ACTL3182-Formula-Sheet PDF

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Utility Theory 

Risk Aversion



Absolute Risk Aversion



Relative Risk Aversion



Downside semi-variance

U ( E [W ] )> E [ U ( W ) ] −u'' ( w ) −d A ( w )= ' ( log u' ( w ) ) = dw u (w ) R ( w)=wA ( w ) μ

semivar ( X )= ∫ ( x−μ )2 f X ( x ) dx −∞

I



ES=∫ ( I −x ) f X ( x ) dx

Expected shortfall

∞

 Portfolio Theory

σ 2P =w' Σ w



Portfolio variance



Portfolio Lagrangian



Minimum Variance Frontier

γ=

1 L ( w , λ , γ ) = σ P2 + λ ( 1−w ' 1 )+ γ (μ−w' z ) 2 2 C−μB A μ P−2 B μ P + C , 2 λ= σ P = λ+γ μ P= Δ Δ

μA −B Δ w



First order condition in



Global Minimum Variance



Tangency/market portfolio weights

w=λ Σ−1 1+γ Σ−1 z 1 2 Σ−1 1 , σ P= w g= A A 1 −1 Σ ( z−r f 1 ) w t= B− A r f

Capital Asset Pricing Model

Capital Market Line



Market portfolio variance



Portfolio-portfolio covariance



Asset-market covariance



Security market line



Efficient portfolio



CAPM pricing



Risk decomposition

β

(

)

μ M −r f σP σM 1 2 σ M= ( z −r ) B− A r f m f ' σ i , j=wi Σ w j 1 σ ¿ , M =Σ w M = ( z−r f 1) B− A r f z i=r f + β i ( z M −r f ) σi β i= σM E[ X ] p= 1+r f + β ( E [ r M ] −r f ) μP =r f +



σ i2 =β2i σ2M +σ ξ2

i

Factor Models 

Single Factor Model

r i=α i + β i f +ε i

/

E [ r i ]=α i + β i μ f

,

σ i2 =β2i σ2f +σ ε2



Single Factor variance



Single Factor covariance



Single Factor risk decomposition



Single Factor portfolio parameters



Diversification ratio



Multi Factor Model

i

2 f

σ i , j=β i β j σ σ i2 =β2i + σ 2ε ' ' ' α P =w α , β P =w β , ε P ,t =w ε t β2P σ2f Systematic Risk 2 RP = 2 = Total Risk σP K

K

r i=α i + ∑ β i ,k f k + ε i

E [ r i ]=α i + ∑ β i ,k E [ f k ]

/

k=1

k=1

Arbitrage Pricing Theory 

Single Factor APT Model



Single Factor APT Factor Price



Multi Factor APT Factor Price



Solving for

λ

r i=ai+b1 ,i f 1+ ε i , E [ f 1]=0 E [ r i ]=λ 0 +b1 ,i λ1 E [ r i ]=λ 0 +λ1 b1 ,i + λ 2 b 2 ,i +…

and

[ ] [][ ]

1 b1 c1 B= 1 b2 c2 1 b3 c3

s

[] [ ]

E [r 1 ] λ0 B λ1 = E [r 2 ] E [r 3 ] λ2

,

σ f =1 1

,

so

E[ r 1 ] λ0 −1 λ1 =B E[ r 2 ] λ2 E[ r 3 ]



[ w 0 w1 w2 ] B=[ 1 b c ]

Replicating portfolios

[ w 0 w1 w2 ]=[1 b c ] B−1

,

Discrete Time Derivative Pricing

c t + K e−r



Put-Call Parity



Discrete time:

ϕ



Discrete time:

ψ



Discrete time:

f

f now



Discrete time:

q

q now



Discrete time:

v

v now

( T−t )

ϕnow

= pt + S t f −f ¿ up down s up −sdown

1 e−rδt ( f up −ϕ sup ) B ( now ) ¿ ϕnow s now +ψ now B(now) s erδt −s down ¿ now sup−s down B (0 ) ¿ f now=E Q X B( T) ¿

ψ now

[

]

Continuous Time Derivative Pricing

dS ( t )=μ St dt +σ St dW (t )



GBM dynamics



GBM solution at time

t



GBM solution at time

T



Ito’s Lemma

(e μ−21σ )t +σW ( t ) 2

d ( f ( X ( t ) , t) ) =

S ( t ) =S 0 given

t

(μ− 12 σ )√T−t +σW ( T −t ) 2

S ( T ) =S ( t ) e

2 2 1 ∂ f 1 ∂2 f ∂f ∂f ( dt )2 ( ) + dX (t) dX (t )+ dt+ 2 ∂t 2 2 ∂ x2 ∂t ∂x



d W Q ( t ) =dW ( T )+γ ( t ) dt

Girsanov Theorem

T

1

T



Girsanov Theorem Radon-Nikodym



Martingale Representation theorem

dQ −∫γ ( t) dW (t ) −2 ∫ γ (t )dt =e dP dN (t )=ϕ( t ) dM (t )



Standard Normal trick

E [ e√ v Z 1 ( Z ∈ B) ] =exp

0

2

0

( v2 ) P [ ~Z +√ v ∈ B ] S( t ) 1 + r + σ (T −t ) ln ( K ) ( 2 ) d= 2



d1

and

d2

notation

,

1

d 2=d 1−σ √T −t

σ √ T −t...