the strike price (a.k.a. exercise price) of the option
• T −t – the time to maturity
• N (x) =
´ x
−∞
1√
2π
e−z2
dz – the CDF of the standard normal distribution
C(S,t) = SN (d1)−KN (d2)e−r(T−t)
P(S,t) = Ke−r(T−t)
−S+C(S,t) = −SN (−d1)+KN (−d2)e−r(T−t)
d1 ≡
1
σ
√
T −t
ln
S
K
+ r +
1
2
σ2
(T −t)
d2 ≡
1
σ
√
T −t
ln
S
K
+ r −
1
2
σ2
(T −t) = d1 −σ
√
T −t
2.2.1 Shorthand
• τ ≡ T −t
• D ≡ e−rτ – the discount factor.
• F ≡ S/D – the forward price.
C(F,τ) = D[FN (d+)−KN (d−)]
d± ≡
1
σ
√
τ
ln
F
K
±
1
2
σ2
τ
d ≡ d± σ
√
τ
These, conjointly, are the Black-Scholes formula.
8 Compiled on December 12, 2015