Pricing Model of Financial Engineering Fang-Bo Yeh System Control Group Department of Mathematics Tunghai University

2 葉芳柏 教授 英國 Glasgow 大學 數學博士 專長 控制工程理論、科學計算模擬、飛彈導引、泛函分析、財務金融工程 現任 東海大學數學系教授 國立交通大學應用數學研究所, 財務金融研究所兼任教授 亞洲控制工程學刊編輯. 歷任 1. 英國 Glasgow 大學數學系客座教授 2. 英國 Newcastle 大學數學統計系客座教授 3. 英國 Oxford 大學財務金融中心研究 4. 荷蘭國立 Groningen 大學資訊數學系客座授 5. 日本國立大阪大學電子機械控制工程系客座教授 6. 成功大學航空太空研究所兼任教授 7. 航空發展中心顧問 8. 東海大學數學系主任、所長、理學院院長、教務長 9. 國科會中心學門審議委員、諮議委員 10. 教育部大學評鑑委員 11. 國際數學控制學刊編輯委員 學術獎勵 1. 國際電機電子工程師學會獎 IEEE M. Barry Carlton Award 2. 國際航空電子系統傑出論文獎 3. 國科會傑出研究獎

Contents 1. Classic and Derivatives Market 2. Derivatives Pricing 3. Methods for Pricing 4. Numerical Solution for Pricing Model

Classic and Derivatives Market Underlying Assets Cash Commodities ( wheat, gold ) Fixed income ( T-bonds ) Stock Equities ( AOL stock ) Equity indexes ( S&P 500 ) Currency Currencies ( GBP, JPY ) Contracts Forward & Swap : FRAs, Caps, Floors, Interest Rate Swaps Futures & Options : Options, Convertibles Bond Option, Swaptions

5 Derivative Securities Forward Contract : is an agreement to buy or sell. Call Option : gives its owner the right but not the obligation to buy a specified asset on or before a specified date for a specified price. European, American, Lookback, Asian, Capped, Exotics…..

6 Call Option on AOL Stock on Sep. 8, you buy one Nov.call option contract written on AOL  contract size: 100 shares  strike price: 80  maturity: December 26  option premium: 7 1 / 8 per share on Sep. 8,… you pay the premium of $ at maturity on December 26,… if you exercise the option, you take delivery of 100 shares of AOL stock and pay the strike price of $8,000 otherwise, nothing happens

7 Call Option on AOL Stock denote by S T the price of AOL stock on December 26 date Sep. 8 December 26 scenario (if S T < 80) (if S T  80) exercise option? no yes cash flows (on per-share basis) pay option premium   receive stock   S T pay strike price   -80

8 Call Option on AOL Stock 0 AOL stock price on December pay-off profit pay-offnet profit Fang-bo Yeh

9 Maximal Losses and Gains on Option Positions Mathematics Finance 2003 Option Markets Fang-Bo Yeh Tunghai Mathematics 0 long call maximal gain: unlimited maximal loss: premium short call maximal gain: premium maximal loss: unlimited long put maximal gain: strike minus premium maximal loss: premium 0 00 short put maximal gain: premium maximal loss: strike minus premium

10 Simple Option Strategies: Covered Call covered call: the potential loss on a short call position is unlimited the worst case occurs when the stock price at maturity is very high and the option is exercised the easiest protection against this case is to buy the stock at the same time as you write the option this strategy is called “covered call” covered call pay-offs: Cost of strategy: you receive the option premium C while paying the stock price S the total cost is hence S-C Mathematics Finance 2003 Option Markets Fang-Bo Yeh Tunghai Mathematics cash flows at maturity case: S T < K S T  K Short call - K-S T long stock S T S T total: S T K

11 Simple Option Strategies: Covered Call Mathematics Finance 2003 Option Markets Fang-Bo Yeh Tunghai Mathematics short call long stock covered call K + = K pay-off profit K STST premium 0

12 Simple Option Strategies: Protective Put protective put: suppose you have a long position in some asset, and you are worried about potential capital losses on your position to protect your position, you can purchase an at-the-money put option which allows you to sell the asset at a fixed price should its value decline this strategy is called “protective put” protective put pay-offs: cost of strategy: the additional cost of protection is the price of the option, P the total cost is hence S+P Mathematics Finance 2003 Option Markets Fang-Bo Yeh Tunghai Mathematics cash flows at maturity case: S T < K S T  K long stock S T S T long put K-S T - total: K S T

13 Simple Option Strategies: Protective Put Mathematics Finance 2003 Option Markets Fang-Bo Yeh Tunghai Mathematics long stock long put protective put K + = K profit K STST 0 pay-off premium

14 Financial Engineering Bond + Single Option S&P500 Index Notes Bond + Multiple Option Floored Floating Rate Bonds, Range Notes Bond + Forward (Swap) ;Structured Notes Inverse Floating Rate Note Stock + Option Equity-Linked Securities, ELKS

Main Problem : What is the fair price for the contract? Ans: (1). The expected value of the discounted future stochastic payoff (2). It is determined by market forces which is impossible have a theoretical price

Main result: It is possible have a theoretical price which is consistent with the underlying prices given by the market But is not the same one as in answer (1).

Methods Assume efficient market Risk neutral valuation and solving conditional expectation of the random variable The elimination of randomness and solving diffusion equation

Problem Formulation Contract F : Underlying asset S, return Future time T, future pay-off f(S T ) Riskless bond B, return Find contract value F(t, S t )

Differentiable Not differentiable Deterministic Stochastic

20 Deterministic Function

21 Stochastic Brownian Motion

22 From Calculus to Stochastic Calculus Calculus Stochastic Calculus Differentiation Ito Differentiation Integration Ito Integration Statistics Stochastic Process Distribution Measure Probability Equivalent Probability

Assume 1). The future pay-off is attainable: (controllable) exists a portfolio such that 2). Efficient market: (observable) If then

By assumptions (1)(2) Ito’s lemma The Black-Scholes-Merton Equation:

European Call Option Price:

Martingale Measure CMG Drift Brownian Motion Brownian Motion

Where

Main Result The fair price is the expected value of the discounted future stochastic payoff under the new martingale measure.

29 From Real world to Martingale world Discounted Asset Price & Derivatives Price Under Real World Measure is not Martingale But Under Risk Neutral Measure is Martingale

30 Numerical Solution Methods Finite Difference Monte Carlo Simulation Idea: Approximate differentials Monte Carlo Integration by simple differences via Generating and sampling Taylor series Random variable

31 Introduction to Financial Mathematics (1) Topics for 2003 : 1. Pricing Model for Financial Engineering. 2. Asset Pricing and Stochastic Process. 3. Conditional Expectation and Martingales. 4. Risk Neutral Probability and Arbitrage Free Principal. 5. Black-Scholes Model : PDE and Martingale and Ito’s Calculus. 6. Numerical method and Simulations.

32 References M. Baxter, A. Rennie, Financial Calculus,Cambridge university press, 1998 R.J. Elliott and P.E. Kopp, Mathematics of Financial Markets, Springer Finance, 2001 N.H. Bingham and R. Kiesel, Risk Neutral Evaluation, Springer Finance, P. Wilmott, Derivatives, John Wiley and Sons, J.C. Hull, Options, Futures and other derivatives, Prentice Hall R. Jarrow and S. Turnbull, Derivatives Securities, Southern College Publishing, 1999.