Mathematical Finance

MA3269

Ryan Heng

November 15, 2014

Chapter 1

Expected Utility

1.1 Definitions

Discrete X

EU(X + w0) =

n∑i=1

piU(xi + w0)

Continuous X

EU(X + w0) =

∫ b

a

f(x)U(x+ w0) dx

1.2 Utility function

A utility function U is strictly increasing. So,

U(x) > U(y), ∀x > y

andU(x) = U(y) =⇒ x = y .

U is concave (strictly concave) if

U(αx+ (1− α)y) ≥ (>) αU(x) + (1− α)U(y)

and convex (strictly convex) if

U(αx+ (1− α)y) ≤ (<) αU(x) + (1− α)U(y) .

If U is strictly concave (e.g. U(x) =√x), the individual is risk averse. Also,

U ′′ < 0, and by Jensen’s Inequality,

E(U(X)) < U(E(X)) .

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1.3 Certainty Equivalent

The certainty equivalent is c = CE(X;U), where

U(c) = E(U(w0 +X)) .

The individual is indifferent between investing in X and having a final wealthof c.

1.4 Risk Premium

The risk premium is r = RP(X;U), where

U(w0 − r) = E(U(w0 +X)) .

Clearly, r = w0 − c.

1.5 Investment decision

Invest EU(X + wo) > U(w0) c > w0 r < 0Avoid EU(X + wo) < U(w0) c < w0 r > 0Indifferent EU(X + wo) = U(w0) c = w0 r = 0

1.6 Arrow-Pratt Risk Aversion

The ARA of a utility function U is

UARA = −U′′

U ′= −(lnu′)′ .

A risk averse individual has UARA > 0. If ∀w, UARA > VARA, we say that U isglobally more risk averse than V. This is true if and only if there is an increasingand strictly concave function g such that

U(w) = g(V (w)) .

1.7 Positive affine transformation

V is a positive affine transformation of U if

V = αU + β, ∀α > 0, β ∈ R .

Let V be a positive affine transformation of U. Then U is concave (convex) ifand only if V is concave (convex). Thus they have the same certainty equivalentand make the same investment decisions. Also, UARA = VARA if and only if Uand V are positive transformations of each other.

1.8 Portfolio selection

An individual will choose to invest a portion of his initial wealth in a riskyinvestment X that has a random rate of return, r, so as to maximise

E(U(W )) = E(U(w0(1 + αr))), α ∈ [0, 1] .

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Chapter 2

Markowitz’s PortfolioTheory & CAPM

2.1 Definitions

cov(ri, rj) = σij , cov(rp, rq) = wTpCwq

corr(ri, rj) = ρij =σijσiσj

µp =

n∑i=1

wiµi = µTw = wTµ

σ2p =

n∑i=1

n∑j=1

wiwjσij = wTCw

2.2 Useful identities

∂

∂wi(σ2p) = 2

n∑j=1

wjσij ,∂

∂w(wTCw) = 2Cw

∂

∂w(wT v) = v, ∀v

cov(rp, rg) =1

a

2.3 Minimum-variance frontier

σ2 =a(µ− b

a )2 + (c− b2

a )

ac− b2(a bb c

)=

(1TC−11 1TC−1µ1TC−1µ µTC−1µ

)wg =

C−11

1TC−11, σg =

1√a, µg =

b

a

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2.4 Two Fund Theorem

Let

w1 = wg =C−11

1TC−11and w2 =

C−1µ

1TC−1µ.

If b = 1TC−1µ > 0, then Fund 2 is efficient. So, any portfolio αw1+(1−α)w2,where α < 1, is efficient.

2.5 One Fund Theorem

If portfolio p lies on the CML, then µp = αrm + (1− α)rf , for some α > 0. So,

α =µp − rfrm − rf

= βp, σp = ασm .

wp =(µp − rf )C−1(µ− rf1)

(µ− rf1)TC−1(µ− rf1)=

(µp − rf )C−1(µ− rf1)

c− 2brf + ar2f

rf < µg =b

a

wtan =C−1(µ− rf1)

b− rfa

µtan =c− rfbb− rfa

σ2tan =

c− 2rfb+ ar2f(b− rfa)2

σ =|µ− rf |√

c− 2brf + ar2f

(CML), µp =µtan − rfσtan

σp + rf (efficient CML)

SRp =µp − rfσp

2.6 Capital Asset Pricing Model

We take the market portfolio wm to be wtan. So, the CML can be written as

µp − rf =σpσm

(µm − rf ) .

µp−rf is the risk premium of portfolio p. In general, CAPM holds for any asseti or porfolio p.

µi − rf =σimσ2m

(µm − rf ) = βi(µm − rf ), rf =µi − βiµm

1− βi

µp − rf =σpmσ2m

(µm − rf ) = βp(µm − rf )

βp =

n∑i=1

wiβi = wTβ, σpm = cov(

n∑i=1

wiri, rm)

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Chapter 3

European Option Theory

3.1 Bounds on Option Prices

max{0, S0 −Ke−rT } ≤ C ≤ S0

max{0,Ke−rT − S0} ≤ P ≤ Ke−rT

Let K2 > K1. Then,C(K2) ≤ C(K1)

C(K2)− C(K1) ≤ e−rT (K2 −K1)

P (K2) ≥ P (K1)

P (K2)− P (K1) ≤ e−rT (K2 −K1)

Let 0 < K1 < K2. P (K) is convex in K. So,

P (αK1 + (1− α)K2) ≤ αP (K1) + (1− α)P (K2), ∀α ∈ (0, 1)

3.2 Put-Call Parity

C +Ke−rT = P + S0

3.3 Binomial Option Pricing

The replicating portfolio consists of ∆ units of shares and $B of risk-free asset.

∆ =Fu1 − F d1S0(u− d)

, B =u · F d1 − d · Fu1er·δt(u− d)

F0 = ∆ · S0 +B = e−r·δt(qFu1 + (1− q)F d1 )

q =er·δt − du− d

(single period t)

F0 = e−2r·δt(q2Fuu2 + 2q(1− q)Fud2 + (1− q)2F dd2 ) (path independent)

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