Derivative 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 SerqT T K 2
qT rT BSCallS,T, K,r,q, e SN(d1) e KN(d2 ) rT rqT rT e e SN(d1) e KN(d2 )
BSCallS,T, K,r,q, erT BSCallF,T, K,0,0,
• Similarly,
BSPutS,T, K,r,q, erT BSPutF,T, K,0,0, Symmetry between puts and calls in Black’s formula
ln(F / K) T ln(F / K) T rT CallF,T, K,r, e F N K N T 2 T 2
ln(K / F) T ln(K / F) T rT PutF,T, K,r, e K N F N T 2 T 2
CallF,T, K,r, PutK,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, volatility: 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) BSPut1.36648,0.25,1.40,0,0,0.15 1 0.250.0041 Use Black with 1 USD rate 0.062903 $0.062839 1.001025
10,000,000 Notional adjusted at strike price, Option 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 futures contract. 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: call option on a Eurodollar futures contract with strike 97 in 1 month. Futures settle in 3 months.
-- in or before 1 month: can exercise and get long a futures. The payoff will be max(f-97,0) times the notional amount, cash-settled.
• Similar for a put option, 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 ot 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 rCt C f f Derivation (II)
• Notice that the futures position requires no financing (except for margin, which we do not take into account). Hence,
C C 1 2C C PNL rCt t f f 2 f o(t) t f 2 f 2 f
C 1 2C rCt t f 2 o(t) t 2 f 2
2 C f 2 2 2C f 2 2C f rCt t t 2t o(t) t 2 f 2 2 f 2 f
2 C f 2 2 2C f 2 2C f rC t 2t o(t) t 2 f 2 2 f 2 f Derivation (III)
• If we assume that
2 f 2 E t ot 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 expiration. 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 dividend 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 rt 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 Fx 0 x X F 0, then
F wxFx F xX
for some positive weights wx, 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 FS 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,n1 FS t1 t2 tn n 1 r t tn1 tn The probability pn,n1
• 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,n1(X ) pn,n1St X | St ,dt ,rt ,i n n1 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 tn1 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,n1 S F E pn,n1 S F 1 r t tn1 tn ,tn1 tn1 tn ,tn1 tn
forward price= 1 r t E pn,n1 S tn S expectation of tn1 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
PS , 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
n1 1 qt t j j0 T S , T S tn n1 tn t0 t0 1 r t t j j0 • Then, E P T |T E pn,n1 T tn1 tn tn1
n n 1 qt t 1 qt t j j 1 r t j0 pn,n1 j0 tn E S S n tn1 n 1 q t tn 1 r t 1 r t tn t j t j j0 j0
n1 1 qt t j j0 P S T E T |T T n1 tn tn tn1 tn tn 1 r t t j j0
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.,
n1 1 qt t setting j j0 T S , T S tn n1 tn t0 t0 1 r t t j j0
E P T |T T we have t n 1 t n tn .