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BOUNDS AND OTHER NO ARBITRAGE

CONDITIONS ON OPTIONS PRICES

First we review the topics:

Risk-free borrowing and lending and

Short sales

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Risk-free lending and borrowing

Arbitrage: A market situation in which an investor can make a

profit with: no equity and no risk.

Efficiency: A market is said to be efficient if prices are such that

there exist no arbitrage opportunities. Alternatively, a market is said to be inefficient if prices present arbitrage

opportunities for investors in this market.

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Risk-free lending and borrowing

PURE ARBITRAGE PROFIT:A PROFIT MADE

1.WITHOUT EQUITY INVESTMENTand

2. WITHOUT ANY RISK

We will assume that

the options market is efficient.

This assumption implies that one cannot make arbitrage profits

in the options markets

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We are now ready to analyze upper and lower

BOUNDS AND OTHER NO ARBITRAGE CONDITIONS

ON OPTIONS PRICES.

The basic assumptions and notations that underlie the

analysis are:

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ASSUMPTIONS:

1. The market is frictionless:No transaction cost nor taxes exist. Trading are executed instantly. There exists no

restrictions to short selling.

2. Market prices are synchronous across assets.

If a strategy requires the purchase or sale of several assets in

different markets, the prices in these markets are simultaneous.

Moreover,No bid-ask spread existonly one market price.

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ASSUMPTIONS:

3. Risk-free borrowing and lending exists at the unique

risk-free rate. Risk-free borrowing and lending is

done via selling short and purchasing T-bills

4. There exist no arbitrage opportunities

in the options market.

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NOTATIONS:

t = the current date.St = the current market price of the underlying asset. X = the option’s exercise (strike) price. K in the text book.T = the option’s expiration date.T-t = the time remaining to the

option’s expiration.r = the annual risk-free rate. = the annual standard deviation of the returns on the underlying asset. D = cash dividend per share.

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NOTATIONS:Ct = the current market premium of an American call. ct = the current market premium of an European call.

Pt = the current market premium of an American call.

pt = the current market premium of an European call. In general, we express the options’ premiums as the following functions:

Ct , ct = c{St , X, T-t, r, , D },

Pt , pt = p{St , X, T-t, r, , D }.

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FINAL REMARK:Many strategies described below use lending or borrowing capital at the risk-free rate. Mostly, the amount borrowed or lent is the discounted value of the option’s exercise price.

Namely, Xe-r(T-t).The assumption here is that the holder of the strategy can always buy T-bills (lend) or sell short T-bills (borrow) for exactly the amount of Xe-r(T-t). It follows that upon terminating the strategy at the option’s expiration time, the lender will receive this amount’s face value, namely, a cash flow of X. If borrowed, the borrower will pay this amount’s face

value, namely, a cash flow of – X.

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RESULTS FOR CALLS:

1. Call values at expiration:(3.1) page 102

CT = cT = Max{ 0, ST – X }.

Proof: At expiration the call is either exercised, in which case CF = ST –

X, or you let the option expire worthless, in which case, CF = 0.

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RESULTS FOR CALLS(P104):

2. Minimum call value:A call premium cannot be negative.At any time t, prior to expiration,

Ct , ct 0.

Proof: The current market price of a call is the NPV[Max{ 0, ST – X }] 0.

3. Maximum Call value: Ct St.

Proof: The call is a right to buy the stock. Investors will not pay for this right a price that is higher than what the value the right to buy gives them, I.e., the stock itself.

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RESULTS FOR CALLS:

4. Lower bound: American call value:

At any time t, prior to expiration,

Ct Max{ 0, St - X}.

Proof: Assume to the contrary that

Ct < Max{ 0, St - X}.

Then, buy the call and immediately exercise it for an arbitrage profit of: St – X – Ct > 0. Contradiction of the no arbitrage profits assumption.

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RESULTS FOR CALLS:11. The money value of calls:The higher the exercise price, the lower is the value of a call. Proof: Let X1 < X2 be the exercise prices for two calls on the same underlying asset and the same time to expiration. To show that c2 < c1 assume, to the contrary, that c2 > c1 or, c2 - c1 > 0. Then,At expiration

Strategy ICF ST < X1 X1<ST < X2 ST >X2

Sell c(X2 ) c2 0 0 -(ST –X2)

Buy c(X1 ) -c1 0 ST –X1 ST–X1

Total ? 0 ST –X1 X2 - X1

In the absence of arbitrage the initial cash flow cannot be positive.

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RESULTS FOR CALLS:12. The money value of calls:Let X1 < X2 then C1 - C2 X2 - X1

Proof: Let X1 < X2 be the exercise prices or

two American calls on the same underlying

asset and the same time to expiration. Assume that C1 - C2 > X2 - X1 or,

equivalently: C1 - C2 – (X2 - X1) > 0. Then,

At expirationStrategy ICF ST < X1 X1<ST < X2 ST >X2

Sell C(X1 ) C1 0 -(ST –X1) -(ST –X1)

Buy C(X2 )-C2 0 0 ST–X2

Lend – (X2 - X1) X2-X1+i X2- X1+i X2- X1+i

Total ? X2-X1+i X2-ST+i i

i = interestEven if the sold call is exercised before Expiration, the total value in hand is >0.

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RESULTS FOR CALLS:13. The money value of calls:Let X1<X2 then c1-c2 (X2-X1)e-r(T-t)

Proof: Let X1 < X2 for two European calls

on the same underlying asset and the same

time to expiration. Assume that c1-c2 > (X2-X1)e-r(T-t) or,

c1-c2 -(X2-X1)e-r(T-t) >0. Then,

At expirationStrategy ICF ST < X1 X1<ST < X2 ST >X2

Sell c(X1 ) c1 0 -(ST –X1) -(ST –X1)

Buy c(X2 )-c2 0 0 ST–X2

Lend -(X2-X1)e-r(T-t) X2-X1 X2-X1 X2-X1

Total ? X2-X1 X2- ST 0

In the absence of arbitrage, the initial cash flow cannot be positive.

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RESULTS FOR CALLS:5. Lower bound: European call value: At any time t, prior to expiration,

ct Max{ 0, St - Xe-r(T-t)}.

Proof: If, to the contrary,ct < Max{ 0, St - Xe-r(T-t)}, then,

0 < St - Xe-r(T-t) - ct At expiration

Strategy I.C.F ST < X ST > X

Sell stock short St -ST -ST

Buy call - ct 0 ST - XLend funds - Xe-r(T-t) X XTotal ? X – ST 0 In the absence of arbitrage, the

initial cash flow cannot be positive.

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RESULTS FOR CALLS:6. The market value of an American call is at least as high as the market value of a European call.

Ct ct Max{ 0, St - Xe-r(T-

t)}.

Proof: An American call may be exercised at any time, t, prior to expiration, t<T, while the European call holder may exercise it only at expiration.

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RESULTS FOR CALLS:7. The time value of calls:The longer the time to expiration, the higher is the value of a call.

Proof: Let T1 < T2 for two calls on the same underlying asset and the same exercise price. To show that

c2 > c1 assume, to the contrary, that c1 > c2 or, c1 - c2 > 0.

At expiration T1

Strategy I.C.F ST1 < X ST1 > X

Sell c(T1 ) c1 0 -(ST1 –X)

Buy c(T2 ) - c2 c(TV) C(T2-T1)

Total ? c(TV) ?

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RESULTS FOR CALLS:Case 1. The calls are American

style.The question is whether

C(T2-T1) - (ST1 –X) ?

If > 0 the proof is completed. If < 0, the open call can be exercised immediately for (ST1 –X) and the proof

is complete.

Case 2. The calls are European style.

In this case, result 5. Guarantees that the

cash flow above is >0 because

ct1 St1 - Xe-r(T2-T1).

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RESULTS FOR CALLS:8. Cash dividends and calls:

Cash dividends and calls:It is not optimal to exercise an American call prior to its expiration if the underlying stock does not pay out any dividend during the life of the option.

Proof: If an American call holder wishes to rid of the option at any time prior to its expiration, the market premium is greater than the intrinsic value because the time value is always positive.

9. The American feature is worthless if theunderlying stock does not pay out any dividend during the life of the call. Mathematically: Ct = ct.

Proof: Follows from result 8. above.

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RESULTS FOR CALLS:10. Early exercise of

Unprotected American calls on a cash dividend paying stock.

(Section 8.7):Consider an American call on a cashdividend paying stock. It may be optimal to exercise this American call an instant before the stock goes X dividend.Two condition must hold for the early exercise to be optimal:First, the call must be in-the-money. Second, the $[dividend/share], D,

must exceed the time value of the call at

the x dividend instant. To see this result Consider:

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RESULTS FOR CALLS:

FACTS:1. The share price drops by

$D/share when the stock goes x-dividend.

2. The call value decreases when the price per share falls.

3. The exchanges do not compensate call holders for the loss of value that ensues the price drop on the x-dividend date.

Time linetA tXDIV tPAYMENT

SCUMD SXDIV

4. SXDIV = SCDIV - D.

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Early exercise of Unprotected American calls on a

cash dividend paying stock:The call holder goal is to maximize the Cash flow from the call. Thus, at any moment in time, exercising the call isinferior to selling the call. This

conclusion may change, however, an instant

before the stock goes x dividend:

Exercise Do not exercise

Cash flow: SCD – X; c{SXD, X, T-tXD-}

Substitute: SCD = SXD + D.Cash flow: SXD –X + D; SXD – X + TV.Conclusion: Early exercise of American calls may be optimal. If the call is in the money and D > TV, early exercise is optimal.

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Early exercise of Unprotected American calls on a

cash dividend paying stock:This result means that an

investor is indifferent to exercising the call

an instant before the stock goes x dividend if the x dividend stock

price S*XD satisfies:

S*XD –X + D = c{S*

XD , X, tXD-t}.It can be shown that this implies

that the price price S*XD exists if:

D > X[1 – e-r(T – t)].

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RESULTS FOR CALLS:14. The money value of calls:Let X1 < X2 < X3 and

X2 = X1 + (1 - )X3 for 0 < < 1.

The premiums on the three calls must

satisfy: c2 c1 + (1 - )c3 At Expiration

STRATEGY ICF ST < X1 X1<ST < X2 X2 <ST < X3 ST > X3

Buy calls X1 -c1 0 (ST – X1) (ST – X1) (ST – X1)

Sell one call X2 c2 0 0 -(ST – X2) -(ST – X2)

Buy 1- calls X3 -(1-)c3 0 0 0 (1-)(ST–X3)

Total c2-c1-(1-)c3 0 (ST–X1) (1-)(X3-ST) 0

All the cash flows at expiration are non negative.Hence, the Initial Cash Flow cannot be positive!

I.e.,c2-c1-(1-)c3 0.

Or, c2 c1 + (1 - )c3.

Remark: When = 1/2 , the strategy is called a Butterfly. In this case, 2X2 = X1 + X3

and the result asserts: 2c2 c1 + c3

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RESULTS FOR CALLS:Example: Let $80, $95 and $100 be the exercise prices of three calls on the same underlying asset and the same expiration. Observe that: $95 = (.25)$80 + (.75)$100.

Thus, in this case, = ¼. Result 14. Asserts that c(95) ¼c(80) + ¾c(100). Or,

4c(95) c(80) + 3c(100).

If the latter inequality does not hold, then 4c(95) > c(80) + 3c(100) or,4c(95) - c(80) - 3c(100) > 0

and arbitrage profit can be made by the Strategy:

Buy the $80 callSell four $95 callsBuy three $100 calls.

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RESULTS FOR CALLS:15. Volatility:The higher the price volatility of theUnderlying asset, the higher is the Call value.Proof: The call holder never loses more than the initial premium. The upside gain, however, is unlimited. Thus, higher volatility increases the potential gain while the potential loss remains Unchanged.

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RESULTS FOR CALLS:16. The interest rate:The Higher the risk-free rate, theHigher is the call value. Proof: The result follows from result

6:

Ct ct Max{ 0, St - Xe-r(T-t)}.

With increasing risk-free rates, the difference St - Xe-r(T-t) increases and

the call value must increase as well.

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RESULTS FOR PUTS:

17. Put values at expiration:

PT = pT = Max{ 0, X - ST}.

Proof: At expiration the put is either exercised, in which case CF = X - ST, or it is left to expire worthless, in which case CF = 0.

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RESULTS FOR PUTS:

18. Minimum put value:A put premium cannot be negative.At any time t, prior to expiration

Pt , pt 0.

Proof: The current market price of a put is the

NPV[Max{ 0, X - ST}] 0.

19a.Maximum American Put value:At any time t < T, Pt X.

Proof: The put is a right to sell the stock for X, thus, the put’s price cannot exceed the maximum value it will create: X, which occurs if S drops to zero.

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RESULTS FOR PUTS:

19b.Maximum European Put value:

Pt Xe-r(T-t).

Proof: The European put may be exercised

only at expiration. The maximum revenue it can create at that time is X, ( in case S drops to zero), thus, at any time point before expiration, the

European put cannot exceed the NPV{X}.

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RESULTS FOR PUTS:

20.Lower bound: American put value:

At any time t, prior to expiration,

Pt Max{ 0, X - St}.

Proof: Assume to the contrary that

Pt < Max{ 0, X - St}.

Then, buy the put and immediately exercise it for an arbitrage profit of: X - St – Pt > 0. A contradiction of the

no arbitrage profits assumption.

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RESULTS FOR PUTS(P116):21. Lower bound: European put value: At any time t, t < T,

pt Max{ 0, Xe-r(T-t) - St}.

Proof: If, to the contrary,pt < Max{ 0, Xe-r(T-t) - St} then,

0 < Xe-r(T-t) - St - pt

At expirationStrategy I.C.F ST < X ST > X

Buy stock -St ST ST

Buy put - pt X - ST 0

Borrow Xe-r(T-t) - X - XTotal ? 0 ST- X

In the absence of arbitrage, the initial cash flow cannot be positive.

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RESULTS FOR PUTS:22. The market value of an American put is at least as high as the market value of a European put.

Pt pt Max{0, Xe-r(T-t) - St}.

Proof: An American put may be exercised at any time, t, prior to expiration, t<T, while the European put holder may exercise it only at expiration.

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RESULTS FOR PUTS:23. The time value of puts:

The longer the time to expiration, thehigher is the value of an American

put.

Proof: Let T1 < T2 for two American

puts on the same underlying asset and the same exercise price.To show that P2 > P1 assume, to the

contrary, that P1 > P2 or, P1 - P2 > 0.

At expiration T1

Strategy I.C.F ST1 < X ST1 < X

Sell P(T1 ) P1 -(ST1 –X) 0

Buy P(T2 ) -P2 P(T2-T1) P(TV) Total ? ? P(TV)You may exercise the put. The result follows.

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RESULTS FOR PUTS:24. American put is always priced higher than its European counterpart.

Pt pt

Proof: An American put may beexercised at any time, t, prior to expiration, t<T, while the European put holder may exercise it only at expiration. Moreover, ifThe price of the underlying asset fall below a certain threshold price, it becomes Optimal to exercise the American put and earn X – St. At that very same moment the European put holder wants to (optimally) Exercise the put but cannot because it is a European put.The next figure demonstrates the relationship Between the American and European puts premiums.

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RESULTS FOR PUTS(P176,7):24. American put is always priced higher than its European counterpart.Pt pt

S* S** X

P/L

X

S

Xe-r(T-t)

Pp

For S< S** the European put premium is less than the put’s intrinsic value.

For S< S* the Americanan put premium coincides with the put’s intrinsic value.

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RESULTS FOR PUTS:25. The money value of puts:

The higher the exercise price, the higher is the value of a put. Proof: Let X1 < X2 for two puts on the same underlying asset and the same time to expiration. To show that p2 > p1 assume, to the contrary, that p2 < p1 or, p1 - p2 > 0.

Then,

At expirationStrategy ICF ST < X1 X1<ST < X2 ST >X2

Sell p(X1 ) p1 ST –X1 0 0

Buy p(X2 ) -p2 X2 –ST X2 - ST 0

Total ? X2 - X1 X2 - ST

0

In the absence of arbitrage, the initial cash

flow cannot be positive.

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RESULTS FOR PUTS:26. The money value of puts:Let X1 < X2 then P2 - P1 X2 - X1

Proof: Let X1 < X2 be the exercise prices or two American puts on the same

underlying asset and the same time to expiration. Assume that P2 - P1 > X2 - X1 or,

equivalently: P2 - P1 – (X2 - X1) > 0. Then,At expiration

Strategy ICF ST < X1 X1<ST < X2 ST >X2

Sell P(X2 ) P2 ST –X2 ST –X2) 0

Buy P(X1 )-P1 X1 – ST 0 0

Lend – (X2 - X1) X2-X1+i X2- X1+i X2- X1+i Total ? i ST-X1 +i X2- X1+i i = interestEven if the sold put is exercised before expiration, the total value in hand is >0.

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RESULTS FOR PUTS:27. The money value of puts:Let X1<X2 then p2-p1 (X2-X1)e-r(T-t)

Proof: Let X1 < X2 for two European puts

on the same underlying asset and the same

time to expiration. Assume that p2-p1 > (X2-X1)e-r(T-t) or,

p2- p1 -(X2-X1)e-r(T-t) >0. Then,

At expirationStrategy ICF ST < X1 X1<ST < X2 ST > X2

Sell p(X2 ) p2 ST –X2 ST –X2 0

Buy p(X1 )-p1 X1 – ST 0 0

Lend -(X2-X1)e-r(T-t) X2-X1 X2-X1 X2-X1

Total ? 0 X2- ST X2-X1

In the absence of arbitrage, the initial cash flow cannot be positive.

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RESULTS FOR PUTS:28. Volatility:

The higher the price volatility of theunderlying asset, the higher is the put value.Proof: The put holder never loses more than the initial premium. The upside gain, however, is increasing from zero to X. Thus, higher volatility increases the potential gain while the potential loss remains Unchanged.

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RESULTS FOR PUTS:29. The interest rate:The higher the risk-free rate, the lower is the put value. Proof: Follows from result

22: Ct ct Max{ 0, Xe-r(T-t) -St}.

With increasing risk-free rates, the difference Xe-r(T-t) - St

decreases and the put value decrease too.

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RESULTS for PUTS and CALLS: (Section 8.4)

30. The put-call parity.

European options:

The premiums of European calls and puts written on the same non dividend

paying stock for the same expiration must

satisfy:

ct - pt = St - Xe-r(T-t).

The parity may be rewritten as:

ct + Xe-r(T-t) = St + pt.

Below, we prove the latter version of the parity.

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RESULTS for PUTS and CALLS:Proof. Consider the following two portfolios:

A: Long the stock and the put.B: Long the call and lend Xe-r(T-t) .

Portfolio A: At expirationStrategyI.C.F ST < X ST > X

Buy stock-St ST ST

Buy put - pt X - ST 0

Total -(St+pt) X ST

Portfolio B: At expirationStrategyI.C.F ST < X ST > X

Buy call - ct 0 ST-X

Lend - Xe-r(T-t) X XTotal -(ct+ Xe-r(T-t) ) X

ST

Portfolios A and B have identical cash flows At expiration for all possible prices. Thus,their initial values are equal:

ct + Xe-r(T-t) = St + pt.

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RESULTS for PUTS and CALLS:

31. Synthetic European options:

The put-call parityct - pt = St - Xe-r(T-t)

can be rewritten as a synthetic call:

ct = pt + St - Xe-r(T-t),

Or as a synthetic put:

pt = ct - St + Xe-r(T-t).

Observe that the parity implies that at-the-

money calls are priced higher than their counterpart puts. For at-the-money

options, ct = pt + St - Xe-r(T-t)

becomes:ct = pt + X - Xe-r(T-t) > pt.

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RESULTS for PUTS and CALLS:32. The put-call parity. European options:Suppose that European puts and calls are written on a dividend paying stock and suppose that there will be two dividend Payments: D1 at t1 and D2 at t2. The option’s premiums must satisfy the following equation:

ct-pt = St-Xe-r(T-t) – D1e-r(t1-t) – D2 e-r(t2-t)

The proof follows the same path of result 30.But the stock will pay dividend twice at t1 and t2. Thus, borrowing the discounted value of the dividends and paying them upon receiving the dividends will leave the strategy unaffected and the result follows.

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RESULTS for PUTS and CALLS:

34. The put-call parity.

American options:The put-call parity for European

options asserts that:

ct - pt = St - Xe-r(T-t).

This result does not necessarily hold for

American options. The premiums on American

options satisfy the following inequalities:

St - X < Ct - Pt < St - Xe-r(T-t).