Derivative Securities: Lecture 4 Introduction to option pricing
Sources: J. Hull, 7th edition Avellaneda and Laurence (2000) Yahoo!Finance & assorted websites
Option Pricing
• In previous lectures, we covered forward pricing and the importance of cost-of carry
• We also covered Put-Call Parity, which can be viewed as relation that should hold between European-style puts and calls with the same expiration
• Put-call parity can be seen as pricing conversions relative to forwards on the same underlying asset
• What other relations exist between options and spreads on the same underlying asset? Call Spread
Call Spread: Long a call with strike K, short a call with strike L (L>K)
K L S Since the payoff is non-negative, the value of the spread must be positive
K L CallK,T CallL,T
CSK, L,T CallK,T CallL,T 0
Spread makes money if the price of the underlying goes up Put Spread
Put Spread: Long a put with strike L, short a put with strike L (L>K)
K L S Since the payoff is non-negative, the value of the spread must be positive
K L PutK,T PutL,T
PSK, L,T PutL,T PutK,T 0
Spread makes money if the price of the underlying goes down Butterfly Spread
Butterfly spread: Long call with strike K, long call with strike L, short 2 calls with strike (K+L)/2
K (K+L)/2 L S
Since the spread has non-negative payoff, it must have positive value
K L BK,(K L) / 2, L,T CallK,T CallL,T 2Call ,T 0 2
Butterflies make $ if the stock price is near (K+L)/2 at expiration. Straddle
Straddle: Long call and long put with the same strike
-1 +1
K Straddles make money if the stock price moves away from the strike and ends far from it Strangle
Strangle: Long 1 put with strike K, long 1 call with strike L, L>K.
K L S
Strangle is also non-directional, like a straddle, but makes $ only if the stock moves very far away.
Straddles and strangles are often used to express views about volatility of the underlying stock and are non-directional. Risk-reversal
Risk-reversal : Long 1 put with strike K, short 1 call with strike L, L>K.
K L
Directional spread. Can be seen as financing a put by selling an upside call. Calendar Spread
Calendar Spread: Short 1 call with maturity T1, Long 1 call with maturity T2, T1 Maturity T1 Maturity T2 K K • If the underlying pays no dividends between T1 and T2, then the longer maturity call is above intrinsic value at time T1. Calendars have positive value. • For American options, calendars always have positive value Reconstructing Call prices from Butterfly Spreads Assume for simplicity a countable number of strikes, Kn nK, n 0,1,2,3,..... and that the stock price can only take values on the lattice ST mK, m 1,2,3,.... CallKn ,T CallK j ,T CallK j1,T CSK j , K j1,T jn jn • A call can be viewed as a portfolio of call spreads • A call spread can be viewed as a portfolio of butterfly spreads CSK j , K j1,T BKi , Ki1, Ki2 ,T i j K +1 K Calls as super-positions of butterfly spreads CallKn ,T BKi , Ki1, Ki2 ,T jn i j j 1 nBK j , K j1, K j2 ,T jn BK j , K j1, K j2 ,T j 1 nK jn K BK j , K j1, K j2 ,T K j1 Kn wK j1,T wK j1,T jn K K j1 Kn wK j1,T j0 The weights correspond to values of Butterfly spreads centered at each Kj. In particular, 1 they are positive K j From weights to probabilities wK,T 0 1 1 wK j ,T BK j1, K j , K j1,T CS0,K,T 1 K 1 K PV$1 erT (assuming that the stock can only take values K) rT wK j ,T e pK j ,T ; pK j ,T 1, pK j ,T 0. j rT CallK,T e maxK j K,0pK j ,T j rT p e E maxST K,0 First moment of p is the forward price E p S erT Call0,T rT qT rqT e e S0 S0e F0,T • A call with strike 0 is the option to buy the stock at zero at time T Its value is therefore the present value of the forward price (pay now, get stock later). • It follows that the first moment of p is the forward price. • It also follows that put prices are given by a similar formula, namely Put(K,T ) CallK,T KerT SeqT rT p rT rT e E ST K e K e F rT p rT p e E ST K e E K ST rT p e E K ST General Payoffs • Any twice differentiable function f(S) can be expressed as a combination • of put and call payoffs, using the formula S ' • f S f 0 f 0 S F S Y f ' ' Y dY (this is just Taylor expansion) F F f 0 f ' 0S F Y S f ''Y dY S Y f ''Y dY 0 F • Thus, a European-style payoff can be viewed as a spread of puts and calls. By linearity of pricing, Fair value of a claim with payoff f ST F erT f 0 0 Put(Y,T ) f '' Y dY Call(Y,T ) f '' Y dY 0 F F rT rT ' p rT p '' '' e f 0 e f 0E ST F e E (Y ST ) f Y dY (ST Y ) f Y dY 0 F rT p e E f ST Fundamental theorem of pricing (one period model) • An arbitrage opportunity is a portfolio of derivative securities and cash which has the following properties: - The payoff is non-negative in all future states of the market - The price of the portfolio is zero or negative (a credit) Assume that each security has a unique price (i.e. assume bid-offer). • If there are no arbitrage opportunities, then there exists a probability distribution of future states of the market such that, for any function f(S), the price of a security with payoff f(ST) is rT p Pf e E f ST • Conversely, if such a probability exists there are no arbitrage opportunities Practical Application to European Options • A pricing measure is a probability of future prices of the underlying asset with the property that p E ST F0,T • If we determine a suitable pricing measure, then all European options with expiration date T should have value given by rT rT CallK,T e EST K , PutK,T e EK ST • The main issue is then to determine a suitable pricing measure in the real, practical world. What does a pricing measure achieve in the case of options? T F0,T S Call(S;K,T) Option fair value is a smoothed out version of the payoff K S The pricing measure gives a model to compute the option’s fair value as a function of the price of the underlying asset, the strike and the maturity The Black-Scholes Model Assume that the pricing measure is log-normal, i.e. log-returns are normal X 2 ST S0e , X ~ NT, T 2 2 yT 2T T 2 dy T 2 ES S e ye 2 T S e 2 S e T 0 2 0 0 2 T 2 2 r q r q - 2 2 2 X Z T T r qT, Z ~ N0,1 2 Call pricing with the Black-Scholes model z 2 2 dz Call S, K,T erT E S K erT Sez T T / 2rqT K e 2 T 2 z 2 z 2 2 2 dz dz 1 K T rT z T T / 2rqT 2 rT 2 e Se e e K e A ln r qT A 2 A 2 T S 2 z 2 z 2 2 dz dz eqT S ez T T / 2e 2 erT K e 2 A 2 A 2 2 z T z 2 dz dz eqT S e 2 erT K e 2 2 2 A A z 2 z 2 dz dz eqT S e 2 erT K e 2 A T 2 A 2 A T z 2 A z 2 dz dz eqT S e 2 erT K e 2 2 2 Black-Scholes Formula qT rT BSCallS,T, K,r,q, Se Nd1 Ke Nd2 1 F 2T 1 F 2T d ln 0,T , d ln 0,T , F SerqT 1 2 0,T T K 2 T K 2 x z 2 1 Nx e 2 dz cumulative normal distribution 2 Black-Scholes Formula at work S=$48, K=$50, r=6%, sigma=40%, q=0 Multi-period asset model • Derivative securities may depend on multiple expiration/cash-flow dates. Furthermore, the 1-period model described above is rigid in the sense that it cannot price American-style options. • We consider instead a more realistic approach to pricing based on the statistics of stock returns over short periods of time (e.g. 1 day). • We assume that the underlying price has returns satisfying Stt St St 2 ~ Nt, t St St • We also assume that successive returns are uncorrelated. Modeling the return of a price time- series (OHLC) Ln St t Model closing prices, for example. The % changes between closing prices are normal and uncorrelated St= closing price of period (t-1,t) Parameterization 2 2 S 2 S S t E t , t E t E t S S S t t t annualized expected return annualized standard deviation 1% daily standard deviation => 15.9% annualized standard deviation 1 t , 252 15.9 252 Pricing Derivatives • Let us model the value of a derivative security as a function of the underlying asset price and the time to expiration Vt CSt ,t 0 t T Change in market value over one period : Vt CSt ,t 2 2 CS ,t CS ,t 1 CS ,t 1 CS ,t 2 t t t S t t 2 t S .... t S 2 t 2 2 S 2 t 2 2 CS ,t CS ,t 1 CS ,t 2 S t t t S t S t o t 2 t t S 2 S St 2 2 2 CS ,t 1 CS ,t 2 CS ,t 1 CS ,t 2 S t t S 2 t t S t S t 2t ot t 2 S 2 t S t 2 S 2 t S t t St t The hedging argument • Consider a portfolio which is long 1 derivative and short stocks. • Assume derivative does not pay dividends CSt ,t t St Profit and loss, including financing and dividends : PNL Vt rt Vt St St rt St qt Vt rt t St t St St rt St qt Vt rt t St r qt t CS ,t CS ,t 1 2CS ,t C S ,t r t t S r q t S 2 2 t t 2 t t t S 2 S Analyzing the residual term t 2 2CS ,t S S 2 t t 2t ot t t S 2 S t 2 Conditional 2 2 CSt ,t St 2 expectation E t | St St E t | St ot S 2 S of epsilon t 2CS ,t S 2 t 2t 2 ot t S 2 ot • The residual term has essentially zero expected return (vanishing exp. return in the limit Dt_>0.) The fair value of our derivative security is… • The PNL for the long short portfolio of 1 derivative and – beta shares has expected value EPNL t o(t) CS ,t CS ,t 1 2CS ,t C S ,t r t t S r q t S 2 2 t o(t) t 2 t t S 2 S • This portfolio has no exposure to the stock price changes. Therefore, if C(St,t) represents the ``fair value’’ of the derivative, the portfolio should have zero rate of return (we already took into acct its financing). Thus: 2 CS ,t 1 CS ,t CS ,t 2S 2 t r qS CS,tr 0 t 2 S 2 S • This is the Black-Scholes partial differential equation (PDE). American-style calls & puts • Consider a call option on an underlying asset paying dividends continuously. Since the option can be exercised anytime, we have CS,t maxS K,0, t T. (1) • The terminal condition at t=T corresponds to the final payoff CS,T maxS K,0. • Thus, the function C(S,t) should satisfy the Black-Scholes PDE in the region of the (S,t)-plane for which strict inequality holds in (1), and it should be equal to max(S-K,0) otherwise. • The solution of this problem is done numerically and will be addressed in the next lecture.