Financial Derivatives Section 5
|
|
- Brice Dennis
- 5 years ago
- Views:
Transcription
1 Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr University of Piraeus Spring 2018 M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
2 Outline 1 A Process for the Price of the Underlying Assets The Model A Simple Idea for Volatility Estimation 2 Itô s Lemma 3 The Black-Scholes Model The Black-Scholes-Merton partial differential equation The risk-neutral valuation The Black-Scholes pricing formula Binomial model and B-S model M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
3 Outline 1 A Process for the Price of the Underlying Assets The Model A Simple Idea for Volatility Estimation 2 Itô s Lemma 3 The Black-Scholes Model The Black-Scholes-Merton partial differential equation The risk-neutral valuation The Black-Scholes pricing formula Binomial model and B-S model M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
4 Building a Model for the Price of the Underlying Assets Constant expected rate of return We assume that the expected rate of return of an underlying asset price remains constant and is equal to a number µ. In other words, we assume: [ ] 1 S(t + τ) S(t) τ E = µ S(t) This means that on average (or when there is no random noise) it holds that: S(t) S(t) = µ τ S(t) = µs(t) τ, for every time t > 0 and time interval τ, where S(t) denotes the difference S(t + τ) S(t). For t dt, this implies that on average: ds(t) S(t) = µdt ds(t) = µs(t)dt. Hence for every t > 0, on average we have S(t) = S(0)e µt. This is the solution of the equation S (t) = µs(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
5 Building a Model for the Price of the Underlying Assets cont d Including the volatility Of course, underlying assets do exhibit variability. This is included in the model in the following way: S(t) S(t) = µ τ + σε t τ where ε t N(0, 1) (normal distribution with mean 0 and variance 1). This means that [ ] 1 S(t + τ) S(t) τ Var = σ 2 S(t) which implies that the model assumes that the variance of the rate of return is equal to a constant σ 2. The parameter σ is called the volatility of the underlying asset price. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
6 Building a Model for the Price of the Underlying Assets cont d Letting τ become very small The price model can be written as S(t) S(t) = µ τ + σε t τ S(t) = µs(t) τ + σs(t)ε t τ. When we consider the changes of the price to happen in any moment of time (continuously), we basically assume that τ dt. Then, the model is usually written as: ds(t) = µs(t)dt + σs(t)dz(t) ( ) where dt stands for a very small (tiny) time change and z(t) is a random variable that follows the Normal Distribution with mean zero and variance equal to dt. The model ( ) is the famous Black-Scholes(-Merton-Samuelson) model for underlying assets prices without dividend. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
7 Interpretation of the B-S Model The model implies that the rate of return of the price follows the Normal distribution: S(t) N(µ τ, σ 2 τ). S(t) The parameter µ, can be considered as the expected continuously compounded return (per annum) earned by the underlying asset price. The fact that we consider it constant is a big assumption. The B-S model for the asset prices is keeping with the weak-form of efficient market hypothesis. Equation ( ) is a stochastic differential equation and for the solution we need a tool that is called Ito s Lemma. The solution is ( S(t) = S(0)e µ σ2 2 ) t+σz(t) ( ) which is called the Geometric Brownian Motion (GBM). Notice how the volatility comes into the model... M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
8 Log-normality of the Asset Prices Equation ( ) implies that for every time t > 0 or ln ln ( ) S(t) = S(0) ( ) S(t) = S(0) ) (µ σ2 t + σdz(t) 2 ) (µ σ2 t + σε t t 2 which means that the logarithmic return of the asset follows normal distribution: ( ) ) ) S(t) ln(s(t)) ln(s(0)) = ln N ((µ σ2 t, σ 2 t, S(0) 2 or equivalently, ( ln (S(t)) N ln (S(0)) + ) ) (µ σ2 t, σ 2 t. 2 M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
9 Lognormality of the Asset Prices, cont d The asset price at time t obeys the Log-normal distribution, with E[S(t)] = S(0)e µt and ( ) Var(S(t)) = S 2 (0)e 2µt e σ2t 1.
10 Lognormality of the Asset Prices, cont d M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
11 A Way to Estimate the Volatility One way to estimate the volatility of the stock price is the following: 1 Choose the frequency of stock price data (every hour, every day etc) (denote this time length by τ). 2 Choose the number of observations of stock prices that is going to be used for the estimation (denote this number by n). 3 If S i denotes the stock price at the end of the i-th interval, calculate the series: ( ) Si u i = ln, for i = 1, 2,..., n. S i 1 4 Then estimate the standard deviation of u i s by: s = 1 n (u i u) n 1 2 where u is the mean of u i s. i=1 5 Estimate the volatility of the stock price by: ˆσ = s τ. 6 Standard error= ˆσ 2n.
12 Outline 1 A Process for the Price of the Underlying Assets The Model A Simple Idea for Volatility Estimation 2 Itô s Lemma 3 The Black-Scholes Model The Black-Scholes-Merton partial differential equation The risk-neutral valuation The Black-Scholes pricing formula Binomial model and B-S model M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
13 The Itô s Lemma This lemma answers questions like the following: If the stock price follows the B-S model, which is the model that the price of a derivative on this stock follows? More precisely, if C(S(t), t) is the price of such a derivative at time t, we want to find how dc(s(t), t) looks like (the dynamics of the derivative price). dc(s(t), t) =(??? )dt+(??? )dz(t) The answer is given by the Itô s lemma. Itô s lemma Suppose that S(t) follows the model ds(t) = µs(t)dt + σs(t)dz(t) and C(S, t) is a twice differentiable function. Then, the dynamics of the stochastic process C(S(t), t) are the following: ( ) ( ) dc(s(t), t) = µs(t) C S + C t σ2 S 2 (t) 2 C S 2 dt + σs(t) C dz(t) S M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
14 The Itô s Lemma, an example As you have seen, the price of the futures contract when a stock does not pay dividends is given by F (t, T ) = S(t)e r(t t), when there is no dividend payments from the underlying asset. We ask the following: What is the dynamics of F (t, T ) if the stock price follows a GBM: ds(t) = µs(t)dt + σs(t)dz(t)? The answer is given by the Itô s lemma. F S = er(t t), F t df (t, T ) = = rs(t)e r(t t) = rf (t, T ) and 2 C S 2 = 0 ( ) µs(t)e r(t t) rf (t, T ) dt + σs(t)e r(t t) dz(t) df (t, T ) = (µ r)f (t, T )dt + σf (t, T )dz(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
15 The Itô s Lemma, a second example How the function G(S(t)) = ln(s(t)) evolves? We apply Itô s lemma: Hence, G S = 1 S, 2 G S 2 = 1 G and S 2 t = 0. d ln(s(t)) = Something that we have already seen. ) (µ σ2 dt + σdz(t). 2 M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
16 Outline 1 A Process for the Price of the Underlying Assets The Model A Simple Idea for Volatility Estimation 2 Itô s Lemma 3 The Black-Scholes Model The Black-Scholes-Merton partial differential equation The risk-neutral valuation The Black-Scholes pricing formula Binomial model and B-S model M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
17 The imposed assumptions Throughout this section, we are going to use the following assumptions: 1 The underlying stock price follows a GBM (constant volatility, no jumps). 2 There is no arbitrage opportunities. 3 The underlying asset does not give any dividend until the maturity of the option (we will withdraw this assumption later). 4 The short selling is allowed. 5 There is no transaction costs (no bid/ask spread and no short selling cost). 6 The trading is continuous. 7 The underlying asset is divisible (we can buy and short sell fractions of the asset). 8 The risk-free internet rate refers to continuously discounting and remains constant until the option s maturity: db(t) = rb(t)dt in other words B(t) = B(0)e rt, t (0, T ]. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
18 The general ideas The aim of the B-S model is to derive option prices and a way to dynamically hedge the involved risk. The basic idea is described in the following steps: 1 The Black-Scholes-Merton analysis is analogous to the no-arbitrage analysis of the binomial model. We are asking for example how many stocks we should buy/sell in order to hedge the risk of selling a call option. 2 Consider a derivative written on the stock price S(t), with payoff C(T ) = C(S(T ), T ) at maturity. 3 We denote the price of this derivative at time t, by C(t) = C(S(t), t). 4 As in the case of the binomial model, we want to construct a portfolio which consists at time t of (t) shares of the stock and an amount B(t) invested at free-risk interest rate. such that the value of the portfolio to be equal to C(t), at any time t. 5 The outcome is an equation that is satisfied by the function C(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
19 Derivation of the B-S-M pde Our goal is to find (t) and B(t) such that: (t)s(t) + B(t) = C(S(t), t) for every time t T. We have assumed that: ds(t) = µs(t)dt + σs(t)dz(t) and db(t) = rb(t)dt. The changes of the portfolios in the small period of time give: (t)ds(t) + db(t) = dc(s(t), t) (t)ds(t) + rb(t)dt = dc(s(t), t). How can we find the dynamics of C(S(t), t)? This is exactly where the Itô s lemma helps! ( C C dc(s(t), t) = µs(t) + S t ) C 2 S 2 σ2 S 2 (t) dt + C S σs(t)dz(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
20 Derivation of the B-S-M pde cont d The last equation can be written as: dc(s(t), t) = C (µs(t)dt + σs(t)dz(t)) + S which is equal to: ( dc(s(t), t) = C S ds(t) + In other words, we have to solve the equation: (t)ds(t) + rb(t)dt = C S ds(t) + C t ( C t ) C 2 S 2 σ2 S 2 (t) dt, ) 2 C S σ 2 S 2 (t) dt 2 ( C t C 2 S 2 σ2 S 2 (t) ) dt. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
21 Derivation of the B-S-M pde cont d One solution of the above equation is when (t) = C S This means that B(t) = C(t) C S S(t) By replacing these findings in the desired equation, we get: rc(t) = rs(t) C C S (t) + t (t) + σ2 S 2 (t) 2 The above equation is written simplier as: 2 C S 2 (t) No stochastic term!!! rc(t) = rs(t)c S (t) + C t (t) + σ2 S 2 (t) 2 C SS (t) This is the famous Black-Scholes-Merton partial differential equation for the derivative prices. If we follow the dynamic hedging of (t), we perfectly replicate the option payoff. Hedging all the risk involved! Exactly what we have succeeded in the binomial model. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
22 Discussion of the BSM pde The BSM pde is satisfied by (almost) all the derivatives written on the stock. The BSM pde is going with boundary conditions determined by the derivative payoff. For the call option for example, the boundary condition is C(S(T ), T ) = max{s(t ) K, 0}. The BSM pde can be solved in a closed-form only for European options. For the American ones, numerical methods are used to approximate the solution. A very important notice: r appears in the equation instead of µ. Why? This means that the solution of the pde (or equivalently the derivative value) is independent on µ!! Another important notice: The replication portfolio should be changed frequently since (t) and B(t) change as t and S(t) change. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
23 An example of the BSM pde Assume that an investor has a long position in a futures contract on the stock price S(t) with maturity T and strike price K. The value of this contract today is f (t) = S(t) Ke r(t t). Why? Since the future is a derivative on the stock, it should satisfy the BSM pde. Indeed, f t (t) = rke r(t t), f S = 1 and f SS = 0 and hence which holds true. rke r(t t) + rs(t) σ2 S 2 0 = r(s Ke r(t t) ), M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
24 Risk-neutral valuation As in the case of binomial model, the subjective expected rate of return does not influence the derivative valuation. This is because the perfect replication of derivative payoff is possible and hence any position on a derivative written on the stock could be thought as a position with no risk. In sequel, this implies that for the derivative valuation, we can assume that any security (the stock and the derivatives on this stock) has expected return equal to the risk-free interest rate. In other words, Ê[S(T )] = S(0)e rt or Ê[S(T ) S(t)] = S(t)e r(t t), where Ê is the expectation under the risk-neutral probability. Hence, if we want to find the price of a derivative, we need to calculate: C(0) = e rt Ê[C(T )] or C(t) = e r(t t) Ê[C(T ) S(t)] If we calculate C(t), we can use the formula (t) = C S position on the derivative. to fully hedge the M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
25 A sketch of the proof of the B-S formula In 1973, Fischer Black and Myron Scholes assumed the GBM model for the stock price and derived a closed form solution for the call option price. We can have a look on a way to derive the B-S formula by following the steps below: (1) As we have seen, according to GBM: E[S(T )] = S(0)e µt. Since the perfect replication of the option payoff is possible, we can assume that we live in a risk-neutral world: Ê[S(T )] = S(0)e rt. (2) Under the risk-neutral probabilities, the stock price follows a log-normal distribution but with different mean: ( ) ) ln (S(T )) N ln (S(0)) + (r σ2 T, σ 2 T. 2 (3) In order to price the call option with strike price K, it would be enough to calculate the expectation: e rt Ê[max{S(T ) K, 0}], where (under the risk-neutral probabilities) S(T ) has the above log-normal distribution.
26 A sketch of the proof of the B-S formula cont d (4) Note that Ê[max{S(T ) K, 0}] = + (s K)g(s)ds, where g(s) is the K corresponding probability density function of the lognormal distribution. (5) The next step is to define a new variable: x = ln(s) ln(s(0)) + (r σ2 /2)T σ T. (6) The desired expectation becomes: + ln(k) ln(s(0))+rt T σ 2 /2 σ T (e xσ T +ln(s(0))+rt σ2 T 2 K)h(x)dx, where h(x) is the probability density function of the standard normal distribution. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
27 The Black & Scholes formula The price of the European call option with strike price K is given by: c(t) = S(t)N(d 1 ) Ke r(t t) N(d 2 ) where d 1 = ln(s(t)/k)+(r+σ2 /2)(T t) σ, T t d 2 = ln(s(t)/k)+(r σ2 /2)(T t) σ = d T t 1 σ T t. Where N(.) is the standardized cumulative Normal distribution. Using the put-call parity, one can directly derive the price of the corresponding European put option: p(t) = Ke r(t t) N( d 2 ) S(t)N( d 1 ) It can be checked that the price c(t) solves the BSM pde. (What a nice exercise!) M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
28 B-S formula, an example Consider the following inputs for a European call option on a stock price with the following data: S(0) = 9.30, K = 10, r = 1%, σ = 35.32% and T t = 90 trading days. We calculate: d 1 = ln( )+( )( ) / = and d 2 = d = N( 0.223) = and N( 0.434) = Hence, and c(0) = e = p(0) = e = 1.18 M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
29 The Black & Scholes formula with dividend payments In the case where the stock pays dividend, the option price changes. (Why?) Assume that the amount and the timing of the dividends during the life of the option is known (for short time, this is not an unreasonable assumption). On the day that the dividend is paid, the stock price declines by this exact amount. Denote by D(t) the present value at time t of the dividends paid in period [t, T ]. Under these assumptions the BS formula becomes: c(t) = (S(t) D(t))N(d 1 ) Ke r(t t) N(d 2 ) and where p(t) = Ke r(t t) N( d 2 ) (S(t) D(t))N( d 1 ), d 1 = ln((s(t) D(t))/K) + (r + σ2 /2)(T t) σ T t and d 2 = d 1 σ T t. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
30 The Black & Scholes formula with dividend yield When the underline asset is an index, it is much more preferable to consider the dividend payments in the form of a continuous dividend yield q. Then, The payment of a dividend yield at rate q causes the reduction of the price growth of the underling asset. If dividend is paid, the price of the underlying asset grows from S(0) to S(T ), then in the absence of dividend it would grow from S(0) to S(T )e qt. Alternatively, without dividend it would grow from S(0)e qt to S(T ). This is like we continuously discount the dividend payments and reduce the current price accordingly. Under these assumptions the BS formula becomes: and where, c(t) = S(t)e q(t t) N(d 1 ) Ke r(t t) N(d 2 ) p(t) = Ke r(t t) N( d 2 ) S(t)e q(t t) N( d 1 ), d 1 = ln(s(t)/k) + (r q + σ2 /2)(T t) σ T t and d 2 = d 1 σ T t. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
31 Discussion on the B-S formula The Black-Scholes portfolio can be constructed only in the case where the underlying asset is tradable. Non-arbitrage relations such as put-call parity and arbitrage bounds are satisfied by the BS formula. The BS formula provides a clear relationship between the option prices and the affecting parameters. For the case of American options, one can use approximation arguments from the BSM pde or from the BS price. There is a number of problems/inconsistencies in the GBM model, which are reflected in the BS formula. The constant volatility is the most problematic one (it has been shown that the volatility changes with time). The validity of the formula is based on the continuous trading, which can only be approximated (the accumulated error of this approximation may be significant). The BS formula tends to overvalue deep OTM calls and undervalue deep ITM calls. M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
32 The implied volatility Volatility The only parameter in the BS pricing formula that can not be directly observed is the volatility σ. Estimation of volatility An alternative way to estimate the volatility of a particular stock price is to consider the market prices of the options as inputs and use the BS formula to extract the volatility. In other words, we solve the following equation with respect to σ: c(t, σ) = Market call option price at time t. The solutions Since the c(t, σ) is a continuous and increasing function of σ, there is one-to-one correspondence between the marketed call option prices and the implied volatility.
33 B-S model as a limiting case of binomial model The CRR model, Cox-Ross-Rubinstein Assuming that we have estimated the volatility of a stock price. The CRR binomial model imposes that: Taking the limit u = e σ t and d = e σ t Cox, Ross and Rubinstein (1979) proved that as the time interval t goes to zero, or equivalently as the number of periods of the binomial models goes to infinity, the CRR model for the stock price tends to the GBM. Symmetric random walk Brownian Motion and Exponential random walk Geometric Brownian Motion Note also that the hedging strategy for a derivative in the BS model: (t) = C S (t) can be seen as the limit of the corresponding hedging strategy in the binomial model: (t) = cu c d us(0) ds(0). M. Anthropelos (Un. of Piraeus) B-S Model Spring / 33
Reading: You should read Hull chapter 12 and perhaps the very first part of chapter 13.
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Asset Price Dynamics Introduction These notes give assumptions of asset price returns that are derived from the efficient markets hypothesis. Although a hypothesis,
More informationBlack-Scholes-Merton Model
Black-Scholes-Merton Model Weerachart Kilenthong University of the Thai Chamber of Commerce c Kilenthong 2017 Weerachart Kilenthong University of the Thai Chamber Black-Scholes-Merton of Commerce Model
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationFinancial Derivatives Section 3
Financial Derivatives Section 3 Introduction to Option Pricing Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un.
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More information1 Geometric Brownian motion
Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is
More informationValuing Stock Options: The Black-Scholes-Merton Model. Chapter 13
Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 1 The Black-Scholes-Merton Random Walk Assumption l Consider a stock whose price is S l In a short period of time of length t the return
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
More informationLecture 11: Ito Calculus. Tuesday, October 23, 12
Lecture 11: Ito Calculus Continuous time models We start with the model from Chapter 3 log S j log S j 1 = µ t + p tz j Sum it over j: log S N log S 0 = NX µ t + NX p tzj j=1 j=1 Can we take the limit
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationThe Black-Scholes Equation
The Black-Scholes Equation MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will: derive the Black-Scholes partial differential equation using Itô s Lemma and no-arbitrage
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More information4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu
4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationAspects of Financial Mathematics:
Aspects of Financial Mathematics: Options, Derivatives, Arbitrage, and the Black-Scholes Pricing Formula J. Robert Buchanan Millersville University of Pennsylvania email: Bob.Buchanan@millersville.edu
More informationThe Black-Scholes Model
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh The Black-Scholes Model In these notes we will use Itô s Lemma and a replicating argument to derive the famous Black-Scholes formula
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
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
More informationLecture 8: The Black-Scholes theory
Lecture 8: The Black-Scholes theory Dr. Roman V Belavkin MSO4112 Contents 1 Geometric Brownian motion 1 2 The Black-Scholes pricing 2 3 The Black-Scholes equation 3 References 5 1 Geometric Brownian motion
More informationThe Black-Scholes PDE from Scratch
The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to: Come up with some dynamics for the stock returns Discuss Brownian motion
More informationStochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 19 & 6 th, 1 he Black-Scholes-Merton Model for Options plus Applications Where we are Previously: Modeling the Stochastic Process for Derivative
More informationDepartment of Mathematics. Mathematics of Financial Derivatives
Department of Mathematics MA408 Mathematics of Financial Derivatives Thursday 15th January, 2009 2pm 4pm Duration: 2 hours Attempt THREE questions MA408 Page 1 of 5 1. (a) Suppose 0 < E 1 < E 3 and E 2
More informationInvestment Guarantees Chapter 7. Investment Guarantees Chapter 7: Option Pricing Theory. Key Exam Topics in This Lesson.
Investment Guarantees Chapter 7 Investment Guarantees Chapter 7: Option Pricing Theory Mary Hardy (2003) Video By: J. Eddie Smith, IV, FSA, MAAA Investment Guarantees Chapter 7 1 / 15 Key Exam Topics in
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 18 & 5 th, 13 he Black-Scholes-Merton Model for Options plus Applications 11.1 Where we are Last Week: Modeling the Stochastic Process for
More informationDr. Maddah ENMG 625 Financial Eng g II 10/16/06
Dr. Maddah ENMG 65 Financial Eng g II 10/16/06 Chapter 11 Models of Asset Dynamics () Random Walk A random process, z, is an additive process defined over times t 0, t 1,, t k, t k+1,, such that z( t )
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationMixing Di usion and Jump Processes
Mixing Di usion and Jump Processes Mixing Di usion and Jump Processes 1/ 27 Introduction Using a mixture of jump and di usion processes can model asset prices that are subject to large, discontinuous changes,
More informationDynamic Hedging and PDE Valuation
Dynamic Hedging and PDE Valuation Dynamic Hedging and PDE Valuation 1/ 36 Introduction Asset prices are modeled as following di usion processes, permitting the possibility of continuous trading. This environment
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationHomework Assignments
Homework Assignments Week 1 (p. 57) #4.1, 4., 4.3 Week (pp 58 6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15 19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9 31) #.,.6,.9 Week 4 (pp 36 37)
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationDerivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester
Derivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester Our exam is Wednesday, December 19, at the normal class place and time. You may bring two sheets of notes (8.5
More informationLecture 18. More on option pricing. Lecture 18 1 / 21
Lecture 18 More on option pricing Lecture 18 1 / 21 Introduction In this lecture we will see more applications of option pricing theory. Lecture 18 2 / 21 Greeks (1) The price f of a derivative depends
More informationFIN FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Greeks Introduction We have studied how to price an option using the Black-Scholes formula. Now we wish to consider how the option price changes, either
More information(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given:
(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more
More informationRisk Neutral Measures
CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationMSC FINANCIAL ENGINEERING PRICING I, AUTUMN LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK
MSC FINANCIAL ENGINEERING PRICING I, AUTUMN 2010-2011 LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK In this section we look at some easy extensions of the Black
More informationINVESTMENTS Class 2: Securities, Random Walk on Wall Street
15.433 INVESTMENTS Class 2: Securities, Random Walk on Wall Street Reto R. Gallati MIT Sloan School of Management Spring 2003 February 5th 2003 Outline Probability Theory A brief review of probability
More informationlast problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends.
224 10 Arbitrage and SDEs last problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends. 10.1 (Calculation of Delta First and Finest
More informationModule 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.
Stochastic Differential Equation Consider. Moreover partition the interval into and define, where. Now by Rieman Integral we know that, where. Moreover. Using the fundamentals mentioned above we can easily
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationBluff Your Way Through Black-Scholes
Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background
More informationFinancial Risk Management
Risk-neutrality in derivatives pricing University of Oulu - Department of Finance Spring 2018 Portfolio of two assets Value at time t = 0 Expected return Value at time t = 1 Asset A Asset B 10.00 30.00
More informationTEACHING NOTE 98-04: EXCHANGE OPTION PRICING
TEACHING NOTE 98-04: EXCHANGE OPTION PRICING Version date: June 3, 017 C:\CLASSES\TEACHING NOTES\TN98-04.WPD The exchange option, first developed by Margrabe (1978), has proven to be an extremely powerful
More informationThe Black-Scholes Equation using Heat Equation
The Black-Scholes Equation using Heat Equation Peter Cassar May 0, 05 Assumptions of the Black-Scholes Model We have a risk free asset given by the price process, dbt = rbt The asset price follows a geometric
More informationChapter 15: Jump Processes and Incomplete Markets. 1 Jumps as One Explanation of Incomplete Markets
Chapter 5: Jump Processes and Incomplete Markets Jumps as One Explanation of Incomplete Markets It is easy to argue that Brownian motion paths cannot model actual stock price movements properly in reality,
More informationMORNING SESSION. Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quantitative Finance and Investment Core Exam QFICORE MORNING SESSION Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Instructions 1.
More informationCourse MFE/3F Practice Exam 2 Solutions
Course MFE/3F Practice Exam Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 A Chapter 16, Black-Scholes Equation The expressions for the value
More informationB. Combinations. 1. Synthetic Call (Put-Call Parity). 2. Writing a Covered Call. 3. Straddle, Strangle. 4. Spreads (Bull, Bear, Butterfly).
1 EG, Ch. 22; Options I. Overview. A. Definitions. 1. Option - contract in entitling holder to buy/sell a certain asset at or before a certain time at a specified price. Gives holder the right, but not
More informationOptions. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Options
Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Definitions and Terminology Definition An option is the right, but not the obligation, to buy or sell a security such
More informationModels of Option Pricing: The Black-Scholes, Binomial and Monte Carlo Methods
Registration number 65 Models of Option Pricing: The Black-Scholes, Binomial and Monte Carlo Methods Supervised by Dr Christopher Greenman University of East Anglia Faculty of Science School of Computing
More informationComputational Finance. Computational Finance p. 1
Computational Finance Computational Finance p. 1 Outline Binomial model: option pricing and optimal investment Monte Carlo techniques for pricing of options pricing of non-standard options improving accuracy
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 217 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 217 13 Lecture 13 November 15, 217 Derivation of the Black-Scholes-Merton
More information3.1 Itô s Lemma for Continuous Stochastic Variables
Lecture 3 Log Normal Distribution 3.1 Itô s Lemma for Continuous Stochastic Variables Mathematical Finance is about pricing (or valuing) financial contracts, and in particular those contracts which depend
More informationStochastic Modelling in Finance
in Finance Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH April 2010 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes
More information2.3 Mathematical Finance: Option pricing
CHAPTR 2. CONTINUUM MODL 8 2.3 Mathematical Finance: Option pricing Options are some of the commonest examples of derivative securities (also termed financial derivatives or simply derivatives). A uropean
More informationAdvanced Stochastic Processes.
Advanced Stochastic Processes. David Gamarnik LECTURE 16 Applications of Ito calculus to finance Lecture outline Trading strategies Black Scholes option pricing formula 16.1. Security price processes,
More informationProbability in Options Pricing
Probability in Options Pricing Mark Cohen and Luke Skon Kenyon College cohenmj@kenyon.edu December 14, 2012 Mark Cohen and Luke Skon (Kenyon college) Probability Presentation December 14, 2012 1 / 16 What
More informationAdvanced Corporate Finance. 5. Options (a refresher)
Advanced Corporate Finance 5. Options (a refresher) Objectives of the session 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option pricing model 5.
More informationBasic Concepts in Mathematical Finance
Chapter 1 Basic Concepts in Mathematical Finance In this chapter, we give an overview of basic concepts in mathematical finance theory, and then explain those concepts in very simple cases, namely in the
More informationAppendix A Financial Calculations
Derivatives Demystified: A Step-by-Step Guide to Forwards, Futures, Swaps and Options, Second Edition By Andrew M. Chisholm 010 John Wiley & Sons, Ltd. Appendix A Financial Calculations TIME VALUE OF MONEY
More informationFinancial Derivatives Section 1
Financial Derivatives Section 1 Forwards & Futures Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of Piraeus)
More informationMSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013
MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationReplication and Absence of Arbitrage in Non-Semimartingale Models
Replication and Absence of Arbitrage in Non-Semimartingale Models Matematiikan päivät, Tampere, 4-5. January 2006 Tommi Sottinen University of Helsinki 4.1.2006 Outline 1. The classical pricing model:
More informationCourse MFE/3F Practice Exam 1 Solutions
Course MFE/3F Practice Exam 1 Solutions he chapter references below refer to the chapters of the ActuraialBrew.com Study Manual. Solution 1 C Chapter 16, Sharpe Ratio If we (incorrectly) assume that the
More informationCHAPTER 10 OPTION PRICING - II. Derivatives and Risk Management By Rajiv Srivastava. Copyright Oxford University Press
CHAPTER 10 OPTION PRICING - II Options Pricing II Intrinsic Value and Time Value Boundary Conditions for Option Pricing Arbitrage Based Relationship for Option Pricing Put Call Parity 2 Binomial Option
More informationFINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationMerton s Jump Diffusion Model. David Bonnemort, Yunhye Chu, Cory Steffen, Carl Tams
Merton s Jump Diffusion Model David Bonnemort, Yunhye Chu, Cory Steffen, Carl Tams Outline Background The Problem Research Summary & future direction Background Terms Option: (Call/Put) is a derivative
More informationChapter 9 - Mechanics of Options Markets
Chapter 9 - Mechanics of Options Markets Types of options Option positions and profit/loss diagrams Underlying assets Specifications Trading options Margins Taxation Warrants, employee stock options, and
More informationDeriving and Solving the Black-Scholes Equation
Introduction Deriving and Solving the Black-Scholes Equation Shane Moore April 27, 2014 The Black-Scholes equation, named after Fischer Black and Myron Scholes, is a partial differential equation, which
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More informationHomework Assignments
Homework Assignments Week 1 (p 57) #4.1, 4., 4.3 Week (pp 58-6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15-19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9-31) #.,.6,.9 Week 4 (pp 36-37)
More informationLévy models in finance
Lévy models in finance Ernesto Mordecki Universidad de la República, Montevideo, Uruguay PASI - Guanajuato - June 2010 Summary General aim: describe jummp modelling in finace through some relevant issues.
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 14 Lecture 14 November 15, 2017 Derivation of the
More information1 Interest Based Instruments
1 Interest Based Instruments e.g., Bonds, forward rate agreements (FRA), and swaps. Note that the higher the credit risk, the higher the interest rate. Zero Rates: n year zero rate (or simply n-year zero)
More informationOption 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
More informationOption Pricing. Simple Arbitrage Relations. Payoffs to Call and Put Options. Black-Scholes Model. Put-Call Parity. Implied Volatility
Simple Arbitrage Relations Payoffs to Call and Put Options Black-Scholes Model Put-Call Parity Implied Volatility Option Pricing Options: Definitions A call option gives the buyer the right, but not the
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationForwards and Futures. Chapter Basics of forwards and futures Forwards
Chapter 7 Forwards and Futures Copyright c 2008 2011 Hyeong In Choi, All rights reserved. 7.1 Basics of forwards and futures The financial assets typically stocks we have been dealing with so far are the
More informationnon linear Payoffs Markus K. Brunnermeier
Institutional Finance Lecture 10: Dynamic Arbitrage to Replicate non linear Payoffs Markus K. Brunnermeier Preceptor: Dong Beom Choi Princeton University 1 BINOMIAL OPTION PRICING Consider a European call
More informationBlack-Scholes Option Pricing
Black-Scholes Option Pricing The pricing kernel furnishes an alternate derivation of the Black-Scholes formula for the price of a call option. Arbitrage is again the foundation for the theory. 1 Risk-Free
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationSimulation Analysis of Option Buying
Mat-.108 Sovelletun Matematiikan erikoistyöt Simulation Analysis of Option Buying Max Mether 45748T 04.0.04 Table Of Contents 1 INTRODUCTION... 3 STOCK AND OPTION PRICING THEORY... 4.1 RANDOM WALKS AND
More informationθ(t ) = T f(0, T ) + σ2 T
1 Derivatives Pricing and Financial Modelling Andrew Cairns: room M3.08 E-mail: A.Cairns@ma.hw.ac.uk Tutorial 10 1. (Ho-Lee) Let X(T ) = T 0 W t dt. (a) What is the distribution of X(T )? (b) Find E[exp(
More informationAttempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS MTHE6026A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are
More informationEvaluating the Black-Scholes option pricing model using hedging simulations
Bachelor Informatica Informatica Universiteit van Amsterdam Evaluating the Black-Scholes option pricing model using hedging simulations Wendy Günther CKN : 6052088 Wendy.Gunther@student.uva.nl June 24,
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationReplication strategies of derivatives under proportional transaction costs - An extension to the Boyle and Vorst model.
Replication strategies of derivatives under proportional transaction costs - An extension to the Boyle and Vorst model Henrik Brunlid September 16, 2005 Abstract When we introduce transaction costs
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 20 Lecture 20 Implied volatility November 30, 2017
More informationFINANCIAL PRICING MODELS
Page 1-22 like equions FINANCIAL PRICING MODELS 20 de Setembro de 2013 PhD Page 1- Student 22 Contents Page 2-22 1 2 3 4 5 PhD Page 2- Student 22 Page 3-22 In 1973, Fischer Black and Myron Scholes presented
More information