act370 01 m091 v14

Sharp: ACT370 01 M091 v22 of 29

Keith Sharp ACT370 Ch. 9 Lecture Framework Generalized put-call parity

Sharp: ACT370 01 M091 v22 of 29

0 1 ………… T

Time T Now worth ST + max(0, K-ST) = max(ST K)

Time 0 Buy stock and put S0 + P(ST, K, T)

T: exercise date S0 : stock price

1.Already known Put-Call Parity : Portion 1

Sharp: ACT370 01 M091 v22 of 29

0 1 ………… T

Time T Now worth K +max(0, ST - K) = max(K, ST )

Time 0 Buy bond and call PV0,T(K) + C(ST, K, T)

2 Put-Call Parity: Portion 2

T: exercise date S0 : stock price

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Since portfolio is certain to be worth zero at time T, it must be worth zero at time 0 or else we would all want to buy or sell it, an arbitrage argument. Hence S0 + P(ST, K, T) - PV0,T(K) - C(ST ,K, T) = 0 and can reorder terms till looks familiar, eg: S0 + P(ST, K,T) = PV0,T(K) + C(ST ,K, T) :standard put-call parity

0 1 ………… T

Time T Now worth max(ST ,K) - max(K, ST ) = 0

Time 0 Buy stock and put, sell bond and call S0 + P(ST, K, T) -PV0,T(K) - C(ST, K,T)

3. Difference of two portions

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Examp PutCall00

(DM2e-Prob09.03) Suppose the S&R Index is 800, the continuously compounded risk-free rate is 5%, and the dividend yield is 0%. A 1-year 815-strike European call costs $75 and a 1-year 815-strike European put costs $45. Consider the strategy of buying the stock, selling the 815-strike call and buying the 815-strike put. What difference between the call and put prices would eliminate arbitrage?

Solution PutCall00

This S+P-C portfolio is worth a certain K at expiry so should now cost K exp(-rT). This requires C-P= S-Kexp(-rT)=24.748 S 800

K 815

T 1

r 0.05

C‐P= 24.74802

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Examp PutCall01 (DM2e-Prob09.01) A stock currently sells for $32.00. A 6-month call option with strike of $35.00 has a premium of $2.27. Assuming a 4% continuously compounded risk-free rate and a 6% continuous dividend yield, what is the price of the associated put option?

Solution PutCall01

At time zero, need only exp(-δT) of a stock to get 1 stock at expiry since the stock ‘breeds’ (pays dividends which can be reinvested) at continuous rate δ: S0 exp(-δT) + P(K,T) = K exp(-rT) + C(K,T) (RHS and LHS both equal max(S, K) at expiry) 32* exp(-0.06*0.5) + P(35, 0.5)- 2.27 = 35 exp(-0.04*0.5) P(35, 0.5)=5.5227 S0= 32

C0= 2.27

rcont= 0.04

delta= 0.06

T= 0.5

K= 35

S0*exp(‐delta*T)= 31.05426

Kexp(‐rcont*T)= 34.30695

PO= 5.522696

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Examp PutCall02 (DM2e-Prob09.02) A stock currently sells for S0 = $32.00. A T=6-month call option with strike of $30.00 has a premium of $4.29, and a T=6-month put with the same strike has a premium of $2.64. Assume a r=0.04 continuously compounded risk-free rate and a δ continuous dividend yield. Calculate S0 (1- exp(-δT)). (McDonald means ‘calculate S0 (1- exp(-δT)’ but in the text question he asks ‘What is the present value of dividends payable over the next 6 months?’ which is actually a random variable depending on the price path followed by the stock)

Solution PutCall02

At time zero, need only exp(-δT) of a stock to get 1 stock at expiry since the stock ‘breeds’ (pays dividends which can be reinvested) at continuous rate δ: S0 exp(-δT) + P(K,T) = K exp(-rT) + C(K,T) (RHS and LHS both equal max(S, K) at expiry) S0 exp(-δT) +2.64 = 30 exp(-0.04*0.5) + 4.29 S0 - S0 exp(-δT) = 0.94404 C0= 4.29

rcont= 0.04

P00= 2.64

T= 0.5

K= 30

Kexp(‐rcont*T)= 29.40596

S0*exp(‐delta*T)= 31.05596

S0‐S0*exp(‐delta*T)= 0.94404

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Assets ST, QT paying dividends or coupons

Q0 + C(ST ,QT, T) = S0 + P(ST ,QT, T) (generalized put-call parity if no dividends) Now consider the possibility that the assets ST, QT are dividend-paying stocks or coupon-paying bonds or anything paying income, whatever it’s called. Someone owning the asset gets that income, but someone promising to buy it in the future doesn’t. So when considering the portfolio that will end up with an equal value to another portfolio at time T, we need to imagine buying at time 0 a non-income paying version of the asset for which we’ll pay e.g. S0

– PV0,T(Income from S). Hence:

Q0 – PV0,T(Income from Q) + C(ST ,QT, T) = S0 – PV0,T(Income from S) + P(ST ,QT, T) As usual, at time T I have max (QT , ST).

Sharp: ACT370 01 M091 v22 of 29

Currency Options Can regard the ‘foreign’ currency like a dividend-paying stock, with the dividend rate now being the foreign interest rate which lets the foreign currency ‘reproduce’ itself. Or, and less Americentrically, notice that e.g. what looks to a New York dealer like a call option on the yen will look to a Tokyo dealer like a put option on the US dollar with only a ‘scale’ difference. In other words, the New Yorker might talk about option price per yen while the Tokyo dealer might talk about option price per dollar. Be particularly careful with dollar-euro or dollar-loonie situations. Both the loonie and the euro in the last few years have fluctuated above and below the US dollar, from a US perspective. So ‘saying exchange rate of 0.95’ is not clear whether 0.95 or 1/0.95 is meant. With yen (JPY=¥1=about $US 0.01) or UK pounds (‘sterling’, GBP=£1=about $US1.60 in 2011) it’s obvious when the reciprocal is used and we’ll use these for examples. The P.R. Chinese yuan (‘Renminbi’, RMB, = about $0.15) is important but not yet sufficiently freely traded to be used in examples. McDonald uses x as the number of USD per foreign currency unit, r (we might use rU )for the interest rate paid on a USD account and rF for the interest rate on an account denominated in the ‘foreign’ currency.

Sharp: ACT370 01 M091 v22 of 29

Put-Call parity for currency options x0: time 0 spot price of e.g. yen in USD (about x0 =0.01) 1/x0: time 0 spot price of USD in yen rU: interest rate paid on a USD account e.g. in New York rY: interest rate on a yen account e.g. in Tokyo K: USD value of strike price per yen Cost of a New Yorker buying now enough yen to grow to one yen after T years is cost of exp(-rYT) yen, which in $ is x0*exp(-rYT). Spend this and also buy a USD-denominated put on one yen, P(¥, K, T), total x0*exp(-rYT) + P(¥, K, T). Alternative is to hold now an amount of USD K*exp(-rUT) and to hold also a call on one yen, C(¥, K, T), total K*exp(-rUT) + C(¥, K, T). x0*exp(-rYT) + P(¥, K, T)= K*exp(-rUT) + C(¥, K, T). where both sides at T are worth max( ¥1, $US K)=USD max(xT, K). Recall: S0 exp(-δT) + P(S, K, T) = K*exp(-rU T) + C(S,K ,T) S0 +P(ST ,QT, T)= Q0 + C(ST ,QT, T) (no divs) S0exp(-δST)+P(ST ,QT, T)= Q0exp(-δQT) + C(ST ,QT, T) where both sides at time T are worth {max(ST, QT)}

Sharp: ACT370 01 M091 v22 of 29

New York’s Call is Tokyo’s Put x0: time 0 spot price of e.g. yen in USD (about x0 =0.01) 1/x0: time 0 spot price of USD in yen K: USD value of strike price per yen 1/K: yen value of strike price per USD At T the US-based call on one yen, C$ (x, K, T) is worth USD max(0, xT - K)

If you are seeing this from Tokyo and own K yen-denominated puts on 1 USD, with exercise price 1/K K P¥(1/x, 1/K, T) then at time T you will have JPY K max(0, 1/K - 1/xT)=JPY max(0, xT – K)/xT =USD max(0, x(T)-K) So allowing for ‘scale’ difference K the C and P are same. Allow also for P¥ being in yen, and convert now at time 0: C $ (x, K, T) = x0 K P¥(1/x, 1/K, T) (9.7)

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Currency options and scale alternative method:

Another way to think of this is that in New York a call

option on one Yen is the right at time T to receive one Yen

and deliver K dollars.

In Tokyo, a put option on one dollar is the right at time T to

deliver one dollar and to receive (1/K) Yen.

So you need K of the Tokyo puts to be the same deal as the

New York call, thus the standard Tokyo put is worth (1/K)

as much as the New York call. Also, the Tokyo put is

denominated in Yen, so the Yen price of the standard Tokyo

put is (1/K)*(1/x0) times the dollar price of the standard New

York call. This gives us again

C $ (x, K, T) = x0 K P¥(1/x, 1/K, T) (9.7)

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Example CurrOpt01 (McDonald Prob09.07c) Suppose that the dollar-denominated interest rate is 5%, the yen-denominated interest rate is 1% (both rates are continuously compounded), the spot exchange rate is 0.009 $/¥, and the price of a dollar-denominated European call to buy one yen with 1 year to expiration and strike price of $0.009 is $0.0006. Calculate the price in Tokyo of the yen-denominated at-the-money dollar put (repeat, put). P¥(1/x, 1/K, T)=(C $ (x, K, T))/(x0 K) =0.0006/(0.009*0.009) =

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Properties: American versus European (These slides stolen from McDonald)

An American option can be exercised at any time A European option can only be exercised at expiration, So an American option must always be at least as valuable as an otherwise identical European option

CAmer(S, K, T) > CEur(S, K, T) PAmer(S, K, T) > PEur(S, K, T)

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Option price boundaries: Call

Call price cannot be negative exceed stock price be less than price implied by put-call parity using

zero for put price:

0, 0, 0,( , , ) ( , , ) max[0, ( ) ( )]Amer Eur T T TS C S K T C S K T PV F PV K

Sharp: ACT370 01 M091 v22 of 29

Option price boundaries: Put

Put price cannot be more than the strike price be less than price implied by put-call parity

using zero for put price

0, 0, 0,( , , ) ( , , ) max[0, ( ) ( )]Amer Eur T T TK P S K T P S K T PV K PV F

Sharp: ACT370 01 M091 v22 of 29

Early exercise of American options A non-dividend paying American call option should not be exercised early, because

Amer Eur TC C S K

That means, one would lose money by exercising early instead of selling the option If there are dividends, it may be optimal to exercise early e.g. so as to receive a huge (perhaps liquidating) dividend It may be optimal to exercise a non-dividend paying put option early if the underlying stock price is sufficiently low, so you can start getting interest on the K

Sharp: ACT370 01 M091 v22 of 29

Prices vs Time to expiration

An American option (both put and call) with more time to expiration is at least as valuable as an American option with less time to expiration. This is because the longer option can easily be converted into the shorter option by exercising it early

A European call option on a non-dividend paying stock will be more valuable than an otherwise identical option with less time to expiration.

European call options on dividend-paying stock and European puts may be less valuable than an otherwise identical option with less time to expiration – may want to receive a big dividend but cannot exercise early

Sharp: ACT370 01 M091 v22 of 29

Prices vs Different strike prices (K1 < K2 < K3),

For both European and American options: A call with a low strike price is at least as valuable as an otherwise identical call with higher strike price

1 2( ) ( )C K C K (if C(K2) expires in the money, then C(K1) expires in even more money, in fact is worth K2-K1 more) A put with a high strike price is at least as valuable as an otherwise identical put with low strike price

2 1( ) ( )P K P K

Sharp: ACT370 01 M091 v22 of 29


For both European and American options: The premium difference between otherwise identical calls with different strike prices cannot be greater than the difference in strike prices

1 2 2 1( ) ( )C K C K K K

(≤100% probability of K2 - K1 payout difference ) The premium difference between otherwise identical puts with different strike prices cannot be greater than the difference in strike prices

2 1 2 1( ) ( )P K P K K K

(≤100% probability of K2 - K1 payout difference )

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Sharp: ACT370 01 M091 v22 of 29


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Sharp: ACT370 01 M091 v22 of 29

Why Tab 9.7 +4 -10 + 6 pattern of calls at K= 50, 59, 65? If the graph of call price against K were a straight line, then the K=59 call price would be a linear interpolation between K=50 and K=65, hence weights 4/10 and 6/10. In fact the given (must be wrong) K=59 call price is above that straight line. So we can make an arbitrage profit by shorting the overpriced K=59 call and going long the interpolated synthetic call. The correct K=59 call, because of the required convexity, is below.

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