Derivative Securities: Lecture 4 Introduction to pricing

Sources: J. Hull, 7th edition Avellaneda and Laurence (2000) Yahoo! & 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

• Put-call parity can be seen as pricing conversions relative to forwards on the same

• What other relations exist between options and spreads on the same underlying asset? Call Spread

Call Spread: a call with strike K, 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  CallK,T   CallL,T 

CSK, L,T   CallK,T  CallL,T   0

Spread makes money if the 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  PutK,T   PutL,T 

PSK, L,T   PutL,T  PutK,T   0

Spread makes money if the price of the underlying goes down 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  BK,(K  L) / 2, L,T   CallK,T  CallL,T  2Call ,T   0  2 

Butterflies make $ if the price is near (K+L)/2 at expiration.

Straddle: Long call and long put with the same strike

-1 +1

K make money if the stock price moves away from the strike and ends far from it

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 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: Short 1 call with 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 from Butterfly Spreads

Assume for simplicity a countable number of strikes, Kn  nK, n  0,1,2,3,..... and that the stock price can only take values on the lattice ST  mK, m 1,2,3,....

CallKn ,T   CallK j ,T  CallK j1,T  CSK j , K j1,T  jn jn

• A call can be viewed as a portfolio of call spreads

• A call spread can be viewed as a portfolio of butterfly spreads

CSK j , K j1,T   BKi , Ki1, Ki2 ,T  i j

K +1

K Calls as super-positions of butterfly spreads

 CallKn ,T    BKi , Ki1, Ki2 ,T  jn i j

   j 1 nBK j , K j1, K j2 ,T  jn

  BK j , K j1, K j2 ,T    j 1 nK   jn  K   BK j , K j1, K j2 ,T   K j1  Kn wK j1,T  wK j1,T  jn K    K j1  Kn  wK j1,T  j0 The weights correspond to values of Butterfly spreads centered at each Kj. In particular, 1 they are positive

K j From weights to probabilities wK,T   0

 1  1  wK j ,T   BK j1, K j , K j1,T  CS0,K,T  1 K 1 K

 PV$1  erT (assuming that the stock can only take values  K)

rT wK j ,T  e pK j ,T ;  pK j ,T 1, pK j ,T  0. j

rT CallK,T   e  maxK j  K,0pK j ,T  j rT p  e E maxST  K,0 First moment of p is the

E p S erT Call0,T  rT qT rqT  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 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 )  CallK,T  KerT  SeqT 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 ' 0S  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   erT 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 0E 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 opportunity is a portfolio of securities and 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 )

Assume that each 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  CallK,T   e EST  K  , PutK,T   e EK  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 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 ~ NT, T 

2  2    yT   2T     T 2 dy T   2  ES  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  qT, Z ~ N0,1 2 Call pricing with the Black-Scholes model

 z 2 2   dz Call S, K,T  erT E S  K   erT Sez T  T / 2rqT  K e 2    T       2

 z 2  z 2 2 2  dz  dz  1   K   T  rT z T  T / 2rqT 2 rT 2     e  Se e  e K  e  A  ln    r  qT  A 2 A 2   T   S  2 

 z 2  z 2  2  dz    dz   eqT S ez T  T / 2e 2   erT K e 2         A 2   A 2 

2  z T   z 2   dz    dz   eqT S e 2   erT K e 2    2    2   A   A   z 2  z 2   dz    dz   eqT S e 2   erT K e 2         A T 2   A 2 

 A T z 2  A z 2   dz    dz   eqT S e 2   erT K e 2          2    2  Black-Scholes Formula

qT rT BSCallS,T, K,r,q,   Se Nd1  Ke Nd2 

1   F   2T  1   F   2T  d  ln 0,T   , d  ln 0,T   , F  SerqT 1     2     0,T  T   K  2   T   K  2 

x z 2 1  Nx   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

Stt  St St 2  ~ Nt, 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  CSt ,t 0  t  T

Change in market value over one period :

Vt  CSt ,t

2 2 CS ,t CS ,t 1  CS ,t 1  CS ,t 2  t t  t S  t t 2  t S  .... t S 2 t 2 2 S 2 t

2 2 CS ,t CS ,t 1  CS ,t 2  S   t t  t S  t S  t   o t 2 t     t S 2 S  St 

2 2 2    CS ,t 1  CS ,t 2  CS ,t 1  CS ,t 2  S    t  t S  2 t  t S  t S  t   2t  ot  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  .

• Assume derivative does not pay dividends CSt ,t   t  St Profit and loss, including financing and dividends :

PNL  Vt rt  Vt   St  St rt  St qt

 Vt rt t  St   t   St  St rt  St qt

 Vt rt t  St r  qt   t

 CS ,t CS ,t 1  2CS ,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  2CS ,t  S     S 2 t  t   2t  ot t t S 2  S   t  

2 Conditional 2   2  CSt ,t  St  2  expectation E t | St  St E   t | St   ot S 2  S  of epsilon  t  

 2CS ,t  S 2 t  2t 2  ot t S 2

 ot

• 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 – shares has expected value

EPNL t  o(t)  CS ,t CS ,t 1  2CS ,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 (we already took into acct its financing). Thus:

2 CS ,t 1  CS ,t CS ,t   2S 2 t  r  qS  CS,tr  0 t 2 S 2 S

• This is the Black-Scholes partial differential equation (PDE). American-style calls & puts

• Consider a on an underlying asset paying dividends continuously. Since the option can be exercised anytime, we have

CS,t  maxS  K,0, t  T. (1)

• The terminal condition at t=T corresponds to the final payoff

CS,T   maxS  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.