# Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing

Save this PDF as:

Size: px
Start display at page:

Download "Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing"

## Transcription

1 Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common stock by a certain date for a specified price. (In US, a contract involves 100 shares.) Call option: to buy Put option: to sell Specified price: strike price K date: expiration T (measured in years) Note: You can also write a call or put option (underwrite). Factors affecting the price of an option Current stock price: P t time to expiration: T t Risk-free interest rate: r per annum Stock volatility: σ annualized Payoff for European options (exercised at T only) Call option: V (P T ) = (P T K) + = P T K if P T > K 0 if P T K 1

2 The holder only exercises her option if P T > K (buys the stock via exercising the option and sells the stock on the market). Put option: V (P T ) = (K P T ) + = K P T if P T < K 0 if P T K The holder only exercises her option if P T < K (buys the stock from the market and sells it via option). Mathematical framework Stock (log) price follows a diffusion equation, i.e. a continuoustime continuous stochastic process such as dx t = µ(x t, t)dt + σ(x t, t)dw t, where µ(x t, t) and σ(x t, t) are the drift and diffusion coefficient, respectively, and w t is a standard Brownian motion (or Wiener process). In a complete market, use hedging to derive the price of an option (no arbitrage argument). In an incomplete market (e.g. existence of jumps), specify risk and a hedging strategy to minimize the risk. Stochastic processes Wiener process (or Standard Brownian motion) notation: w t initial value: w 0 = 0 small increments are independent and normal time poits: 0 = t 0 < t 1 < < t n 1 < t n = t 2

3 { w i = w ti w ti 1 } property: w t N(0, t) are independent w t = w t+ t w t N(0, t). zero drift and rate of variance change is 1. That is, dw t = 0dt + 1dw t. A simple way to understand Wiener processes is to do simulation. In R or S-Plus, this can be achieved by using: n=5000 at = rnorm(n) wt = cumsum(at)/sqrt(n) plot(wt,type= l ) Repeat the above commands to generate lots of wt series. Generalized Wiener process dx t = µdt + σdw t, where the drift µ & rate of volatility change σ are constant. Ito s process dx t = µ(x t, t)dt + σ(x t, t)dw t, where both drift and volatility are time-varying. Geometric Brownian motion dp t = µp t dt + σp t dw t, so that µ(p t, t) = µp t and σ(p t, t) = σp t with µ and σ being constant. 3

4 Simulated Standard Brownian Motions w(t) time Figure 1: Time plots of four simulated Wiener processes 4

5 Illustration: Four simulated standard Brownian motions. key feature: variability increases with time. Assume that the price of a stock follows a geometric Brownian motion. What is the distribution of the log return? To answer this question, we need Ito s calculus. Review of differentiation G(x): a differentiable function of x. What is dg(x)? Taylor expansion: G 2 Letting x 0, we have How about G(x, y)? G G(x + x) G(x) = G x x x 2 ( x) dg = G x dx. 3 G x 3 ( x)3 +. G = G G x + x y y G 2 x 2 ( x)2 + 2 G x y x y G 2 y 2 ( y)2 +. Taking limit as x 0 and y 0, we have dg = G G dx + x y dy. Stochastic differentiation Now, consider G(x t, t) with x t being an Ito s process. 5

6 G = G G x + x t t G 2 x 2 ( x)2 + 2 G x t x t G 2 t 2 ( t)2 +. A discretized version of the Ito s process is x = µ t + σ ɛ t, where µ = µ(x t, t) and σ = σ(x t, t). Therefore, ( x) 2 = µ 2 ( t) 2 + σ 2 ɛ 2 t + 2µ σ ɛ( t) 3/2 = σ 2 ɛ 2 t + H( t). Thus, ( x) 2 contains a term of order t. E(σ 2 ɛ 2 t) = σ 2 t, Var(σ 2 ɛ 2 t) = E[σ 4 ɛ 4 ( t) 2 ] [E(σ 2 ɛ 2 t)] 2 = 2σ 4 ( t) 2, where we use E(ɛ 4 ) = 3. These two properties show that Consequently, Using this result, we have σ 2 ɛ 2 t σ 2 t as t 0. ( x) 2 σ 2 dt as t 0. dg = G G dx + x t dt G 2 x 2 σ2 dt = G x µ + G t G 2 This is the well-known Ito s lemma. 6 x 2 σ2 dt + G x σ dw t.

7 Example 1. Let G(w t, t) = w 2 t. What is dg(w t, t)? Answer: Here µ = 0 and σ = 1. Therefore, G w t = 2w t, G t = 0, 2 G w 2 t = 2. dw 2 t = (2w t )dt + 2w tdw t = dt + 2w t dw t. Example 2. If P t follows a geometric Brownian motion, what is the model for ln(p t )? Answer: Let G(P t, t) = ln(p t ). we have G P t = 1 P t, G t = 0, 1 2 G = Consequently, via Ito s lemma, we obtain d ln(p t ) = = 1 µp t + 1 P t 2 1 P 2 t P 2 t µ σ2 dt + σdw t Pt 2 σ 2 P 2 dt + 1 σp t dw t P t Thus, ln(p t ) follows a generalized Wiener Process with drift rate µ σ 2 /2 and variance rate σ 2. The log return from t to T is normal with mean (µ σ 2 /2)(T t) and variance σ 2 (T t). This result enables us to perform simulation. Let t be the time interval. Then, the result says ln(p t+ t ) = ln(p t ) + (µ σ 2 /2) t + tσɛ t+ t, 7 t

8 where ɛ t+ t is a N(0,1) random variable. By generating ɛ t+ t, we can obtain a simulated value for ln(p t+ t ). If one repeats the above simulation many times, one can take the average of ln(p t+ t ) as the expectation of ln(p t+ t ) given the model and ln(p t ). More specifically, one can use simulation to generate a distribution for ln(p t+ t ). In mathematical finance, this technique is referred to as discretization of a stochastic diffusion equation (SDE). The distribution of ln(p t+ t ) obtained in this way works when P t follows a geometric Brownian motion. The distribution, in general, is not equivalent to the true distribution of ln(p t+ t ) for a general SDE. Certain conditions must be met for the two distributions to be equivalent. Details are beyond this class. However, some results are avilable in the literature concerning the relationship between discrete-time distribution and continuous-time distribution. In other words, one derives conditions under which a discrete-time model converges to a continuoustime model when the time interval t approaches zero. For the GARCH-type models, GARCH-M has a diffusion limit. Return to the geometric Brownian motion. Even though we have a close-form solution for options pricing when σ is constant. We can simulate ln(p t+ t ) when σ t is time-varying. For instance, σ t may follow a GARCH(1,1) model. For options pricing, under risk neural assumption, µ is replaced by the risk-free interest rate per annum. [Of course, σ t is the annualized volatility.] This is another application of the volatility models discussed in Chapter 3. Example 3. If P t follows a geometric Brownian motion, what is the model for 1 P t? 8

9 Answer: Let G(P t ) = 1/P t. By Ito s lemma, the solution is d 1 P t = ( µ + σ 2 ) 1 P t dt σ 1 P t dw t, which is again a geometric Brownian motion. Estimation of µ and σ Assume that n log returns are available, say {r t t = 1,, n}. Statistical theory: Estimate the mean and variance by the sample mean and variance. n t=1 r t r = n, s 2 1 n r = (r t r) 2. n 1 t=1 Remember the length of time intervals! Let be the length of time intervals measured in years. Then, the distribution of r t is We obtain the estimates r t N[(µ σ 2 /2), σ 2 ]. ˆσ = s r. ˆµ = r + ˆσ2 2 = r + s2 r 2. Example. Daily log returns of IBM stock in The data show r = and s r = Since = 1/252 year, we obtain that ˆσ = s r = , ˆµ = r + ˆσ2 2 =

10 Thus, the estimated expected return was 61.98% and the standard deviation was 30.4% per annum for IBM stock in Example. Daily log returns of Cisco stock in Data show r = and s r = , Also, Q(12) = Therefore, we have ˆσ = s r = = 0.418, 1.0/252.0 Expected return was 92.4% per annum Estimated s.d. was 41.8% per annum. ˆµ = r + ˆσ2 2 = Example. Daily log returns of Cisco stock in Data show r = and s r = Therefore, ˆσ = ˆµ = Time-varying nature of mean and volatility is clearly shown. Distributions of stock prices If the price follows then, ln(p T ) ln(p t ) N Consequently, given P t, ln(p T ) N dp t = µp t dt + σp t dw t, and we obtain (log-normal dist; ch. 1) (µ σ2 2 )(T t), σ2 (T t) ln(p t ) + (µ σ2 2 )(T t), σ2 (T t) E(P T ) = P t exp[µ(t t)], Var(P T ) = P 2 t exp[2µ(t t)]{exp[σ 2 (T t)] 1}. 10.,

11 The result can be used to make inference about P T. Simulation is often used to study the behavior of P T. Black-Scholes equation Price of stock: P t is a Geo. B. Motion price of derivative: G t = G(P t, t) contingent the stock Risk neutral world: expected returns are given by the risk-free interest rate (no arbitrage) From Ito s lemma: dg t = G t µp t + G t P t t A discretized version of the set-up: G t G t = µp t + G t P t t Consider the Portfolio: short on derivative long G t P t 2 G t P 2 t P t = µp t t + σp t w t, shares of the stock. Value of the portfolio is 2 G t P 2 t σ 2 Pt 2 dt + G t σp t dw t. P t σ 2 Pt 2 t + G t σp t w t, P t The change in value is V t = G t + G t P t P t. V t = G t + G t P t P t. 11

12 by substitution, we have V t = G t t G t P 2 t σ 2 P 2 t. No stochastic component involved. The protfolio must be riskless during a small time interval. V t = rv t t where r is the risk-free interest rate. We then have G t t G t P 2 t σ 2 P 2 t = r t t G t G t P t t. P t and G t t + rp G t t + 1 P t 2 σ2 Pt 2 2 G t = rg Pt 2 t, the Black-Scholes differential equ. for derivative pricing. Example. A forward contract on a stock (no dividend). Here G t = P t K exp[ r(t t)] where K is the delivery price. We have G t t = rk exp[ r(t t)], G t P t = 1, Substituting these quantities into LHS yields 2 G t P 2 t = 0. rk exp[ r(t t)] + rp t = r{p t K exp[ r(t t)]}, which equals RHS. Black-Scholes formulas 12

13 A European call option: expected payoff Price of the call: (current value) E [max(p T K, 0)] c t = exp[ r(t t)]e [max(p T K, 0)]. In a risk-neutral world, µ = r so that ln(p T ) N ln(p t ) + Let g(p T ) be the pdf of P T. Then, r σ2 (T t), σ 2 (T t) 2 c t = exp[ r(t t)] K (P T K)g(P T )dp T. After some algebra (appendix) c t = P t Φ(h + ) K exp[ r(t t)]φ(h ) where Φ(x) is the CDF of N(0, 1), h + = ln(p t/k) + (r + σ 2 /2)(T t) σ T t h = ln(p t/k) + (r σ 2 /2)(T t) σ = h + σ T t. T t See Chapter 6 for some interpretations of the formula. For put option: p t = K exp[ r(t t)]φ( h ) P t Φ( h + ). Alternatively, use the put-call parity: p t c t = K exp[ r(t t)] P t. Put-call parity: Same underlying stock, same strike price, same time to maturity. 13.

14 current Expiration date T date t P K K < P Write call c K P Buy put p K P Buy stock P P P Borrow Ke r(t t) K K Total Current amount: c p P + Ke r(t t). On the expiration date: 1. If P K: call is worthless, put is K P, stock P and owe bank K. Net worth = If K < P : call is K P, put is worthless, stock P and owe K. Net worth = 0. Under no arbitrage assumption: The initial position should be zero. That is, c = p + P Ke r(t t). Example. P t = \$80. σ = 20% per annum. r = 8% per annum. What is the price of a European call option with a strike price of \$90 that will expire in 3 months? From the assumptions, we have P t = 80, K = 90, T t = 0.25, σ = 0.2 and r = Therefore, ln(80/90) + ( /2) 0.25 h + =.2 = = h = h It can be found Φ(.9278) = , Φ( ) =

15 Therefore, c t = \$80Φ( ) \$90Φ( ) exp( 0.02) = \$0.73. The stock price has to rise by \$10.73 for the purchaser of the call option to break even. If K = \$81, then c t = \$80Φ( ) \$81 exp( 0.02)Φ( ) = \$3.49. Lower bounds of European options: No dividends. Why? Consider two portfolios: c t P t K exp[ r(t t)]. A: One European call option plus cash K exp[ r(t t)]. B: One share of the stock. For A: Invest the cash at risk-free interest rate. At time T, the value is K. If P T > K, the call option is exercised so that the portfolio is worth P T. If P T < K, the call option expires at T and the portfolio is worth K. Therefore, the value of the portfolio is max(p T, K). For B: The value at time T is P T. Thus, portfolio A must be worth more than portfolio B today; that is, c t + K exp[ r(t t)] P t. See Example 6.7 for an application. Stochastic integral The formula t 0 dx s = x t x 0 15

16 continues hold. In particular, t 0 dw s = w t w 0 = w t. From dw 2 t = dt + 2w t dw t we have Therefore, w 2 t = t + 2 t 0 w sdw s. t 0 w sdw s = 1 2 (w2 t t). Different from t 0 ydy = (y 2 t y 2 0)/2. Assume x t is a Geo. Brownian motion, dx t = µx t dt + σx t dw t. Apply Ito s lemma to G(x t, t) = ln(x t ), we obtain d ln(x t ) = Taking integration, we have Consequently, and t 0 d ln(x s) = Change x t to P t. The price is µ σ2 dt + σdw t. 2 µ σ2 2 t 0 ds + σ t 0 dw s. ln(x t ) = ln(x 0 ) + (µ σ 2 /2)t + σw t, Jump diffusion Weaknesses of diffusion models: x t = x 0 exp[(µ σ 2 /2)t + σw t ]. P t = P 0 exp[(µ σ 2 /2)t + σw t ]. 16

17 no volatility smile (convex function of implied volatility vs strike price) fail to capture effects of rare events (tails) Modification: jump diffusion and stochastic volatility Jumps are governed by a probaility law: Poisson process: X t is a Poisson process if P r(x t = m) = λm t m m! exp( λt), λ > 0. Use a special jump diffusion model by Kou (2002). dp t P t w t : a Wiener process, = µdt + σdw t + d n t : a Poisson process with rate λ, n t i=1 (J i 1) {J i }: iid such that X = ln(j) has a double exp. dist. with pdf, f X (x) = 1 2η e x κ /η, 0 < η < 1. the above three processes are independent. n t = the number of jumps in [0, t] and Poisson(λt). At the ith jump, the proportion of price jump is J i 1. For pdf of double exp. dist., see Figure 6.8 of the text. Stock price under the jump diffusion model: P t = P 0 exp[(µ σ 2 /2)t + σw t ] n t This result can be used to obtain the distribution for the return series. 17 i=1 J i.

18 Price of an option: Analytical results available, but complicated. Example P t = \$80. K = \$81. r = 0.08 and T t = Jump: λ = 10, κ = 0.02 and η = We obtain c t = \$3.92, which is higher than \$3.49 of Example 6.6. p t = \$3.31, which is also higher. Some greeks: option value V, stock price P 1. Delta: = V P 2. Gamma: Γ = P 3. Theta: Θ = V t. 18

### European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

### On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

### 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

### The Black-Scholes Model

The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

### The Black-Scholes Formula

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

### CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

### Monte Carlo Methods in Finance

Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction

### Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

### Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial

### Mathematical Finance

Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

### Properties of Stock Options. Chapter 10

Properties of Stock Options Chapter 10 1 Notation c : European call option price C : American Call option price p : European put option price P : American Put option price S 0 : Stock price today K : Strike

### Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

### LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50

### Put-Call Parity. chris bemis

Put-Call Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain

### CAS/CIA Exam 3 Spring 2007 Solutions

CAS/CIA Exam 3 Note: The CAS/CIA Spring 007 Exam 3 contained 16 questions that were relevant for SOA Exam MFE. The solutions to those questions are included in this document. Solution 3 B Chapter 1, PutCall

### Black-Scholes Equation for Option Pricing

Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there

### Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D

Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS

### 1 Introduction Outline The Foreign Exchange (FX) Market Project Objective Thesis Overview... 1

Contents 1 Introduction 1 1.1 Outline.......................................... 1 1.2 The Foreign Exchange (FX) Market.......................... 1 1.3 Project Objective.....................................

### Understanding Options and Their Role in Hedging via the Greeks

Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200 Options are priced assuming that

### The Black-Scholes Formula

ECO-30004 OPTIONS AND FUTURES SPRING 2008 The Black-Scholes Formula The Black-Scholes Formula We next examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they

### FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing

FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. Put-Call Parity II. One-Period Binomial Option Pricing III. Adding Periods to the Binomial Model IV. Black-Scholes

### MAT 3788 Lecture 8 Forwards for assets with dividends Currency forwards

MA 3788 Lecture 8 Forwards for assets with dividends Currency forwards Prof. Boyan Kostadinov, City ech of CUNY he Value of a Forward Contract as ime Passes Recall from last time that the forward price

### MATH3075/3975 Financial Mathematics

MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the Black-Scholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per

### Chapter 11 Properties of Stock Options. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull

Chapter 11 Properties of Stock Options 1 Notation c: European call option price p: European put option price S 0 : Stock price today K: Strike price T: Life of option σ: Volatility of stock price C: American

### Valuing Options / Volatility

Chapter 5 Valuing Options / Volatility Measures Now that the foundation regarding the basics of futures and options contracts has been set, we now move to discuss the role of volatility in futures and

### Options/1. Prof. Ian Giddy

Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

### Option Valuation. Chapter 21

Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

### Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the

### USING MONTE CARLO SIMULATION AND IMPORTANCE SAMPLING TO RAPIDLY OBTAIN JUMP-DIFFUSION PRICES OF CONTINUOUS BARRIER OPTIONS

USING MONTE CARLO SIMULATION AND IMPORTANCE SAMPLING TO RAPIDLY OBTAIN JUMP-DIFFUSION PRICES OF CONTINUOUS BARRIER OPTIONS MARK S. JOSHI AND TERENCE LEUNG Abstract. The problem of pricing a continuous

### Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the

### Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk

Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day

### Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a

Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history

### CF4103 Financial Time Series. 1 Financial Time Series and Their Characteristics

CF4103 Financial Time Series 1 Financial Time Series and Their Characteristics Financial time series analysis is concerned with theory and practice of asset valuation over time. For example, there are

### Lecture 21 Options Pricing

Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call

### Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

### Lecture 5: Put - Call Parity

Lecture 5: Put - Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible

### ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

### Simulating Stochastic Differential Equations

Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular

### Financial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan

Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction

### The Black-Scholes-Merton Approach to Pricing Options

he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

### Option Pricing. 1 Introduction. Mrinal K. Ghosh

Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified

### Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week

Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics

### Calibration of Stock Betas from Skews of Implied Volatilities

Calibration of Stock Betas from Skews of Implied Volatilities Jean-Pierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of

### Black-Scholes and the Volatility Surface

IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2009 c 2009 by Martin Haugh Black-Scholes and the Volatility Surface When we studied discrete-time models we used martingale pricing to derive

### Merton-Black-Scholes model for option pricing. Peter Denteneer. 22 oktober 2009

Merton-Black-Scholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,

### Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6

Lecture 15 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static

### Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

### Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

### QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for

### Volatility Index: VIX vs. GVIX

I. II. III. IV. Volatility Index: VIX vs. GVIX "Does VIX Truly Measure Return Volatility?" by Victor Chow, Wanjun Jiang, and Jingrui Li (214) An Ex-ante (forward-looking) approach based on Market Price

### Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013

Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed

### Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

### Betting on Volatility: A Delta Hedging Approach. Liang Zhong

Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price

### 1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes

### Notes on Black-Scholes Option Pricing Formula

. Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading

### Option pricing. Vinod Kothari

Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

### Finite Differences Schemes for Pricing of European and American Options

Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes

### Lecture 11: Risk-Neutral Valuation Steven Skiena. skiena

Lecture 11: Risk-Neutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Risk-Neutral Probabilities We can

### Black-Scholes model: Greeks - sensitivity analysis

VII. Black-Scholes model: Greeks- sensitivity analysis p. 1/15 VII. Black-Scholes model: Greeks - sensitivity analysis Beáta Stehlíková Financial derivatives, winter term 2014/2015 Faculty of Mathematics,

### where N is the standard normal distribution function,

The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at

### On Market-Making and Delta-Hedging

On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide

### One Period Binomial Model

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

### Options and Derivative Pricing. U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University.

Options and Derivative Pricing U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University. e-mail: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,

### The Black-Scholes pricing formulas

The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

### Options 1 OPTIONS. Introduction

Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or

### Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

### Chapter 21: Options and Corporate Finance

Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the

### Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

### EXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0

EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the

### Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative

### Monte Carlo Simulation of Stochastic Processes

Monte Carlo Simulation of Stochastic Processes Last update: January 10th, 2004. In this section are presented the steps to perform the simulation of the main stochastic processes used in real options applications,

### Option Pricing with S+FinMetrics. PETER FULEKY Department of Economics University of Washington

Option Pricing with S+FinMetrics PETER FULEKY Department of Economics University of Washington August 27, 2007 Contents 1 Introduction 3 1.1 Terminology.............................. 3 1.2 Option Positions...........................

### American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus

### Binomial lattice model for stock prices

Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

### Contents. iii. MFE/3F Study Manual 9 th edition 10 th printing Copyright 2015 ASM

Contents 1 Put-Call Parity 1 1.1 Review of derivative instruments................................. 1 1.1.1 Forwards........................................... 1 1.1.2 Call and put options....................................

### Introduction to Options. Derivatives

Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived

### Option Pricing Basics

Option Pricing Basics Aswath Damodaran Aswath Damodaran 1 What is an option? An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called

### Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption

### OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION

OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION NITESH AIDASANI KHYAMI Abstract. Option contracts are used by all major financial institutions and investors, either to speculate on stock market

### Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time THIRD EDITION TOMAS BJORK Stockholm School of Economics OXTORD UNIVERSITY PRESS 1 Introduction 1 1.1 Problem Formulation i 1 v. 2 The Binomial Model 5 2.1 The One Period

### FIN 411 -- Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices

FIN 411 -- Investments Option Pricing imple arbitrage relations s to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but

### The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models

780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon

### Models Used in Variance Swap Pricing

Models Used in Variance Swap Pricing Final Analysis Report Jason Vinar, Xu Li, Bowen Sun, Jingnan Zhang Qi Zhang, Tianyi Luo, Wensheng Sun, Yiming Wang Financial Modelling Workshop 2011 Presentation Jan

### Figure S9.1 Profit from long position in Problem 9.9

Problem 9.9 Suppose that a European call option to buy a share for \$100.00 costs \$5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances

### Variance Reduction Three Approaches to Control Variates

Mathematical Statistics Stockholm University Variance Reduction Three Approaches to Control Variates Thomas Lidebrandt Examensarbete 2007:10 ISSN 0282-9169 Postal address: Mathematical Statistics Dept.

### Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

### THE BLACK-SCHOLES MODEL AND EXTENSIONS

THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that

### Martingale Pricing Applied to Options, Forwards and Futures

IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the

### Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12

Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond

### HPCFinance: New Thinking in Finance. Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation

HPCFinance: New Thinking in Finance Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation Dr. Mark Cathcart, Standard Life February 14, 2014 0 / 58 Outline Outline of Presentation

### Brownian Motion and Ito s Lemma

Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process Brownian

### Options: Valuation and (No) Arbitrage

Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial

### 2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

### Black-Scholes. 3.1 Digital Options

3 Black-Scholes In this chapter, we will study the value of European digital and share digital options and standard European puts and calls under the Black-Scholes assumptions. We will also explain how

### Dynamics of exchange rates and pricing of currency derivatives

Dynamics of exchange rates and pricing of currency derivatives by Jens Christian Jore Christensen THESIS for the degree of Master of Science Master i Modellering og Dataanalyse Faculty of Mathematics and