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?????? ??(Option) - ????? ????-

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Title: PowerPoint Author: John Shin (johnshin_at_kfb.co.kr) Last modified by: John Shin Created Date: 11/24/2001 6:29:00 AM Document presentation format – PowerPoint PPT presentation

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Title: ?????? ??(Option) - ????? ????-

1
?????? ??(Option)- ????? ????-

???? ??? ??
? ? ? (johnshin_at_kfb.co.kr)

2
Prologue

? Forwards, futures, ? FX swaps? ?? ??? ???
???????.
? ???? ??? ?? ?? ??? ????? ?? ????? ???? ????.
1?, 2? ???? ??? ????? 02? 8? 15? ??? ???????.
???? URL http//vols.com.ne.kr/fxkorea.html
????? ????. - ??? ?-

3
????(Currency Options)
4
Introductions to Option

Option Market
?? ??? ??? ??? ?? ??? ??? ?? ?? ??? ? ?? ??? ??
???? ???? ??? ??/??/??/????? ?? ??? ?? ??? ???.
?? ????(holder, long) ? ??? ??? ??, ????
(writer, short)??? ???.

5
Introductions to Option (Contd)

Plain Vanilla Option? Risk Profile

6
Introductions to Option (Contd)

Glossary
Call Options Give the holder (buyer) the right,
but not the obligation, to BUY the underlying
asset from the writer (seller) by a given time at
a given price
Put Options Give the holder the right, but not
the obligation, to SELL the underlying asset to
the writer by a given time at a given price
Maturity The given time is called the
Expiration or Maturity date
Strike The given price is called the Strike
or Exercise price
European An option that can be exercised only
at the end of its life
American An option that can be exercised at
any time during its life.
Premium Option contract is a kind of financial
assets, and its price movement depends on its
undelying assets.Therefore, premium is the price
of financial assets.That is, the price paid by
the buyer of the options
Option Value Intrinsic Value Time Value
1)Intrinsic Value Profit an Option Holder would
make from exercising the option immediately,
i.e., Difference Between Exercise Price and Price
of the Underlying
2)Time Value Value of Being able to Postpone
Decision to Exercise. This is, the expected
increase in the options intrinsic value in the
remaining life of the option.

7
Introductions to Option (Contd)

Why Option?
???? ?? ????? ??
????? ?? ??????
?? ???? ?? ??
?????? ????? ????
?? vs. ??? Flexibility
The forward contract exactly matches the existing
FX position There is neither risk of loss nor
potential for gain. The option locks in the
worst case at the premium paid and leaves the
option holder with a potential unlimited gain in
case of favorable spot moves.

8
CASE1 ????? ??? Hedge

S??? ?????? 3??? ??? U?? ????? ?? ????? Hedge???
??, ??? U/Won??? ??? ?? ???? ??? ???? ???? ??
??? ??? ????? ???? ??. ???? ?? ??(Won??)? ????
????? ?? ??? ?? ??? ??? ? ?? Hedge ??? ?????
? ?? ?? Hedge? ??? ?? Risk Profile??

9
CASE1 ????? ??? Hedge (Contd)

Buy U-Call (Won-Put) Option

10
CASE2 ??? ????? Hedge

D ??? ????? ??? ??? ???? ??? ??? ????? ??. ?????
????? 1??? ?? FFR? ???? D ??? ???? ??? ??? 50
????. ??? Margin? ???? ??? ????? Hedge??? ???
??? ??? bidding process? ?? 3??? ???.
D ?? ???? ????? Hedge? ?? ?? ????? ?? ?????
Hedge? ???? ???? ??? ?? ???? ?? ????? ?? ?? ????.
D ??? ??? ??? ??? ????? ????? Hedge? ?????
?? ??? ???? ????? Hedge? ? ?? ????

11
CASE2 ??? ????? Hedge (Contd)

??? Hedge? ???
50? ????? ?? ????? 50 ??? Hedge? ?????? ???,
??? ??? ?? ?? ??? ??? ?? 0
?? 100? ??? ?? (???? Hedge??? ?? ??)
????? ?? Hedge
U-Call(Ffr-Put) Option

12
CASE3 Zero-cost Options

Zero-cost Options
??? ??? ??? ???? Hedge
??? ??? Premium? ??
??? ??? ??/?????? Zero-premium ??? ?? ??
????? ?? ?? Premium ????? ?? ?? Premium
??? ????? ??? ??? Risk Profile? ??? ? ??
Flexibility? ??
--gt ?? ???? Financial Engineering Tool

13
Introductions to Option (Contd)

Option? ??
Option Value ???? (Time Value) ???? (
Intrinsic Value)
Value of (Call) Option at Maturity C max ( S
- X, 0 )

14
Introductions to Option (Contd)

Intrinsic Value(????)
????(S-X)? ????? ?????? ????.
????? ????? 0? ??? ??.

15
Introductions to Option (Contd)

Time Value(????)
????? ?????? ????? ? ????.
?, Time value Option Value-Intrinsic Value.
????? ??? ???? ???? ??? ?? Premium? ???? ???.
????? ATM?? ??? ??.
?? ?? 1?? 800? ?? 2,000?? ? ????

16
Introductions to Option (Contd)

Option?? ????
Spot price (S)
Strike price (X)
Time to expiration ( Maturity, say 182days...)
Volatility of underlying (?)
Risk-free interest rate (r, rf)
For USD/KRW, r KRW, rf USD
Option?? ????? ????
CALL PUT
???? (S) -
???? (X) -
Volatility (?)
Interest Rate (r) /-
Time to Mature (T)

17
Volatility

Volatility? ??
High volatility means you have higher
change(probability) to win the option at the
maturity, so, other things being equal, the
premium also much expensive.

18
Volatility (Contd)

High volatility equals high premium but nobody
can calculate the future volatility
Types of volatilities
Futures volatility
Historical volatility
Implied volatility
Volatility smile
Risk Reversal
Volatility smile is not always uniform in both
directions.
To reflect the preference for upside or downside
protection

??
??
??
Historical Vol.
Implied Vol.
Futures Vol.
19
Volatility (Contd)

Volatility smile
The Black Scholes model used to price options
assumes that future spot rates are lognormally
distributed around the forward rate (A variable
with a lognormal distribution has the property
that its natural logarithm is normally
distributed). In reality, extreme outcomes are
more likely than the lognormal distribution
suggests - The BS model underestimates the
probability of strong directional spot movements
and therefore undervalues options with low deltas
1st Adjustment Traders routinely compensate for
these differences by adjusting the
at-the-money-forward vols for out-of-the-money
strikes to more accurately reflect the perceived
risk The manner in which traders adjust the
at-the-money volatilities gives rise to the
characteristic smile of the vol curve - This is
called the Smile Effect
For example, if the actual distribution shows
fatter tails than that suggested by the lognormal
distribution (what is termed excess kurtosis),
low delta options will have been underpriced
using BS
Traders compensate for this by adding a spread
above the ATMF vols to both the low and high
strike options

20
Volatility (Contd)

Volatility smile (Contd)
2nd Adjustment Also, the BS model does not
take into account any market trends.
Accordingly, option traders have to adjust their
vol prices such that strikes lying in the trend
will be more expensive than the strikes
symmetrical to them compared to the outright.
In theory, all strikes should trade at the same
vol since they are all based on the same
underlying instrument.
The adjustments which traders make to the ATMF
vols in order to quote high strike or low strike
options result in the characteristic smile
profile.
Smile Effect in a neutral market
The market has a neutral bias towards higher or
lower strikes
The price structure is symmetrical
Only extreme strikes are adjusted

21
Volatility (Contd)

Volatility smile (Contd)
Smile Effect in a bullish market
When high strike options are in demand, the
implied volatilities need to be adjusted higher
The price structure is asymmetrical
The market favors higher strikes (OTM Calls)
Smile Effect in a bearish market
When low strike options are in demand, the
implied volatilities need to be adjusted higher
The price structure is asymmetrical
The market favors lower strikes (OTM Puts)
Since the curve may be shaped like a lop-sided
smile or a smirk or a frown, people have been
using the term volatility skew instead of
volatility smile because the term skew doesnt
imply the sort of symmetry that the term
smiledoes.
A smile curve can be defined for every
maturity. We may have a rather neutral sentiment
on the short term but a bullish view on the long
term. Check the concept of volatility surface
(strike x maturity x vol)

22
Volatility (Contd)

Risk Reversal
Now that it is clear how and why high strike and
low strike vols differ from the ATMF vols, it
becomes important to understand how this is
measured or obtained in the market
The risk reversal is the volatility spread
between the level of vol quoted in the market for
a high strike option and the vol for a low strike
option
Risk reversals are collars, where the bought
option and the sold option have the same delta
As options with the same delta have the same
sensitivity to the vol (or same vega), risk
reversals are vega neutral
As a vega neutral structure, the vol spread will
be more important than the actual vol level
R/R are quoted as vol spreads
They will also have to reflect an eventual
asymmetry of the Smile Effect
The market convention is to quote the difference
between 25 delta strikes, however any other delta
may be priced
So, ignoring bid offer, if the vol of a 25 delta
JPY put is 10.80, and if the vol of a 25 delta
JPY call is 11.20, then the risk reversal would
be quoted as 0.40, JPY calls over, indicating
that JPY calls are favored over JPY puts (a
skewness towards a large yen appreciation)

23
Volatility (Contd)

Risk Reversal (Contd)
Instead of quoting exercise prices directly, the
convention in the options market is to quote
prices for options with particular deltas. Like
the practice of quoting implied volatilities, the
rationale for this is to allow comparison of
quotes without needing to take into account
changes in the underlying price. When referring
to the delta of options, market participants also
drop the sign and the decimal point of the delta.
So for example, an OTM put option with a BS
delta of -0.25 is referred to as a 25-delta put.
A 25-delta risk reversal is obtained by buying a
25-delta option and selling a 25-delta option in
the opposite direction.
In this example, the OTM call more expensive than
the equally OTM put (compared with what would be
predicted by the BS model)

24
Volatility (Contd)

Risk Reversal (Contd)
R/R shows what direction the market is favoring.
R/R also gives an indication of the strength of
the markets expectations.
R/R indicates the degree of skewness compared
with the lognormal distribution, which itself is
positively skewed.
Traders need to reach an agreement on the actual
level of volatilities for the call and put when
trading R/R.
To translate risk reversal quotes into actual
vols, one requires information on strangles or
butterflies.

25
Volatility (Contd)

Risk Reversal (Contd)
A 25 delta strangle is obtained by buying (or
selling) a 25 delta call and a 25 delta put.
Strangles are quoted in absolute volatility terms
as the average of call and put volatilities
(often expressed as a spread over ATMF vol).
A long 25 delta butterfly is the combination of a
short ATMF straddle and a long 25 delta strangle.
Butterflies are quoted as a spread between the
strangles and the straddles.
Observing both the risk reversal and the strangle
(or the butterfly) allows the calculation of two
separate volatilities for the call and put.
For example, from the following mid-market
information,
ATMF vol 10.0
Butterfly (or Strangle quoted as a spread over
ATMF vol) 0.6
R/R 1.0 call over
Then Strangle 10.0 0.6 10.6, Vol for the
call 10.6 1.0 / 2 11.1, Vol for the put
10.6 1.0 / 2 10.1

26
Volatility (Contd)

Volatility Smile

27
Volatility (Contd)

Volatility Time value
An increase in volatility does not affect the
intrinsic value of an option, but does have an
interesting effect on the time value of an
option.
The time value of a one-year European call with a
strike of Y100 calculated for a range of spot
prices at different volatility levels result in
the above curve.
For any ATM option, an increase in volatility
will proportionately increase its time value

28
Volatility (Contd)

Volatility Quotation

29
Volatility (Contd)

Historical Volatility ???
??? ???? ??? ????? ???? (Natural Log, Ln)? ????
?? ? ? ?? ?? ????? ??? ?.
Monthly Volatility (Daily Data ?? ?)
Ln(St/St-1)(?? 21??? ????) ?252
Monthly Volatility (Weekly Data ?? ?)
Ln(St/St-1)(?? 4??? ????) ?52
1year 12??, 1year 52?, 1year 252?

30
Volatility (Contd)

Historical Volatility ???(Contd)

LN(B4/B3)
STDEV(C4C8)SQRT(252)
STDEV(C4C24)SQRT(252)
31
Volatility (Contd)
32
Option Valuation by Binomial Trees

Theoretical Option Value
Sum of Expected Values Sum of probability
expected price change
Binomial P/L

33
Option Valuation by Binomial Trees (Contd)

Binomial Tree
The binomial tree displays an assets potential
price outcomes and the probability of occurrence
associated with each specific time intervals.
Assumption There is 5050 chance price will be
moved by 1
Binomial Tree

34
Option Valuation by Binomial Trees (Contd)

Binomial Tree Option
The binomial tree is useful for visualizing how
different variables affect options pricing
All values above(Below) the strike price line
represent outcomes that would produce a payoff
for a Call (Put)option
Buy U Call

35
Option Valuation by Binomial Trees (Contd)

Theoretical Value Sum of MAX(0,Si-K)
Probability
(Y71/32)(Y55/32)(Y35/16)(Y15/16)(05/32
)(0132) Y2.25
Given a time t1T European call option with a
strike price of Y98,
the theoretical value of this call is Y2.25
Option Value

36
Option Valuation by Binomial Trees (Contd)

Concept of Binomial Option Valuation
Method(No-arbitrage or Riskless hedge approach)
??(?)? ?????(Short Call)?? ??? ?????(??-?? ???)?
????? ?? ??? ??.
??? ?? ?? ??-?? ????? ???, ??? ?? ?? ??? ???.
??? ???? r?? ???, ?????? ?????
?? S0? - f ??.

37
Option Valuation by Binomial Trees (Contd)

???? ??? ????? ???(Risk neutral approach)
?????? ???? ?? ??? p(?????? Risk neutral
probability)? ??? ????? (1-p)? ??? ??? ??? ?????,
????? 1?? ?? ????? ??? ??.
??, ??? ?? ?? ??? ??? ??? ????(?? E(x2)-
E(x)2),
??? ?? ??? ? ??.
?? ??? ??? ??, ?? ?? ????? ??? ?? ??? ??? ??.

38
Option Valuation by Binomial Trees (Contd)

???? ??? ????? ???(Risk neutral approach)
(Contd)
?? ??? ???, ?? ??? ??? ?? ??? ? ??.

39
Option Valuation by Binomial Trees (Contd)

????? ???? ??
??? ?? ????? ?????, ??? ??? r?? r-rf? ??? ????,
??? ?? ??? ????.
??? ????? ???, ??? ????? ????? ???? ????? ??? ?
??.
?, ????? ????? Exp(-r?t)?? ???? ??.

40
Option Valuation by Binomial Trees (Contd)

????? ??? ??? ? ?? ????
??(So)1,230
????(X)1,230
????(r)5
????(rf)2
???8
??3??(0.25)
????(N)5 (?t0.05)
????0.9975
????(a)1.0015
????(p)0.5375
u1.0180
d0.9823
?? ?) 92.99115.080.53750.997567.81(1-0.5375)
0.9975

41
Option Valuation by Binomial Trees (Contd)

Currency Option Pricing Application with Visual
Basic Code
?? ?????? ??????? Binomial Tree? ??? ???? ????

Function Binomial_European(S, X, Sday, Mday, Vol,
r, rf, Call_Put, N) Dim St(0 To 200, 0 To
200) As Double Dim optlet_price(0 To 200, 0
To 200) As Double tau (Mday - Sday) / 365
dt tau / N u Exp(Vol Sqr(dt)) d
1 / u a Exp((r - rf) dt) b Exp(-r
dt) P (a - d) / (u - d) For i 0 To N
For j 0 To i St(i, j) S
u j d (i - j) Next j Next i
For j 0 To N If (Call_Put "Call")
Then optlet_price(N, j)
Application.WorksheetFunction.Max(St(N, j) - X,
0) Else optlet_price(N, j)
Application.WorksheetFunction.Max(X - St(N, j),
0) End If Next j For i N - 1 To
0 Step -1 For j 0 To i
optlet_price(i, j) (P optlet_price(i 1, j
1) (1 - P) optlet_price(i 1, j)) b
Next j Next i Binomial_European
optlet_price(0, 0) End Function
42
Option Valuation by Binomial Trees (Contd)

Currency Option Pricing Application with Visual
Basic Code (Contd)
?? ?????? ??????? Binomial Tree? ??? ???? ????

Function Binomial_American(S, X, Sday, Mday, Vol,
r, rf, Call_Put, N) Dim St(0 To 200, 0 To
200) As Double Dim optlet_price(0 To 200, 0
To 200) As Double tau (Mday - Sday) / 365
dt tau / N u Exp(Vol Sqr(dt)) d
1 / u a Exp((r - rf) dt) b
Exp(-r dt) P (a - d) / (u - d) For i
0 To N For j 0 To i St(i,
j) S u j d (i - j) Next j
Next i For j 0 To N If (Call_Put
"Call") Then optlet_price(N, j)
Application.WorksheetFunction.Max(St(N, j) - X,
0) Else optlet_price(N, j)
Application.WorksheetFunction.Max(X - St(N, j),
0) End If Next j For i N - 1 To
0 Step -1 For j 0 To i
optlet_price(i, j) (P optlet_price(i 1, j
1) (1 - P) optlet_price(i 1, j)) b
If (Call_Put "Call") Then
optlet_price(i, j) Application.WorksheetF
unction.Max(St(i, j) - X, optlet_price(i, j))
Else optlet_price(i, j)
Application.WorksheetFunction.Max(X - St(i, j),
optlet_price(i, j)) End If
Next j Next i Binomial_American
optlet_price(0, 0) End Function
43
Financial Variables Movement

Markov Property
???? ???? ??? ??? ?? ??? ?, ?? Stochastic
Process? ???? ??.
Markov Process? Stochastic Process? ? ???, ??? ??
???? ???? ??? ??? ?? ????? ??? ?? ???.
Wiener Process( Brownian Motion)
Markov Process ? ??? 0??, ??? 1? ?? Wiener
Process ?? Brownian Motion?? ??.
?? z? Wiener Process? ?? ?, ?t ??? ??? ??? ??.
Mean of ?z 0
St. dev. of ?z sqrt(?t)
Generalized Wiener Process
Mean of ?x a?t
St. dev. of ?x bsqrt(?t)

44
Financial Variables Movement (Contd)

Process of Stock Price
??? ?? ??(?t)? ??? ???(?S)? ?????(?)? ??(?t)? ?
????,
??(?t)? 0? ????? ??? ?, ?? ?? ????.
???, ??(S)? ?? ???? ?? ??? ?, ??? ?? ?? ????.

45
Financial Variables Movement (Contd)

?? Simulations

46
Financial Variables Movement (Contd)

?? Simulations (Contd)

47
Financial Variables Movement (Contd)

?? Simulations (Contd)

48
Financial Variables Movement (Contd)

Ito Process
Generalized Wiener Process? ? ?? a, b? ???? x? ??
t ? ?? ? ?, ?? Ito Process? ??, ??? ?? ?? ????.
Itos Lemma Option Price of Stock
?? ?? ??? ????? ??? ??? ????.
Itos Lemma? ?? x, t? ??? f? ??? ?? ??? ???? ???.
?? f ?? Ito Process? ???,
Drift(?? ????)?
???

49
Financial Variables Movement (Contd)

Itos Lemma Option Price of Stock (Contd)
?? ?
???????, ??(S)? ??(t)? ??? f? ?? ?? ?? ??? ???.
Itos Lemma ? ??
flnS?? ? ?, ?f/ ?S1S, ?2f/ ?S2-1/S2, ?f/
?t0??.
???,
? ????.
?, ??? (?-?2/2)T, ??? ?2T? ????? ??? ??.

50
Black-Scholes Model

Lognormal Property of Stock price
??? Lognormal??? ???, Geometric Brownian Motion?
???,
???, lnS ?? GBM? ??? ???, lnS? ????? ??? ?? ??
????.

51
Black-Scholes Model (Contd)

?????(Distribution of Rate of Return)
?? ???? T????? ???? ?????? ? ?,
?, ???? ??? (?-?2/2)??, ????? ?/?T ? ????? ???.

52
Black-Scholes Model (Contd)

Black-Scholes-Merton ?? ??? ??
? ?????,
?? ????.
?f/ ?S ??? ??? (-)? ?????? ??? ?????? ??? ??
?????? ???
??? ?? ?? ?t??? ????? ????(??)? ??? ??.
??? ?? ?? ??? ????? ???? (??) ? r??t? ????,
?
??? ????.

53
Black-Scholes Model (Contd)

Black-Scholes? ??

54
Proof of Black-Scholes Model

Key Result
Key Result? ??
??, ??? ?? ????,
H(Q)? Q? ?? ?????? ? ?,

55
Proof of Black-Scholes Model (Contd)

Key Result? ?? (Contd)

56
Currency Options by Black-Scholes Model

Black-Scholes ?????? ??
?? ?? ??? ?? GBM? ???, ??-??????, ??? ?? ?? ????.
??? ???? ??(q)? ?? ??? ?????, S0 ?? S0 Exp(-qT)
?? S0 Exp(-rfT)? ??? ?? ????.(??/? ??? ?? rf?
??? ??)
??, Black-Scholes? ??? ?? ?? ?????, ???? ???
???? ???? ????.

57
Currency Options by Black-Scholes Model (Contd)

Put-Call Parity
??? ???? ? ????? ??? ?? ??? ????.

58
Currency Options by Black-Scholes Model (Contd)

Currency Option Pricing Application with Visual
Basic Code
?? ?????? ??????? Black-Scholes ???? ????

Function EC(S, X, Sday, Mday, vol, r, rf) As
Double Dim t As Double Dim d1 As Double
Dim d2 As Double t (Mday - Sday) /
365 d1 (Log(S / X) (r - rf vol 2 / 2)
t) / (vol t 0.5) d2 d1 - vol t
0.5 EC Exp(-rf t) S
Application.NormSDist(d1) - Exp(-r t) X
Application.NormSDist(d2) End Function Function
EP(S, X, Sday, Mday, vol, r, rf) As Double
Dim t As Double Dim d1 As Double Dim d2
As Double t (Mday - Sday) / 365 d1
(Log(S / X) (r - rf vol 2 / 2) t) /
(vol t 0.5) d2 d1 - vol t 0.5
EP Exp(-r t) X Application.NormSDist(-d
2) - Exp(-rf t) S Application.NormSDist(-d1)
End Function
59
Option Sensitivity (Delta)

????
??? ITM?? ?? ???
????
???? ????? ?? ???
ATM??? ??? 0.50
????
??? ??? ??? ??? ???? ??? ???? ???? ??? ??? ?????
?????? ???? ??? ? ??.
Dynamic Hedging
??? ??,??,Volatility ?? ??? ?? ??? ??? ??? ???
??? ??.
???? ????
???? ???? ?? ???? ?????? ? ??? ?????, ???? ??? ??
?????? carrying cost??? ?? risk? ??? ?.

60
Option Sensitivity (Delta Contd)

??(Delta)
?? ???? ???? ????,
? ????.
??(?)? ???? ????? ?? ???? ????.
? ????? ?????.
??? ??? ??
Black-Scholes ?? ?????? ??,
????? ??
????????? ????? r?? r-rf? ???? ??? ??.

61
Option Sensitivity (Delta Contd)

????(S) ????? ?? ??? ??
?????(Delta Neutral)? ????(Dynamic Delta Hedging)
?????? ?? ???? ???? ????? ??? ?????? ???, ????
??? 0.6? ?? 0.6??? ????? ???? ??? ???(0)??.
???? ?? ? ??? ?? ? ??? ?? ?? ????? ??
???(Rebalancing)? ???? ??. ?? ?? ???? ?????? ??.

OTM
X
62
Option Sensitivity (Delta Contd)

Continuous delta hedging
???? ?? ????? ??? ?? ?? ?? ??????? ??? ????.
?? ???? ? ?????
????(the gamma effect, convexity effects)
?????? ??(change in perceived volatility)
time decay due solely to the passage of time
???? ????? ?? ??
No transaction cost
no jumps
no shift in the volatility.

63
Option Sensitivity (Gamma)

Delta of delta
??? ??? ?? ???? ???? ?????? ?????? ?? ??? ???.
Positive/Negative ??
??? ???? ???????? ???? ??? ???? ???????? ???? ??
Positive ???? ? ??? ??? Negative ?? ????.
At the money?? ??? ???? ?.
??? ??? ?? ?? ???? ??.
?? ?? ???? ??? ???
??
Put?? ??? () ??
???? ?? (-??) (??)
???? ?? (-??) - (??)
Put?? ??? (-) ??
???? ?? (??) (-??)
???? ?? (??) - (-??)

64
Option Sensitivity (Gamma Contd)

??(Gamma)
??? ???? ????? ?? ????? ????. ?,
?? ??? ?? ???? ?? S?S? ? ? ????? C? C? ?? C?
C? ??.
?? C-C? ????? ??? ????, ?? ????? Convexity?
?? ????.

65
Option Sensitivity (Gamma Contd)

??? ????? ??
????? ??
????? ??? ?? ??? ????, ?? ATM? ?? ??? ???????
????.
?? Plain vanilla ????? Exotic(Digital)??? ?? ?
?? ????.
?? ???(Gamma Neutral)
?????? ????? ?????? ??? 0?? ??? ?? ???.
???, ????? ???? ?? ??? 0??? ??????? ??? ?? ? ??,
?? ??? ???? ???? ????.
??-?? ???(Delta-Gamma Neutral)??? ?? ???? ????
??? ??? ??? ??? ??? ????.

66
Option Sensitivity (Theta Vega)

?? ???
????? ?? ????? ??
?????? ??? ????? ????? ?.
?? ??? ??? ??? ??.
At the money?? ??? ???? ?.
At the money?? ????? ?? ??? ??(??)? ?? ?? ?? ???
???? ??? ?. ?? ???????? ??, at the money ?? ?????
?? ?? ???? ?? ????? ??? ?? ???? ???? ???? ??? ???
??..
?????
Implied volatility??? ?? ????? ??
Historical volatility?? ???? ??? ??.
?? ??? ??? ??? ??.
At the money?? ??? ???? ?. Deep out of money? ???
?? ???? ?? ??? ?? ?? ??? volatility? ??? ? ?????
? ??? ?? ???? at the money? ???? ??? ?? ?

67
Option Sensitivity (Theta)

??(Theta)
??? ??? ??? ?? ?? ??? ???? ???. ?,
??? ???? ?? ????? ???? ??? Time Decay??? ??.
??? ????? ??
??? ??
????? ????? ?? ????(-)??.
ATM? ?? Time decay? ?? ??.

68
Option Sensitivity (Vega)

??(Vega)
??? ???? ?? ????? ??? ???. ?,
???? ?? ???? ?? ?? ???? ??.
????? ????? ?????? ????? ??? ??.
??? ????? ??

69
Option Sensitivity (Sensitivity Factor
Relationship)

??, ??, ??? ??
Black-Scholes Merton? ??????? ??,
????? ???? ??, ??? ?? ??? ????.
?, ??? ?? ?()? ??, ??? ? ?(-)? ??.

70
Option Sensitivity (????? ??? ??)

??? ??? ?? ?????.
?? ?? ?? ??? ???.
Delta-gamma replication.
Volatility hedging
B/S ? ???? ??? volatility? ???? ??? ?? volatility
??? ?? ??? ??? ?? ???? ?? ??? B/S????? ?? ??.
????? vs. ????
?? ??? ???????? ?? ?? ??? ???? ?? ?, ?? ?? ???
???? ??? ????? ??? ??? ??? ?.

71
Option Sensitivity (Application)

Currency Option Pricing Application with Visual
Basic Code
?? ?????? ??????? Black-Scholes ????? Greek??

Function Delta_Call(S, X, Sday, Mday, vol, r, rf)
As Double Dim t As Double Dim d1 As
Double t (Mday - Sday) / 365 d1
(Log(S / X) (r - rf vol 2 / 2) t) / (vol
t 0.5) Delta_Call Exp(-rf t)
Application.NormSDist(d1) End Function Function
Delta_Put(S, X, Sday, Mday, vol, r, rf) As
Double Dim t As Double Dim d1 As Double
t (Mday - Sday) / 365 d1 (Log(S / X)
(r - rf vol 2 / 2) t) / (vol t 0.5)
Delta_Put Exp(-rf t) (Application.NormSDist(
d1) - 1) End Function Function Gamma(S, X, Sday,
Mday, vol, r, rf) Dim t As Double Dim d1
As Double Dim N1 As Double t (Mday -
Sday) / 365 d1 (Log(S / X) (r - rf vol
2 / 2) t) / (vol t 0.5) N1 (1 / (2
Application.Pi()) 0.5) Exp((-d1 2) / 2)
Gamma (N1 Exp(-r t)) / (S vol t
0.5) End Function
72
Option Sensitivity (Application Contd)

Currency Option Pricing Application with Visual
Basic Code(Contd)
?? ?????? ??????? Black-Scholes ????? Greek??

Function Vega(S, X, Sday, Mday, vol, r, rf)
Dim t As Double Dim d1 As Double Dim N1
As Double t (Mday - Sday) / 365 d1
(Log(S / X) (r - rf vol 2 / 2) t) / (vol
t 0.5) N1 (1 / (2 Application.Pi())
0.5) Exp((-d1 2) / 2) Vega S (t
0.5) N1 Exp(-rf t) End Function Function
Theta_Call(S, X, Sday, Mday, vol, r, rf) Dim
t As Double Dim d1 As Double Dim N1 As
Double Dim d2 As Double t (Mday - Sday)
/ 365 d1 (Log(S / X) (r - rf vol 2 /
2) t) / (vol t 0.5) d2 d1 - vol t
0.5 N1 (1 / (2 Application.Pi()) 0.5)
Exp((-d1 2) / 2) Theta_Call -(S N1
vol Exp(-rf t)) / (2 t 0.5) rf S
Application.NormSDist(d1) Exp(-rf t) - r X
Exp(-r t) Application.NormSDist(d2) End
Function Function Theta_Put(S, X, Sday, Mday,
vol, r, rf) Dim t As Double Dim d1 As
Double Dim d2 As Double Dim N1 As Double
t (Mday - Sday) / 365 d1 (Log(S / X)
(r - rf vol 2 / 2) t) / (vol t 0.5)
d2 d1 - vol t 0.5 N1 (1 / (2
Application.Pi()) 0.5) Exp((-d1 2) / 2)
Theta_Put -(S N1 vol Exp(-rf t)) / (2
t 0.5) - rf S Application.NormSDist(-d1)
Exp(-rf t) r X Exp(-r t)
Application.NormSDist(-d2) End Function
73
Option Valuation Sensitivity Practice
74
Exotic Options

Barrier Options
Barrier Options? Payoffs? ????? ???????
????(Hurdle Rate)? ?????? ?? ????.
Barrier Options? ?? Knock-In(K/I)?
Knock-Out(K/O)?? ?? ? ???,
K/I? ????? ????? ??? ??? ???, K/O? ??? ???? ???.
Barrier Options? ??? ?? Plain Vanilla???? ?????
???? ??? ?? ?????, ??? ????? ? ??? ????? ???
????.

75
Exotic Options (Contd)

Barrier Options? ????(Down Barrier Call)
H lt X ? ??,
cdocdic???, cdoc-cdi
H gt X ? ??,

76
Exotic Options (Contd)

Barrier Options? ????(Up Barrier Call)
Hlt X ? ??, cuo0, cuic
Hgt X ? ??,

77
Exotic Options (Contd)

Barrier Options? ????(Down Barrier Put)
HltX? ??,
H gt X ? ??, pdo0, pdip
Barrier Options? ????(Up Barrier Put)
Hlt X ? ??,
Hgt X ? ??,

78
Exotic Options (Contd)

Binary Options(Digital Options)
Binary Options? ?? Payoffs? ?????.
???? ?? Cash-or-nothing ???? Cash-or-nothing
Call? ??,
Cash-or-nothing Put?
??.
? ?? ??? Binary Options? Asset-or-nothing ??
????? ???? ??,
???? ??.

79
Exotic Options (Contd)

Non-standard American Options
Bermudan option? ???? American option? ?? ???
???? ???? ? ? ??.
?? ?? ??? ????(Binomial Tree)? ????(Trinomial
Tree)? ?? ?? ????.
Forward Start Options
Forward Start Options? ?? ?????? ???? ????.
??? ATM Forward Start ?Options? ???
??.
??? c? ??????? T2-T1??? ????
Compound Options
Compound Options? ??? ?? ????.
???, Call on Call, Put on Call, Call on Put, Put
on Put? ?? ? ??.
Chooser Options
Chooser Options? ???? ?? ??? ??? ? ?? ????.
??? ??? Payoffs? max(c,p)??.

80
Exotic Options (Contd)

Chooser Options (Contd)
?? Put-Call Parity? ?? ??? ? ?? ???.
?? Chooser Options? ????? X? ???? ?????
Xe-(r-rf)(T2-T1)? ???? ????.
Lookback Options
Lookback Options? Payoffs? ?? ????? ?? ?? ???? ??
????.
??? ???? ??? ????? ????? ?????? ??? ????.
??? ???? ??? ????? ????? ?????? ??? ????.
?? Lookback Options? ??? ?? ? ?? ??.
Shout Options
Shout Options? ???? ????? ?????? ????? ??? ????,
??? ?? Payoff? ??? ? ??. Lookback Option? ??? ???
?? ???, ????? ??.

81
Exotic Options (Contd)