Chapter 14: Advanced Derivatives and
Strategies
We look at everything. We don’t get scared because of the
complexity involved. But we examine it to death.
Arvind Sodhani, treasury, Intel
Business Week, October 21, 1994, p. 95
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 1
Important Concepts in Chapter 14




The concept of portfolio insurance and its execution using
puts, calls, futures and t-bills
New and advanced derivatives and strategies such as
equity forwards, warrants, equity-linked debt, structured
notes, and mortgage securities
Exotic options such as digital options, chooser options,
Asian options, lookback options, and barrier options
Derivatives on electricity and weather
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 2
Advanced Equity Derivatives and Strategies

Portfolio Insurance
 We can insure a portfolio by holding one put for each
share of stock. For a portfolio worth V, we should hold
 N = V/(S0 + P) puts and shares
 This will establish a minimum of
 Vmin = XV/(S0 + P) where X is the exercise price
 Example: On Sept. 26, market index is 445.75 and Dec
485 put is $38.57. Expiration is Dec. 19. Risk-free rate
is 2.99 % continuously compounded. Volatility is .155.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 3
Advanced Equity Derivatives and Strategies
(continued)

Portfolio Insurance (continued)
 We hold 100,000 units of the index portfolio for V =
$44,575,000. We have
• Vmin = (485)(44,575,000)/(445.75 + 38.57) =
44,637,585
• N = 44,575,000/(445.75 + 38.57) = 92,036
 This guarantees a minimum return of 1.0014(365/84) 1 = .0061 per year, which must be below the riskfree rate.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 4
Advanced Equity Derivatives and Strategies
(continued)

Portfolio Insurance (continued)
 Outcomes
• Index is 510 at expiration
– 92,036 shares worth 510 = $46,938,360
– 92,036 puts worth $0 = $0
– Total value = $46,938,360 (> Vmin)
• Index is 450 at expiration
– Sell stock by exercising puts so you have
92,036(485) = $44,637,460 ( Vmin)
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 5
Advanced Equity Derivatives and Strategies
(continued)

Portfolio Insurance (continued)
 See Figure 14.1, p. 503.
 If calls and t-bills used,
 NB = Vmin/BT (number of bills)
 NC = V/(S0 + P) (number of calls)
 So NB = 44,637,585/100 = 446,376
 NC = 92,036
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 6
Advanced Equity Derivatives and Strategies
(continued)

Portfolio Insurance (continued)
 Outcomes
• Index is 510 at expiration
– Bills worth $44,637,600
– 92,036 calls worth $25 = $2,300,900
– Total value = $46,938,500 (> Vmin)
• Index is 450 at expiration
– Bills worth $44,637,600
– 92,036 calls worth $0
– Total value = $44,637,600 ( Vmin)
• See Figure 14.2, p. 505.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 7
Advanced Equity Derivatives and Strategies
(continued)

Portfolio Insurance (continued)
 Dynamic hedging: A dynamically adjusted
combination of stock and futures or stock and t-bills
that can replicate the stock-put or call-tbill.
 This can be easier because the futures and t-bill
markets are more liquid than the options markets
 The number of futures required is
 Vmin 
 V   rcT 
N f  
 N(d1 )   e 
 S0 
 X 

 See Appendix
D. M. Chance
14.A for derivation.
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 8
Advanced Equity Derivatives and Strategies
(continued)

Portfolio Insurance (continued)
 Alternatively, use stock and t-bills (see Appendix 14.A
again for derivation).
V  N SS0
NB 
t - bills
B
 Vmin 
NS  
 N(d1 ) shares of stock
 X 
 See
D. M. Chance
Table 14.1, p. 507 for example of dynamic hedge
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 9
Advanced Equity Derivatives and Strategies
(continued)

Equity Forwards
 Forward contracts on stock or stock indices
 Precisely like all other forward contracts we have
covered.
 Break forward is similar to an ordinary call but has no
up-front cost. At expiration, however, its value can be
negative, unlike an ordinary call.
 See Table 14.2, p. 510. Note that K = compound
future value of call with exercise price F plus
compound future value of stock, which is forward
price of stock.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 10
Advanced Equity Derivatives and Strategies
(continued)

Equity Forwards (continued)
 Example using AOL: S0 = 125.9375, T = .0959, rc = .0446,
volatility = .83.
• F = 125.9375e.0446(.0959) = 126.48
• Ordinary call with X = 126.48 is worth 12.88. K = 126.48
+ 12.88e.0446(.0959) = 139.41
• See Figure 14.3, p. 511.
 Note similarity to forward contract and call option.
 To determine the value of a break forward at time t during its
life, we simply value it as a call and a loan:
Ce (S t , T  t, F)  (K  F)e rc (T  t)
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 11
Advanced Equity Derivatives and Strategies
(continued)

Equity Forwards (continued)
 For example, 15 days later, AOL is at 115.75, T – t = 20/365 =
0.0548, and the other inputs are unchanged. We obtain
C e (115.75,0. 0548,126.4 8)  (139.41  126 .48)e .0446(0.0548)
 5.05  12 .90  7.85

A more general version of a break forward is a pay-later option. In
this case, the buyer simply borrows the premium and has to pay it
back at expiration. This option is just an ordinary call plus a loan
of the call premium Ce(S0,T,X). At expiration, the buyer decides
whether to exercise the call and in either case pays back
Ce ( S 0 , T , X )e rcT
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 12
Advanced Equity Derivatives and Strategies
(continued)

Equity Warrants
 Warrants issued by firm
 Warrants trading on over-the-counter markets and
American Stock Exchange based on various securities
and indices.
 Many of these are quantos, which pay off based on the
performance of a foreign stock index but payment is
made in a different currency than the one associated
with the country of the foreign stock index.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 13
Advanced Equity Derivatives and Strategies
(continued)

Equity-Linked Debt
 A bond that usually pays a minimum return plus a
percentage of any increase in a stock index
 Example: One-year zero coupon bond paying 1%
interest and 50 percent of any gain on the S&P 500.
 Currently one-year zero coupon bond offers 5 %
compounded annually. S&P 500 is at 1500 with a
volatility of .12 and a yield of 1.5%.
• If you invest $10 you receive $10(1.01) =
$10.10 for sure. The present value of this is
10.10/1.05 = 9.62 (5% is opportunity cost).
• This amounts to a loss of $0.38.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 14
Advanced Equity Derivatives and Strategies
(continued)

Equity-Linked Debt (continued)
 Option payoff is $10(.5)Max(0,(ST - 1500)/1500).
This can be written as
• (5/1500)Max(0,ST - 1500), which is 5/1500th of
a European call with exercise price 1500.
 Plugging values into Black-Scholes model gives call
value of $96.81. Multiplying by 5/1500 gives a
value of $0.32. This is less than the amount given
up by accepting the lower rate on the bond ($0.38)
but might be worthwhile to some investors.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 15
Advanced Interest Rate Derivatives

Structured Notes
 Definition: an intermediate term debt security issued
by a corporation with a good credit rating in which the
coupon is altered by the use of a derivative. Examples:
 Floating coupon indexed usually to LIBOR or the
CMT rate (e.g., 1.5 times the rate).
 Range floater, which pays interest only if a reference
rate (e.g., LIBOR) stays within a given range over a
period of time. If rate stays within range, coupon
will be higher than otherwise.
 Reverse (inverse) floater, where coupon moves
opposite to interest rates, such as 12 - LIBOR
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 16
Advanced Interest Rate Derivatives
(continued)

Structured Notes (continued)
• Example: An issuer could hedge it by a swap
paying LIBOR and receiving fixed rate
– LIBOR < 12: -(12 - LIBOR) (note) + Fixed
rate - LIBOR (swap) = Fixed rate - 12
– LIBOR  12: 0 (note) + Fixed rate - LIBOR
(swap) = Fixed rate - LIBOR. Issuer could
buy a cap to pay it LIBOR while it pays the
strike rate if it wanted to make it risk-free.
• Many inverse floaters are extremely volatile due
to leverage in the rate adjustment formula.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 17
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities
 Securities constructed by offering claims on a portfolio
of mortgages, a process called securitization.
 Mortgage-backed securities are subject to prepayment
risk.
 Mortgage pass-throughs and strips
 Mortgage pass-through: a security in which the
holder receives the principal and interest payments
made on a portfolio of mortgages.
 Mortgage strip: a claim on either the principal or
interest on a mortgage pass-through. Called
principal only (PO) or interest only (IO).
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 18
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities (continued)
 Example: Assume a mortgage-backed security
representing a single $100,000 mortgage at 9.75 % for
30 years. Assume annual payments for simplicity.
 See Table 14.3, p. 516 for amortization schedule.
Annual payment would be
• $100,000/[(1-(1.0975)-30)/.0975] = $10,387.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 19
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities (continued)
 Assume a 7 percent discount rate and that the
mortgage is paid off in year 12.
• Value of IO strip = 9,750(1.07)-1 + 9,688(1.07)-2
+ … + 8,614(1.07)-12 = 74,254.
• Value of PO strip = 637(1.07)-1 + 699(1.07)-2 +
… + (1,773 + 86,574)(1.07)-12 = 46,690.
• Value of pass-through = $74,254 + $46,690 =
$120,944
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 20
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities (continued)
 Let discount rate drop to 6 % and assume
homeowner pays off two years from now.
• Value of IO = $9,750(1.06)-1 + $9,688(1.06)-2 =
$17,820, loss of 76%
• Value of PO = $637(1.06)-1 + ($699 +
$98,663)(1.06)-2 = $89,033, gain of 91%
• Value of pass-through = $17,820 + $89,034 =
$106,854, loss of 12%
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 21
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities (continued)
 If the discount rate rises to 8% and there is no
change in the payoff date of year 12,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + .
. . + $8,614(1.08)-12 = $70,532, a 5% loss
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . .
+ ($1,773 + $86,574)(1.08)-12 = $42,128, a 10%
loss
• Value of pass-through = $70,532 + $42,128 =
$112,660, a loss of almost 7%.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 22
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities (continued)
 If rate goes to 8 % and prepayment moves back to
year 14,
• Value of IO = $9,750(1.08)-1 + $9,688(1.08)-2 + .
. . . + $8,250(1.08)-14 = $76,445, a gain of 3%
• Value of PO = $637(1.08)-1 + $699(1.08)-2 + . . .
+ ($2,136 + $82,492)(1.08)-14 = $37,276, a loss
of 20%.
• Value of pass-through = $76,445 + $37,276 =
$113,721, a loss of about 6%.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 23
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities (continued)
 Mortgage-backed security values are typically very
volatile.
 Collateralized Mortgage Obligations (CMOs)
 Mortgage-backed security in which payments are
split into pieces called tranches with different claims
reflecting different risks.
 Some tranches are paid first, some receive only
interest and some receive any residual after other
tranches have been repaid.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 24
Advanced Interest Rate Derivatives
(continued)

Mortgage-Backed Securities (continued)
 Collateralized Mortgage Obligations (CMOs)
(continued)
 The different tranches receive interest, principal and
prepayments according to different priorities.
 Some CMO tranches are extremely volatile and
others have low volatility.
 A CMO is generally a fairly complex security.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 25
Exotic Options

Digital and Chooser Options
 Digital options, sometimes called binary options, are of
two types:
 Asset-or-nothing options pay the holder the asset if
the option expires in the money and nothing
otherwise.
 Cash-or-nothing options pay the holder a fixed
amount of cash (usually $1) if the option expires in
the money and nothing otherwise.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 26
Exotic Options (continued)

Digital and Chooser Options (continued)
 See Table 14.4, p. 520 for example of a portfolio
long cash-or-nothings and short X asset-or-nothings.
This combination is equivalent to an ordinary
European call. The values of the options are
O aon  S0 N(d 1 )
O con  e  rc T N(d 2 )
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 27
Exotic Options (continued)

Digital and Chooser Options (continued)
 Example: Asset-or-nothing option written on S&P 500
Total Return Index, at 1440. Exercise price of 1440.
Risk-free rate is 4.88%, standard deviation is .11 and
time to expiration is 0.5 years. We obtain
 d1 = .3526, N(.35) = .6368
 Oaon = 1440(.6368) = 917
 For 1,440 cash-or-nothing options,
 d2 = .2748, N(.27) = .6064
 (1,440)Ocon = 1440e-.0488(.5)(.6064) = 852.17.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 28
Exotic Options (continued)

Digital and Chooser Options (continued)
 A variation of the previously covered pay-later option is
the contingent-pay option. Here the premium is paid at
expiration but only if the option expires in-the-money.
Table 14.5, p. 521 shows that this option is a
combination of a standard option and Ccp cash-ornothing calls. The value must be zero today so
Ce (S0 , T, X)  Ccp e rcT N(d2 )  0
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 29
Exotic Options (continued)

Digital and Chooser Options (continued)
 Solving for Ccp gives
C cp 
 For
C e (S 0 , T, X)
e  rc T N(d 2 )
the example we have been using
C cp 
64.83
e
.0488(0.5)
0.6064
 109.55
 Now
move forward two months where St = 1440 and Tt = 4/6 = 0.333. The value is
Ce ( S t , T  t , X )  Ccp e  rc (T t ) N (d 2 )
 Ce (1400 ,0.333 ,1440 )  109 .55 e .0488( 0.333) 0.4131
 44 .53
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 30
Exotic Options (continued)

Digital and Chooser Options (continued)
 Chooser Options: Also called as-you-like-it options,
they enable the investor to decide at a specific time
after purchasing the option but before expiration that
the option will be a call or a put.
 Assume the decision must be made at time t < T
 The chooser option is identical to
• an ordinary call expiring at T with exercise price
X plus
• an ordinary put expiring at t with exercise price
X(1+r)-(T-t)
 Compare and contrast chooser with straddle.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 31
Exotic Options (continued)

Digital and Chooser Options (continued)
 Example: AOL chooser in which choice must be made
in 20 days. Call/put expires in 35 days. S0 = 125.9375,
X = 125,  = .83, rc = .0446. T = 35/365 = .0959, t =
20/365 = .0548 so T - t = .0959 - .0548 = .0411.
Exercise price on put used to price the chooser is
125(1.0456)-.0411 = 124.77.
 Using Black-Scholes model, put is worth 7.80 and call
is worth 13.21 for a total of 21.01. Straddle is worth
13.21 (call) + 12.09 (put) = 25.30.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 32
Exotic Options (continued)

Path-Dependent Options
 Path-dependent options are options in which the payoff
is determined by the sequence of prices followed by the
asset and not just by the price of the asset at expiration.
 We shall price these options using a binomial
framework. See Table 14.6, p. 523 which shows a
three-period problem. Note eight paths, and the
average, maximum, and minimum prices of each path
are computed.
 Note how the probabilities are calculated.
 In practice the binomial model is difficult to use for
path-dependent options. Monte Carlo simulation (see
Appendix 15.B) is often used.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 33
Exotic Options (continued)

Path-Dependent Options (continued)
 Asian option: an option in which the final payoff is
determined by the average price of the asset during the
option’s life. Some are average price options because
the average price substitutes for the asset price at
expiration. Others are average strike options because
the average price substitutes for the exercise price at
expiration. Can be calls or puts. Useful for hedging or
speculating when the average is acceptable as a
measure of the underlying risk. Also useful for cases
where market can be manipulated.
 See Table 14.7, p. 525 for example of pricing Asian
options.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 34
Exotic Options (continued)

Path-Dependent Options (continued)
 Lookback option: Also called a no-regrets option, it
permits purchase of the asset at its lowest price during
the option’s life or sale of the asset at its highest price
during the option’s life.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 35
Exotic Options (continued)

Path-Dependent Options (continued)
 Lookback options (continued):
 Four different types.
• lookback call: exercise price set at minimum
price during option’s life
• lookback put: exercise price set at maximum
price during option’s life
• fixed-strike lookback call: payoff based on
maximum price during option’s life (instead of
final price) compared to fixed strike
• fixed-strike lookback put: payoff based on
minimum price during option’s life (instead of
final price) compared to fixed strike
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 36
Exotic Options (continued)

Path-Dependent Options (continued)
 Lookback options (continued):
 See Table 14.8, p. 526 for example of pricing
lookback options.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 37
Exotic Options (continued)

Path-Dependent Options (continued)
 Barrier Options: Options that either terminate early if
the asset price hits a certain level, called the barrier, or
activate only if the asset price hits the barrier. The
former are called knock-out options (or simply outoptions) and the latter are called knock-in options (or
simply in-options). If the barrier is above the current
price, it is called an up-option. If the barrier is below
the current price, it is called a down-option.
 See Table 14.9, p. 528 for example of pricing.
 Barrier options are normally cheaper than ordinary
options because they provide payoffs for fewer
outcomes than ordinary options.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 38
Exotic Options (continued)

Path-Dependent Options (continued)
 Other Exotic Options:
 compound and installment options
 multi-asset options, exchange options, min-max
options (rainbow options), alternative options,
outperformance options
 shout, cliquet and lock-in options
 contingent premium, pay-later and deferred strike
options
 forward-start and tandem options
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 39
Some Important New Derivatives


Electricity Derivatives
 Electricity is a non-storable asset
 These derivatives are difficult to price
Weather Derivatives
 Measures of weather activity
 Heating degree days and cooling degree days
 Quantity of rain or snow
 Financial loss caused by weather
 Pricing is difficult but not impossible; a lot of data are
available on weather
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 40
Summary
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 41
Appendix 14A.: Derivation of the Dynamic
Hedge Ratio for Portfolio Insurance

Stock-Futures Dynamic Hedge
 Portfolio of N shares and N puts is worth
 V = N(S + P)
 So N = V/(S+P).
 Change in portfolio value for a small change in stock
price is
V
 P   V  P 
1  
 N1    
S
 S   S  P  S 
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 42
Appendix 14.A: Derivation of the Dynamic
Hedge Ratio for Portfolio Insurance
(continued)

Stock-Futures Dynamic Hedge (continued)
 A portfolio of NS shares and Nf futures is worth today
V = NSS + NfVf
 where Vf is value of futures, which starts at zero. It
follows that NS = V/S
 Set change in portfolio value for small change in S to
V
 f 
 NS  N f  
 S 
S
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 43
Appendix 14.A: Derivation of the Dynamic
Hedge Ratio for Portfolio Insurance
(continued)

Stock-Futures Dynamic Hedge (continued)
 Assuming no dividends, the futures price is
f  Se rcT
 So
f
 er T
S
c
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 44
Appendix 14.A: Derivation of the Dynamic
Hedge Ratio for Portfolio Insurance
(continued)

Stock-Futures Dynamic Hedge (continued)
 After substituting, setting the two partial derivatives of
V with respect to S equal to other, recognizing that 1 +
P/ S is C/ S and N(d1) is C/ S, we obtain the
number of futures contracts as
 Vmin
N f  
 X
D. M. Chance

 V   rcT
 N(d1 )    e

 S 
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Appendix 14.A: Derivation of the Dynamic
Hedge Ratio for Portfolio Insurance
(continued)

Stock-Tbill Dynamic Hedge
 A portfolio of stock and tbills is worth
V  N SS  N B B
 Its
sensitivity to a change in S is
V
 NS
S
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 46
Appendix 14.A: Derivation of the Dynamic
Hedge Ratio for Portfolio Insurance
(continued)

Stock-Tbill Dynamic Hedge (continued)
 The t-bill price is not sensitive to the stock price.
Setting the sensitivity of the stock-tbill portfolio to that
of the stock-futures portfolio gives
 Vmin 
NS  
 N(d1 )
 X 
 This
is the number of shares of stock to hold with t-bills
to replicate the stock and put.
D. M. Chance
An Introduction to Derivatives and Risk Management, 6th ed.
Ch. 14: 47
Appendix 14.B: Monte Carlo Simulation



A method of using random numbers designed to simulate
the random observations of prices of an asset. The
simulated series of asset prices at expiration is then
converted to an equivalent series of option prices at
expiration.
Then the current option price is the discounted average of
the option prices obtained at expiration from the
simulation.
Random prices can be simulated by drawing a standard
normal random variable, e, and inserting into the formula
S  Src t  Se t
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Ch. 14: 48
Appendix 14.B: Monte Carlo Simulation
(continued)
t is the length of the time interval over which
the stock price change occurs.
 Note: simulating a standard normal random variable
can be done approximately as the sum of twelve unit
uniform random numbers (in Excel, “=Rand( )”) minus
6.0.
Each simulated stock price is treated as the stock price at
expiration; thus, t is the maturity in years of the option.
 For each simulated stock price, compute the option
price at expiration using the intrinsic value.
 where

D. M. Chance
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Ch. 14: 49
Appendix 14.B: Monte Carlo Simulation
(continued)
 Take

the average of all of the option prices at
expiration.
 Discount the average over the life of the option at the
risk-free rate. This is the estimate of the current option
price.
This procedure will probably require at least 50,000
random numbers for a standard option and more for exotic
and complex options and derivatives.
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