Example: a digital option; for a digital call with strike K , 1 - - PowerPoint PPT Presentation

Different Payoffs Discontinuous Payoffs

• Same basic model, with two assets:

– The cash bond {Bt}t≥0; if the risk-free interest rate is a constant r and B0 = 1, then Bt = ert, t ≥ 0. – A risky asset with price {St}t≥0; we assume that under the market probability measure P, {St}t≥0 is geometric Brownian motion: dSt = µStdt + σStdWt where {St}t≥0 is P-Brownian motion.

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• Recall:

– the value at time t of a European option whose payoff at time T is CT = f(ST) is Vt = F(t, St), where Vt = F(t, St) = EQ e−r(T−t)f(ST)

• Ft
• = e−r(T−t)

∞

−∞f

• x exp
• r − σ2

2

• (T − t) + σy

√ T − t

• × exp(−y2/2)

√ 2π dy – The replicating portfolio consists of φt = ∂F ∂x (t, x)

• x=St

shares and ψt = e−rt(Vt − φtSt) cash bonds.

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• The Feynman-Kac representation shows that F is continuous

and differentiable for t ∈ [0, T) and x ∈ R.

• But F(t, x) → f(x) as t → T, which is often not differentiable

(calls and puts) and may not be continuous.

• So F may behave badly as t → T.

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• Example: a digital option; for a digital call with strike K,

CT =

  

1 if ST ≥ K if ST < K

• Calculus shows that

Vt = e−r(T−t)Φ(d2), where Φ is the standard normal cumulative distribution func- tion, and d2 = 1 σ√T − t

• log

St

K

• +
• r − σ2

2

• (T − t)
• is the same constant that arises in the price of a normal call.

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• More calculus gives

φt =e−r(T−t) × 1 St × 1

• 2π(T − t)σ2

× exp

 −

1 2(T − t)σ2

• log

St

K

• +
• r − σ2

2

• (T − t)

2 

• This is proportional to a lognormal density function in St,

centered approximately at K, with scale

• (T − t)σ2.
• As t approaches T, this is large for St close to K, and small
• therwise, so the hedge may need large adjustments if St is

close to K; digital options are often booked as spreads to smooth the delta.

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Multistage Options

• Some option definitions involve a maturity T1 and an inter-

mediate time T0, 0 < T0 < T1.

• Example:

forward start call option; payoff at t = T1 is

• ST1 − K
• +, where the strike K = ST0.
• Work backwards: for T0 ≤ t ≤ T1, it is a standard European

call with (known) strike ST0, and Vt = e−r(T1−t)EQ

• ST1 − ST0
• +
• Ft
• = StΦ(d1) − ST0e−r(T1−t)Φ(d2)

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• Note that d1 and d2 depend on St, as well as K = ST0, r, σ,

and T1.

• But at t = T0, the option is “at the money”, and d1 and d2

depend only on r, σ, T0, and T1.

• So

VT0 = ST0

• Φ(d1) − e−r(T1−T0)Φ(d2)
• = c(r, σ, T0, T1)ST0.
• So for 0 ≤ t < T0, Vt = c(r, σ, T0, T1)St.

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Compound options

• Example: “Call on call”; the underlying is a call option on

some asset, initiated at time T0 > 0 and maturing at time T1 > T0, with strike K1.

• The compound option is the option to buy the underlying

call at time T0, with a strike of K0.

• The value of the underlying call, at time T0, is given by

Black-Scholes, as a function of ST0: say C

• ST0, T0; K1, T1
• .

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• The value of the compound option at time T0 is

V

• ST0, T0
• =
• C
• ST0, T0; K1, T1
• − K0
• + .
• The value of the compound option at a time t < T0 is the dis-

counted expected value of C under the risk-neutral measure:

EQ

e−r(T0−t)V

• ST0, T0
• .
• No closed form expression–the integral is computed numeri-

cally.

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