Securities: Lecture 7 Further applications of Black-Scholes and Arbitrage Pricing Theory

Sources: J. Hull Avellaneda and Laurence Black’s Formula

• Sometimes its easier to think in terms of forward prices. Recalling that in Black-Scholes

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

qT rT BSCallS,T, K,r,q,   e SN(d1)  e KN(d2 ) rT rqT rT  e e SN(d1)  e KN(d2 )

BSCallS,T, K,r,q,   erT  BSCallF,T, K,0,0, 

• Similarly,

BSPutS,T, K,r,q,   erT  BSPutF,T, K,0,0,  Symmetry between puts and calls in Black’s formula

  ln(F / K)  T   ln(F / K)  T  rT     CallF,T, K,r,   e F  N    K  N      T 2    T 2 

  ln(K / F)  T   ln(K / F)  T  rT     PutF,T, K,r,   e K  N    F  N      T 2    T 2 

CallF,T, K,r,   PutK,T, F,r, 

Call= ``exchanging cash for stock’’, Put=``exchanging stock for cash’’ Options on Foreign Exchange

• Example: Price a 90-day USD call/EUR put on 10,000,000 USD with strike 1.40. US rate=0.41%, EUR rate=1.50%. Current exchange rate: EUR= USD 1.3702, : 16%.

 90  1  0.0041 360 F  (1.3702)   1.36648 Forward with USD  90  as domestic currency 1  0.0150  360  Put on 1 EUR 1 notional amt. V ( per EUR notional)  BSPut1.36648,0.25,1.40,0,0,0.15 1 0.250.0041 Use Black with 1 USD rate  0.062903  $0.062839 1.001025

10,000,000 Notional adjusted at , Value  $0.062839  $448,847.10  EUR 327,577.8 option value in EUR converted 1.40 at spot price Options on Futures

• Consider an option on a . Typically, the option gives the right, but not the obligation to enter/exit a futures contract at a specified price and is cash-settled.

• Example: on a Eurodollar futures contract with strike 97 in 1 month. Futures settle in 3 months.

-- in or before 1 month: can and get long a futures. The payoff will be max(f-97,0) times the notional amount, cash-settled.

• Similar for a , with cash settlement of max(97-f,0). Derivation (I)

• Assuming that the option fair value depends on futures price and time-to- maturity,

C  C( f ,t), t  calendar time, T  maturity

2 C C 1  C 2 C  t  f  2 f   ot t f 2 f

C • Exposure to futures : f f

C • Assume that you `hedge’ by taking a position in  contracts and in one option, the latter financed at rate r. f The 1-day PNL is C PNL  rCt  C  f f Derivation (II)

• Notice that the futures position requires no financing (except for , which we do not take into account). Hence,

C C 1  2C C PNL  rCt  t  f  f 2  f  o(t) t f 2 f 2 f

C 1  2C  rCt  t  f 2  o(t) t 2 f 2

2 C f 2 2  2C f 2  2C  f    rCt  t  t     2t  o(t) t 2 f 2 2 f 2  f    

2  C f 2 2  2C  f 2  2C  f     rC   t     2t  o(t)  t 2 f 2  2 f 2  f       Derivation (III)

• If we assume that

 2   f  2 E     t  ot  f    

the profit/loss from the hedged strategy would be on average

 C  2 f 2 2C  PNL     rC t  o t  2     t 2 f 

• We conclude that the fair value of the option should satisfy the PDE

C f ,t  2 f 2 2C f ,t   rC f ,t  0, 0  t  T t 2 f 2 Black vs. Black Scholes PDE

• The PDE is a special case of the Black-Scholes PDE with r=q and therefore no first-order term.

• For European-style options, this leads to the boundary-value problem

C f ,t  2 f 2  2C f ,t   rC f ,t  0, 0  t  T t 2 f 2

  C f ,T   F f , [F( f )  ( f  K) (call) , F( f )  (K  f ) (put)]

This has Black’s formula as an exact solution.

• For American options, the PDE is solved numerically, e.g. with the trinomial scheme.

• If the interest rate is positive, American-style options which are deep-in-the- money should be exercised before . Multi-period models & beyond Black-Scholes

• In a previous lecture, we presented option pricing as an expected value over a probability measure.

• Using a hedging argument, we also presented a PDE approach for pricing options and showed (numerically) that this lead to the same prices in the case of European options.

• To connect both methods and increase our understanding of pricing of derivatives, we need to consider multi-period models and a more general setting. A multi-period model for asset-pricing Price level

time

Assumptions • N trading dates • On each trading date, we know the interest rate for the next period, the yield for the next period and the price level, i.e.

St ,qt ,rt , t  t1,t2 ,...,tN

• S and q can represent vectors, i.e., multiple securities Arbitrage opportunities

• An arbitrage opportunity consists of a trading strategy which has zero initial cost and which gives rise to non-negative cash-flows in all future states of the market with at least one state with positive cash flows.

• In a 1-period model, we saw that the absence of arbitrage opportunities implies the existence of a probability measure in the set of states at time T such that

1 all traded securities V  E p V  satisfy 0 1 rt T

p In particular, F0,T  E ST  The Hahn-Banach Theorem

• Let X be a finite space representing the future states of a market, and let  ( F ) be a linear functional defined on functions of future states:

F  G  F  G Here, the functional represents the price of a derivative with payoff F(x) outcome x is realized.

• If    has the property that Fx  0 x  X  F   0, then

F    wxFx F xX

for some positive weights  wx, x  X  Absence of arbitrage opportunities: the formalization of ``fair value’’

• In the one-period model, we found that the absence of arbitrage opportunities implied that there exists a probability measure on future states such that

any derivative security with value VF at time 0 and value F(S) at time T

should have a price V F satisfying

1 V S   E P FS  F 0 1 rT T

• In a multi-period model, the same equation should hold conditionally on the

current state of the market, i.e. if a security pays a cash-flow F(S) at time t n  1 ,

its value at time t n should satisfy

1 V S , S ,...., S ;t  E pn,n1 FS  t1 t2 tn n 1 r t tn1 tn The probability pn,n1

• The probability can be seen as a function of the observations up

to time t n forecasting the price level at time t n  1 :

Observations pn,n1(X )  pn,n1St  X | St ,dt ,rt ,i  n n1 i i i up to time n

X

X • Main idea: The pricing probability from date n to date n+1 must satisfy a constraint, associated with forward pricing. Past observations

t n tn1 Forwards and probability constraints

• If interest and dividend are known at date n, then the forward price is known as well, and we must have

1 0  E pn,n1 S  F   E pn,n1 S   F 1 r t tn1 tn ,tn1 tn1 tn ,tn1 tn

forward price= 1 r t E pn,n1 S   tn S expectation of tn1 1 q t tn tn future spot

``currency-type model for the dividend’’ (for simplicity) Probabilities for prices defined over paths

Price level (S)

St S 2 t4

S t1 S t3 time

PS , S , S ,..., S  p S  p S  p S ...p S  t1 t2 t3 tN 0,1 t1 1,2 t2 2,3 t3 N 1,N tN

• Given the 1-step probabilities p n , n  1 , this formula defines a probability over paths Characterization of P( )

• Define the discounted total return process T by

 n1   1 qt t  j  j0  T   S , T  S tn  n1  tn t0 t0  1 r t    t j    j0  • Then, E P T |T  E pn,n1 T  tn1 tn tn1

 n   n   1 qt t  1 qt t  j  j 1 r t  j0  pn,n1  j0  tn  E S  S  n  tn1  n 1 q t tn  1 r t   1 r t  tn   t j     t j    j0   j0 

 n1   1 qt t  j  j0  P  S  T E T |T  T  n1  tn tn tn1 tn tn  1 r t    t j    j0 

No-arbitrage theorem for multi-period asset pricing

• If there are no arbitrage opportunities, there exists a probability measure over paths of traded securities, P(), such that the discounted total-return process of any traded security is a martingale under P, i.e.,

 n1   1 qt t setting  j  j0  T   S , T  S tn  n1  tn t0 t0  1 r t    t j    j0 

E P T |T  T we have t n  1 t n tn .