FEC
FINANCIAL ENGINEERING CLUB
PRICING EUROPEAN OPTIONS
AGENDA
 Stochastic Processes
 Stochastic Calculus
 Black-Scholes Equation
STOCHASTIC PROCESSES
A SIMPLE PROCESS
 Let 𝑋𝑡 = 1 with probability 𝑝 = 0.5 and 𝑋𝑡 = −1 with probability 1 − 𝑝 = .5 (for
all t) and consider the symmetric random walk, 𝑀𝑡 = 𝑡𝑖=0 𝑋𝑖
 Assume that 𝑋𝑖 ’s are i.i.d.
 𝑋0 = 0
 Both 𝑋 = (𝑋𝑡 : 𝑡 ∈ ℕ ∪ 0 ) and 𝑀 = (𝑀𝑡 : 𝑡 ∈ ℕ ∪ 0 ) are random processes
 A random/stochastic process is (vaguely) just a collection of random variables
 They could be i.i.d.
 They may be correlated—they may even have different distributions
 There is no general theory/application for random processes until more context and
structure is applied
A SIMPLE PROCESS
 Note that 𝑋𝑡 ’s are iid with 𝐸 𝑋𝑡 = .5 ∗ 1 + .5 ∗ −1 = 0
, ∀𝑡 and
Var 𝑋𝑡 = 𝐸 𝑋𝑡 2 = .5 ∗ 1
 Then 𝐸 𝑀𝑡 = 𝐸
𝑡
𝑖=0 𝑋𝑖
Var 𝑀𝑡 = Var
=
2
+ .5 ∗ −1
𝑡
𝑖=0 𝐸[𝑋𝑖 ]
𝑡
𝑖=0 𝑋𝑖
=
2
= 1, ∀𝑡
= 0 and
𝑡
𝑖=0 𝑉𝑎𝑟(𝑋𝑖 )
=𝑡
A SIMPLE PROCESS
 Generally, we care about the increments of a process:
𝑡
𝑀𝑡 − 𝑀𝑠 =
𝑋𝑖
𝑖=𝑠+1
So that 𝐸 𝑀𝑡 − 𝑀𝑠 = 𝐸
𝑡
𝑖=𝑠+1 𝑋𝑖
= 0 , and 𝑉𝑎𝑟 𝑀𝑡 − 𝑀𝑠 = 𝑉𝑎𝑟
𝑡
𝑖=𝑠+1 𝑋𝑖
=𝑡−𝑠
 The symmetric random walk is defined to have independent increments
 A process X is said to have independent increments if, for 0 = 𝑡0 < 𝑡1 < … < 𝑡𝑚 the increments
𝑋𝑡1 − 𝑋𝑡0 , 𝑋𝑡2 − 𝑋𝑡1 , … , 𝑋𝑡𝑚 − 𝑋𝑡𝑚−1 are independent
QUADRATIC VARIATION
 Define the quadratic variation of a sequence 𝑋 up to time 𝑡 as [𝑋, 𝑋]𝑡
= 𝑡𝑖=1(𝑋𝑖 − 𝑋𝑖−1 )2
 This is a path-dependent measure of variation (thus it is random)
 For some unique processes, it may not be random
 For our symmetric random walk, note that a one step increment, 𝑀𝑖 − 𝑀𝑖−1 , is either 1
or −1. Thus
[𝑀, 𝑀]𝑡 =
𝑡
(𝑀𝑖 − 𝑀𝑖−1
𝑖=1
)2
=
𝑡
𝑖=1
1=𝑡
SCALED SYMMETRIC RANDOM WALK
Let 𝑊
𝑛
𝑡 =
𝑀𝑛𝑡
𝑛
be a scaled symmetric random walk
𝑛
 If 𝑛 ∗ 𝑡 is not an integer, 𝑊
integers of 𝑛 ∗ 𝑡
𝑡 is interpolated between the two neighboring
 Like a the symmetric r.w., the scaled symmetric r.w. has independent increments
𝐸𝑊

𝑛
𝑡 −𝑊
[𝑊 (𝑛) , 𝑊 (𝑛) ]𝑡
=
𝑛
𝑠
𝑛𝑡
𝑖=1
=0
𝑊
𝑛
𝑉𝑎𝑟 𝑊
𝑖
𝑛
−𝑊
𝑛
𝑖−1
𝑛
𝑛
2
𝑡 −𝑊
=
𝑛𝑡
𝑖=1
𝑛
𝑠
𝑋𝑖 2
𝑛
=𝑡−𝑠
=
𝑛𝑡 1
𝑖=1 𝑛
=𝑡
BROWNIAN MOTION
 By the central limit theorem 𝑊
Brownian motion
𝑛
𝑡
𝑎.𝑠.
𝑊 𝑡 as 𝑛 → +∞, where 𝑊 𝑡 is a
Properties of B.M.
1) 𝑃 𝑊 0 = 0 = 1
2) 𝑊 has independent increments
3) 𝑊 𝑡 − 𝑊 𝑠 ~𝑁(0, 𝜎 2 𝑡 − 𝑠 ) for 𝑡 ≥ 𝑠 (we have been using B.M. with 𝜎 = 1)
4) 𝑃 𝑊 𝑡 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 = 1
BROWNIAN MOTION
Ex) What is 𝐸 𝑊(𝑡) 𝑊(𝑠)] assuming 𝑠 ≤ 𝑡 (suppose W has parameter σ)
𝐸𝑊 𝑡
𝑊 𝑠 ] = 𝐸 (𝑊(𝑡) − 𝑊 𝑠 ) + 𝑊 𝑠 𝑊 𝑠 ]
= 𝐸 𝑊 𝑡 − 𝑊 𝑠 ) |𝑊 𝑠 ] + 𝐸[𝑊(𝑠) 𝑊 𝑠 ]
=
0
+
𝑊(𝑠) = 𝑊(𝑠)
Ex) What is 𝑅𝑊 𝑠, 𝑡 = 𝐸 𝑊 𝑠 𝑊(𝑡) ?
𝑅𝑊 𝑠, 𝑡 = 𝐸 𝑊 𝑠 𝑊(𝑡) = 𝐸 𝑊 𝑠 − 𝑊 0
= 𝐸
𝑊 𝑠 −𝑊 0
= 𝑠𝜎 2 + 0 ∗ 0 = 𝑠𝜎 2
2
+ 𝐸 𝑊 𝑠 −𝑊 0
𝑊𝑡 − 𝑊𝑠 , 𝑊𝑠 independent
→ 𝑊 is a martingale
𝑊 𝑡 −𝑊 𝑠 +𝑊 𝑠 −𝑊 0
𝑊 𝑡 −𝑊 𝑠
BROWNIAN MOTION
 Note that B.M. is a function and not a sequence of random
variables and so our definition of quadratic variation must be
altered:
Let 𝑃 be a partition of the interval 0, 𝑇 : 𝑡0 , 𝑡1 , … , 𝑡𝑛 with 0
= 𝑡0 < 𝑡1 < … < 𝑡𝑛 = 𝑇. Let 𝑃 = max 𝑡𝑗+1 − 𝑡𝑗 . For a
𝑗
function 𝑓 𝑡 , the quadratic variation of 𝑓 up to time T is
𝑓, 𝑓 𝑇 = lim
𝑃 →0
𝑛−1
𝑗=0
[𝑓 𝑡𝑗+1 − 𝑓(𝑡𝑗 )]2
BROWNIAN MOTION AND QUADRATIC
VARIATION
 Note if 𝑓 has a continuous derivative,
𝑛−1
𝑗=0 [𝑓
𝑡𝑗+1 − 𝑓(𝑡𝑗
)]2
Then 𝑓, 𝑓 𝑇 ≤ lim
=
𝑛−1
𝑗=0
≤
𝑃
𝑃 →0
𝑃
𝑓′
𝑛−1
𝑗=0
𝑛−1
𝑗=0
= lim 𝑃 ∗ lim
𝑃 →0
=0 ∗
𝑃 →0
𝑇 ′
|𝑓
0
2
𝑥𝑗∗
𝑡𝑗+1 − 𝑡𝑗
2
𝑓 ′ 𝑥𝑗∗
𝑓′
𝑥𝑗∗
𝑛−1
𝑗=0
2
𝑡 | 𝑑𝑡 = 0
2
2
𝑓′
(by MVT)
𝑡𝑗+1 − 𝑡𝑗
𝑡𝑗+1 − 𝑡𝑗
𝑥𝑗∗
2
𝑡𝑗+1 − 𝑡𝑗
BROWNIAN MOTION AND QUADRATIC
VARIATION
 For a B.M. 𝑊(𝑡), consider the random variable 𝑊 𝑡𝑗+1 − 𝑊 𝑡𝑗
𝐸
𝑉𝑎𝑟
𝑊 𝑡𝑗+1 − 𝑊 𝑡𝑗
𝑊 𝑡𝑗+1 − 𝑊 𝑡𝑗
− 2(𝑡𝑗+1 − 𝑡𝑗 ) 𝐸
2
2
= 𝑡𝑗+1 − 𝑡𝑗
2
=𝐸
𝑊 𝑡𝑗+1 − 𝑊 𝑡𝑗
𝑊 𝑡𝑗+1 − 𝑊 𝑡𝑗
2
2 −(𝑡
𝑗+1 − 𝑡𝑗 )
2
= 𝐸
𝑊 𝑡𝑗+1 − 𝑊 𝑡𝑗
4
+ (𝑡𝑗+1 − 𝑡𝑗 )2 = 3(𝑡𝑗+1 − 𝑡𝑗 )2 − 2(𝑡𝑗+1 − 𝑡𝑗 )2 + (𝑡𝑗+1 − 𝑡𝑗 )2 = 𝟐(𝒕𝒋+𝟏 − 𝒕𝒋 )𝟐
BROWNIAN MOTION AND QUADRATIC
VARIATION
𝑊 𝑡𝑗+1 −𝑊 𝑡𝑗
. Choose 𝑛 large so that 𝑡𝑗
𝑡𝑗+1 − 𝑡𝑗
2
𝑌𝑗+1
2
𝑡𝑗+1 − 𝑊(𝑡𝑗 )] = 𝑇
𝑛
 Let 𝑌𝑗+1 =
[𝑊
 Then 𝑊, 𝑊 𝑇 = lim
lim
𝑛→∞
2
𝑛−1 𝑌𝑗+1
𝑗=0 𝑛
𝑃 →0
𝑛−1
𝑗=0 [𝑊
𝑡𝑗+1 − 𝑊(𝑡𝑗
=
)]2
𝑗𝑇
. Then 𝑡𝑗+1
𝑛
= lim 𝑇 ∗
𝑛→∞
= 1 by LLN.
 Conclusion 𝑊, 𝑊 𝑇 = lim
𝑃 →0
 𝒅𝑾 𝒕 𝒅𝑾 = 𝒅𝒕
 Similarly, 𝑑𝑡𝑑𝑡 = 0 and 𝑑𝑊𝑑𝑡 = 0
𝑛−1
𝑗=0 [𝑊
𝑡𝑗+1 − 𝑊(𝑡𝑗 )]2 = 𝑇
− 𝑡𝑗 =
2
𝑛−1 𝑌𝑗+1
𝑗=0 𝑛
𝑇
and thus
𝑛
= 𝑇 since
STOCHASTIC CALCULUS
ITO INTEGRAL

𝑇
𝑊
0
Let 𝑊
𝑡 𝑑𝑊 𝑡 = lim
𝑛→∞
𝑗𝑇
𝑛
𝑗𝑇
𝑛−1
𝑊
𝑗=0
𝑛
= 𝑊𝑗 and note that
1
2
𝑊
𝑗+1 𝑇
𝑛
𝑛−1
𝑗=0 (𝑊𝑗+1
−𝑊
=
=
=
𝑛−1
𝑗=0 𝑊𝑗
1
2
𝑊𝑗+1 − 𝑊𝑗 = 𝑊𝑛2 −
1
2
1 𝑛−1
1 𝑛−1
𝑛−1
2
2
𝑊
−
𝑊
𝑊
+
𝑊
𝑗
𝑗+1
𝑗=0
2 𝑗=0 𝑗+1
2 𝑗=0 𝑗+1
1 𝑛
1 𝑛−1
𝑛−1
2
2
𝑊
−
𝑊
𝑊
+
𝑊𝑗+1
𝑗
𝑗+1
𝑗=1
𝑗
𝑗=0
𝑗=0
2
2
1
1 𝑛−1
1 𝑛−1
𝑛−1
2
2
𝑊𝑛2 +
𝑊
−
𝑊
𝑊
+
𝑊𝑗+1
𝑗
𝑗+1
𝑗=0
𝑗
𝑗=0
𝑗=0
2
2
2
1
𝑊𝑛2 + 𝑛−1
𝑊𝑗2 − 𝑛−1
𝑗=0
𝑗=0 𝑊𝑗 𝑊𝑗+1
2
1
𝑊𝑛2 + 𝑛−1
𝑗=0 𝑊𝑗 𝑊𝑗 − 𝑊𝑗+1
2
− 𝑊𝑗 )2 =
=
Thus
𝑗𝑇
𝑛
𝑛−1
𝑗=0 (𝑊𝑗+1
− 𝑊𝑗 )2
ITO INTEGRAL

𝑇
𝑊
0
𝑡 𝑑𝑊 𝑡 = lim
𝑛→∞
=
𝑗𝑇
𝑛−1
𝑊
𝑗=0
𝑛
1
lim 𝑊 2
𝑛→∞ 2
1
2
= 𝑊2 𝑇 −
𝑇 −
1
2
1
2
𝑊
𝑛−1
𝑗=0
𝑗+1 𝑇
𝑛
𝑊
−𝑊
𝑗+1 𝑇
𝑛
𝑗𝑇
𝑛
−𝑊
𝑗𝑇
𝑛
2
𝑊, 𝑊 𝑇
Quadratic Variation
1
2
= 𝑊2 𝑇 −
1
𝑇
2
ITO’S LEMMA
 We seek an approximation 𝑓 𝑇, 𝑊 𝑇
− 𝑓 0, 𝑊 0
By Taylor’s formula we have 𝑓 𝑡𝑗+1 , 𝑥𝑗+1 − 𝑓 𝑡𝑗 , 𝑥𝑗 = 𝑓𝑡 𝑡𝑗 , 𝑥𝑗 𝑡𝑗+1 − 𝑡𝑗 + 𝑓𝑥 𝑡𝑗 , 𝑥𝑗 𝑥𝑗+1
ITO’S LEMMA
 𝑓 𝑇, 𝑊 𝑇
+
+
− 𝑓 0, 𝑊 0
𝑛−1
𝑗=0 𝑓𝑥 𝑡𝑗 , 𝑊 𝑡𝑗
𝑛−1
𝑗=0 𝑓𝑥𝑡 𝑡𝑗 , 𝑊 𝑡𝑗
=
𝑛−1
𝑗=0 𝑓
𝑡𝑗+1 , 𝑊 𝑡𝑗+1
𝑊 𝑡𝑗+1 − 𝑊(𝑡𝑗 ) +
1
2
− 𝑓 𝑡𝑗 , 𝑊 𝑡𝑗
𝑛−1
𝑗=0 𝑓𝑥𝑥
𝑡𝑗+1 − 𝑡𝑗 𝑊 𝑡𝑗+1 − 𝑊(𝑡𝑗 ) +
𝑡𝑗 , 𝑊 𝑡𝑗
1
2
𝑛−1
𝑗=0 𝑓𝑡𝑡
=
𝑛−1
𝑗=0 𝑓𝑡
𝑡𝑗 , 𝑊 𝑡𝑗
𝑊(𝑡𝑗+1 ) − 𝑊(𝑡𝑗 )
𝑡𝑗 , 𝑊 𝑡𝑗
2
𝑡𝑗+1 − 𝑡𝑗
𝑡𝑗+1 ) − 𝑊(𝑡𝑗 )
2
 Now taking limits, lim 𝑓 𝑇, 𝑊 𝑇 − 𝑓 0, 𝑊 0 = 𝑓 𝑇, 𝑊 𝑇 − 𝑓 0, 𝑊 0
𝑛 →∞ 𝑇
𝑇
𝑇
= 0 𝑓𝑡 𝑡, 𝑊(𝑡) 𝑑𝑡 + 0 𝑓𝑥 𝑡, 𝑊 𝑡 𝑑𝑊 𝑡 + 0 𝑓𝑥𝑥 𝑡, 𝑊 𝑡 𝑑𝑡 since 𝑑𝑡𝑑𝑊 𝑡 = 0 and 𝑑𝑡𝑑𝑡 = 0
 In differential form, Ito’s formula is 𝑑𝑓(𝑡, 𝑊 𝑡 = 𝑓𝑡 𝑡, 𝑊 𝑡 𝑑𝑡 + 𝑓𝑥 𝑡, 𝑊 𝑡 𝑑𝑊 𝑡
1
1
+ 2 𝑓𝑥𝑥 𝑡, 𝑊 𝑡 𝑑𝑊 𝑡 𝑑𝑊 𝑡 + 𝑓𝑥𝑡 𝑡, 𝑊 𝑡 𝑑𝑊 𝑡 𝑑𝑡 + 2 𝑓𝑡𝑡 𝑡, 𝑊 𝑡 𝑑𝑡𝑑𝑡
with the last two terms cancelling out to zero
ITO’S LEMMA
 Ex) Suppose 𝑓 𝑡, 𝑥 = 𝑡𝑥 2 . What is 𝑑𝑓(𝑡, 𝑊 𝑡 )?
𝑓𝑡 𝑡, 𝑥 = 𝑥 2
𝑓𝑥 𝑡, 𝑥 = 2𝑡𝑥
𝑓𝑥𝑥 𝑡, 𝑥 = 2𝑡
Then 𝑑𝑓 𝑡, 𝑊 𝑡
= 𝑊 2 𝑡 𝑑𝑡 + 2𝑡𝑊 𝑡 𝑑𝑊 𝑡 +
1
2𝑡
2
𝑑𝑊 𝑡
2
= 𝑾𝟐 (𝒕)
ITO’S LEMMA
 Ex) Suppose 𝑓 𝑡, 𝑥 = sin 𝑡𝑥 . What is 𝑑𝑓(𝑡, 𝑊 𝑡 )?
𝑓𝑡 𝑡, 𝑥 = 𝑥 ∗ cos 𝑡𝑥
𝑓𝑥 𝑡, 𝑥 = 𝑡 ∗ cos(𝑡𝑥)
𝑓𝑥𝑥 𝑡, 𝑥 = −𝑡 2 ∗ sin 𝑡𝑥
Then 𝑑𝑓 𝑡, 𝑊 𝑡 = (𝑥 ∗ cos 𝑡𝑥 )𝑑𝑡 + 𝑡 ∗ cos 𝑡𝑥 𝑑𝑊 𝑡 +
∗ 𝐜𝐨𝐬 𝒕𝒙 − 𝒕𝟐 ∗ 𝐬𝐢𝐧 𝒕𝒙 )𝒅𝒕 + 𝒕 ∗ 𝐜𝐨𝐬 𝒕𝒙 𝒅𝑾 𝒕
−𝑡 2
∗ sin 𝑡𝑥
𝑑𝑊 𝑡
2
= (𝒙
ITO’S LEMMA
 More generally, if 𝑋 𝑡 is a stochastic process
𝑑𝑓 𝑡, 𝑋 𝑡
1
= 𝑓𝑡 𝑡, 𝑋 𝑡 𝑑𝑡 + 𝑓𝑥 𝑡, 𝑋 𝑡 𝑑𝑋 𝑡 + 𝑓𝑥𝑥 𝑡, 𝑋 𝑡
2
𝑑𝑋 𝑡
2
We have been using Ito’s formula to construct stochastic differential equations (SDE’s)—that is,
differential equations with a random term.
Consider the SDE: 𝑑𝑋 𝑡 = 𝜎 𝑡 𝑑𝑊 𝑡 + 𝛼 𝑡 −
If 𝑆 𝑡 = 𝑆 0 exp 𝑋 𝑡 , what is 𝑑𝑆(𝑡)?
1 2
𝜎 (𝑡)
2
𝑑𝑡
ITO’S LEMMA
 Here, 𝑆 𝑡 = 𝑓 𝑡, 𝑥 = 𝑆 0 exp 𝑥
 Note that this is actually just a function of a single variable x
𝑓𝑡 𝑡, 𝑥 = 0
𝑓𝑥 𝑡, 𝑥 = 𝑆 0 exp{𝑥}
𝑓𝑥𝑥 𝑡, 𝑥 = 𝑆 0 exp{𝑥}
Then 𝑑𝑆 𝑡 = 𝑆 0 𝑒 𝑋
𝑡
𝑑𝑋 𝑡 +
1
𝑆
2
0 𝑒𝑋
𝑡
𝑑𝑋 𝑡
2
ITO’S LEMMA
 Note that 𝑑𝑋 𝑡
2
= 𝜎 𝑡 𝑑𝑊 𝑡 + 𝛼 𝑡 −
1 2
𝜎
2
𝑡
𝑑𝑡
2
=
𝜎2
𝑡 𝑑𝑊 𝑡
2
+ 2𝜎 𝑡 𝛼 𝑡
BLACK-SCHOLES EQUATION
BLACK-SCHOLES
 Let the underlying follow this SDE with constant rate and volatility: 𝑑𝑆 𝑡 = 𝛼𝑆 𝑡 𝑑𝑡 + 𝜎𝑆 𝑡 𝑑𝑊 𝑡
 The only variable inputs to an options price are the time until maturity and the price of the stock, so
we start by considering the function
𝑐(𝑡, 𝑆(𝑡))
Ito’s formula tells us 𝑑𝑐 𝑡, 𝑆 𝑡
= 𝑐𝑡 𝑡, 𝑆 𝑡 𝑑𝑡 + 𝑐𝑥 𝑡, 𝑆 𝑡 𝑑𝑆 𝑡 +
1
1
𝑐
2 𝑥𝑥
𝑡, 𝑆 𝑡 𝑑𝑆 𝑡 𝑑𝑆 𝑡
= 𝑐𝑡 𝑡, 𝑆 𝑡 𝑑𝑡 + 𝑐𝑥 𝑡, 𝑆 𝑡 (𝛼𝑆 𝑡 𝑑𝑡 + 𝜎𝑆 𝑡 𝑑𝑊 𝑡 ) + 𝑐𝑥𝑥 𝑡, 𝑆 𝑡 𝜎 2 𝑆 2 𝑡 𝑑𝑡
2
1 2
= 𝑐𝑡 𝑡, 𝑆 𝑡 + 𝛼𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡 + 𝜎 𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡 𝑑𝑡 + 𝜎𝑆(𝑡)𝑐𝑥 𝑡, 𝑆 𝑡 𝑑𝑊(𝑡)
2
BLACK SCHOLES
 We need to take the present value of this so we consider the function:
𝑒 −𝑟𝑡 𝑐(𝑡, 𝑆(𝑡))
= −𝑟𝑒 −𝑟𝑡 𝑐 𝑡, 𝑆 𝑡 𝑑𝑡 + 𝑒 −𝑟𝑡 𝑑𝑐 𝑡, 𝑆 𝑡
1
= 𝑒 −𝑟𝑡 −𝑟𝑐 𝑡, 𝑆 𝑡 + 𝑐𝑡 𝑡, 𝑆 𝑡 + 𝛼𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡 + 𝜎 2 𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
2
−𝑟𝑡
+ 𝑒 𝜎𝑆(𝑡)𝑐𝑥 𝑡, 𝑆 𝑡 𝑑𝑊(𝑡)
Again, by Ito’s formula 𝑑𝑐 𝑡, 𝑆 𝑡
𝑑𝑡
BLACK SCHOLES
 Meanwhile, we try to replicate the option contract as we did in the binomial option pricing model. That is, by
investing some money in a stock position and some in some money market account (a bond):
 Let 𝑋 𝑡 be the value of our portfolio at time 𝑡
 At time 𝑡 we invest a necessary amount ∆ 𝑡 into the stock and the remainder, 𝑋 𝑡 − ∆(𝑡), into the money
market instrument.
 Then we gain ∆ 𝑡 𝑑𝑆(𝑡) from our investment in the stock
 And 𝑟 𝑋 𝑡 − ∆ 𝑡 𝑑𝑡 from our investment in the money market instrument
 Thus 𝑑𝑋 𝑡 = ∆ 𝑡 𝑑𝑆 𝑡 + 𝑟 𝑋 𝑡 − ∆ 𝑡 𝑑𝑡 = ∆ 𝑡 𝛼𝑆 𝑡 𝑑𝑡 + 𝜎𝑆 𝑡 𝑑𝑊 𝑡
 By Ito’s lemma, the differential of the PV(stock) is 𝑑 𝑒 −𝑟𝑡 𝑆 𝑡
+ 𝑟 𝑋 𝑡 − ∆ 𝑡 𝑆 𝑡 𝑑𝑡
= 𝛼 − 𝑟 𝑒 −𝑟𝑡 𝑆 𝑡 𝑑𝑡 + 𝜎𝑒 −𝑟𝑡 𝑆 𝑡 𝑑𝑊 𝑡
 Likewise, the differential of our discounted portfolio is 𝑑 𝑒 −𝑟𝑡 𝑋 𝑡
+ ∆ 𝑡 𝜎𝑒 −𝑟𝑡 𝑆 𝑡 𝑑𝑊 𝑡
= ∆ 𝑡 𝛼 − 𝑟 𝑒 −𝑟𝑡 𝑆 𝑡 𝑑𝑡
BLACK SCHOLES
 At each time 𝑡, we want the replicating portfolio 𝑋 𝑡 to match the value of the option 𝑐 𝑡, 𝑆 𝑡
 We do this by ensuring that 𝑑 𝑒 −𝑟𝑡 𝑋 𝑡
= 𝑑 𝑒 −𝑟𝑡 𝑐 𝑡, 𝑆 𝑡
∆ 𝑡 𝛼 − 𝑟 𝑒 −𝑟𝑡 𝑆 𝑡 𝑑𝑡 + ∆ 𝑡 𝜎𝑒 −𝑟𝑡 𝑆 𝑡 𝑑𝑊 𝑡
= 𝑒 −𝑟𝑡 −𝑟𝑐 𝑡, 𝑆 𝑡
+ 𝑐𝑡 𝑡, 𝑆 𝑡
+ 𝑒 −𝑟𝑡 𝜎𝑆(𝑡)𝑐𝑥 𝑡, 𝑆 𝑡 𝑑𝑊(𝑡)
+ 𝑟𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
+
for all 𝑡 and that 𝑋 0 = 𝑐 0, 𝑆 0 :
1 2
𝜎 𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
2
𝑑𝑡
BLACK SCHOLES
 At each time 𝑡, we want the replicating portfolio 𝑋 𝑡 to match the value of the option 𝑐 𝑡, 𝑆 𝑡
 We do this by ensuring that 𝑑 𝑒 −𝑟𝑡 𝑋 𝑡
= 𝑑 𝑒 −𝑟𝑡 𝑐 𝑡, 𝑆 𝑡
∆ 𝑡 𝛼 − 𝑟 𝑆 𝑡 𝑑𝑡 + ∆ 𝑡 𝜎𝑆 𝑡 𝑑𝑊 𝑡
= −𝑟𝑐 𝑡, 𝑆 𝑡
+ 𝑐𝑡 𝑡, 𝑆 𝑡
+ 𝛼𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
+
for all 𝑡 and that 𝑋 0 = 𝑐 0, 𝑆 0 :
1 2
𝜎 𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
2
𝑑𝑡 + 𝜎𝑆(𝑡)𝑐𝑥 𝑡, 𝑆 𝑡 𝑑𝑊(𝑡)
BLACK-SCHOLES
 At each time 𝑡, we want the replicating portfolio 𝑋 𝑡 to match the value of the option 𝑐 𝑡, 𝑆 𝑡
 We do this by ensuring that 𝑑 𝑒 −𝑟𝑡 𝑋 𝑡
= 𝑑 𝑒 −𝑟𝑡 𝑐 𝑡, 𝑆 𝑡
∆ 𝑡 𝛼 − 𝑟 𝑆 𝑡 𝑑𝑡 + ∆ 𝒕 𝝈𝑺 𝒕 𝒅𝑾 𝒕
= −𝑟𝑐 𝑡, 𝑆 𝑡
+ 𝑐𝑡 𝑡, 𝑆 𝑡
+ 𝛼𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
Need ∆ 𝒕 𝝈𝑺 𝒕 = 𝝈𝑺 𝒕 𝒄𝒙 𝒕, 𝑺 𝒕
+
for all 𝑡 and that 𝑋 0 = 𝑐 0, 𝑆 0 :
1 2
𝜎 𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
2
→ ∆ 𝒕 = 𝒄𝒙 𝒕, 𝑺 𝒕
𝑑𝑡 + 𝝈𝑺(𝒕)𝒄𝒙 𝒕, 𝑺 𝒕 𝒅𝑾(𝒕)
BLACK-SCHOLES
 At each time 𝑡, we want the replicating portfolio 𝑋 𝑡 to match the value of the option 𝑐 𝑡, 𝑆 𝑡
 We do this by ensuring that 𝑑 𝑒 −𝑟𝑡 𝑋 𝑡
= 𝑑 𝑒 −𝑟𝑡 𝑐 𝑡, 𝑆 𝑡
∆ 𝒕 𝜶 − 𝒓 𝑺 𝒕 𝒅𝒕 + ∆ 𝑡 𝜎𝑆 𝑡 𝑑𝑊 𝑡
= −𝒓𝒄 𝒕, 𝑺 𝒕
+ 𝒄𝒕 𝒕, 𝑺 𝒕
+ 𝜶𝑺 𝒕 𝒄𝒙 𝒕, 𝑺 𝒕
Need ∆ 𝑡 𝜎𝑆 𝑡 = 𝜎𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
+
for all 𝑡 and that 𝑋 0 = 𝑐 0, 𝑆 0 :
𝟏 𝟐
𝝈 𝒕 𝑺𝟐 𝒕 𝒄𝒙𝒙 𝒕, 𝑺 𝒕
𝟐
𝒅𝒕 + 𝜎𝑆(𝑡)𝑐𝑥 𝑡, 𝑆 𝑡 𝑑𝑊(𝑡)
→ ∆ 𝑡 = 𝑐𝑥 𝑡, 𝑆 𝑡
1
Need∆ 𝑡 𝛼 − 𝑟 𝑆 𝑡 = −𝑟𝑐 𝑡, 𝑆 𝑡 + 𝑐𝑡 𝑡, 𝑆 𝑡 + 𝛼𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡 + 2 𝜎 2 𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
BLACK-SCHOLES
 At each time 𝑡, we want the replicating portfolio 𝑋 𝑡 to match the value of the option 𝑐 𝑡, 𝑆 𝑡
 We do this by ensuring that 𝑑 𝑒 −𝑟𝑡 𝑋 𝑡
= 𝑑 𝑒 −𝑟𝑡 𝑐 𝑡, 𝑆 𝑡
∆ 𝒕 𝜶 − 𝒓 𝑺 𝒕 𝒅𝒕 + ∆ 𝑡 𝜎𝑆 𝑡 𝑑𝑊 𝑡
= −𝒓𝒄 𝒕, 𝑺 𝒕
+ 𝒄𝒕 𝒕, 𝑺 𝒕
+ 𝜶𝑺 𝒕 𝒄𝒙 𝒕, 𝑺 𝒕
Need ∆ 𝑡 𝜎𝑆 𝑡 = 𝜎𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
Need 𝑐𝑥 𝑡, 𝑆 𝑡
𝟏 𝟐
𝝈 𝒕 𝑺𝟐 𝒕 𝒄𝒙𝒙 𝒕, 𝑺 𝒕
𝟐
𝒅𝒕 + 𝜎𝑆(𝑡)𝑐𝑥 𝑡, 𝑆 𝑡 𝑑𝑊(𝑡)
→ ∆ 𝑡 = 𝑐𝑥 𝑡, 𝑆 𝑡
𝛼 − 𝑟 𝑆 𝑡 = −𝑟𝑐 𝑡, 𝑆 𝑡
Simplifying this we need, 𝑟𝑐 𝑡, 𝑆 𝑡
+
for all 𝑡 and that 𝑋 0 = 𝑐 0, 𝑆 0 :
+ 𝑐𝑡 𝑡, 𝑆 𝑡
= 𝑐𝑡 𝑡, 𝑆 𝑡
+ 𝛼𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
+ 𝑟𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
1
+
1 2
𝜎
2
𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
+ 2 𝜎 2 𝑡 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
BLACK-SCHOLES
𝑟𝑐 𝑡, 𝑆 𝑡
With
= 𝑐𝑡 𝑡, 𝑆 𝑡
+ 𝑟𝑆 𝑡 𝑐𝑥 𝑡, 𝑆 𝑡
1
+ 𝜎 2 𝑆 2 𝑡 𝑐𝑥𝑥 𝑡, 𝑆 𝑡
2
∀𝑡 ∈ 0, 𝑇 , 𝑥 ≥ 0
𝑐 𝑇, 𝑥 = max{𝑥 − 𝐾, 0}
Is the Black-Scholes-Merton partial differential equation. Its is a backward parabolic equation, which are known to
have solutions. Using the fact that, 𝑐𝑡 𝑡, 0 = 𝑟𝑐 𝑡, 0 , we solve this ODE: 𝑐 𝑡, 0 = 𝑒 −𝑟𝑡 𝑐 0,0 = 𝑒 −𝑟𝑡 ∗ 0 = 0. This
gives us our first boundary condition at 𝑥 = 0:
𝑐 𝑡, 0 = 0, ∀𝑡
Additionally, lim 𝑐 𝑡, 𝑥 − 𝑥 − 𝑒 −𝑟
𝑥→∞
𝑇−𝑡
𝐾
= 0, ∀𝑡 ∈ 0, 𝑇
That is, the fact that as the underlying approaches ∞, the call option begins to look like the underlying minus the
discounted strike. This serves as the second boundary condition.
BLACK-SCHOLES
 Solving the Black-Scholes-Merton PDE gives us the familiar results:
𝑆, 𝑡 = 𝑁 𝑑1 𝑆 − 𝑁 𝑑2 𝐾𝑒 −𝑟(𝑇−𝑡)
𝑃 𝑆, 𝑡 = 𝑁 −𝑑2 𝐾𝑒 −𝑟(𝑇−𝑡) − 𝑁 −𝑑1 𝑆
1
𝑆
𝑑1 =
∗ 𝑙𝑛
+
𝐾
𝜎 𝑇−𝑡
1
𝑆
𝑑2 =
∗ 𝑙𝑛
+
𝐾
𝜎 𝑇−𝑡
𝜎2
𝑟+
2
𝜎2
𝑟−
2
𝑇−𝑡
𝑇−𝑡
𝑁 𝑥 is the
standard-normal
CDF of x
BLACK-SCHOLES
 Why doesn’t this method work for American options?
 Early exercise is not modeled!
 Pros
 Gives an analytical (no algorithms necessary!) solution to the value of a European option
 This is simple enough to be extended
 The resulting PDE’s can be solved numerically
 Cons
 Some unrealistic assumptions about rates and volatilities does not match data
 Normal distribution has thin tales under-approximates large returns in stocks
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President
Greg Pastorek
gfpastorek@gmail.com
Internal Vice President
Matthew Reardon
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