AMF271: FINANCIAL DERIVATIVES
First Semester, SY 2012-2013
Emmanuel Cabral
Finite Difference Method
Let 𝑢 be a function on [𝑎, 𝑏] such that
𝑑𝑢
𝑑𝑥
and
𝑑2 𝑢
𝑑𝑥 2
exists for all 𝑥 ∈ (𝑎, 𝑏)
Subdivide [𝑎, 𝑏] using the partition points 𝑎 = 𝑥0 , 𝑥1 , 𝑥2 … , 𝑥𝑛 = 𝑏
Let 𝑢 = 𝑓(𝑥)
𝑢𝑖 = 𝑓(𝑥𝑖 ) for 𝑖 = 1,2, … , 𝑛
The first and second-order finite difference formulas that approximate the first and second-order derivatives are given by the
following:
𝑏−𝑎
∆𝑥 =
𝑛
First-order Derivatives
Forward Difference
𝑢𝑖+1 − 𝑢𝑖
𝐷𝑢𝑖 =
∆𝑥
Remark: Slope of tangent is approximated by secant
Backward Difference
𝐷𝑢𝑖 =
𝑢𝑖 − 𝑢𝑖−1
∆𝑥
Central Difference
Second-order Derivatives
Forward Difference
1 𝑢𝑖+1 − 𝑢𝑖 𝑢𝑖 − 𝑢𝑖−1
𝐷𝑢𝑖 = [
+
]
2
∆𝑥
∆𝑥
𝑢𝑖+1 − 𝑢𝑖−1
=
2∆𝑥
𝐷𝑢2𝑖
𝑢𝑖+2 − 𝑢𝑖+1 𝑢𝑖+1 − 𝑢𝑖
−
∆𝑥
∆𝑥
=
∆𝑥
𝑢𝑖+2 − 2𝑢𝑖+1 + 𝑢𝑖
=
∆𝑥 2
CHAPTER 20: BASIC NUMERICAL PROCEDURES
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AMF271: FINANCIAL DERIVATIVES
Backward Difference
𝐷𝑢2𝑖
Centered Difference
Emmanuel Cabral
𝑢𝑖 − 𝑢𝑖−1 𝑢𝑖−1 − 𝑢𝑖−2
−
∆𝑥
∆𝑥
=
∆𝑥
𝑢𝑖 − 2𝑢𝑖−1 + 𝑢𝑖−2
=
∆𝑥 2
𝑢𝑖+1 − 𝑢𝑖 𝑢𝑖 − 𝑢𝑖−1
−
∆𝑥
∆𝑥
∆𝑥
𝑢𝑖+1 − 2𝑢𝑖 + 𝑢𝑖−1
=
∆𝑥 2
𝐷𝑢2𝑖 =
Now let 𝑓 be a function of variables 𝑡 and 𝑆. Let 𝑡 ∈ [0, 𝑇] and 𝑆 ∈ [0, 𝑆max ]
Remark: The graph of 𝑓 is a surface above the 𝑆max × 𝑇 plane.
1
Subdivide [0, 𝑇] into 𝑁 subintervals each of length
𝑁
So [0, 𝑇] is partitioned into
0 = 𝑡0 < 𝑡1 < 𝑡2 < ⋯ < 𝑡𝑛 = 𝑇
Subdivide [0, 𝑆max ] into 𝑀 + 1 equally spaced stock prices
0, 𝑆1 , 𝑆2 , … , 𝑆𝑀 = 𝑆max
where 𝑆𝑗+1 − 𝑆𝑗 = ∆𝑆
CHAPTER 20: BASIC NUMERICAL PROCEDURES
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AMF271: FINANCIAL DERIVATIVES
∆𝑆 =
Emmanuel Cabral
𝑆max − 0
𝑇
,
∆𝑡 =
𝑀+1
𝑁
𝑆1 = 1 ∙ ∆𝑆
𝑆𝑗 = 𝑗∆𝑆
𝑓𝑖,𝑗 = 𝑓(𝑡𝑖 , 𝑆𝑗 ) = 𝑓(𝑖∆𝑡, 𝑗∆𝑆)
𝑓𝑖,𝑗+1 − 𝑓𝑖,𝑗−1
𝜕𝑓
(𝑡𝑖 , 𝑆𝑗 ) =
⏟ 2∆𝑆
𝜕𝑆
centered
difference
𝑓𝑖+1,𝑗 − 𝑓𝑖,𝑗
𝜕𝑓
(𝑡 , 𝑆 ) =
⏟ ∆𝑡
𝜕𝑡 𝑖 𝑗
forward
difference
𝑓𝑖,𝑗+1 − 𝑓𝑖,𝑗 𝑓𝑖,𝑗 − 𝑓𝑖,𝑗−1
−
∆𝑆
∆𝑆
∆𝑆
𝑓𝑖,𝑗+1 − 2𝑓𝑖,𝑗 + 𝑓𝑖,𝑗−1
=
∆𝑆 2
𝜕2𝑓
(𝑡 , 𝑆 ) =
𝜕𝑆 2 𝑖 𝑗
Substitute the above finite difference expressions in the Black-Scholes PDE
𝜕𝑓
𝜕𝑓 1 2 2 𝜕 2 𝑓
+ (𝑟 − 𝑞)𝑆
+ 𝜎 𝑆
= 𝑟𝑓
𝜕𝑡
𝜕𝑆 2
𝜕𝑆 2
where 𝑞 is the continuous dividend yield.
Let 𝑓 be the value of a put option.
First assume European.
The boundary conditions are as follows:
𝑓(𝑇, 𝑆) = (𝐾 − 𝑆𝑇 , 0)+
Implicit Method
𝑓𝑖+1,𝑗 − 𝑓𝑖,𝑗
𝑓𝑖,𝑗+1 − 𝑓𝑖,𝑗−1 1 2 2 2 𝑓𝑖,𝑗+1 − 2𝑓𝑖,𝑗 + 𝑓𝑖,𝑗−1
+ (𝑟 − 𝑞) 𝑗∆𝑆
+ 𝜎 𝑗⏟∆𝑆
= 𝑟𝑓𝑖,𝑗
⏟
∆𝑡
2∆𝑆
2
∆𝑆 2
2
𝑆𝑗
𝑆𝑗
1
1
(𝑓𝑖+1,𝑗 − 𝑓𝑖,𝑗 ) + (𝑟 − 𝑞)𝑗∆𝑡(𝑓𝑖,𝑗+1 − 𝑓𝑖,𝑗−1 ) + 𝜎 2 𝑗 2 ∆𝑡(𝑓𝑖,𝑗+1 − 2𝑓𝑖,𝑗 + 𝑓𝑖,𝑗−1 ) = 𝑟∆𝑡𝑓𝑖,𝑗
2
2
CHAPTER 20: BASIC NUMERICAL PROCEDURES
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AMF271: FINANCIAL DERIVATIVES
First Semester, SY 2012-2013
𝑓𝑖+1,𝑗
Emmanuel Cabral
1
1
1
1
= −𝑓𝑖,𝑗−1 [ 𝜎 2 𝑗 2 ∆𝑡 − (𝑟 − 𝑞)𝑗∆𝑡] + 𝑓𝑖,𝑗 [1 + 𝜎 2 𝑗 2 ∆𝑡 + 𝑟∆𝑡] + 𝑓𝑖,𝑗+1 [− (𝑟 − 𝑞)𝑗∆𝑡 − 𝜎 2 𝑗 2 ∆𝑡]
2
2
2
2
𝑓𝑖+1,𝑗 = 𝑎𝑗 𝑓𝑖,𝑗−1 + 𝑏𝑗 𝑓𝑖,𝑗 + 𝑐𝑗 𝑓𝑖,𝑗+1
where
1
1
(𝑟 − 𝑞)𝑗∆𝑡 − 𝜎 2 𝑗 2 ∆𝑡
2
2
𝑏𝑗 = 1 + 𝜎 2 𝑗 2 ∆𝑡 + 𝑟∆𝑡
1
𝑐𝑗 = − [(𝑟 − 𝑞)𝑗∆𝑡 + 𝜎 2 𝑗 2 ∆𝑡]
2
𝑎𝑗 =
At the boundary nodes,
𝑓𝑁,𝑗 = max(𝐾 − 𝑗∆𝑆, 0) ,
𝑗 = 0,1, … , 𝑀
𝑓𝑖,0 = 𝐾,
𝑖 = 0,1, … , 𝑁
𝑓𝑖,𝑀 = 0,
𝑖 = 0,1, … , 𝑁
Remark: Values of 𝑓 at the inner nodes are unknown.
𝑗
𝑗
𝑗
⋮
𝑗
𝑗
= 0:
= 1:
= 2:
𝑓𝑁−1,0 = 𝑓𝑁,0 = 𝐾 (BC)
𝑎1 𝑓𝑁−1,0 + 𝑏1 𝑓𝑁−1,1 + 𝑐1 𝑓𝑁−1,2 = 𝑓𝑁,1
𝑎2 𝑓𝑁−1,1 + 𝑏2 𝑓𝑁−1,2 + 𝑐2 𝑓𝑁−1,3 = 𝑓𝑁,2
= 𝑀 − 1:
= 𝑀:
𝑎𝑀−1 𝑓𝑁−1,𝑀−2 + 𝑏𝑀−1 𝑓𝑁−1,𝑀−1 + 𝑐𝑀−1 𝑓𝑁−1,𝑀 = 𝑓𝑁,𝑀−1
𝑓𝑁−1,𝑀 = 𝑓𝑁,𝑀 = 𝐾 (BC)
⟹We have a system of 𝑀 + 1 equations with 𝑀 − 1 unknown (values of 𝑓 at interior nodes)
So, to get the values of 𝑓𝑁−1,𝑗 for 𝑗 = 1,2, … , 𝑀 − 1, we need to solve the system
𝑓𝑁−1,0
𝑓𝑁,0
𝑀+1
⏞𝑎1 𝑏1 𝑐1 0 …
𝑓
𝑓𝑁,1
0
0
𝑁−1,1
𝑓
𝑓𝑁,2
0 𝑎2 𝑏2 𝑐2 …
0
0
𝑁−1,2
⋮
⋮
𝑀 − 1 0 0 𝑎3 𝑏3 …
=
0
0
𝑓𝑁−1,𝑀−2
𝑓𝑁,𝑀−2
⋮
⋮
⋮
⋮ ⋱
⋮
⋮
[ 0 0 0 0 … 𝑏𝑀−1 𝑐𝑀−1 ] 𝑓𝑁−1,𝑀−1
𝑓𝑁,𝑀−1
{
[ 𝑓𝑁−1,𝑀 ] [ 𝑓𝑁,𝑀 ]
Add the two boundary conditions to make an 𝑀 + 1 × 𝑀 + 1 matrix
𝐴
⏞1
𝑎1
0
0
⋮
0
[0
0
𝑏1
𝑎2
0
⋮
0
0
0
𝑐1
𝑏2
𝑎3
⋮
0
0
0
0
𝑐2
𝑏3
⋮
0
0
…
…
…
…
⋱
…
…
0
0
0
0
⋮
0
0
0
0
⋮
𝑎𝑀−1
0
𝑏𝑀−1
0
0
0
0
0
⋮
𝑓𝑁−1,0
𝑓𝑁−1,1
𝑓𝑁−1,2
⋮
𝑓𝑁−1,𝑀−2
𝑐𝑀−1 𝑓𝑁−1,𝑀−1
1 ] [ 𝑓𝑁−1,𝑀 ]
=
𝑓𝑁,0
𝑓𝑁,1
𝑓𝑁,2
⋮
𝑓𝑁,𝑀−2
𝑓𝑁,𝑀−1
[ 𝑓𝑁,𝑀 ]
That is,
CHAPTER 20: BASIC NUMERICAL PROCEDURES
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AMF271: FINANCIAL DERIVATIVES
𝑓 𝑁−1 =
and 𝐴𝑓 𝑁−1 = 𝑓 𝑁 , where 𝑓 𝑁 is known.
⟹ 𝑓 𝑁−1 = 𝐴−1 𝑓 𝑁 assuming 𝐴 is invertible
𝑓𝑁−1,0
𝑓𝑁−1,1
𝑓𝑁−1,2
⋮
, 𝑓𝑁 =
𝑓𝑁−1,𝑀−2
𝑓𝑁−1,𝑀−1
[ 𝑓𝑁−1,𝑀 ]
Emmanuel Cabral
𝑓𝑁,0
𝑓𝑁,1
𝑓𝑁,2
⋮
𝑓𝑁,𝑀−2
𝑓𝑁,𝑀−1
[ 𝑓𝑁,𝑀 ]
Proceeding backwards, we have
𝑓 𝑁−2 = 𝐴−1 𝑓 𝑁−1
⋮
𝑓 1 = 𝐴−1 𝑓 2
Algorithm for European (Implicit Method)
1. Input values of 𝑓(𝐵, 𝐶), 𝐴 (given from parameter)
2. For 𝑖 = 𝑁 − 1 to 1
𝑓 𝑖 = 𝐴−1 𝑓 𝑖
Algorithm for American (Implicit Method)
1. Input values of 𝑓(𝐵, 𝐶), 𝐴 (given from parameter)
2. For 𝑖 = 1 to 𝑁 − 1
𝑓 𝑖 = 𝐴−1 𝑓 𝑖
For 𝑗 = 1 to 𝑀 − 1
𝑓(𝑗) = max(𝑓(𝑗), 𝐾 − 𝑗∆𝑆)
Control Variate Technique
𝑓𝐴 = 𝑓𝐴∗ − 𝑓𝐵∗ + 𝑓𝐵
where 𝑓𝐴∗ : Value of American option obtained from simulation (e.g., finite difference)
𝑓𝐵∗ : Value of European option obtained from simulation (e.g., finite difference)
𝑓𝐵 : Value of European option obtained analytically (Black-Scholes)
Example
𝑓𝐴∗ = 4.07
𝑓𝐵∗ = 3.91
𝑓𝐵 = 4.08
⟹ 𝑓𝐴 = (4.07 − 3.91) − 4.08
Convertible Bonds
Convertible Bond: Bondholder has the option to convert the bond into equity
Callable Bond
Example 26.1
Modify the problem by letting 𝜆 = 0 (no default risk)
1
𝑢 = 𝑒 𝜎√∆𝑡 , 𝑑 =
𝑎 = 𝑒 𝑟∆𝑡 , ∆𝑡 =
𝑝=
𝑎−𝑑
𝑢−𝑑
𝑢
3
,𝑇=
12
9
12
,𝑞 = 1−𝑝
Homework:
CHAPTER 20: BASIC NUMERICAL PROCEDURES
5
First Semester, SY 2012-2013
1.
2.
AMF271: FINANCIAL DERIVATIVES
Emmanuel Cabral
Construct a binomial tree to price the convertible bond
Price the same convertible bond using finite difference.
Non-Dividend Stock
where 𝑓 is the price of the derivative
𝜕𝑓
𝜕𝑓 1 2 2 𝜕 2 𝑓
+ 𝑟𝑆
+ 𝜎 𝑆
= 𝑟𝑓
𝜕𝑡
𝜕𝑆 2
𝜕𝑆 2
Boundary Condition
𝑓(𝑇, 𝑆𝑇 ) = max(2𝑆𝑇 , 100)
Finite Difference Methods (Summary)
Implicit Finite Difference Method
Explicit Finite Difference Method
Comparison Between Implicit and Explicit Methods
Other Remarks
- d
The Explicit Method and its Relationship with the Trinomial Model
Change of Variable
For better computational efficiency, we can let Z = ln S so that the BS PDE becomes
Let 𝑍 = ln 𝑆
𝜕𝑓 𝜕𝑓 𝜕𝑍
=
𝜕𝑆 𝜕𝑍 𝜕𝑆
𝜕𝑓 1
=
𝜕𝑍 𝑆
𝜕𝑓
𝜕𝑓
1
𝜕2 𝑓
BS: + 𝑟𝑆 + 𝜎 2 𝑆 2 2 = 𝑟𝑓
𝜕𝑡
𝜕𝑆
2
𝜕𝑆
Implied Volatility
Let 𝑐𝐵𝑆 be the price (Black-Scholes) of a European call
𝑐𝑚 be the market price of the same European call
where 𝑆0 , 𝐾, 𝑇, 𝑟 are given
Implied volatility is the volatility implied from observed option prices.
Given 𝑐𝑚 , 𝑆0 , 𝐾, 𝑇, 𝑟, what is 𝑐𝑚 ? (Not the same as historical volatilitly)
In practice, traders use BS model to estimate 𝜎imp from 𝑐𝑚 .
Historical volatility is backward looking.
Implied volatility is forward looking, i.e., it reflects the market’s opinion about the volatility of a particular stock (market
outlook)
CHAPTER 20: BASIC NUMERICAL PROCEDURES
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AMF271: FINANCIAL DERIVATIVES
Emmanuel Cabral
In theory, 𝜎 should be independent of 𝐾 and 𝑇.
In practice, 𝜎 changes with 𝐾 and 𝑇.
𝜎 vs 𝐾: volatility smile
𝜎 vs 𝑇: volatility term structure
𝜎 vs 𝐾, 𝑇 → volatility surface
Option prices are usually quoted in terms of volatility.
(Derivagem can be used to estimate implied volatility for a given 𝐾, 𝑆0 )
To get the volatility smile for many 𝐾 values, we use a program, e.g., scilab
Numerical Method for Implied Volatility
Newton’s Method of Root Finding
Suppose we want to solve the equation 𝑓(𝑥) = 𝑘, where 𝑘 is a constant. We want the root of this equation.
First, let 𝐹(𝑥) = 𝑓(𝑥) − 𝑘. We are solving 𝐹(𝑥) = 0
Assuming that 𝐹 is differentiable at 𝑥0 , we get the tangent line to 𝐹 at 𝑥0 :
slope of tangent = 𝐹 ′ (𝑥0 ) =
𝑦−𝐹(𝑥0 )
𝑥−𝑥0
(𝑥, 𝑦) is an arbitrary point on the tangent line.
This tangent line crosses the 𝑥 − axis at
𝑥1 = 𝑥0 −
𝐹(𝑥0 )
𝐹 ′ (𝑥0 )
So 𝑥1 becomes the new estimate of the root.
Repeat the process:
𝑥𝑛+1 = 𝑥𝑛 −
𝐹(𝑥𝑛 )
𝐹 ′ (𝑥𝑛 )
Now, let 𝑐 = market price of a European call with given 𝑆0 , 𝐾, 𝑇, 𝑟.
Let 𝑓(𝜎) = 𝑐 where 𝑓 is now the BS formula
𝐹(𝜎) = 𝑓(𝜎) − 𝑐.
𝑓(𝜎) − 𝑐
𝑓(𝜎) − 𝑐
𝜎𝑛+1 = 𝜎𝑛 −
= 𝜎𝑛 −
′
(𝜎)
𝐹
𝑓 ′ (𝜎)
Also,
𝑓(𝜎) = 𝑆0 ℕ(𝑑1 ) − 𝐾𝑒 −𝑟𝑇 ℕ(𝑑2 )
𝑆
𝜎2
𝑆
𝜎2
ln ( 0 ) + (𝑟 + 𝑇)
ln ( 0 ) + (𝑟 − 𝑇)
𝐾
2
𝐾
2
𝑑1 =
, 𝑑2 =
𝜎√𝑇
𝜎√𝑇
CHAPTER 20: BASIC NUMERICAL PROCEDURES
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𝑓 ′ (𝜎) = 𝑆0
1
√2𝜋
𝑑12
𝑒− 2
AMF271: FINANCIAL DERIVATIVES
Emmanuel Cabral
𝜕𝑑1
1 −𝑑22 𝜕𝑑2
∙
−𝐾
𝑒 2 ∙
⏟
⏟
𝜕𝜎
𝜕𝜎
√2𝜋
?
?
Next exercise: Plot a volatility smile. (Currency option or equity option)
Implied Volatility
Let 𝑐𝐵𝑆 , 𝑝𝐵𝑆 be the European call and put prices respectively from Black-Scholes formula. Let 𝑐𝑚 and 𝑝𝑚 be the corresponding
call and put market prices.
In the absence of arbitrage opportunities, we have
𝑐𝑚 + 𝐾𝑒 −𝑟𝑇 = 𝑝𝑚 + 𝑆0 𝑒 −𝑞𝑇
where 𝑞 = continuous dividend yield
Put-call parity can also be verified for the Black-Scholes model. That is
𝑐𝐵𝑆 + 𝐾𝑒 −𝑟𝑇 = 𝑝𝐵𝑆 + 𝑆0 𝑒 −𝑞𝑇
⟹ 𝑝𝐵𝑆 − 𝑝𝑚 = 𝑐𝐵𝑆 − 𝑐𝑚
Discrepancy between BS and MKT prices should be the same for both call and put with the same strike and maturity.
Remark: 𝜎imp is the volatility, which when substituted into the BS model gives 𝑐𝐵𝑆 = 𝑐𝑚 .
 If 𝜎 = 𝜎imp , we have 𝑐𝐵𝑆 − 𝑐𝑚 = 𝑝𝐵𝑆 − 𝑝𝑚 = 0. Note that 𝑝𝐵𝑆 can be obtained from 𝑐𝐵𝑆 via BS model using 𝜎 = 𝜎imp . So
𝑝𝑚 = 𝑝𝐵𝑆 with 𝜎 = 𝜎imp
 𝜎imp should be the same for both put and call with the same time to maturity 𝑇, and strike price 𝐾.
 ∀𝑇, 𝐾, the correct volatility to use with BS model to price the European call should be the same as that used to price
the European put.
Volatility Smile: 𝝈𝐢𝐦𝐩 vs 𝑲
The smile shifts with 𝑆0 , the current asset price.
The smile is more stable if it is taken as the relationship with 𝐾⁄𝑆 , i.e., 𝜎imp vs 𝐾⁄𝑆
0
0
Volatility Term Structure: 𝝈𝐢𝐦𝐩 vs 𝑻
Traders allow implied volatility to depend on the remaining time to maturity.
Homework
1. For a given maturity 𝑇, construct a volatility smile, i.e., 𝜎imp vs 𝐾⁄𝑆
0
2.
3.
4.
Do this for other maturities 𝑇2 , 𝑇3 , … , 𝑇𝑛
Steps 1 and 2 will help create your volatility surface (similar to table in p.417)
Use your volatility surface to price a call or put with intermediate maturity and intermediate 𝐾⁄𝑆 value using 20
dimensional linear interpolation. (Create your own hypothetical example.)
Remarks
1. BS model is a sophisticated interpolation tool used by traders for ensuring that an option is priced consistently with
market prices of other actively traded options.
2. Replacing the BS model may change the volatility surface but actual prices quoted in the market would not change
appreciably.
The Implied Risk-Neutral Probability
(Implied Probability Distribution of an Underlying Asset Price)
BS Model assumes lognormal risk-neutral probability distribution of asset prices
𝑑𝑆𝑡
= 𝑟𝑑𝑡 + 𝜎𝑑𝑊
𝑆𝑡
1 2
)𝑇+𝜎𝑊
𝑆𝑡 = 𝑆0 𝑒 (𝑟−2𝜎
𝑡
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First Semester, SY 2012-2013
AMF271: FINANCIAL DERIVATIVES
Emmanuel Cabral
1
ln(𝑆𝑇 ) ~𝒩 (ln 𝑆0 + (𝑟 − 𝜎 2 ) 𝑇 , 𝜎 2 𝑇)
2
Actual market prices give a slightly different risk-neutral probability distribution.
Determining Implied Risk-Neutral Probability Distributions from Volatility Smile
The price of a European call on an asset with strike of 𝐾 and maturity 𝑇 is given by
𝑐 = 𝑒 −𝑟𝑇 𝔼[(𝑆
⏟ 𝑇 − 𝐾)+ ]
∞
under risk neutral
prob distribution
= ∫ (𝑆𝑇 − 𝐾)𝑔(𝑆𝑇 )𝑑𝑆𝑇
𝐾
where 𝑔(𝑆𝑇 ) = risk-neutral pdf of 𝑆𝑇 . What is 𝑔?
∞
𝜕𝑐
= −𝑒 −𝑟𝑇 ∫ 𝑔(𝑆𝑇 )𝑑𝑆𝑇
𝜕𝐾
𝐾
𝜕2𝑐
−𝑟𝑇
= 𝑒 𝑔(𝐾),
by First Fundamental Theorem of Calculus
𝜕𝐾 2
𝜕2𝑐
𝑔(𝐾) = 𝑒 𝑟𝑇
𝜕𝐾 2
Let 𝑐 = 𝑐(∙, 𝐾)
𝜕2𝑐
=
𝜕𝐾 2
𝑐(∙, 𝐾 + 𝛿) − 𝑐(∙, 𝐾) 𝑐(∙, 𝐾) − 𝑐(∙, 𝐾 − 𝛿)
−
𝛿
𝛿
𝛿
𝑐
𝑐
𝑐
1
2
3
⏞
⏞ 𝐾) + ⏞
𝑐(∙, 𝐾 + 𝛿) − 2 𝑐(∙,
𝑐(∙, 𝐾 − 𝛿)
=
𝛿2
where 𝑐1 , 𝑐2 , 𝑐3 are three European call prices with strike 𝐾 + 𝛿, 𝐾, 𝐾 − 𝛿 respectively.
Then,
𝑐1 − 2𝑐2 + 𝑐3
𝑔(𝐾) = 𝑒 𝑟𝑇 [
]
𝛿2
is the pdf evaluated at 𝐾
Read Example 19A.1
HW: From one of the volatility smiles that you constructed, obtain a risk-neutral probability distribution for the asset
price.
Example 19A.1
𝑔(𝐾) = 𝑒 𝑟𝑇
Let 𝑆𝑇 be in one of the intervals [𝐾 − 𝛿, 𝐾 + 𝛿]
𝛿 = 0.5
Assume that 𝑔 is constant between 𝐾 = 6 and 𝐾 = 7
Let 𝑆𝑇 be inside [6,7]
𝑔(𝑆𝑇 ) = 𝑒 𝑟𝑇
𝜕2𝑐
𝜕𝐾 2
(𝑐1 − 2𝑐2 + 𝑐3 )
0.52
Exercise 17.20
Spot call option 𝑐 =?
3
𝑇=
12
𝐹0 = 12
CHAPTER 20: BASIC NUMERICAL PROCEDURES
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First Semester, SY 2012-2013
AMF271: FINANCIAL DERIVATIVES
Emmanuel Cabral
𝐾 = 13
𝑟 = 4%
𝜎 = 25%
Black’s Model for futures call: 𝑐 = [𝐹0 ℕ(𝑑1 ) − 𝐾ℕ(𝑑2 )]𝑒 −𝑟𝑇
where 𝑑1 =
ln(
12
0.252 3
)+(4+
)( )
13
2
12
0.25√
3
12
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