A glimpse into mathematical finance -the realm of option pricing models

Istvan Redl

Department of MathematicsUniversity of Bath

PS Seminar

8 Oct, 2013

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Contents

1 Introduction

2 Fundamental Theorem of Asset Pricing (FTAP)

3 Hierarchy of models

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Introduction

What is math finance about? - examples

‘Interdisciplinary subject’? - problems stem from finance and are treatedwith mathematical tools

Math finance traces back to the early ’70s, since then it has become awidely accepted area within mathematics (Hans Föllmer’s view)

Examples(i) Optimization - Portfolio/Asset management(ii) Risk management(iii) Option pricing

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Fundamental Theorem of Asset Pricing (FTAP)

Setup

Consider a financial market, say with d + 1 assets, (e.g. bonds, equities,commodities, currencies, etc.)

We discuss a simple one-period model, in both periods the assets on themarket have some prices, at t = 0 their prices are denoted by

π = (π0, π1, . . . , πd ) ∈ Rd+1+

π is called price system

The t = 1 asset prices are unknown at t = 0. This uncertainty is modeledby a probability space (Ω,F ,P). Asset prices at t = 1 are assumed to benon-negative measurable functions

S = (S0, S1, . . . , Sd )

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Fundamental Theorem of Asset Pricing (FTAP)

Setup cont.

π0 is the riskless bond, i.e.

π0 = 1 S0 ≡ 1 + r ,

with the assumption r ≥ 0.

Consider a portfolio ξ = (ξ0, ξ1, . . . , ξd ) ∈ Rd+1. At time t = 0 the priceof a given portfolio is

π · ξ =d∑

i=0

πiξi

At time t = 1

ξ · S(ω) =d∑

i=0

ξiS i (ω)

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Fundamental Theorem of Asset Pricing (FTAP)

Arbitrage and martingale measure

Definition (Arbitrage opportunity)

A portfolio ξ ∈ Rd+1 is called arbitrage opportunity if π · ξ ≤ 0, butξ · S ≥ 0 P-a.s. and P(ξ · S > 0) > 0.

Definition (Risk neutral measure)

P∗ is called a risk neutral or martingale measure, if

πi = E∗[

S i

1 + r

], i = 0, 1, . . . , d .

Theorem (FTAP)

A market model is free of arbitrage if and only if there exists a risk neutralmeasure.

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Hierarchy of models

Structure of models and associated PDEs

PDEs are typically of type (parabolic)

∂tV +AV − rV = 0

different models lead to different operator A

Black-Scholes: r and σ are constants

dSt = rStdt + σStdWt S0 = s

(ABSV )(s) =12σ2s2∂2

ssV (s) + rs∂sV (s)

CEV - constant elasticity of variance: r and σ are constants

dSt = rStdt + σSρt dWt S0 = s 0 < ρ < 1

(ACEV V )(s) =12σ2s2ρ∂2

ssV (s) + rs∂sV (s)

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Hierarchy of models

Structure cont.

Local volatility models: r is a constant, but volatility is adeterministic function σ : R+ → R+

dSt = rStdt + σ(St)StdWt S0 = s

(ALV V )(s) =12s2σ2(s)∂2

ssV (s) + rs∂sV (s)

Stochastic volatility: r is constant, but volatility is given by anotherstochastic process

dSt = rStdt +√

YtdWt S0 = s

dYt = α(m − Yt)dt + β√

YtdWt

(ASV V )(x , y) =12y∂2

xxV (x , y) + βρy∂xyV (x , y) +12β2y∂2

yyV (x , y)

+

(r − 1

2y)∂xV (x , y) + α(m − y)∂yV (x , y)

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Hierarchy of models

Structure cont. - models with jumps

Jump models: r and σ are constants

dSt = rStdt + σStdLt S0 = s

(AJV )(s) =12σ2∂2

ssV (s) + γ∂sV (s)

+

∫R

(V (s + z)− V (s)− z∂sV (s))ν(dz)

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α-stable process

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Thank you for your attention! - Questions

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Happy? - Good!

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