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If denominated in cash, we could make the bet fair by setting the strike price equal to
K = Se(r −δ−.5σ )T , which is the median (50% of the probability is above this value). This will
make d2 = 0 and the bets worth e−rT /2 which is not a surprise since the sum of the two bets must
be worth e−rT . Using T = 1, r = 6%, σ = 30%, we have K = 100e.06−.3 /2 = 101.51.
If denominated in shares, we could make the bet fair by setting the strike price equal to
K = Se(r −δ+.5σ )T = 100e.06+.3 /2 = 111.07, which is above the forward price. This makes d1 = 0
and the bets worth S/2 = 50.
If we purchase one unit of the claim, −VS shares, and invest W in the risk free bond, our investment
is worth I = V (S, t) − VS S + W = 0. By purchasing one claim, we will receive a dividend of
dt that will be added to dI . The change in the investment value is
dI = dt + Vt dt + VS dS +
= σ 2 S 2 VSS dt
− VS dS − VS δSdt + rW dt
+ Vt + σ 2 S 2 VSS − VS δS + rW d t.
(13) Since this is risk free and is (initially) a zero investment, both the drift and I must be zero. This
implies W = VS S − V and
+ Vt + σ 2 S 2 VSS − VS δS + r (VS S − V ) = 0,
2 (14) 1
+ Vt + σ 2 S 2 VSS + (r − δ) VS S = rV .
2 (15) hence Note if we assume is a continuous yield of the claim (rather than a $ per unit rate), the ﬁrst term
would be V rather than .
Setting b = −1 and using Proposition 21.1, we change the dividend yield of S to η = .02 −
.2 (.3) (.5) = −.01. The prepaid forward price, i.e. V in equation (21.35), is S0 e−ηT . Letting
δ ∗ = .06 + (.06 − .01) − .52 = −.14, we have the value of the claim being
50e.01(2) = 0.8455.
134 (16) Chapter 21 The Black-Scholes Equation Using Proposition 20.4, the claim should be worth
S0 e−δT Qb e
b(r −δQ )+.5b(b−1)σQ 2 ebρσ σQ T (17) e.03(2) = 0.8455. (18) which equals
50e−.04 90−1 e −.05+.52 2 Note that Proposition 20.4 derives the forward price; upon discounting, the forward price of S
becomes S0 e−δT and the forward price of Qb terms does not get discounted.
Using Proposition 21.1, since b = 1, the insurance payoff should be worth
Qe(r −δQ )T V (S, K, σS , r, T , δ − ρσ σS ) (19) hence we should use a dividend yield of .02 + .2 (.3) (.5) = .05 making the put relatively more
valuable. For K = 50, V = 7.09 hence the insurance is worth 90e(.06−.01)2 (7.09) = 705.21. If we
wanted to insure 90e(.06−.01)2 = 99.465 units, it would cost 90e(.06−.01)2 (6.05) = 601.77. This is
intuitive since ln (S) and ln (Q) are negatively correlated. When Q is high, S is more likely to be
low making the insurance payout larger (the holder has the right to sell more units for K ). 135 Chapter 22
Exotic Options: II
In the same way as the COD, the paylater is priced initially using
0 = BSP ut (S0 , K, σ, r, T , δ) − P × DR(S0 , K, σ, r, T , δ, H )
Thus, the amount to be paid if the barrier is hit is
BSP ut (S0 , K, σ, r, T , δ)
DR(S0 , K, σ, r, T , δ, H )
0.7590 At subsequent times prior to hitting the barrier, the value of the paylater put is
BSP ut (S0 , K, σ, r, T − t, δ) − P × DR(S0 , K, σ, r, T − t, δ, H )
The paylater premium has the potential to be much lower than the COD premium. Compare a
paylater with H = K to a COD. You’ll ﬁnd in many cases that the COD premium is approximately
twice as great. This is a consequence of the reﬂection principle—once you have hit the barrier, there
is approximately a 50% chance that the option will move out of the money, which means that half
the time, you’ll pay the premium without the option paying off. Thus, the premium is half that of
the COD, where the premium is always paid when and only when the option is in the money.
The initial delta will be −.1903 − 3.0436 (−.0439) = −0.0567. The DR has a gamma very close
to zero hence, initially, there is little difference between the paylater’s gamma and a regular put
As time evolves the behavior of delta and gamma becomes similar to the COD, since in each case
a small move can trigger a discrete payment; the main difference being that the discrete payment
is likely to occur before expiration when St gets close to the barrier.
We must show the formula is a solution to e−r(T −t) P S T ≥ H and ST < K where P stands for
risk neutral probability.
We begin with the case when H ≥ K (i.e. the top equation). If the barrier is hit, i.e. S = H ,
−d4 = −d2 implying the probability is N (−d2 ) = P (ST < K) (i.e. the risk neutral probability of
receiving one dollar). If at time T , S T ≥ H and ST < K , the barrier has been hit and the probability
is equal to N (−d2 ) = 1. Lastly, if at time T , S T < H or ST > K we must check the probability is
zero. If ST > K and the barrier has been hit, the probability becomes N (−d2 ) = 0. If the barrier
136 Chapter 22 Exotic Options: II has not been hit (as a reminder ST > K ), then ST < H and H 2 / (ST K) > 0 implying d4 → ∞.
The probability will be H N (−d4 ) = 0.
For the case when H < K , if the barrier is hit d6 = d8 and H drops out of the probability leaving
N (−d2 ) = P (ST < K). If at time T , S T ≥ H and ST < K , the barrier has been hit and the
probability is equal to N (−d...
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- Spring '14