Option Price Basics Exercise

8/7/2019 Option Price Basics Exercise

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PRICING & HEDGING OF A EUROPEAN STYLE OPTION :

( Para 10.1.1. Of A.V.Rajwade's book 3rd edition )

ASSUMPTIONS :

YOU ARE WRITING A USD CALL OPTION.

CURRENT SPOT : 46.5

6 MONTH FORWARD : 47.5

STRIKE PRICE : 46.5

MATURITY : 6MTHS

VIEWS : FUTURE SPOT WOULD BE SOMEWHEREBETWEEN 44 & 49.

LOWEST : 44

HIGHEST : 49

WHAT PREMIUM SHOULD BE CHARGED ?

YOU BUY 1 USD @ CURRENT FWD RATE I.E. 47.5THE OPTION WILL NOT BE EXERCISED IF THE SPOT RATE

ON MATURITY IS BELOW 46.5

OUR ASSUMED CHEAPEST RATE IS : 44

IF THE OPTION IS NOT EXERCISED, IN THE WORST CASE, W

MAY HAVE TO SELL THE USD BOT @ 47.5AT THE CURR

RATE OF 44.

IF THE FEES CHARGED ARE 3.5RS PER $, THEN NO

WOULD BE INCURRED IF THE OPTION IS NOT EXERCISED &

HEDGE IS TO BE UNWOUND AT THE WORST END OF THE RA

EXPECTED.


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IF, THEREFORE, ON MATURITY THE SPOT IS AT THE WORST

YOU DO NOT MAKE ANY PROFIT OR LOSS.

IF, ON MATURITY, THE SPOT IS AT THE BEST END I.E.

THE OPTION WOULD BE EXERCISED & YOU MAKE A PROFIT

2.5RS PER $.

IF THE SPOT IS ANYWHERE BETWEEN THE STRIKE PRICE O

AND THE WORST END I.E. 44 , YOU MAKE A PROFI

LOSS INCURRED ON CANCELLATION OF THE HEDGE WOUL

THAN THE FEES RECEIVED.

THE FEES CHARGED I.E. 3.5WOULD BE NEEDED ONLY 6

HENCE. IT WOULD, THEN, BE SUFFICIENT IF YOU RECEIVE F

HAS A FUTURE VALUE OF RS 3.5

THE PRESENT VALUE OF RS 3.5IS THUS THE PRICE

COMPETITIVE PRICING STRATEGY :

FEES : RS 3.5

HEDGE : 1USD BOT @ 47.5

IF FUTURE SPOT IS BELOW THE STRIKE OF 46.5THEN

HEDGE AND THE FEES ENSURE A BREAKEVEN.

CAN MORE THAN 1 OPTION BE WRITTEN ?

LET US ASSUME THAT THIS NUMBER n = 2


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THE BUYER OF THE OPTION WOULD PAY 93ON M

BESIDES, THE FEES RECEIVED I.E. RS 3.5ARE AVAILAB

1USD WOULD HAVE TO BE BOUGHT AT SPOT RATE ON

49

THE EQUATION IS :

Receipt : Payment :

46.5n + 3.5 = 47.5 + 49x (n-1)

which is 96.5 which is 96.5

n = 2 Surplus / (deficit ) : -

TO SUMMARIZE :

FOR WRITING AN OPTION CONTRACT TO SELL 2

STRIKE RATE OF RS 46.5FOR A MATURITY OF 6 MON

A FEE EQUAL TO PV OF RS 3.5& BUY 1USD F

RATE OF RS 47.5BY WAY OF HEDGE.

IF THE SPOT ON MATURITY IS BETWEEN 44 &

PROFIT. THE WRITER BREAKS-EVEN AT 44 &

ASSUMED RANGE.

MORE GENERALLY, THE FEE PER USD IS RS. 1.75 & TH

IS 0.50

WIDER RANGE :

LET US ASSUME THAT THE EXPECTED RANGE IS WIDER AT

IF THE OTHER ASSUMPTIONS AS REGARDS THE STRIKE, SP


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RATE ARE UNCHANGED, THEN IT CAN BE SHOWN THAT THE

WOULD HAVE TO BE RS _____ PER USD ( AS AGAINST RS 1.

HEDGE RATIO AT _____%.

THE WIDER THE RANGE I.E. HIGHER THE VOLATILITY, HIGHE

BE THE FEES.

MORE FAVORABLE STRIKE PRICE :

IF THE STRIKE PRICE IS CHANGED TO 46.00 WHICH IS MORE

FAVORABLE TO THE BUYER AND IF OTHER FACTORS ARE ACONSTANT, THEN THE PRICE BECOMES RS ______PER $, n

BECOMES _____ & THE HEDGE RATIO OF _____ %


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NT

OSS

HE

GE


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END,

49THEN

OF

46.5

SINCE THE

BE LESS

MONTHS

E WHICH

F THE OPTION.

HE ABOVE


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TURITY.

E.

MATURITY.

USD @

HS, ONE CAN CHARGE

D AT CURRENT FWD

49THE WRITER MAKES

49WHICH IS THE

HEDGE RATIO

3 / 50.

T & FWD


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FEES

5 ) AND THE

R WOULD

SSUMED


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DEFINITIONS :

P ( x, X , t ) = PRICE OF A CALL OPTION ON CURRENCY C2 IN UNITS O

PER UNIT OF C2 1 RS.

WHERE

x = SPOT RATE IN UNITS OF C1 PER UNIT OF C2 (48.50

X = STRIKE RATE IN UNITS OF C1 PER UNIT OF C2 (48.00

t = TIME TO MATURITY OF OPTION 3 MONT

VALUE OF AN AMERICAN OPTION = INTRINSIC VALUE + TIM

INTRINSIC VALUE = PROFIT AVAILABLE ON IMMEDIATE EXERCISE ( x - X FOR

SAY ( 48.50

TIME VALUE = PREMIUM - INTRINSIC VALUE

SAY ( 1.00 -

CALL OPTIONS ARE :

IN-THE-MONEY WHEN x > X SPOT > 48.00

AT-THE-MONEY WHEN x = X SPOT = 48.00

OUT-OF-THE-MONEY x < X SPOT < 48.00

PUT OPTIONS ARE :

IN-THE-MONEY WHEN X > x SPOT < 48.00

AT-THE-MONEY WHEN X = x SPOT = 48.00

OUT-OF-THE-MONEY X < x SPOT > 48.00

AMERICAN OPTIONS WHICH ARE IN THE MONEY ONLY CAN HAVE

INTRINSIC VALUE

OTHERS CAN HAVE ONLY TIME VALUE WHICH IS P( x, X, t )


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PRICE LIMITS ON CALL OPTIONS :

1 P ( x, X, t ) > OR = 0, NEVER NEGATIVE

2 P ( x, X' , t ) > P ( x, X" , t ) WHERE X' < X" SAY X' = 48.00

AND X" = 48.2

3 P ( x, X, t1 ) > P( X, x, t2 ) WHERE t1 > t2

4 P(x, X, t ) < OR = x

5 FOR AN AMERICAN OPTION P(x, X, t ) > x - X

6 FOR EUROPEAN OPTIONS CONSIDER THE FOLLOWING TABLE

X = STRIKE PRICE

x' = SPOT ON MATURITY

F = FORWARD FOR THE GIVEN MATURITY

I1 = INT. RATE FOR THE OFFERED CURRENCY

I2 = INT.RATE FOR THE BASE CURRENCY

F = x*(1+I1*t)/(1+I2*t)

x' < X x' = X

TRANSACTION

BUY CALL OPTION 0 0

SELL PUT OPTION - (X-x') 0

BOTH TOGETHER x' - X x' - X

SELL FORWARD F - x' F - x'

ALL THREE TOGETHER F - X F - X


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COMBINATION OF BUY CALL, SELL PUT ( AT SAME STRIKE PRICE) AND

FORWARD SALE IS EQUAL TO F - X IRRESPECTIVE OF SPOT ON

MATURITY

IN AN EFFICIENT MARKET THE THREE TRANSACTIONS TAKEN TOGETHER

SHOULD GIVE TODAY A CASH FLOW EQUAL TO THE PRESENT VALUEOF ( F - X ) OR ( F - X )/(1+I1*t)

CASH FLOWS ARE AS UNDER :

BUY CALL : OUTFLOW OF CALL PREMIUM

SELL PUT : INFLOW OF PUT PREMIUM

FORWARD CONTRACT = 0

THE DIFFERENCE BETWEEN THE CALL AND PUT PREMIUM HAS TO BE EQ

TO THE NET PRESENT VALUE OF ( F - X )

IF F = X I.E. THE OPTION IS " AT-THE-MONEY " THEN THE PRE

CALL AND PUT OPTIONS SHOULD BE EQUAL

7 ALSO FOR A GIVEN x, X AND t , AN AMERICAN OPTION WOULD BE PRICE

HIGHER THAN AN EUROPEAN OPTION AS THE FORMER PROVIDES GREA

FLEXIBILITY.


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C1

)

S

E VALUE

CALL )

- 48.00 )

0.50 )


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x' > X

x' - X

0

x' - X

F - x'

F - X


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AL

IUM ON

D

ER


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EXAMPLE ON OPTION STRUCTURES :

FOLLOWING PRICES ARE AVAILABLE IN THE MARKET FOR USD/INR 3 MTH

MATURITY

STRIKE CALL PUT

45.50 1.50/1.53 0.09/0.12

46.00 1.10/1.13 0.15/0.18 buy at s46.50 0.70/0.73 0.22/0.25 sell at b

47.00 0.30/0.33 0.32/0.35

47.50 0.20/0.23 0.72/0.75

48.00 0.12/0.15 1.12/1.15

48.50 0.09/0.12 1.52/1.55

RESPOND TO THE FOLLOWING :

1 What would be the approximate forward rate ?

2 Which option should an exporter buy if he wants to have a minimum

realisation of Rs 46.25 per $ ?

3 What range forward can an importer get which will not involve any

receipt / payment of premium ?

4 In the above range forward, if the spot exchange rate on maturity is

Rs. 48 per $ , what will be the cash flow to be exchanged ?

( assume that the the underlying is not due on that day )

5 At a mimimum realization of Rs 46.50 what level of participation can an

exporter expect under a participating structure ?

6 What will be the participation structure if the minimum realization is

reduced to Rs 46 per $ ?

7 What seagull can be structured for an exporter where no premium willbe paid / received ?

8 An importer with an existing range forward of Rs 46.50 - 48.00 wants

be the net inflow / outflow on $ 1 mio ?

to cancel the contract when the above prices are prevailing. What will


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ANSWERS :

1 The premium level for ATM options would be same. An european option with a strike equal to

the current forward would be ATM. The prices for as trike of 46.00 are alo\most same. The forward

rate would be just below 46.00 say 45.95

2 Put option with a strike price of Rs 45.50 would ensure a minimum realization of 45.25

3

Call at a strike of 47.00 would cost the importer 0.15 and a put at a strike of 45.00 would

fetch him a premium of 0.15

The structure would then be 45.00 - 47.00

4 Importer will receive Rs1 per $ ( 48.00 - 47.00 )

However, this will be off-set by the higher price he will have to pay for hedging the underlying.

participation

5 Exporter needs to buy an OOTM put and simultaneously sell a call which by default would be I

If he buys a put at a strike of 46.50, it would cost him 0.25 Writing a call at the saem strike would

earn him a premium of 0.70 He would be required to sell only 0.25/0.70 calls to equalize the premium

The participation would be 1 - 25/70 = 0.643 or 64.3 %

6 Exporter needs to buy a put @ 46.00 and would sell a call @ same etrike i.e. 46.00

For buying option exporter pays a premium of 0.18 while for selling a call @ 46.00, exporter

would receive a premium of 1.10. He would be required to sell only 0.18/1.10 calls to equlaise premiu

The participation would be 1 -18/110 = 0.8363 or 83.63 %

7 A seagull for an exporter would comprise buying an OOTM put, selling an OOTM call and selli

OOTM put.

Buy put 45.00 @ 0.18

Sell call 47.50 @ 0.09

Sell put 44.50 @ 0.09

8 A range forward for an importer would be buying a call and selling a put.

The importer would earlier have bought a call at a strike of 47.00 and sold a put at 45.50

He would now have to reverse these I.e. sell a call at 47.00 strike & earn 0.12 premium

He would also have to buy a put at a strike of 45.50 which would cost him 0.25

Importer needs to buy a call and sell a put. Both the options would be out of the money.


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47

46.5

46-48

M

.

m.

g an


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Option Price Basics Exercise