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Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
Financial engineering3478
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Financial engineering3478

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  • 1. Financial Engineering & Physical Sciences
  • 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. 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. <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. Buy Call
  • 6. Sell Call
  • 7. Buy Put
  • 8. Sell Put
  • 9. Strangle (OTM Call + OTM Put)
  • 10. Condor (DITM Call – ITM Call – OTM Call + DOTM Call)
  • 11. Back Spread (2 OTM Calls – 1 ATM Call)
  • 12. Strap (2 ATM Calls + 1 ATM Put)
  • 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. <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. 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. <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>
  • 19.  
  • 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>
  • 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>
  • 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>
  • 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>
  • 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>
  • 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>
  • 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>
  • 27. Payoff from Call Option Probability Distribution of Stock Price
  • 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>
  • 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>
  • 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>
  • 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>
  • 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>
  • 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>
  • 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>
  • 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>

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