Financial Engineering
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Financial Engineering

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Financial Engineering Financial Engineering Presentation Transcript

  • Financial Engineering & Physical Sciences
  • How to find a good fund manager
    • Monkeys vs. Fund Managers
    • 10-Week Solicitation
    • Pitfall of Survivorship
    • Need 25 years to prove a 6% premium with 95% confidence (assuming 20% volatility).
  • 파생상품 (Derivatives)
    • Derivative’s value depends on the values of other assets.
    • The price of a derivative can be estimated with relatively reliable theories.
    • Financial engineering can be used to design or analyze complex derivative instruments.
    • Examples
      • Forwards
        • Osaka Rice Exchange in early 1700’s
        • Agree on price now, pay later.
      • Swaps
        • Exchange assets now; return them later; pay differential rent in the meantime.
      • Options
        • Calls – Agree on price now; if option buyer wants, he buys asset later.
        • Puts – Agree on price now; if option buyer wants, he sells asset later
  • Buy Call
  • Sell Call
  • Buy Put
  • Sell Put
  • Strangle (OTM Call + OTM Put)
  • Condor (DITM Call – ITM Call – OTM Call + DOTM Call)
  • Back Spread (2 OTM Calls – 1 ATM Call)
  • Strap (2 ATM Calls + 1 ATM Put)
  • Financial Engineering
    • Roles of Derivatives and Financial Engineering
      • To provide a wider set of future states (in a convenient way)
      • To satisfy investors with different expectation on the future
      • To manage risks
      • To cope with legal and tax constraints
    • Two main streams in Financial Engineering industry
      • Econometrics (buy side)
        • Analysis of past information
        • Future prediction based on past information (time series analysis)
      • Applied physics (sell side)
        • Grid-based calculation of PDEs
        • Monte Carlo simulations (variance reduction)
        • Stochastic calculus
        • Theoretical approaches (martingales and measures)
  • Model of the Behavior of Stock Prices
    • Fluctuation of stock prices is modeled with stochastic process .
    • Markov process
      • Only the present value of a variable is relevant for predicting the future (the market is efficient).
    • Wiener process (Brownian motion)
      • A particular type of Markov process with  =0,  2 =1
      • Scales with  t 1/2
      •  z =   t 1/2
      • (  = random drawing from a Gaussian with [0,1])
    • Proof of  z =   t 1/2
    •  z = z(T)-z(0) =  1 N G i (0,f(  t))
    • (G = random drawing from a Gaussian with [0,f(  t)])
    • N steps of  t
    • Var[z(T) - z(0)] = N Var[G(0,f(  t))] = N f(  t) 2
    • 1 step of N  t
    • Var[z(T) - z(0)] = N Var[G(0,f(  t))] = N f(  t) 2
    • N f(  t) 2 = f(N  t) 2
    •  f(x) = x 1/2
    •   z =   t 1/2
  •  
    • Generalized Wiener process
    • Ito process
    • Geometric Ito process
  •  
  • Chain Rule
    • Let S = S(t,z) .
    • In deterministic calculus,
    • In stochastic calculus,
  • Ito’s Lemma
    • Let S follow an Ito process:
    • Then, f(S,t) follows the following process:
  • Justification of Generalized Wiener Process
    • Generalized Wiener process can be obtained from Conditional Probability Density Functions , which have the following properties:
    • When the process is Markovian,
    • This is known as the Chapman-Kolmogorov equation .
    • If we require that the process is continuous, the C-K equation becomes the Fokker-Planck equation , or the Forward Kolmogorov equation :
    • where
    • and the initial condition is
    • The solution to the F-K equation is
    • where the process w 0 has the following properties:
    • The solution to the F-K equation with a =0 & b =1 is
    • These suggest that we will be interested in processes of the form
  • Risk
    • Risk-Free Assets vs. Risky Assets
    • Price of Risk
    • Risk Preference
      • Risk-averse (caused mainly by capital limit)
      • Risk-neutral
      • Risk-loving
  • Risk-Neutral Valuation
      • 1. Assume that the expected return of the underlying asset is the same as the riskless return (  = r ).
      • 2. Calculate the expected payoff from the derivative at its maturity.
      • 3. Discount the expected payoff at the riskless return r .
    •  Won 1997 Nobel Prize in Economics!
  • Payoff from Call Option Probability Distribution of Stock Price
  • Black-Scholes Pricing Formula for European Call & Put Options
    • Call
    • Put
    • Where N(x) is the cumulative standard normal distribution, and
  • The Black-Scholes-Merton Differential Equation
    • Assume the stock has a geometric Brownian motion
    • Let f(S,t) be the price of a derivative contingent on S.
    • Consider a portfolio composed of stocks of amount a and derivatives of amount b . Then the value of the portfolio is:
    • By having , the Wiener process can be eliminated. The value of this portfolio is
    • Since this portfolio is now riskless, there would be riskless arbitrage opportunity unless
    • By equating the last two equations,
    • This is the Black-Scholes-Merton differential equation .
  • Expectation & the B-S-M Equation
    • Consider the following boundary value problem:
    • Let S satisfy the stochastic differential equation:
    • Apply Ito’s lemma:
    • Integrate from t = 0 to T :
    • As far as (i.e., Ito integral is definable), one has
    • If F solves the PDE, taking expectations gives
    • This is the Feynman-Kac formula, a fundamental connection between PDEs and SDEs. It shows that stochastic calculus can solve PDEs for us.
    • Now, by defining , the boundary value problem becomes the B-S-M equation:
    • Thus, solving the B-S-M equation is equivalent to finding the expectation:
  • Summary
    • Ways Derivatives (and Financial Engineering) Are Used
      • To hedge risks
      • To speculate (take a view on the future direction of the market)
      • To lock in an arbitrage profit
      • To change the nature of a liability
      • To change the nature of an investment without incurring the costs of selling one portfolio and buying another