A glimpse into mathematical finance? The realm of option pricing models
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The document presents an overview of mathematical finance with a focus on option pricing models, highlighting its interdisciplinary nature and fundamental theorem of asset pricing (FTAP). It discusses the setup of financial markets, the concept of arbitrage opportunities, and introduces different modeling hierarchies including local and stochastic volatility models. The presentation concludes by inviting questions from the audience.
A glimpse into mathematical finance? The realm of option pricing models
- 1. A glimpse into mathematical finance the realm of option pricing models Istvan Redl Department of Mathematics University of Bath PS Seminar 8 Oct, 2013 Istvan Redl (University of Bath) A glimpse into mathematical finance 1 / 12
- 2. Contents 1 Introduction 2 Fundamental Theorem of Asset Pricing (FTAP) 3 Hierarchy of models Istvan Redl (University of Bath) A glimpse into mathematical finance 2 / 12
- 3. Introduction What is math finance about? - examples ‘Interdisciplinary subject’ ? - problems stem from finance and are treated with mathematical tools Math finance traces back to the early ’70s, since then it has become a widely accepted area within mathematics (Hans Föllmer’s view) Examples (i) Optimization - Portfolio/Asset management (ii) Risk management (iii) Option pricing Istvan Redl (University of Bath) A glimpse into mathematical finance 3 / 12
- 4. 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 the market have some prices, at t = 0 their prices are denoted by π = (π 0 , π 1 , . . . , π d ) ∈ Rd+1 ¯ + π is called price system ¯ The t = 1 asset prices are unknown at t = 0. This uncertainty is modeled by a probability space (Ω, F, P). Asset prices at t = 1 are assumed to be non-negative measurable functions ¯ S = (S 0 , S 1 , . . . , S d ) Istvan Redl (University of Bath) A glimpse into mathematical finance 4 / 12
- 5. Fundamental Theorem of Asset Pricing (FTAP) Setup cont. π 0 is the riskless bond, i.e. π0 = 1 S0 ≡ 1 + r, with the assumption r ≥ 0. ¯ Consider a portfolio ξ = (ξ 0 , ξ 1 , . . . , ξ d ) ∈ Rd+1 . At time t = 0 the price of a given portfolio is d π·ξ = ¯ ¯ πi ξi i=0 At time t = 1 d ¯ ¯ ξ · S(ω) = ξ i S i (ω) i=0 Istvan Redl (University of Bath) A glimpse into mathematical finance 5 / 12
- 6. 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, but ¯ ¯ ¯ · S ≥ 0 P-a.s. and P(ξ · S > 0) > 0. ¯ ¯ ¯ ξ Definition (Risk neutral measure) P∗ is called a risk neutral or martingale measure, if π i = E∗ Si , 1+r i = 0, 1, . . . , d . Theorem (FTAP) A market model is free of arbitrage if and only if there exists a risk neutral measure. Istvan Redl (University of Bath) A glimpse into mathematical finance 6 / 12
- 7. Hierarchy of models Structure of models and associated PDEs PDEs are typically of type (parabolic) ∂t V + AV − rV = 0 different models lead to different operator A Black-Scholes: r and σ are constants dSt = rSt dt + σSt dWt S0 = s 1 2 (ABS V )(s) = σ 2 s 2 ∂ss V (s) + rs∂s V (s) 2 CEV - constant elasticity of variance: r and σ are constants dSt = rSt dt + σStρ dWt S0 = s 0<ρ<1 1 2 (ACEV V )(s) = σ 2 s 2ρ ∂ss V (s) + rs∂s V (s) 2 Istvan Redl (University of Bath) A glimpse into mathematical finance 7 / 12
- 8. Hierarchy of models Structure cont. Local volatility models: r is a constant, but volatility is a deterministic function σ : R+ → R+ dSt = rSt dt + σ(St )St dWt S0 = s 1 2 (ALV V )(s) = s 2 σ 2 (s)∂ss V (s) + rs∂s V (s) 2 Stochastic volatility: r is constant, but volatility is given by another stochastic process Yt dWt S0 = s dYt = α(m − Yt )dt + β Yt d W t dSt = rSt dt + 1 2 1 2 (ASV V )(x, y ) = y ∂xx V (x, y ) + βρy ∂xy V (x, y ) + β 2 y ∂yy V (x, y ) 2 2 1 + r − y ∂x V (x, y ) + α(m − y )∂y V (x, y ) 2 Istvan Redl (University of Bath) A glimpse into mathematical finance 8 / 12
- 9. Hierarchy of models Structure cont. - models with jumps Jump models: r and σ are constants dSt = rSt dt + σSt dLt S0 = s 1 2 (AJ V )(s) = σ 2 ∂ss V (s) + γ∂s V (s) 2 (V (s + z) − V (s) − z∂s V (s))ν(dz) + R Istvan Redl (University of Bath) A glimpse into mathematical finance 9 / 12
- 10. α-stable process Istvan Redl (University of Bath) A glimpse into mathematical finance 10 / 12
- 11. Thank you for your attention! - Questions Istvan Redl (University of Bath) A glimpse into mathematical finance 11 / 12
- 12. Happy? - Good! Istvan Redl (University of Bath) A glimpse into mathematical finance 12 / 12