Financial Engineering


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Financial Engineering

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