1 MAFS5030 Quantitative Modeling of Derivative Securities
Tutorial Note 4
MAFS5030 Quantitative Modeling of Derivative Securities
Tutorial Note 4
Review on pricing technique in continuous time model (with multiple
assets)
In this tutorial, we shall review some important techniques in pricing derivatives
involving more than one stochastic random variables/ underlying assets. One difficulty
in pricing such derivatives is that the calculation involves the computation of
expectation involving several correlated random variables. One needs to employ various
methods/theorems so that one can compute the price in an easier way. The following
summarizes some useful theorems/methods for this purpose:
Girsanov Theorem (Describe the price dynamic of the underlying asset when the
probability measure is changed from π to οΏ½ΜοΏ½)
Numeraire invariance theorem (Describe the pricing formula of a contingent
claim when there is a change in numeraire)
Quanto prewashing technique (A method to determine the drift rate of the
target state variable under some probability measure in pricing quanto options).
Example 1 (Pricing of exchange options: A quick review)
We consider an exchange options (European) which the holder has the right to
exchange πΎ units of asset π for one units of asset π. We let π be the maturity date
of the options. The terminal payoff of the options is seen to be
ππ(ππ, ππ) = max(ππ β πΎππ , 0), where ππ and ππ are prices of assets π and π at maturity date π respectively. We
assume that the price processes of two assets are governed by
πππ‘ = (π β ππ)ππ‘ππ‘ + ππππ‘πππ,π‘π ,
πππ‘ = (π β ππ)ππ‘ππ‘ + ππππ‘πππ,π‘π ,
where ππ,π‘π and ππ,π‘
π are π-Brownian with πππ,π‘π πππ,π‘
π = πππ‘.
Given the current prices of two assets ππ‘ and ππ‘, compute the current price of the
exchange options.
Solution
We take οΏ½ΜοΏ½π‘ = ππππ‘ππ‘ as the numeraire (recall that the asset π pays continuous dividend
at the yield rate ππ). Using numeraire invariance theorem, the current price of the
exchange options can be expressed as
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ππ‘ = πβπ(πβπ‘)πΌπ[ππ(ππ , ππ)|β±π‘] = ππ‘πΌ
π [ππ(ππ, ππ)
ππ|β±π‘] = οΏ½ΜοΏ½π‘πΌ
ππ [ππ(ππ, ππ)
οΏ½ΜοΏ½π|β±π‘]
= ππππ‘ππ‘πΌππ [
max(ππ β πΎππ , 0)
ππππππ|β±π‘]
= ππ‘ πβππ(πβπ‘)πΌππ [max (
ππππβ πΎ) |β±π‘]
β πππππ ππ π‘βπ ππ₯πβππππ πππ‘ππππ ππ‘ π‘πππ π‘
(ππππ π’πππ ππ π’πππ‘π ππ ππ π ππ‘ π)
β¦β¦(β)
To calculate the expectation πΌππ [max (ππ
ππβ πΎ) |β±π‘], we need to know the
βdistributionβ of the random variable ππ
ππ under ππ.
By applying Itoβs lemma on the function π(ππ‘, ππ‘) =ππ‘
ππ‘, we get
π (ππ‘ππ‘) =
[
(π β ππ)ππ‘ππ
πππ‘+ (π β ππ)ππ‘
ππ
πππ‘β
β πππππππ
ππ=1
+1
2ππ2ππ‘
2π2π
πππ‘2 + πππππππ‘ππ‘
π2π
πππ‘πππ‘+1
2ππ2ππ‘
2π2π
πππ‘2
β 12β β ππππ
π2πππππππ
ππ=1
ππ=1 ]
ππ‘ + ππππ‘ππ
πππ‘πππ,π‘
π
+ ππππ‘ππ
πππ‘πππ,π‘
π .
β π (ππ‘ππ‘) = [(π β ππ)ππ‘ (β
ππ‘
ππ‘2) + (π β ππ)ππ‘ (
1
ππ‘) +
1
2ππ2ππ‘
2 (2ππ‘
ππ‘3)
+ πππππππ‘ππ‘ (β1
ππ‘2) +
1
2ππ2ππ‘
2(0)] ππ‘ + ππππ‘ (βππ‘
ππ‘2)πππ,π‘
π
+ ππππ‘ (1
ππ‘)πππ,π‘
π
β π (ππ‘ππ‘) = [(π β ππ) β (π β ππ) + ππ
2 β πππππ]ππ‘ππ‘ππ‘ β ππ
ππ‘ππ‘πππ,π‘
π + ππππ‘ππ‘πππ,π‘
π
By taking ππ‘ =ππ‘
ππ‘, we get
Unfortunately, the above equation is not useful because the random variables ππ,π‘π and
ππ,π‘π are no longer to be the Brownian motion under ππ.
Step 1: Compute the price dynamic of ππ‘/ππ‘
πππ‘ = [(π β ππ) β (π β ππ) + ππ2 β πππππ]ππ‘ππ‘ β ππππ‘πππ,π‘
π + ππππ‘πππ,π‘π
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To do this, we shall use Girsanov theorem. Firstly, the Radon-Nikodym derivative is
given by
πΏπ‘ =πππππ
|β±0 =οΏ½ΜοΏ½π‘/οΏ½ΜοΏ½0ππ‘/π0
=β
οΏ½ΜοΏ½π‘=ππ‘ππππ‘
ππ‘=πππ‘ ππ‘π
πππ‘
π0πππ‘
1
=
π0π(πβππβ
ππ2
2)π‘+ππππ,π‘
π
ππππ‘
π0πππ‘
= πβππ2
2π‘+ππππ,π‘
π
= πβ« β(βππ)πππ,π‘ππ‘
0β12β«
(βππ)2ππ
π‘0 .
By taking πΎ(π‘) = βππ, one can deduce from Girsanovβs theorem that the process
ππ,π‘ππ = ππ,π‘
π +β« πΎ(π )ππ π‘
0
= ππ,π‘π β πππ‘
is a ππ-Brownian.
It remains to get a corresponding ππ-Brownian for ππ,π‘π . From the result in the Lecture
Note (I skipped the details here), we get that the process
ππ,π‘ππ = ππ,π‘
π β ππππ‘,
is also a ππ-Brownian with πππ,π‘ππ(πππ,π‘
ππ) = πππ‘.
Using ππ, the price dynamic of πππ‘ can now be expressed as
πππ‘ = [(π β ππ) β (π β ππ) + ππ2 β πππππ]ππ‘ππ‘ β ππππ‘π (ππ,π‘
π + πππ‘)β
ππ,π‘π
+ ππππ‘π (ππ,π‘ππ + ππππ‘)β
ππ,π‘π
Hence, we get
Note that the sum βππππ‘πππ,π‘ππ + ππππ‘πππ,π‘
ππ is normally distributed with mean 0 and
variance (ππ2 β 2πππππ + ππ
2)ππ‘2ππ‘. So we can write the above equation as
where ππ‘ππ is ππ-Brownian.
Step 2: Replace ππ,π‘π and ππ,π‘
π by two ππ-Brownians
πππ‘ = (ππ β ππ)ππ‘ππ‘ β ππππ‘πππ,π‘ππ + ππππ‘πππ,π‘
ππ
πππ‘ = (ππ β ππ)ππ‘ππ‘ + β(ππ2 β 2πππππ + ππ
2)β
ππ
ππ‘πππ‘ππ
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Given the current value of ππ‘, ππ‘ (so that ππ‘ =ππ‘
ππ‘ is known), the value of ππ at maturity
date π can now be expressed as
ππ = ππ‘π(ππβππβ
ππ2
2)(πβπ‘)+ππππβπ‘
ππ
.
Note that
ππ =ππππ> πΎ β ππβπ‘
ππ >lnπΎππ‘β (ππ β ππ β
ππ2
2 )(π β π‘)
ππβ πππππ‘ππ ππ¦ π
From equation (β), the time-π‘ price of the exchange options can now be expressed as
ππ‘ = ππ‘πβππ(πβπ‘)πΌππ [max (
ππππβ πΎ) |β±π‘] = ππ‘π
βππ(πβπ‘)πΌππ[max(ππ β πΎ) |β±π‘]
= ππ‘πβππ(πβπ‘) [β« (ππ‘π
(ππβππβππ2
2)(πβπ‘)+πππ§
βπΎ)(1
β2π(π β π‘)πβ
π§2
2(πβπ‘))ππ§β
π
+β« (0) (1
β2π(π β π‘)πβ
π§2
2(πβπ‘))ππ§π
ββ
]
= ππ‘πβππ(πβπ‘) [β« ππ‘π
(ππβππβππ2
2)(πβπ‘)+πππ§
(1
β2π(π β π‘)πβ
π§2
2(πβπ‘))ππ§β
π
β πΎβ« (1
β2π(π β π‘)πβ
π§2
2(πβπ‘))ππ§β
π
]
= ππ‘πβππ(πβπ‘) [ππ‘π
(ππβππ)(πβπ‘)β« (1
β2π(π β π‘)πβ(π§βππ(πβπ‘))
2
2(πβπ‘) )ππ§β
π
β πΎβ« (1
β2π(π β π‘)πβ12(
π§
βπβπ‘)2
)ππ§β
π
]
=β
π§1=π§βππ(πβπ‘)
βπβπ‘
π§2=π§
βπβπ‘
ππ‘πβππ(πβπ‘) [ππ‘π
(ππβππ)(πβπ‘)β« (1
β2ππβ
π§12
2 )ππ§1
β
πβππ(πβπ‘)
βπβπ‘
β πΎβ« (1
β2ππβ
π§22 )ππ§2
β
π
βπβπ‘
]
Step 3: Obtain the price process of ππ‘ under ππ
Step 4: Obtain the pricing formula
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= ππ‘πβππ(πβπ‘)ππ‘π
(ππβππ)(πβπ‘) [1 β π (π β ππ(π β π‘)
βπ β π‘)]
β πΎππ‘πβππ(πβπ‘) [1 β π (
π
βπ β π‘)]
= ππ‘πβππ(πβπ‘) (
ππ‘ππ‘) π(ππβππ)(πβπ‘) [π (β
π β ππ(π β π‘)
βπ β π‘)]
β πΎππ‘πβππ(πβπ‘) [π (β
π
βπ β π‘)]
= ππ‘πβππ(πβπ‘)π(π1) β πΎππ‘π
βππ(πβπ‘)π(π2),
where
π1 = βπ β ππ(π β π‘)
βπ β π‘=ln (
ππ‘πΎππ‘
) + (ππ β ππ +ππ2
2 )(π β π‘)
ππβπ β π‘
π2 = βπ
βπ β π‘=ln (
ππ‘πΎππ‘
) + (ππ β ππ βππ2
2 )(π β π‘)
ππβπ β π‘
Example 2 (Quantos digital options)
(a) We consider a quanto on a foreign currency denominated asset which pays
holder an amount πΉπ (1 unit of foreign currency) at the maturity date π if the
asset price ππ is above ππ and nothing if otherwise. So the terminal payoff of
the options can be expressed as
ππ(ππ, πΉπ) = {πΉπ ππ ππ β₯ ππ
0 ππ ππ < ππ.
Here, πΉπ denotes the exchange rate for the foreign currency and ππ is the
price (measured in foreign currency) of the underlying asset. We assume that
both πΉπ and ππ are governing by Geometric Brownian motion.
Given the value of πΉπ‘ and ππ‘, what is the time-π‘ price of the options?
(b) We consider a quanto on a foreign currency denominated asset which pays
holder an amount πΉπ (1 unit of foreign currency) at the maturity date π if the
asset price ππ is above ππ and nothing if otherwise. So the terminal payoff of
the options can be expressed as
ππ(ππ, πΉπ) = {1 ππ ππ β₯ ππ
0 ππ ππ < ππ.
Here, πΉπ denotes the exchange rate for the foreign currency and ππ is the
price (measured in foreign currency) of the underlying asset. What is the
corresponding time-π‘ price of the options?
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Some notations:
We let ππ and ππ be the risk neutral probability measure under domestic
currency and foreign currency respectively.
We let ππ and ππ be the risk-free interest rate of domestic currency and foreign
currency respectively.
Solution of (a)
Assuming the options is attainable, the time-π‘ price of the quanto options can be
expressed, using risk neutral valuation principle, as
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ[ππ(ππ, πΉπ)|β±π‘] = π
βππ(πβπ‘)πΌππ [πΉππ{ππ>ππ}|β±π‘].
We take π(π‘) = ππππ‘πΉπ‘ as the numeraire (*Here, ππ‘ is the value of one unit of foreign
currency bought at time 0). Using numeraire invariance theorem, we deduce that
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ [πΉππ{ππ>ππ}|β±π‘] =β
ππ(π‘)=ππππ‘
ππ(π‘)
ππ(π)πΌππ [πΉππ{ππ>ππ}|β±π‘]
= ππ(π‘)πΌππ [
πΉππ{ππ>ππ}
ππ(π)|β±π‘] =
(β) π(π‘)πΌππ [πΉππ{ππ>ππ}
π(π)|β±π‘]
= ππππ‘πΉπ‘πΌππ [
πΉππ{ππ>ππ}
πππππΉπ|β±π‘] = π
βππ(πβπ‘)πΉπ‘πΌππ [π{ππ>ππ}|β±π‘].
So we get
To calculate the expectation, we need to obtain the price dynamic of ππ under the
probability measure ππ.
Since ππ‘ is assumed to follow Geometric Brownian Motion and its value is in foreign
currency, so under risk neutral probability measure ππ (in foreign currency world), ππ‘
should follow:
Here, π is dividend yield rate of the asset (π = 0 if the asset is non-dividend paying) and
ππ is the volatility of the asset.
IDEA: It may be difficult to compute the expectation directly since we need to
handle two correlated random variables (πΉπ , ππ) simultaneously. The computation
would be easier if we apply change of numeraire technical by taking the foreign
currency as the numeraire.
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΉπ‘πΌ
ππ [π{ππ>ππ}|β±π‘]
πππ‘ = (ππ β π)ππ‘ππ‘ + ππππ‘πππ‘ππ .
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Given the current value of ππ‘, the asset price at the maturity date can be expressed as
ππ = ππ‘π(ππβπβ
ππ2
2)(πβπ‘)+ππππβπ‘
ππ
.
Together with the fact that ππβπ‘ππ is normally distributed with mean 0 and variance π β π‘
and the fact that
ππ > ππ β ππβπ‘ππ >
lnππππ‘β (ππ β π β
ππ2
2 )(π β π‘)
ππβ π
the expectation can be computed as
πΌππ [π{ππ>ππ}|β±π‘] = β« (1)1
β2π(π β π‘)πβ
π§2
2(πβπ‘)ππ§β
π
=β
π¦=π§
βπβπ‘
β«1
β2ππβ
π¦2
2 ππ¦β
π
βπβπ‘
= 1 β π (π
βπ β π‘) = π (β
π
βπ β π‘)
= π
(
lnππ‘ππ+ (ππ β π β
ππ2
2 )(π β π‘)
ππβπ β π‘β =π1 )
.
Hence, the time-π‘ price of the quanto options is
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΉπ‘πΌ
ππ [π{ππ>ππ}|β±π‘] = πβππ(πβπ‘)πΉπ‘π(π1).
Solution of (b)
Using risk neutral valuation principle, the time-π‘ price of the quanto options can be
expressed, using risk neutral valuation principle, as
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ[ππ(ππ, πΉπ)|β±π‘] = π
βππ(πβπ‘)πΌππ [π{ππ>ππ}|β±π‘].
Recall that the price dynamic of ππ under ππ is
πππ‘ππ‘= (ππ β π)ππ‘ + πππππ,π‘
ππ .
Using Girsanovβs theorem, the corresponding price process of ππ under ππ is given by πππ‘ππ‘= πΏπ
πππ‘ + πππππ,π‘ππ .
Here, ππ,π‘ππ is standard Brownian motion under ππ.
IDEA: To calculate the expectation, we need to obtain the price dynamic of ππ
under ππ (in domestic world). To do so, we shall apply the quanto prewashing
technique.
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We note the followings:
Using risk neutral probability measure ππ, the price process ππ‘β = ππ‘πΉπ‘ (price of
the asset under domestic currency) should have a drift rate πΏπβπ = ππ β π
Under risk neutral probability measure ππ, the price process πΉπ‘ should satisfy ππΉπ‘πΉπ‘= (ππ β ππ)ππ‘ + ππΉπππΉ,π‘
ππ .
Here, ππΉ,π‘ππ is standard Brownian motion under ππ. We assume that
(πππ,π‘ππ)(πππΉ,π‘
ππ) = πππ‘.
By applying Itoβs lemma on the function π(ππ‘, πΉπ‘) = ππ‘πΉπ‘, we obtain
π(ππ‘πΉπ‘) = (0 + πΉπ‘(πΏππππ‘) + ππ‘(ππ β ππ)πΉπ‘ + πππππΉπΉπ‘ππ‘)ππ‘ + ππ‘(ππΉπΉπ‘)πππΉ,π‘
ππ
+ πΉπ‘(ππππ‘)πππ,π‘ππ
β π(ππ‘πΉπ‘) = ππ‘πΉπ‘[(πΏππ + ππ β ππ + πππππΉ)ππ‘ + ππΉπππΉ,π‘
ππ + πππππ,π‘ππ]
βπππ‘
β
ππ‘β =
π(ππ‘πΉπ‘)
ππ‘πΉπ‘= (πΏπ
π + ππ β ππ + πππππΉ)β πΏπβπ
ππ‘ + ππΉπππΉ,π‘ππ + πππππ,π‘
ππ .
By comparing the drift rates, we have
πΏππ + ππ β ππ + πππππΉ = ππ β π β πΏπ
π = ππ β π β πππππΉ.
The price process of the ππ‘ under ππ is
Given the value of ππ‘, the asset price at the maturity date π is
ππ = ππ‘π(ππβπβπππππΉβ
ππΉ2
2)(πβπ‘)+ππππ,πβπ‘
ππ
.
Using similar method as in (a), the expectation can be computed as
πΌππ [π{ππ>ππ}|β±π‘] = β« (1)1
β2π(π β π‘)πβ
π§2
2(πβπ‘)ππ§β
πβ²= β― = π(β
πβ²
βπ β π‘) = π(π1
β² ),
where πβ² =lnππ
ππ‘β(πβπβ
ππ2
2)(πβπ‘)
ππ and π1
β² =lnππ‘ππ+(ππβπβπππππΉβ
ππ2
2)(πβπ‘)
ππβπβπ‘.
Hence, the time-π‘ price of this quanto options is given by
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ [π{ππ>ππ}|β±π‘] = π
βππ(πβπ‘)π(π1β² ).
How to determine πΏππ?
πππ‘ππ‘= (ππ β π β πππππΉ)ππ‘ + πππππ,π‘
ππ
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Example 3 (Two more examples on quanto options: A quick review)
(a) We consider a foreign equity call struck which the terminal payoff is given by
ππ = πΉπmax(ππ β ππ, 0).
Find the time-π‘ price of the quanto options.
(b) We consider a fixed exchange rate foreign equity call which the terminal
payoff is given by
ππ = πΉ0max(ππ β ππ, 0),
Where πΉ0 is the predetermined fixed exchange rate. Find the time-π‘ price of
the quanto options.
(*The notations used in this example is same as those used in Example 2)
Solution of (a)
The time-π‘ price of the quanto options can be expressed, using risk neutral valuation
principle, as
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ[ππ(ππ , πΉπ)|β±π‘] = π
βππ(πβπ‘)πΌππ[πΉπmax(ππ β ππ, 0) |β±π‘].
Since the expectation involves two random variables, one shall simplify the expectation
using change of numeraire technique (take foreign currency πΉπ‘ as numeraire).
We take ππ‘ = ππππ‘πΉπ‘ as our numeraire. By numeraire invariance theorem, we deduce
that
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ[πΉπmax(ππ β ππ , 0) |β±π‘]
=βππ‘=π
ππ‘ππ‘ππ
πΌππ[πΉπmax(ππ β ππ, 0) |β±π‘]
= ππ‘πΌππ [
πΉπmax(ππ β ππ , 0)
ππ|β±π‘]
= ππ‘πΌππ [
πΉπmax(ππ β ππ , 0)
ππ|β±π‘] = π
πππ‘πΉπ‘πΌππ [
πΉπmax(ππ β ππ , 0)
πππππΉπ|β±π‘]
= πβππ(πβπ‘)πΉπ‘πΌππ[max(ππ β ππ , 0) |β±π‘].
So we get
To calculate the above expectation, we observe that the term πβππ(πβπ‘)πΌππ[max(ππ β
ππ , 0) |β±π‘] is simply the time-π‘ price of the European call option on the foreign asset in
foreign currency. So we have
πβππ(πβπ‘)πΌππ[max(ππ β ππ , 0) |β±π‘] = ππ(π, π‘)
= ππ‘πβπ(πβπ‘)π(π1) β πππ
βππ(πβπ‘)π(π2),
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΉπ‘πΌ
ππ[max(ππ β ππ , 0) |β±π‘]
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where
π1 =
lnππ‘ππ+ (ππ β π +
ππ2
2 )(π β π‘)
ππβπ β π‘, π2 = π1 β ππβπ β π‘.
Solution of (b)
The time-π‘ price of the quanto options can be expressed, using risk neutral valuation
principle, as
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ[ππ(ππ , πΉπ)|β±π‘] = π
βππ(πβπ‘)πΌππ[πΉ0max(ππ β ππ , 0) |β±π‘].
Since there is only one random variable in the expectation, we can compute the
expectation directly without using change of numeraire.
To compute the expectation, one needs to obtain the price dynamic of ππ‘ under risk
neutral probability measure ππ (not ππ).
Using similar method as in Example 2(b), we get
Given the current value of ππ‘, the value of ππ can be expressed as
ππ = ππ‘π(πΏπ
πβππ2
2)(πβπ‘)+ππππ,πβπ‘
ππ
. Note that
ππ > π β ππ,πβπ‘ππ >
lnπππ‘β (πΏπ
π βππ2
2 )(π β π‘)
ππβ π
.
Together with the fact that ππ,πβπ‘ππ is normally distributed with mean 0 and variance π β
π‘, the fair price can be computed as
ππ‘(ππ‘, πΉπ‘) = πβππ(πβπ‘)πΌππ[πΉ0max(ππ β ππ, 0) |β±π‘]
= πΉ0πβππ(πβπ‘) [β« (ππ‘π
(πΏππβππ2
2)(πβπ‘)+πππ§
β ππ)(1
β2π(π β π‘)πβ
π§2
2(πβπ‘))ππ§β
π
]
= β― = πΉ0πβππ(πβπ‘) [πππΏπ
π(πβπ‘)π(π1) β πππ(π2)],
where
π1 =
lnππ‘ππ+ (πΏπ
π +ππ2
2 )(π β π‘)
ππβπ β π‘, π2 = π1 β ππβπ β π‘.
πππ‘ππ‘= πΏπ
πππ‘ + πππππ,π‘ππ = (ππ β π β πππππΉ)ππ‘ + πππππ,π‘
ππ
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Example 4 (Problem 4 of 2013 Final)
We let πΉπ\π = πΉπ\π(π‘) denote the Singaporean currency price of 1 unit of US
currency and πΉπ»\π denote the Hong Kong currency price of 1 unit of Singaporean
currency.
Assume that πΉπ\π is governed by the following dynamics (GBM) under the risk
neutral measure ππ in the Singaporean currency world: ππΉπ\π
πΉπ\π= (πππΊπ· β ππππ·)ππ‘ + ππΉπ\ππππΉπ\π
ππ ,
where ππΉπ\πππ is the Brownian motion under ππ.
We consider a quanto option that pays πΉπ»\π Hong Kong dollars if πΉπ\π is above the
strike price π. Find the value of the quanto option in Hong Kong currency in terms of
the riskless interest rates of different currency worlds and volatility values ππΉπ\π and
ππΉπ»\π .
Solution
We shall calculate the price of the quanto options by the following steps:
Using risk neutral valuation principle, the price of the quanto price (in HKD currency) is
given by
ππ‘ = πβππ»πΎπ·(πβπ‘)πΌπ
π»[ππ|β±π‘] = π
βππ»πΎπ·(πβπ‘)πΌππ»[πΉπ»\π(π)π{πΉπ\π(π)>π}|β±π‘].
Since the expectation involves two random variables, one needs to simplify the
expectation by applying βchange of numeraireβ technique.
Note that πΉπ»\π represents the price of 1 unit of Singaporean currency in HKD currency.
We shall choose Singaporean currency as our new numeraire.
We take ππ‘ = ππππΊπ·π‘πΉπ»\π(π‘) as the numeriare. By numeraire invariance theorem, the
pricing formula can be expressed as
ππ‘ = πβππ»πΎπ·(πβπ‘)πΌπ
π»[πΉπ»\π(π)π{πΉπ\π(π)>π}|β±π‘]
=βππ‘=π
ππ»πΎπ·π‘ππ‘ππ
πΌππ»[πΉπ»\π(π)π{πΉπ\π(π)>π}|β±π‘]
= ππ‘πΌππ» [
πΉπ»\π(π)
πππ{πΉπ\π(π)>π}|β±π‘] = ππ‘πΌ
ππ [πΉπ»\π(π)
πππ{πΉπ\π(π)>π}|β±π‘]
= ππππΊπ·π‘πΉπ»\π(π‘)πΌππ [
πΉπ»\π(π)
ππππΊπ·ππΉπ»\π(π)π{πΉπ\π(π)>π}|β±π‘]
= πβπππΊπ·(πβπ‘)πΉπ»\π(π‘)πΌππ [π{πΉπ\π(π)>π}|β±π‘]
Step 1: Choose the βcorrectβ measure (HKD/SGD/USD) for calculating the price
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So we get
Recall that πΉπ\π denotes the price of 1 unit of US currency in Singaporean currency and
it can be treated as price of an asset in Singaporean currency. Under risk neutral
probability measure ππ, πΉπ\π should follow (as given in the question):
(*Note: Here, ππππ· is seen to be continuous yield rate (risk-free interest of USD
currency) of the underlying asset (USD currency).)
Given the current exchange rate πΉπ\π = πΉπ\π(π‘) at time π‘, the exchange rate at the
maturity date π can be expressed as
Here, ππΉπ\πππ = ππΉπ\π,πβπ‘
ππ is normally distributed with mean 0 and variance π β π‘ under
ππ.
Note that
πΉπ\π(π) > π β ππΉπ\πππ >
lnππΉπ\π
β (πππΊπ· β ππππ· βππΉπ\π2
2 ) (π β π‘)
ππΉπ\πβ π
Then the expectation can be computed as
πΌππ[π{πΉπ\π(π)>π}|β±π‘] = β« (1) (
1
β2π(π β π‘)πβ
π§2
2(πβπ‘))ππ§β
π
=β
π¦=π§
βπβπ‘
β«1
β2ππβ
π¦2
2 ππ¦β
π
βπβπ‘
= 1 β π (π
βπ β π‘) = π (β
π
βπ β π‘)
ππ‘ = πβπππΊπ·(πβπ‘)πΉπ»\π(π‘)πΌ
ππ [π{πΉπ\π(π)>π}|β±π‘]
Step 2: Predict the price dynamic of πΉπ\π
ππΉπ\π
πΉπ\π= (πππΊπ· β ππππ·)ππ‘ + ππΉπ\ππππΉπ\π
ππ .
Step 3: Compute the price of the options
πΉπ\π(π) = πΉπ\ππ(πππΊπ·βππππ·β
ππΉπ\π2
2)(πβπ‘)+ππΉπ\πππΉπ\π
ππ
.
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= π
(
lnπΉπ\ππ + (πππΊπ· β ππππ· β
ππΉπ\π2
2 ) (π β π‘)
ππΉπ\πβπ β π‘β =π1 )
.
Then the current price of the derivative is given by
ππ‘ = πβπππΊπ·(πβπ‘)πΉπ»\π(π‘)π(π1).
Example 5 (Problem 9 of HW3)
We let πΉπ\π denote the Singaporean currency price of one unit of US currency and
πΉπ»\π denote the Hong Kong currency price of one unit of Singaporean currency.
Suppose we assume πΉπ\π to be governed by the following dynamics under the risk
neutral measure ππ in the Singaporean currency world: ππΉπ\π
πΉπ\π= (πππΊπ· β ππππ·)ππ‘ + ππΉπ\ππππΉπ\π
ππ ,
Where πππΊπ· and ππππ· are the Singaporean and US riskless interest rates, respectively.
Similar Geometric Brownian motion assumption is made of other exchange rate
processes.
We consider a digital quanto option pays 1 US dollar at maturity if πΉπ\π is above
πΌπΉπ»\π for some constant value πΌ. Find the value of the digital quanto option in Hong
Kong currency.
Solution
We shall calculate the price of the quanto options by the following steps:
Using risk neutral valuation principle, the price of the quanto price (in HKD currency) is
given by
ππ‘ = πβππ»πΎπ·(πβπ‘)πΌπ
π»[ππ|β±π‘] = π
βππ»πΎπ·(πβπ‘)πΌππ»[πΉπ»\π(π)π{πΉπ\π(π)>πΌπΉπ»\π(π)}|β±π‘]
= πβππ»πΎπ·(πβπ‘)πΌππ»[πΉπ»\π(π)π
{πΉπ\π(π)
πΉπ»\π(π)>πΌ}|β±π‘]
=β
πΉπ\π»=1
πΉπ»\π
πβππ»πΎπ·(πβπ‘)πΌππ»[πΉπ»\π(π)π{πΉπ\π(π)πΉπ\π»(π)>πΌ}|β±π‘]
=β
πΉπ\π»=πΉπ\ππΉπ\π»
πβππ»πΎπ·(πβπ‘)πΌππ»[πΉπ»\π(π)π{πΉπ\π»(π)>πΌ}|β±π‘].
Step 1: Choose the βcorrectβ measure (HKD/SGD/USD) for calculating the price
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Since the expectation involves two random variables, one needs to simplify the
expectation by applying βchange of numeraireβ technique.
Note that πΉπ»\π represents the price of 1 unit of USD currency in HKD currency. We shall
choose USD currency as our new numeraire.
We take ππ‘ = πππππ·π‘πΉπ»\π(π‘) as the numeriare. By numeraire invariance theorem, the
pricing formula can be expressed as
ππ‘ = πβππ»πΎπ·(πβπ‘)πΌπ
π»[πΉπ»\π(π)π{πΉπ\π»(π)>πΌ}|β±π‘]
=βππ‘=π
ππ»πΎπ·π‘ππ‘ππ
πΌππ»[πΉπ»\π(π)π{πΉπ\π»(π)>πΌ}|β±π‘]
= ππ‘πΌππ» [
πΉπ»\π(π)
πππ{πΉπ\π»(π)>πΌ}|β±π‘] = ππ‘πΌ
ππ [πΉπ»\π(π)
πππ{πΉπ\π»(π)>πΌ}|β±π‘]
= πππππ·π‘πΉπ»\π(π‘)πΌππ [
πΉπ»\π(π)
πππππ·ππΉπ»\π(π)π{πΉπ\π»(π)>πΌ}|β±π‘]
= πβππππ·(πβπ‘)πΉπ»\π(π‘)πΌππ [π{πΉπ\π»(π)>πΌ}|β±π‘]
So we get
Since πΉπ\π» is assumed to follow GBM, so the governing equation for πΉπ\π» is
where ππΉπ\π»ππ is ππ-Brownian.
However, πΉπ\π» denotes the price of 1 HKD currency in SGD currency (not USD currency).
So we CANNOT simply say that the drift rate πΏπΉπ\π»π equals πππΊπ· β ππ»πΎπ·.
Here, one has to apply quanto prewashing technique to find the drift rate πΏπΉπ\π»π .
By treating πΉπ\π» as the foreign asset (price in SGD currency) and taking πΉπ\π as the
exchange rate (SGD β USD), we observe that the product πΉπ\ππΉπ\π» = πΉπ\π» is simply the
price of 1 HKD currency in USD currency.
So under ππ, the price dynamic of πΉπ\π» should satisfy
ππΉπ\π»
πΉπ\π»= (ππππ· β ππ»πΎπ·)β
πππππ‘ πππ‘π
ππ‘ + ππΉπ\π»πππΉπ\π»ππ β¦β¦(2).
ππ‘ = πβππππ·(πβπ‘)πΉπ»\π(π‘)πΌ
ππ [π{πΉπ\π»(π)>πΌ}|β±π‘].
Step 2: Predict the price dynamic of πΉπ\π» in ππ
ππΉπ\π»
πΉπ\π»= πΏπΉπ\π»
π ππ‘ + ππΉπ\π»πππΉπ\π»ππ , β¦β¦ (1)
.
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On the other hand, the price dynamic of πΉπ\π under ππ is given by
ππΉπ\π
πΉπ\π= (ππππ· β πππΊπ·)β
πππππ‘ πππ‘π
ππ‘ + ππΉπ\ππππΉπ\πππ β¦β¦(3).
applying Itoβs lemma on π = πΉπ\ππΉπ\π», we get (*Note: We assume πππΉπ\πππ πππΉπ\π»
ππ = πππ‘)
π(πΉπ\ππΉπ\π»)
πΉπ\ππΉπ\π»β
=ππΉπ\π»πΉπ\π»
= [(ππππ· β πππΊπ·) + πΏπΉπ\π»π + πππΉπ\πππΉπ\π»] ππ‘ + ππΉπ\ππππΉπ\π
ππ + ππΉπ\π»πππΉπ\π»ππ .
By comparing the drift rates [with equation (2)], we have
(ππππ· β πππΊπ·) + πΏπΉπ\π»π + πππΉπ\πππΉπ\π» = ππππ· β ππ»πΎπ·
Given the current value of πΉπ\π» = πΉπ\π»(π‘), the exchange rate at the maturity date can
be expressed as (from equation (1)):
πΉπ\π»(π) = πΉπ\π»π(πΏπΉπ\π»
π βππΉπ\π»2
2)(πβπ‘)+ππΉπ\π»ππΉπ\π»
ππ
,
where ππΉπ\π»ππ = ππΉπ\π»
ππ (π β π‘) is normally distributed with mean 0 and variance π β π‘
under ππΆ. Note that
πΉπ\π»(π) > πΌ β ππΉπ\π»ππ
>
lnπΌπΉπ\π»
β (πΏπΉπ\π»π β
ππΉπ\π»2
2 ) (π β π‘)
ππΉπ\π»β π
So the expectation (in step 1) can be computed as
πΌππ[π{πΉπ\π»(π)>πΌ}|β±π‘] = β« (1) (
1
β2π(π β π‘)πβ
π§2
2(πβπ‘))ππ§β
π
=β
π¦=π§
βπβπ‘
β«1
β2ππβ
π¦2
2 ππ¦β
π
βπβπ‘
= 1 β π (π
βπ β π‘) = π (β
π
βπ β π‘)
= π
(
lnπΉπ\π»πΌ+ (πΏπΉπ\π»
π βππΉπ\π»2
2) (π β π‘)
ππΉπ\π»βπ β π‘β =π1 )
.
Then the current price of the derivative is given by
ππ‘ = πβππππ·(πβπ‘)πΉπ»\π(π‘)π(π1),
where πΏπΉπ\π»π = πππΊπ· β ππ»πΎπ· β πππΉπ\πππΉπ\π» .
β πΏπΉπ\π»π = πππΊπ· β ππ»πΎπ· β πππΉπ\πππΉπ\π»
Step 3: Compute the price of the options
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Appendix β Some useful theorem
Girsanov Theorem
Let ππ(π‘) be a Brownian motion under the probability measure π. Consider a non-
anticipative function πΎ(π‘) with respect to ππ(π‘) that satisfies the Novikov condition:
πΌ [πβ«12πΎ(π )2ππ
π‘0 ] < β
We let οΏ½ΜοΏ½ be another probability measure and consider the Radon-Nikodym derivative
ποΏ½ΜοΏ½
ππ= π(π‘) = exp(β« βπΎ(π )πππ(π )
π‘
0
β1
2β« πΎ(π )2ππ π‘
0
).
Then under the new probability measure οΏ½ΜοΏ½, the stochastic process
ποΏ½ΜοΏ½(π‘) = ππ(π‘) + β« πΎ(π )ππ π‘
0
.
is a Brownian motion under οΏ½ΜοΏ½.
Remark
Suppose that a stochastic process ππ‘ satisfies
πππ‘ = π(π‘, ππ‘)ππ‘ + π(π‘, ππ‘)πππ‘π,
where ππ‘π is the Brownian motion under π.
Suppose that the probability measure π is changed to οΏ½ΜοΏ½, then the dynamics of the
process ππ‘ becomes
πππ‘ = π(π‘, ππ‘)ππ‘ + π(π‘, ππ‘)π (ππ‘οΏ½ΜοΏ½ ββ« πΎ(π )ππ
π‘
0
) = π(π‘, ππ‘)ππ‘ + π(π‘, ππ‘)[πππ‘οΏ½ΜοΏ½ β πΎ(π‘)ππ‘]
β πππ‘ = [π(π‘, ππ‘) β πΎ(π‘)π(π‘, ππ‘)]ππ‘ + π(π‘, ππ‘)πππ‘οΏ½ΜοΏ½.
(*Note: Remember that πΎ(π‘) is non-anticipative and the value is fixed up to time π‘).
Numeraire Invariance Theorem
Let π(π‘) be a numeraire and assume that there is an equivalent probability measure ππ
such that
πΌππ [π(π)
π(π)|β±π‘] =
π(π‘)
π(π‘).
Here, π(π‘) is the price of an asset. Suppose that a contingent claim π is attainable, then
we have
π(π‘)πΌπ [π
π(π)|β±π‘] = π(π‘)πΌ
ππ [π
π(π)|β±π‘].
Here, π is risk neutral probability measure and π(π‘) = πππ‘ is the money market
account.