Financial Engineering — lecture notes

• How do we price and hedge financial derivatives in a mathematically rigorous yet practically useful manner?

Pillar of financial engineering:

• No-arbitrage pricing: Prices are constrained by the absence of "free lunches"

• Replication: Synthesising payoffs from traded instruments

• Risk-neutral valuation: Computing expectations under the "right" probability measure

History of quant finance

Derivatives

Forward Contract

Option

Replication Padigram

Martingale: it is a fair game, discounted expected return = 0

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Binomial Model

Risk neutral measure Q: Martingale that returns risk free rate

Multiperiod

• Backward induction through the tree ⇒binomial pricing formula

• N →∞yields Black-Scholes—coin flips become Brownian motion

• PDE: Taylor expansion of the tree reproduces the Black-Scholes equation

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Probability Theory

• Lévy process: Stationary and independent increments

  • Standard BM is the canonical Lévy process with continuous paths and Gaussian increments

• Markov property: Future depends on the present

• Martingale: Fair game

Brownian Motion

• Lévy, markov and martingale

• Conditions:

  • $W_0=0$

  • Independent normal increments

  • Continuous trajectories

Random Walk

Quadratic variation

Scaling law

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Gaussian Shift Theorem: $E!\left[e^{cZ} f(Z)\right] = e^{c^{2}/2} E!\left[f(Z+c)\right] $

Exotic Option:

• up-down type first-order binary option $V_K^{\,j}(S_T,\tau=0)= f(S_T)\,\mathbf{1}_{\{j S_T > j K\}},\qquad j\in\{-1,+1\}$

• Second order option: can have sth happen in between

• Rainbow Options

  • Two-asset binaries

  • Exchange options

  • Options on the min/max

  • Product/quotient options

  • Executive options

• Professor ideas: power options, trend options