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Abstract: After introducing one of the most important concepts of mathematical finance, the fundamental theorem of asset pricing (FTAP) and the related no arbitrage pricing theory (NAPT), I will briefly discuss the main techniques and tools extensively used in option pricing, namely Monte Carlo, Fourier Transform and PDE methods. In order to give a fairly well-structured overview of a great chunk of currently preferred models, through a simple example the hierarchy of the mathematical models will be demonstrated by going from the basic Black-Scholes to some more advanced models, e.g. Stochastic Volatility with jumps. (Even those people, who are familiar with these concepts, might find the main focus, i.e. structured overview, of this talk beneficial).

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- 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

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