金融隨機計算:第一章

Black-Scholes-Merton Theory of Pricing and Hedging

CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. Derivative Contracts

• Derivatives, also called contingent claims, are contracts based on some underlying assets. • A typical option is a contract that gives its holder the right, but not the obligation, to but one unit of a financial (underlying) asset for a predetermined strike (or exercise) price K by the maturity date T. Option

• The type of option: the option to buy is called a call option; the option to sell is called a put option. • The underlying asset: it can be a stock, bond, currency, etc. • The expiration date: if the option can be exercised at any time before maturity, it is called an American option; if it can only be exercised at maturity, it is called a European option. • The amount of an underlying asset to be purchased or sold. • The price of an option is also called the premium. European Options

• Let denote the underlying risky asset such as a stock price. 푡 • A European푆 option is a contract to pay at a maturity time T, which is path 푇 independent. ℎ 푆 Call and Put

• Call payoff: = = , 0 That means if >+ , the holder will exercise • 푇 푇 푇 theℎ 푆 option푆 to −make퐾 a profit푚푚푚 푆 − 퐾 by buying the stock for the푆푇 price퐾 K and selling it immediately at the market price푆푇 − 퐾 ; otherwise the option is not worthy to be exercised. 푇 • Put payoff: 푆 = = , 0 + ℎ 푆푇 퐾 − 푆푇 푚푚푚 퐾 − 푆푇 American Options

• A holder of an American option can exercise the option at any time before maturity T. • Let denote the time to exercise the option, which is a stopping time because up to time 휏 , the holder has the right to hold or exercise the option based on the information flow. • 푡Call≤ payoff:푇 = = , 0 Put payoff: + • 휏 휏 휏 ℎ 푆 = 푆 − 퐾 = 푚푚푚 푆 − 퐾 , 0 + ℎ 푆푇 퐾 − 푆휏 푚푚푚 퐾 − 푆휏 Exotic Option

• Exotic options are those options for which their contracts are not of European or American style defined previously. • Barrier options. Ex, for a down-and-out call option, = + • Lookbackℎ options.푆푇 푆 Ex,푇 − 퐾= 퐼 푖푖푖푡≤inf푇푆푡>퐵 = inf + 푇 푡 ℎ 푆 − 푡≤푇 푆 푇 푡 • Asian푆 − 푡options.≤푇 푆 Ex, = 1 푇 + ℎ 푆푇 − 푇 ∫0 푆푡 푑� Underlying Risky Asset

• The stock price is governed by the geometric Brownian motion under the 푡 historical filtered푆 probability space , , , : = + (1) 푡 훺 ℱ ℱ 풫 = 푑푆푡 휇푆푡푑� 휎푆푡푑푊푡 푆0 � Replication of Trading Strategy

• A dynamic trading strategy is a pair , of adapted processes specifying the number of 푡 푡 units held at time t of the stock (underlying훼 훽 risky asset) and bond (riskless asset) , where r is the risk-free interest rate. 푟� 푡 • At the maturity푆 T, the value this portfolio푒 will replicate the terminal payoff ; namely + = . 푇 푟� ℎ 푆 훼푇푆푇 훽푇푒 ℎ 푆푇 Self-Financing Portfolio

• That is the variations of portfolio value are due only to the variations of the stock price and bond price. • + = + • In integral form푟� + = 푟� + + 푑 훼푡푆푡 훽푡푒 훼푡푑푆푡 푟훽푡푒 푑� + , 0 푟� 훼푡푆푡 훽푡푒 훼0푆0 훽0 푡 푡 푟� ∫0 훼푠 푑푆푠 ∫0 훽푠 푑푒 ≤ 푡 ≤ 푇 Perfect Replication

Based on the arbitrage pricing theory, the fair price of a derivatives is equal to the value of its portfolio: + = , 푟� where , 푡denotes푡 푡 the option price푡 at time t and stock price훼 푆 =훽 푒. 푃 푡 푆 푃 푡 � 푆푡 � Ito’s Formula

• + + = + +푟� + 훼푡휇푆푡 훽푡푒 푑� 훼푡2휎푆푡푑푊푡 휕휕 휕휕 1 2 2 휕 � 휕휕 2 • 휕�= 휇푆푡 ,휕휕 2 휎 푆푡 휕� 푑� 휎푆푡 휕휕 푑푊푡 휕휕 • 푡 = , 푡 훼 휕휕 푡 푆 −푟� 훽푡 푃 푡 푆푡 − 훼푡푆푡 푒 Black-Scholes PDE

, = 0

With the terminal 퐵condition� , = , where the operatorℒ 푃 푡 � 1 푃 푇 � ℎ � = + + 2 2 휕 2 2 휕 휕 is defineℒ퐵 �on the domain휎 � of 02, 푟×� 0, − 푟. 휕� 휕� 휕� 푇 ∞ Remarks

• The Black-Scholes PDE can be solved either by numerical PDE methods like finite difference methods or by Monte Carlo simulations. • The dynamic trading strategy , is also called hedging strategy by emphasizing the elimination of risks. 훼푡 훽푡 • This strategy requires a holding number of risky asset = , , which is called the Delta in financial휕휕 engineering. 푡 푡 훼 휕휕 푡 푆 Independence of

• By inspecting the Black-Scholes PDE,휇 the rate of returns in (1) does not appear at all. This is a remarkable feature of the Black-Scholes theory that휇 two investors may have completely different speculative view for the rate of returns of the risky asset, but they have to agree that the no-arbitrage price P of the derivative does not depend on .

휇 Black-Scholes Formula

The price of a European call option admits a closed-form solution. , = , where −푟 푇−푡 퐶퐵� 푡 � �풩 푑1 − 퐾1푒 풩 푑2 ln + + 2 = 2 �⁄퐾 푟 휎 푇 − 푡 푑1 = , 1 휎 푇 − 푡 푑2 = 푑1 푡 � − 휎 푇 − 푡 2 푑 2 −푢 ⁄2 풩 푑 � 푒 푑푑 휋 −∞ Put-Call Parity

• , , = • , = 0 −푟 푇−푡 퐶퐵� 푡 푆푡 − 푃퐵� 푡 푆푡 푆푡 − 퐾푒 with퐵 �the퐵 terminal� 퐵� condition = , admitsℒ 퐶a simple− 푃 solution푡 � , = ℎ � � − 퐾 퐵� 퐵� −푟 푇−푡 퐶 − 푃 푡 � � − 퐾푒 Parameter Estimation

• Within the Black-Scholes formula, there is only one non-observable parameter, namely the volatility . 1. the direct method: given a set of historical stock prices, one 휎can use either (1) the approximation of the quadratic variation of log returns or (2) the maximum likelihood of log returns of estimate . 2. the implied method: given the quoted European call or put option prices, one can invert an “implied”휎 volatility from each quoted option. Change of

• By Girsanov’s Theorem, one can construct a new probability measure which is equivalent to so that the ∗drifted Brownian motion + becomes풫 a standard 풫 Brownian motion,휇−푟 denoted by . 푡 휎 = 푊+ 푡 ∗ 푊푡 = + + + 푑푆푡 휇푆푡푑� 휎푆푡푑푊푡 = + 휇 − 푟 휇 푟 − 휇 푆푡푑� 휎푆푡푑 푊푡 푡 ∗ 휎 푟푆푡푑� 휎푆푡푑푊푡 Remarks

1. the discounted stock price becomes a martingale under . This is because of = ∗ . 2. the discounted−푟� value풫 of− portfolio푟� becomes∗ a 푡 푡 푡 martingale푑 under푒 푆 . This푒 is because푆 휎푑푊 of + ∗ = . We call− 푟such� probability풫 푟� measure− 푟� as the risk∗ - 푡 푡 푡 푡 푡 푡 neutral푑푒 measure훼 푆 and훽 푒 the shifted훼 푒 drift∗푆 휎푑 푊as the market price of risk. 풫 휇−푟 휎 Risk-Neutral Evaluation (I)

• The trading strategy must replicate the terminal payoff , + = . 푇 Sinceℎ 푆 + 푟 푇is a martingale under , • 푇 푇 푇 푇 −푟�+ 훼 푆 =훽푟�푒 ℎ 푆 + | ∗ 푡 푡 푡 The−푟 �value푒 of 훼the푆 portfolio푟� 훽 푒 ∗ is equal−푟� to 푟� 풫 • 푡 푡 푡 푇 푇 푇 푡 푒 훼 푆 +훽 푒 = 피 푒 ( 훼 )푆 훽| 푒 ℱ = 푟� ( ∗ )−푟 푇−푡| 훼푡푆푡 훽푡푒 피 푒 ℎ 푆푇 ℱ푡 by the Markov property∗ −푟 푇of− 푡 . 피 푒 ℎ 푆푇 푆푡 푆푡 Risk-Neutral Evaluation (II)

• Based on the definition of the fair price, the option price can be defined as a conditional expectation of the discounted payoff under the risk-neutral probability measure : , = | ∗. ∗ −푟 푇−푡 풫 푃 푡 푆푡 피 푒 ℎ 푆푇 푆푡 Derivation of The Black-Scholes Formula

= + 22 휎 푆푇 푆푡푒�푒 푟 − 푇 − 푡 휎 푇 − 푡푍 , = + + 22 −푟 푇−푡 휎 푃 푡 푆푡 피� 푒 푆푡푒�푒 푟 − 푇 − 푡 휎 푇 − 푡푍 − 퐾 = + 22 −푟 푇−푡 휎 � 푒 �푆푡푒�푒 푟 − 푇 − 푡 휎 푇 − 푡푍 1 + 2 2 −푧 ⁄2 −= 퐾� 푒 푑푑 = 휋 ⋯ −푟 푇−푡 푆푡풩 푑1 − 퐾푒 풩 푑2 Hedging with One Stock

• , = −푟� + + , + 푑 푒 푃 푡 푆푡 2 −푟� 휕휕 1 2 2 휕 � 휕휕 , 푡 2 푡 푡 푒 휕� 2 휎 푆 휕� 푟푆 휕휕 − 푟� 푡 푆 푑� 휕휕 −푟� , = ∗ • 푒 휕휕 푡 푆푡 휎푆푡푑푊푡 −0푟�, + , 푒 푃 푇 푆푇 푇 −푟� 휕휕 ∗ • P 0, 0 = |푡 푡 푡 푃 푆 ∫0 푒 휕휕 푡 푆 휎푆 푑푊 ∗ −푟� 푆0 피 푒 ℎ 푆푇 푆0 Martingale Representation Theorem

• Theorem Let ,0 be a Brownian motion on a probability space , , , and the filtration is generated by this푊푡 Brownian≤ 푡 ≤ 푇 motion. Let , 0 be a martingale with respect훺 ℱ toℱ 푡this0≤ 푡filtration.≤푇 풫 Then there is an adapted process , 0 , such that푀 푡 ≤ 푡 ≤ 푇 = 푢 + , 0 훤 ≤ �푡 ≤ 푇 • is typically푀 푡related푀0 to �the훤 푢hedging푑푊푢 strategy≤ 푡 ≤ 푇 associated with a trading martingale0 . This theorem only guarantees theΓ푢 existence condition of the hedging strategy. One can use Ito formula to compute푀 푡 explicitly. Γ Arbitrage

• An arbitrage is a way of trading so that one starts with zero capital and at some later time T is sure not to have lost money and furthermore has a positive probability of having made money • Definition An arbitrage is a portfolio value process satisfying = 0 and also satisfying for some time > 0 푉푡 0 = 1푉,0 > 0 > 0 푇 풫 푉푇 ≥ 풫 푉푇 Fundamental Theorems of Asset Pricing • First Theorem If a market model has a risk- neutral probability measure, then it does not admit arbitrage. • Definition A market model is complete if every derivative security can be hedged. • Second Theorem Consider a market model that has a risk-neutral probability measure. The model is complete if and only if the risk- neutral probability measure is unique. Forward Contract

• Definition A forward contract is an agreement to pay a specified delivery price L at a delivery date T for the asset price at time t. The T-forward price , of this asset, is the value of K that makes the forward contract have푆푡 no-arbitrage price zero at time퐹퐹 t.� 푆 푡 푇 • Theorem The T-forward price is , = , , 푆푡 where the price 퐹퐹of a�푆 zero푡 푇-coupon is 1 퐵 푡 푇 , = 1| 푇 ∗ − ∫푡 푟푢푑� 퐵 푟 푇 피 푒 푟푡 퐷푡 Futures

• Definition The futures price of an asset whose value at time T is is given by the formula , = | , 0 푇 푆 ∗ A long position푆 in the futures푇 푡contract is an agreement퐹𝐹 to푡 receive푇 피 as푆 a cashℱ flow≤ 푡the≤ changes푇 in the futures price during the time the position is held. A short position in the futures contract receives the opposite cash flow. Forward-Futures Spread

• When the (instantaneous) interest-rate is random, the forward-futures spread is defined by 푟푡 0, 0, = 0 1 푆 ∗ 퐹퐹�푆 푇 − 퐹𝐹푆 푇 ∗ − 피 푆푇 = 피 퐷푇 1 ∗ ∗ ∗ ∗ 피 퐷푇푆푇 − 피 퐷푇피 푆푇 = 피 퐷푇 , ∗ where ∗, denotes푇 the푇 covariance of 푇 퐶�퐶 퐷 푆 and under∗ 피the퐷 risk-neutral probability measure. 퐶�퐶 퐷푇 푆푇 퐷푇 푆푇 Appendix: Girsanov Theorem

Let be a standard Brownian motion under a probability space , , , . Let be an adapted process. Define 푊푡 1 푡 0≤푡≤푇 = 푡 훺 ℱ ℱ 풫 휃 푡 2 푡 2 푍푡 푒�푒 − � 휃푠 푑푊푠 − � 휃푠 푑푑 =0 + 0 푡 ∗ and assume that 푡 푡 푠 푊 푊 �0 휃 푑푑 < 푇 2 2 Set = . Then = 1 and under푠 푠 the probability measure defined 피 �0 휃 푍 푑푑 ∞ by = , the process ,0 , is a Brownian motion. 푇 � 푑푍풫� 푍 피푍 ∗ 풫 푑풫 푍푇 푊푡 ≤ 푡 ≤ 푇