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8/28/14
Dynamic Equity Models 1
Dynamic Equity Models
Learning Objec>ves
¨ Simula>on ¤ Daily, monthly, annual sta>s>cal rela>onships
¨ Lognormal probability density ¨ Stochas>c differen>al equa>on ¨ Con>nuous >me price process ¨ Exact solu>on ¨ Price and return probabili>es in con>nuous >me ¨ Probability basics for op>on deriva>ves
2
More Simula>on 3
Perform a stock price simula>on for which current stock price, S0 = $40.00, the expected monthly con>nuously compounded mean rate of return, u, is 1%, and the expected standard devia>on, s, is 5%. Perform the simula>on with daily >me increments for one year. Use floa>ng point >me, annualized, µ and σ, sta>s>cs. Run the simula>on 10,000 >mes.
years 000.1T
years 004.2521t
%321.1712s
%000.1212u
=
==Δ
=⋅=σ
=⋅=µ
004.17321.z 004.12.t004.t
tσz tμttt
eS S
eS S
⋅⋅+⋅⋅+
Δ⋅⋅+Δ⋅⋅Δ+
=
=
Simula>on: 4
$0
$5
$10
$15
$20
$25
$30
$35
$40
$45
$50
$55
$60
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Stock Price
Time [years]
004.17321.z 004.12.t004.t
tσz tμttt
eS S
eS S⋅⋅+⋅
⋅+
Δ⋅⋅+Δ⋅⋅Δ+
=
=
8/28/14
Dynamic Equity Models 2
Simula>on: 5
-‐6% -‐5% -‐4% -‐3% -‐2% -‐1% 0% 1% 2% 3% 4% 5% 6%Natural Log Daily Return Rate
From Simulation DailyMean rate: u 0.04859%Standard deviation: s 1.09460%
Simula>on: 6
$20 $25 $30 $35 $40 $45 $50 $55 $60 $65 $70 $75 $80 $85 $90 $95Stock Price At 1 Year
M[ST] 45.09$ E[ST] 45.91$ Min[ST] 23.95$ Max[ST] 93.91$
From input
$45.78 e$40.00
eS]E[S
$45.10 e$40.00
eS]M[S
1.0.135
Tμ0T
1.0.12
Tμ0T
*
=
⋅=
⋅=
=
⋅=
⋅=
⋅
⋅
⋅
⋅
The median price is the 5,000th in an ordered list of 10,000 simulated prices at T=1.0 years. The expected price is the average of the 10,000 prices.
From simula>on
Lognormal PDF 7
The lognormal pdf is • Asymmetric
• Mode, median, and mean not equal
• Never nega>ve • Over >me the mode,
median, and mean driZ further apart
• Over >me the distribu>on skews more posi>vely
In the standard price theory Simple rates, future value factors, and asset prices are distributed lognormal
Return Rate and Future Value Factor PDFs 8
[ ][ ] uvEvM
==
( )2su,N~ v
( ) s ,N~ 2µ
[ ][ ]µ=
µ
EM
( ) e~e2u,sNv
[ ][ ] *uv
uv
eeE
eeM
=
=
( )2,Nv12 e~e σµ⋅
[ ] [ ] *
eeE eeM v12v12 µ⋅µ⋅ ==
8/28/14
Dynamic Equity Models 3
[ ]tt,μN~SS
ln
tσztμSS
ln
tσztμ)ln(S)ln(S
wdσtμ)ln(S)ln(S
dwσdt dln(S)
2
0
t
0
t
t0t
0t
⋅σ⋅⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅⋅+⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅⋅+⋅+=
⋅+⋅+=
⋅+⋅µ=
Exact Solu>on 9
The differen>al equa>on for dln(S) is
The solu>on with ini>al condi>on is
At >me t the natural log of price ln(St) is distributed normally as Therefore
[ ]
[ ]
[ ]
[ ][ ] t
t
tt
tσ t,μN0t
tσ t,μ)ln(SNt
0t
*
*
*0
eSE
eSM
eS~S
e~S
tσ t,μ)ln(SN~)ln(S
⋅µ
⋅µ
⋅⋅
⋅⋅+
=
=
⋅
⋅⋅+
Simula>on 10
( )
[ ]
[ ][ ]
[ ] [ ] [ ]( ) f for Variance fEfEfVar
f for Median eM[f]
f for moment 2 e fE
f for moment 1 efE
f for moment k efE
e~er1f
22
u
nds2u22
st2su
th2skukk
)N(u,sv
2
2
22
2
−=
=
=
=
=
=+≡
⋅+⋅
+
⋅+⋅
1ea
)a1ln(u2suu
*u
*
2*
−=
+=
+=
1eg
)g1ln(uu −=
+=
[ ] [ ] [ ] [ ]( ) fEfEfVarrVard 222 −===
Specified Rate of return: u 1.0%Standard deviation, s 5.0%Annual frequency, m 12
Computed
Variance, s2 0.00250 Expected rate of return, u* 1.12500%Expected first moment of f 1.01131 Expected second moment of f 1.02532 Simple mean rate, a 1.13135%Geometric rate, g 1.00502%Simple standard deviation, d 5.05973%
Monthly Statistics
Simula>on 11
[ ]
[ ][ ][ ] [ ] [ ]( )22
222
2
2kkk
fEfEfVar
e fE
efE
efE
2
2
22
−=
=
=
=
σ⋅+µ⋅
σ+µ
σ⋅+µ⋅
1e
)1ln(2
*
*
2*
−=α
α+=µ
σ+µ=µ
µ
1e
)1ln(
−=γ
γ+=µµ
[ ] [ ] [ ] [ ]( ) fEfEfVarVar 222 −==α=δ
Specified Rate of return: u 1.0% µ 12.00000%Standard deviation, s 5.0% σ 17.32051%Annual frequency, m 12
Computed
Variance, s2 0.00250 σ2 0.03000 Expected rate of return, u* 1.12500% µ* 13.50000%Expected first moment of f 1.01131 1.14454 Expected second moment of f 1.02532 1.34986 Simple mean rate, a 1.13135% α 14.45368%Geometric rate, g 1.00502% γ 12.74969%Simple standard deviation, d 5.05973% δ 19.97357%
Monthly Statistics Annual Statistics
Computed
Computed
Simula>on 12
Specified Computed Rate of return: u 1.0% µ 12.00000% µ Δt 0.04762%Standard deviation, s 5.0% σ 17.32051% σ √Δt 1.09109%Annual frequency, m 12 m 252
Computed Computed
Variance, s2 0.00250 σ2 0.03000 σ2 t 0.00012 Expected rate of return, u* 1.12500% µ* 13.50000% µ
∗ Δt 0.05357%Expected first moment of f 1.01131 1.14454 1.00054 Expected second moment of f 1.02532 1.34986 1.00119 Simple mean rate, a 1.13135% α 14.45368% 0.05359%Geometric rate, g 1.00502% γ 12.74969% 0.04763%Simple standard deviation, d 5.05973% δ 19.97357% 1.09171%
Monthly Statistics Annual Statistics
Daily Statistics
Computed
Computed
8/28/14
Dynamic Equity Models 4
Daily Sta>s>cs 13
( )
( ).04763%g
10.45$g140$]M[S
.04762%u 10.45$e40eS]M[S
%05356.u
78.45$e40$eS]E[S
%.05357a 45.78$a140$]E[S
252T
252umu0T
*
252umu0T
mT
**
=
=+⋅=
=
=⋅=⋅=
=
=⋅=⋅=
=
=+⋅=
⋅⋅
⋅⋅
Price as a Stochas>c Diff Eqn 14
ΔwσΔtμSΔS * ⋅+⋅= ΔtzΔw
SSS ttt
⋅=
−=Δ Δ+
dwσSdtμSdS * ⋅⋅+⋅⋅= dtzdw ⋅=
Difference eqn for price as geometric Brownian mo>on with posi>ve expected rate of return
Transform to a differen>al eqn as Δt -‐> dt with the goal to solve the eqn for price, S
( )SfF =
To understand stochas>c differen>al, dS, introduce F which is a func>on of stochas>c process, S. S is dependent on Weiner process, w.
µ: con>nuously compounded natural log mean rate of return µ*: con>nuously compounded simple mean or expected rate of return
Stochas>c Differen>al, dF 15
terms order higher dSSF
21dS
SFdt
tFdF 2
2
2
+∂
∂⋅+
∂
∂+
∂
∂=
2*2
2* dw)SσdtS(μ
SF
21 dw)SσdtS(μ
SF dt
tF dF ⋅⋅+⋅⋅
∂
∂⋅+⋅⋅+⋅⋅
∂
∂+
∂
∂=
dwSσSFdtSσ
SF
21
tFSμ
SFdF 22
2
2* ⋅⋅
∂
∂+⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅⋅
∂
∂⋅+
∂
∂+⋅⋅
∂
∂=
Ignore dt2 and dw·∙dt terms and subs>tute dw2 = dt which will be explained on the next slide.
Write dF as a Taylor series expansion
Subs>tute dS into dF
Stochas>c Differen>al, dF 16
[ ] [ ] [ ] 0zEdtdtzEdwE =⋅=⋅= [ ] [ ] [ ] dtzEdt)dtz(EdwE 222 =⋅=⋅=
[ ] [ ]( )[ ]
[ ] dt1dtzEdt
0dtzE
dwEdwE)dw(VAR
2
2
22
=⋅=⋅=
−⋅=
−= ( )[ ] [ ]( )[ ] ( )
[ ]0dt3dt
dtzEdt
dtdtzE
dwEdwE)(dw VAR
22
242
224
22222
=−⋅=
−⋅=
−⋅=
−=
dw ∼ N(0,dt) dw2 ∼ N(dt,0) Stochas>c Determinis>c
Determine: E[dW], E[dW2], VAR[dW], VAR[dW2] to resolve dw2 = dt
8/28/14
Dynamic Equity Models 5
Probability Distribu>ons Related to dw and dw2 17
-‐4 -‐3 -‐2 -‐1 0 1 2 3 4
0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 20 25
Z distribu>on
Z2 distribu>on
Z4 distribu>on
Solve For Price 18
dwσdtμ
dwσdt2σμ
dwSσS1 dtS)σ
S1(
210Sμ
S1
dwSσS
ln(S) dtSσSln(S)
21
tlnSSμ
Sln(S) dln(S)
2*
222
*
222
2*
⋅+⋅=
⋅+⋅⎟⎟⎠
⎞⎜⎜⎝
⎛−=
⋅⋅+⋅⎟⎠⎞
⎜⎝⎛ ⋅−++⋅⋅=
⋅⋅∂
∂+⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅
∂
∂⋅+
∂
∂+⋅
∂
∂=
dwSσSFdtSσ
SF
21
tFSμ
SFdF 22
2
2* ⋅⋅
∂
∂+⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅⋅
∂
∂⋅+
∂
∂+⋅⋅
∂
∂=
( )SlnF =
Price differen>al eqn
This differen>al equa>on cannot be solved analy>cally, but can be solved under a change of variable, S. Ln(S) can be solved for
.
Solu>on For Price 19
[ ]
[ ]t
2σμ
0tμ
0t
tσ , tμNt
tσz tμt
tσz tμtt
2
*
*
0
*
0
t
eSeSSE
eS~ S
eS S
eS S
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛+
⋅
⋅⋅⋅
⋅⋅+⋅⋅
Δ⋅⋅+Δ⋅⋅Δ+
⋅=⋅=
=
=
[ ]
t2σμ
0tμ
0t
2
0
t
0
t
0t
ttt
2*
eSeS]M[S
tt,μN~SS
ln
tσztμSS
ln
tσztμ)ln(S)ln(S
tσztμ)ln(S)ln(S
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛−
⋅
Δ+
⋅=⋅=
⋅σ⋅⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅⋅+⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅⋅+⋅+=
Δ⋅⋅+Δ⋅+=
Log and Expecta>on Operators 20
[ ] tμ)ln(S)ln(SE
tμSS
lnE
tσztμSS
ln
0t
0
t
t0
t
⋅+=
⋅=⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅⋅+⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛ [ ]
[ ]( ) ( ) tμSlnSEln
eSS
E
eSSE
*t
tμt
tμ0t
0
*
0
*
⋅+=
=⎥⎦
⎤⎢⎣
⎡
⋅=
⋅
⋅
[ ]( ) [ ]
[ ]( ) [ ] ( )
[ ]( ) [ ])ln(SE SEln
2tσ
tμ-‐μ)ln(SESEln
tμ)ln(SEtμSEln
tt
2
*tt
t*
t
>
⋅=
⋅=−
⋅+=⋅+
2σμμ
2σμμ
2*
2*
=−
+=
Note nonlinearity of expecta>on and natural log Start with natural log of price, Start with price expecta>on, then take expected value then take natural log
8/28/14
Dynamic Equity Models 6
Simula>on: Probability of Median and Mean Price
21
[ ] %4995.0SSPr
0011.0 1.12
1.0.12$40.00$45.09ln
T
TSSln
z
MEDT
0
MEDT
0
=<
−=⋅
⋅−⎟⎠
⎞⎜⎝
⎛
=⋅σ
⋅µ−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
[ ] %071.54SSPr
1022.0 1.12
1.0.12$40.00$45.91ln
T
TSSln
z
EXPT
0
EXPT
0
=≤
=⋅
⋅−⎟⎠
⎞⎜⎝
⎛
=⋅σ
⋅µ−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
Simula>on: Probability of Min and Max Price 22
[ ] %013.SSPr
6547.3 1.12
1.0.12$40.00$23.95ln
T
TSSln
z
MINT
0
minT
0
=≤
−=⋅
⋅−⎟⎠
⎞⎜⎝
⎛
=⋅σ
⋅µ−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
[ ] %001.SSPr
2347.4 1.12
1.0.12$40.00$93.91ln
T
TSSln
z
MAXT
0
MAXT
0
=≤
=⋅
⋅−⎟⎠
⎞⎜⎝
⎛
=⋅σ
⋅µ−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
Probability of a Price Decline 23
16064.4 52125.0
52108.
21.10175.87ln
Tσ
TμSSln
z 0
T
0
−=
⋅
⋅−⎟⎠⎞
⎜⎝⎛
=⋅
⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
Using the IBM equity price sta>s>cs of µ=8% and σ = 25% (Topic 9) , what was the probability of the drop in IBM price during the week ending October 10, 2008? IBM stock opened Monday October 6th at $101.21, ST, and closed Friday October 10th at $87.75, S0. Recall that the IBM return sta>s>cs were computed from January 1962 to September 2008.
That weekly decline was expected once in 1,212 years
[ ] %00159.)16064.4(N~)z(N~SSPr 00T =−==≤
[ ]( )0
0T
zN~SSPr =≤
Probability of Not Exceeding a Cri>cal Value 24
An investor owns 100 shares of an equity with a current price per share of $40.00. The equity has an expected rate of return µ*=16% and annual standard devia>on σ = 20%. What is the probability that the investor’s $4,000, S0, will grow to no more than $6,000, K, aZer 5 years?
14.0% 2
20%16.0% 2σμμ
22* =−=−=
[ ] %51.25)0.65860(N~)z(N~KSPr
0.65860 5.0.2
5.0.14$4,000$6,000ln
T
TSKln
z
0T
00
=−==≤
−=⋅
⋅−⎟⎠
⎞⎜⎝
⎛
=⋅σ
⋅µ−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
[ ]( )0
T
zN~KSPr =≤ [ ]
( )2T
zN~KSPr =>
8/28/14
Dynamic Equity Models 7
Probability of a Loss of Value 25
What is the probability that the investor will have a loss aZer 5 years? ( S0 = K = $4,000 )
The probability of a loss is 5.88%
[ ] 5.88%1.56525)(N~)(zN~KS Pr
1.56525 5.0.2
5.0.14$4,000$4,000ln
Tσ
TμSKln
z
0T
00
=−==≤
−=⋅
⋅−⎟⎠
⎞⎜⎝
⎛
=⋅
⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
[ ]( )0
T
zN~KSPr =≤ [ ]
( )2T
zN~KSPr =>
Probability of Exceeding a Cri>cal Value 26 26
An investor owns 100 shares of an equity with a current price per share of $40.00. The equity has an expected rate of return µ*=16% and annual standard devia>on σ = 20%. What is the probability that the investor’s $4,000, S0, will grow to more than $6,000, K, aZer 5 years?
The probability that the value of the shares exceeds $6,000 is 74.49%
[ ] %49.74)Z(N~)Z(N~)Z(N~1KSPr 200T ==−=−=>
( )
0.65860 5.0.2
5.0.14$6,000$4,000ln
Tσ
Tσ5.μKSln
Z
2*0
2
=⋅
⋅+⎟⎠
⎞⎜⎝
⎛
=⋅
⋅⋅−+⎟⎠⎞
⎜⎝⎛
≡
0.65860 5.0.2
5.0.14$4,000$6,000ln
Tσ
TμSKln
Z 00
−=⋅
⋅−⎟⎠
⎞⎜⎝
⎛
=⋅
⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛
= [ ]( )0
T
zN~KSPr =≤ [ ]
( )2T
zN~KSPr =>
Simple Binary Op>on 27
A security, C, is offered as follows: If an equity, S, currently priced at $40, S0, exceeds $45, $K, aZer one year (T=1.0), then the buyer of this security, C, will receive $K, if the equity, S, is less than or equal to K, then the buyer will receive nothing. The annual standard devia>on of the equity, σ, is 20% and the annual expected risk free rate of return, r*, is 6%. If ST > K, then CT = K If ST ≤ K, then CT = 0
[ ] ( )( )
78.14$34867.45$e
.38892-‐N~45$e
dN~KeCEeC
06.
106.
2Tr
TTr
0
**
=⋅⋅=
⋅⋅=
⋅⋅=⋅=
−
⋅−
⋅−⋅−
( )
( )38892.
12.
12..5.064540ln
Tσ
Tσ.5rKSln
d
2
2*0
2
−=⋅
⋅⋅−+⎟⎠⎞
⎜⎝⎛
=
⋅
⋅⋅−+⎟⎠⎞
⎜⎝⎛
=
[ ] [ ]( )2
TT
dN~K
KSPrKCE
⋅=
>⋅=
The fair value of this security known as a “cash or nothing call op>on” is $14.78
[ ]( )0
T
dN~KSPr =≤ [ ]
( )2T
dN~KSPr =>
Confidence Intervals 28
$32.84 e$40.00
eSS
$57.17 e$40.00
eSS
0.50.21.959960.5.16
Tσ1.95996TμT
0.50.21.959960.5.16
Tσ1.95996TμT
*
0
*
0
=
=
=
=
=
=
⋅⋅−⋅⋅
⋅⋅−⋅⋅
−
⋅⋅+⋅⋅
⋅⋅+⋅⋅
+
Confidence Level (1-‐α)
α α/2 -‐Z +Z
90% 10% 5.00% -‐1.64485 1.6448595% 5% 2.50% -‐1.95996 1.9599699% 1% 0.50% -‐2.57583 2.57583
What are the upper and lower bounds on a future stock price for which one is 95% (=1-‐α) confident? St+ and St-‐ are the upper and lower bounds at >me T = 0.5 years
( )95996.1N~ −
8/28/14
Dynamic Equity Models 8
Value at Risk (VaR) 29
What is the maximum loss that an investor would expect over some >me period t ? For example, what is the maximum loss expected with 95% confidence from owning an equity over a 10 day period? The equity has µ*= 16%, σ = 20%, and S0 = $40.00. Unlike the confidence interval, which uses a two tailed confidence , VaR is a one-‐tail interval.
Confidence Level (1-‐α)
α -‐Z
90% 10% -‐1.2815595% 5% -‐1.6448599% 1% -‐2.32635
70.37$ e00.40$
eSS
252102.01.64485
2521016.
Tσ1.64485TμT
*
0
=
=
=
⋅⋅−⋅⋅
⋅⋅−⋅⋅−
( )64485.1N~ −
Value at Risk (VaR) 30
The minimum 95% confident price is $37.67, thus the 95% maximum expected loss is $3.63 or value at risk, VaR
And commonly approximated for short >me periods as follows
$5.6634.34$00.40$VaR =−=
( )
$2.30
e100.40$
e1SVaR
252102.01.64485
2521016.
TσzTμ0
*
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−⋅=
−⋅=
⋅⋅−⋅
⋅⋅+⋅
$2.54
e100.40$
e1SVaR
252102.01.64485
0
TσzT*μ
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−⋅=
⎟⎠⎞⎜
⎝⎛ −⋅=
⋅⋅−
⋅⋅+⋅
VaR is computed directly as follows
Expected Value Exceeding Cri>cal Value 31
The same problem as last slide, but now -‐ what is the expected value of the equity posi>on given that the cri>cal value, K, has been exceeded?
[ ] [ ] ( )( )
( )( )2
1Tμ0
2
1TTT
zN~zN~eS
zN~zN~SEKS|SE
*
⋅⋅=
⋅=>
⋅
( )
( )Tσ
Tσ5.μKSln
z
Tσ
Tσ5.μKSln
z
2*0
2
2*0
1
⋅
⋅⋅−+⎟⎠⎞
⎜⎝⎛
=
⋅
⋅⋅++⎟⎠⎞
⎜⎝⎛
=
0.658605.0.2
5.0.14$6,000$4,000ln
z
10581.15.0.2
5.0.18$6,000$4,000ln
z
2
1
=⋅
⋅+⎟⎠
⎞⎜⎝
⎛
=
=⋅
⋅+⎟⎠
⎞⎜⎝
⎛
=
[ ]
344,10$ 074492.086560.16.8902$
074492.086560.e000,4$000,6$S|SE 516.
TT
=
⋅=
⋅=> ⋅
The deriva>on details are not included in this course.
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
$0 $20 $40 $60 $80 $100 $120 $140 $160 $180 $200
Example: Price Distribu>on at >me T (5Yrs) 32
[ ]
[ ]447214. , .3888794N
Tσ , Tμ)ln(SNT
e~
e~S 0 ⋅⋅+
K=$60
E[ST]=$89.02
Median[ST]=$80.55
E[ST|ST>K]=$103.44
Mode[ST]=$65.95
S0
8/28/14
Dynamic Equity Models 9
Another Simple Binary Op>on 33
A security, C, is offered as follows: If an equity currently priced at $40, S0, exceeds $45, K, aZer exactly one year (T=1.0), then the buyer of this security will receive the price of the equity, ST, if the equity, S, is less than or equal to K, then the buyer will receive nothing. If ST > K, then CT = ST If ST ≤ K, then CT = 0
[ ]( ) ( )
00.17$42509.40$ .18892-‐N~40$ dN~S
CEeC
10
TTr
0
*
=⋅=
⋅=⋅=
⋅= ⋅−
[ ] [ ] [ ]
( ) [ ] ( )( )
( )1Tr0
2
1T2
TTTT
dN~eS
dN~dN~SEdN~
KS|SEKSPrCE
*
⋅⋅=
⋅⋅=
>⋅>=
⋅ The fair value of this security known as a “asset or nothing call op>on” is $14.78
( )
( )18892.
12.
12..5.064540ln
Tσ
Tσ.5rKSln
d
2
2*0
1
−=⋅
⋅⋅++⎟⎠⎞
⎜⎝⎛
=
⋅
⋅⋅++⎟⎠⎞
⎜⎝⎛
=
[ ]( )0
T
dN~KSPr =≤ [ ]
( )2T
dN~KSPr =>
0
0.01
0.02
0.03
0.04
0.05
$10 $20 $30 $40 $50 $60 $70 $80 $90
Comparing the Two Binary Op>ons
¨ cash or nothing call op>on ¨ asset or nothing call op>on 34
[ ] [ ] [ ]
( ) [ ] ( )( )
( )
[ ]( )10
Tr
0
1Tr
0
2
1T2
TTTT
dN~S
CEe C
dN~eS
dN~dN~SEdN~
KS|SEKSPrCE
T*
*
⋅=
⋅=
⋅⋅=
⋅⋅=
>⋅>=
⋅−
⋅
[ ] [ ]( )
[ ]
( )2Tr0
T
2
TT
dN~KeC
KS|KEK
dN~K
KSPrKCE
*
⋅⋅=
>=
⋅=
>⋅=
⋅−
[ ]( ) ( )20
T
d-‐N~dN~KSPr
=
=≤ [ ]( )2
T
dN~KSPr =>
KST >KST ≤
[ ]KS|SE TT >
K
Essen>al Concepts
35
Appendix: Probability and Expecta>on Summary 36
[ ]( ) ( ) ( )
[ ] [ ] ( )( )
[ ] ( )
[ ] [ ] ( )( )2
1TTT
2T
2
1TTT
002
T
dN~dN~SEKS|SE
dN~KSPr
zN~zN~SEKS|SE
zN~1z-‐N~zN~KSPr
⋅=>
=>
⋅=>
−==
=>[ ]( ) ( )
( )
[ ]( )2
T
2
20
T
d-‐N~KSPr
zN~1
z-‐N~zN~KSPr
=≤
−=
=
=≤
Risk Neutral
Risk Neutral
[ ][ ] E Pr Risk neutral probability
Risk neutral expecta>on
8/28/14
Dynamic Equity Models 10
Appendix: 1 Tail Confidence 37
90% 95% 99% Confidence
95% confident that return rate lies above the shaded area
Appendix: 2 Tail Confidence 38
90% 95% 99% Confidence
95% confident that return rate lies between the shaded areas
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