quanto lecture note

8/12/2019 Quanto Lecture Note

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Dynamic replication strategy

How to replicate an option dynamically by a portfolio of the risklessasset in the form of money market account and the risky underlyingasset?

The cost of constructing the replicating portfolio gives the fair price

of an option.

Risk neutrality argument

The two tradeable securities, option and asset, are hedgeable witheach other. Hedgeable securities should have the same market priceof risk.

2



dΠ =

−dc + ∆ dS

= −∂c∂t −σ2

2 S 2 ∂ 2 c

∂S 2dt + ∆ − ∂c

∂S dS

= −∂c∂t −

σ2

2 S 2

∂ 2 c∂S 2

+ ∆ − ∂c∂S

ρS dt + ∆ − ∂c∂S

σS dZ.

Why the differential Sd∆ does not enter into dΠ? By virtue of the assumption of following a self-nancing trading strategy, the

contribution to dΠ due to Sd∆ is offset by the accompanying pur-chase/sale of units of options.

5



If we choose ∆ = ∂c

∂S , then the portfolio becomes a riskless hedge in-

stantaneously since ∂c∂S

changes continuously with time . By virtueof “no arbitrage”, the hedged portfolio should earn the riskless in-terest rate.

By setting dΠ = rΠ dt

dΠ =

−∂c

∂t −σ2

2 S 2

∂ 2 c

∂S 2 dt = rΠ dt = r

−c + S

∂c

∂S dt.

Black-Scholes equation: ∂c∂t

+ σ2

2 S 2

∂ 2 c∂S 2

+ rS ∂c∂S −rc = 0.

Terminal payoff: c( S, T ) = max( S −X, 0).

The parameter ρ (expected rate of return) does not appear in thegoverning equation and the auxiliary condition.

6



• 5 parameters in the option model: S ,T,X,r and σ; only σ isunobservable.

Deciencies in the model

1. Geometric Brownian motion assumption? Actual asset pricedynamics is much more complicated.

2. Continuous hedging at all times

— trading usually involves transaction costs.

3. Interest rate should be stochastic instead of deterministic.

7



Dynamic replication strategy (Merton’s approach)

QS ( t ) = number of units of asset

QV ( t ) = number of units of option

M S ( t ) = dollar value of QS ( t ) units of asset

M V ( t ) = dollar value of QV ( t ) units of option

M ( t ) = value of riskless asset invested in money market account

• Construction of a self-nancing and dynamically hedged portfo-lio containing risky asset, option and riskless asset.

8



• Dynamic replication: Composition is allowed to change at all

times in the replication process.

• The self-nancing portfolio is set up with zero initial net invest-ment cost and no additional funds added or withdrawn after-wards.

The zero net investment condition at time t is

Π( t ) = M S ( t ) + M V ( t ) + M ( t )= QS ( t ) S + QV ( t ) V + M ( t ) = 0 .

Differential of option value V :

dV = ∂V ∂t

dt + ∂V ∂S

dS + σ2

2 S 2 ∂

2V

∂S 2 dt

=∂V ∂t

+ ρS ∂V ∂S

+ σ2

2 S 2

∂ 2 V ∂S 2

dt + σS ∂V ∂S

dZ.

9



Formally, we writedV V = ρV dt + σV dZ

where

ρV =∂V ∂t + ρS ∂V

∂S + σ2

2 S 2 ∂ 2 V ∂S 2

V and σV =

σS ∂V ∂S

V .

dΠ( t ) = [ QS ( t ) dS + QV ( t ) dV + rM ( t ) dt ]+ [ SdQ S ( t ) + V dQS ( t ) + dM ( t )]

zero due to self-nancing trading strategy

10



The instantaneous portfolio return dΠ( t ) can be expressed in termsof M

S ( t ) and M

V ( t ) as follows:

dΠ( t ) = QS ( t ) dS + QV ( t ) dV + rM ( t ) dt

= M S ( t )dS S

+ M V ( t )dV V

+ rM ( t ) dt

= [( ρ −r ) M S ( t ) + ( ρV −r ) M V ( t )] dt+ [ σM S ( t ) + σV M V ( t )] dZ.

We then choose M S ( t ) and M V ( t ) such that the stochastic term

becomes zero.

From the relation:

σM S ( t ) + σV M V ( t ) = σSQ S ( t ) +

σS ∂V ∂S

V V QV ( t ) = 0 ,we obtain

QS ( t )QV ( t )

= −∂V ∂S

.

11



Taking the choice of QV ( t ) = −1, and knowing

0 = Π(t) =

−V + ∆

S +

M (t)

we obtain

V = ∆ S + M ( t ) , where ∆ = ∂V ∂S

.

Since the replicating portfolio is self-nancing and replicates theterminal payoff, by virtue of no-arbitrage argument, the initial costof setting up this replicating portfolio of risky asset and riskless assetmust be equal to the value of the option being replicated.

12



The dynamic replicating portfolio is riskless and requires no netinvestment, so dΠ( t ) = 0.

0 = [( ρ

−r ) M S ( t ) + ( ρV

−r ) M V ( t )] dt.

Putting QS ( t )QV ( T )

= −∂V ∂S

, we obtain

( ρ

−r ) S

∂V

∂S = ( ρV

−r ) V.

Replacing ρV by∂V ∂t

+ ρS ∂V ∂S

+ σ2

2 S 2

∂ 2 V ∂S 2

V , we obtain the Black-

Scholes equation

∂V ∂t

+ σ2

2 S 2

∂ 2 V ∂S 2

+ rS ∂V ∂S −rV = 0 .

13



Alternative perspective on risk neutral valuation

From ρV =∂V ∂t + ρS ∂V

∂S + σ2

2 S 2 ∂ 2 V ∂S 2

V , we obtain

∂V ∂t

+ σ22

S 2 ∂ 2 V ∂S 2

+ ρS ∂V ∂S −ρV V = 0 .

We need to calibrate the parameters ρ and ρV , or nd some other

means to avoid such nuisance.

Combining σV = σS ∂V

∂S V

and ( ρ −r ) S ∂V ∂S

= ( ρV −r ) V , we obtain

ρV −rσV

λ V

= ρ −rσ

λ S

Black-Scholes equation .

14



• The market price of risk is the rate of extra return above r perunit risk.

• Two hedgeable securities should have the same market price of

risk.

• The Black-Scholes equation can be obtained by setting ρ =ρV = r (implying zero market price of risk).

• In the world of zero market price of risk, investors are said to berisk neutral since they do not demand extra returns on holdingrisky assets.

• Option valuation can be performed in the risk neutral world byarticially taking the expected rate of returns of the asset andoption to be r .

15



Arguments of risk neutrality

• We nd the price of a derivative relative to that of the underlyingasset mathematical relationship between the prices is invariantto the risk preference.

• Be careful that the actual rate of return of the underlying as-

set would affect the asset price and thus indirectly affects theabsolute derivative price.

• We simply use the convenience of risk neutrality to arrive at the

mathematical relationship but actual risk neutrality behaviors of the investors are not necessary in the derivation of option prices.

16



“How we came up with the option formula?” — Black (1989)

It started with tinkering and ended with delayed recognition.

The expected return on a warrant should depend on the risk of the warrant in the same way that a common stock’s expectedreturn depends on its risk.

I spent many, many days trying to nd the solution to that (dif-ferential) equation. I have a PhD in applied mathematics, buthad never spent much time on differential equations, so I didn’tknow the standard methods used to solve problems like that. Ihave an A.B. in physics, but I didn’t recognize the equation asa version of the heat equation, which has well-known solutions.

17



Continuous time securities model

• Uncertainty in the nancial market is modeled by the lteredprobability space (Ω , F , (F t ) 0≤t≤T , P ), where Ω is a sample space,

F is a σ-algebra on Ω, P is a probability measure on (Ω , F ) , F tis the ltration and F T = F .• There are M + 1 securities whose price processes are modeled

by adapted stochastic processes S m ( t ) , m = 0 , 1 ,

· · · , M .

• We dene hm ( t ) to be the number of units of the m th securityheld in the portfolio.

• The trading strategy H ( t ) is the vector stochastic process ( h0 ( t )h1 ( t ) · · ·hM ( t )) T , where H ( t ) is a ( M +1)-dimensional predictableprocess since the portfolio composition is determined by the in-vestor based on the information available before time t.

18



• The value process associated with a trading strategy H ( t ) isdened by

V ( t ) =

M

m =0 hm ( t ) S m ( t ) , 0 ≤ t ≤ T,

and the gain process G( t ) is given by

G ( t ) =

M

m =0 t

0 hm ( u ) dS m ( u ) , 0 ≤ t ≤ T.

• Similar to that in discrete models, H ( t ) is self-nancing if andonly if

V ( t ) = V (0) + G ( t ) .

19



• We use S 0 ( t ) to denote the money market account process thatgrows at the riskless interest rate r ( t ), that is,

dS 0 ( t ) = r ( t ) S 0 ( t ) dt.

• The discounted security price process S m ( t ) is dened as

S m

( t ) = S m

( t ) /S 0

( t ) , m = 1 , 2 ,

· · · , M.

• The discounted value process V ( t ) is dened by dividing V ( t )by S 0 ( t ). The discounted gain process G ( t ) is dened by

G ( t ) = V ( t ) −V (0) .

20



Theorem

Let Y be an attainable contingent claim generated by some tradingstrategy H and assume that an equivalent martingale measure Qexists, then for each time t, 0

≤ t

≤ T , the arbitrage price of Y is

given by

V ( t ; H ) = S 0 ( t ) E Q Y

S 0 ( T ) F t .

The validity of the Theorem is readily seen if we consider the dis-counted value process V ( t ; H ) to be a martingale under Q. Thisleads to

V ( t ; H ) = S 0 ( t ) V ( t ; H ) = S 0 ( t ) E Q [V ( T ; H ) |F t ].Furthermore, by observing that V ( T ; H ) = Y /S 0 ( T ), so the riskneutral valuation formula follows.

22



Change of numeraire

• The choice of S 0 ( t ) as the numeraire is not unique in order thatthe risk neutral valuation formula holds.

• Let N ( t ) be a numeraire whereby we have the existence of an

equivalent probability measure QN such that all security pricesdiscounted with respect to N ( t ) are QN -martingale. In addition,if a contingent claim Y is attainable under ( S 0 ( t ) , Q ), then it isalso attainable under ( N ( t ) , Q N ).

23



Bl k S h l d l i i d



Black-Scholes model revisited

The price processes of S ( t ) and M ( t ) are governed by

dS ( t )S ( t )

= ρ dt + σ dZ

dM ( t ) = rM ( t ) dt.

The price process of S ( t ) = S ( t ) /M ( t ) becomesdS ( t )S ( t )

= ( ρ −r ) dt + σ dZ.

We would like to nd the equivalent martingale measure Q suchthat the discounted asset price S is Q-martingale. By the GirsanovTheorem, suppose we choose γ ( t ) in the Radon-Nikodym derivativesuch that

γ ( t ) = ρ −rσ

,

then

Z is a Brownian motion under the probability measure Q and

dZ = dZ + ρ

−r

σ dt.

25



The arbitrage price of a derivative is given by

V ( S, t ) = e−r ( T −t ) E t,S Q [h ( S T )]

where E t,S Q is the expectation under the risk neutral measure Q

conditional on the ltration F t and S t = S . By the Feynman-Kacrepresentation formula, the governing equation of V ( S, t ) is givenby

∂V

∂t+

σ2

2 S 2

∂ 2 V

∂S 2 + rS

∂V

∂S −rV = 0 .

Consider the European call option whose terminal payoff is max( S T −X, 0). The call price c( S, t ) is given by

c( S, t ) = e−r ( T −t ) E Q [max( S T −X, 0)]= e−r ( T −t )E Q [S T 1 S T ≥X ] −XE Q [1 S T ≥X ].

27



• We may treat the domestic money market account and the for-eign money market account in domestic dollars (whose valueis given by F M f ) as traded securities in the domestic currencyworld.

• With reference to the domestic equivalent martingale measure,M d is used as the numeraire.

• By Ito’s lemma, the relative price process X ( t ) = F ( t ) M f ( t ) /M d( t )is governed by

dX ( t )

X ( t ) = ( r

f −r + µ) dt + σ dZ

F .

29



• With the choice of γ = ( r f −r + µ) /σ in the Girsanov Theorem,

we denedZ d = dZ F + γ dt,

where Z d is a Brownian process under Qd.

• Under the domestic equivalent martingale measure Qd, the pro-cess of X now becomes

dX ( t )

X ( t )

= σ dZ d

so that X is Qd-martingale.

• The exchange rate process F under the Qd-measure is given by

dF ( t )F ( t ) = ( r −r f ) dt + σ dZ d.

• The risk neutral drift rate of F under Qd is found to be r −r f .

30



Green function approach

The innite domain Green function is known to be

φ( y, τ ) = 1

σ √ 2πτ exp −

[y + ( r − σ2

2 )τ ]2

2σ2 τ .

Here, φ( y, τ ) satises the initial condition:

limτ →0 +

φ( y, τ ) = δ( y) ,

where δ( y) is the Dirac function representing a unit impulse at theorigin.

The initial condition can be expressed as

w( y, 0) = ∞−∞w( ξ, 0) δ( y −ξ) dξ,

so that w( y, 0) can be considered as the superposition of impulseswith varying magnitude w( ξ, 0) ranging from ξ

→ −∞ to ξ

→ ∞.

32



• Since the Black-Scholes equation is linear, the response in po-sition y and at time to expiry τ due to an impulse of magnitudew( ξ, 0) in position ξ at τ = 0 is given by w( ξ, 0) φ( y −ξ, τ ).

• From the principle of superposition for a linear differential equa-tion, the solution is obtained by summing up the responses dueto these impulses.

c( y, τ ) = e−rτ w( y, τ )= e−rτ ∞−∞

w( ξ, 0) φ( y −ξ, τ ) dξ

= e−rτ ∞ln X ( eξ −X )

1σ √ 2πτ

exp −[y + ( r −σ2

2 )τ −ξ]2

2 σ2 τ dξ.

33



Hence, the price formula of the European call option is found to be

c(S, τ

) = SN

(d1

)

−Xe −rτ

N (d2

),

where

d1 = ln S

X + ( r + σ2

2 )τ σ√ τ

, d2 = d1 −σ√ τ .

The call value lies within the bounds

max( S

−Xe −rτ , 0)

≤c( S, τ )

≤S, S

≥0 , τ

≥0 ,

35



c( S, τ ) = e−rτ E Q [( S T −X ) 1

S T

≥X

]

= e−rτ ∞0max( S T −X, 0) ψ ( S T , T ; S, t ) dS T .

• Under the risk neutral measure,

lnS T S

= r −σ2

2τ + σ Z ( τ )

so that lnS T

S is normally distributed with mean r

−σ2

2τ and

variance σ2 τ, τ = T −t .

• From the density function of a normal random variable, thetransition density function is given by

ψ ( S T , T ; S, t ) = 1

S T σ√ 2πτ exp −

ln S T S − r − σ2

2 τ 2

2 σ2 τ .

37



If we compare the price formula with the expectation representationwe deduce that

N ( d2 ) = E Q [1

S T

≥X

] = Q[S T

≥X ]

SN ( d1 ) = e−rτ E Q [S T 1 S T ≥X ].

• N ( d2 ) is recognized as the probability under the risk neutral

measure Q that the call expires in-the-money, so Xe−rτ N ( d2 )represents the present value of the risk neutral expectation of payment paid by the option holder at expiry.

• SN ( d1 ) is the discounted risk neutral expectation of the terminalasset price conditional on the call being in-the-money at expiry.

38



Variation of the delta of the European call value with respect to the

asset price S . The curve changes concavity at S = Xe− r+ 3 σ2

2 τ .41



Variation of the delta of the European call value with respect to

time to expiry τ . The delta value always tends to one from belowwhen the time to expiry tends to innity. The delta value tends todifferent asymptotic limits as time comes close to expiry, dependingon the moneyness of the option.

42



Continuous dividend yield models

Let q denote the constant continuous dividend yield, that is, theholder receives dividend of amount equal to qS dt within the intervaldt . The asset price dynamics is assumed to follow the GeometricBrownian Motion

dS S = ρ dt + σ dZ.

We form a riskless hedging portfolio by short selling one unit of theEuropean call and long holding

units of the underlying asset. The

differential of the portfolio value Π is given by

43



Martingale pricing approach

Suppose all the dividend yields received are used to purchase addi-tional units of asset, then the wealth process of holding one unit of asset initially is given by

S t = eqt S t ,

where eqt represents the growth factor in the number of units. Thewealth process

S t follows

d S t

S t= ( ρ + q) dt + σ dZ.

We would like to nd the equivalent risk neutral measure Q underwhich the discounted wealth process

S

t is Q-martingale. We choose

γ ( t ) in the Radon-Nikodym derivative to be

γ ( t ) = ρ + q −r

σ.

45

Now Z is Brownian process under Q and



dZ = dZ +

ρ + q −rσ

dt.

Also, S t becomes Q-martingale since

d

S t

S

t

= σ dZ.

The asset price S t under the equivalent risk neutral measure Q be-comes

dS t

S t= ( r

−q) dt + σ dZ.

Hence, the risk neutral drift rate of S t is r −q.

Analogy with foreign currency options

The continuous yield model is also applicable to options on foreigncurrencies where the continuous dividend yield can be considered asthe yield due to the interest earned by the foreign currency at the

foreign interest rate r f .46



Time dependent parameters



Time dependent parameters

Suppose the model parameters become deterministic functions of time, the Black-Scholes equation has to be modied as follows

∂V

∂τ =

σ2 ( τ )

2 S 2

∂ 2 V

∂S 2+[ r ( τ )

−q( τ )] S

∂V

∂S −r ( τ ) V, 0 < S <

∞, τ > 0 ,

where V is the price of the derivative security.

When we apply the following transformations: y = ln S and w =

e τ

0 r ( u ) du V , then∂w∂τ

= σ2 ( τ )

2∂ 2 w∂y 2 + r ( τ ) −q( τ ) −

σ2 ( τ )2

∂w∂y

.

Consider the following form of the fundamental solution

f ( y, τ ) = 1

2πs ( τ )exp −

[y + e( τ )] 2

2 s( τ ),

50



it can be shown that f ( y, τ ) satises the parabolic equation

∂f ∂τ

= 12

s ′( τ ) ∂ 2 f ∂y 2 + e′( τ ) ∂f

∂y.

Suppose we let

s( τ ) = τ

0 σ2 ( u ) du

e( τ ) = τ

0[r ( u ) −q( u )] du −

s ( τ )2

,

one can deduce that the fundamental solution is given by

φ( y, τ ) = 1

2 π

τ 0 σ2 ( u ) du

exp −y + τ 0 [r ( u ) −q( u ) − σ2 ( u )

2 ] du2

2 τ 0 σ2 ( u ) du

.

Given the initial condition w( y, 0), the solution can be expressed as

w( y, τ ) = ∞−∞w( ξ, 0) φ( y −ξ, τ ) dξ.

51



Implied volatilities



The only unobservable parameter in the Black-Scholes formulas is

the volatility value, σ. By inputting an estimated volatility value, weobtain the option price. Conversely, given the market price of anoption, we can back out the corresponding Black-Scholes implied volatiltiy .

• Several implied volatility values obtained simultaneously fromdifferent options (varying strikes and maturities) on the sameunderlying asset provide the market view about the volatility of the stochastic movement of the asset price.

• Given the market prices of European call options with differentmaturities (all have the strike prices of 105, current asset price

is 106 .25 and short-term interest rate over the period is at at5 .6%).

maturity 1-month 3-month 7-monthValue 3 .50 5 .76 7 .97

Implied volatility 21 .2% 30 .5% 19 .4%53

Time dependent volatility



Time dependent volatility

The Black-Scholes formulas remain valid for time dependent volatil-

ity except that 1T −t T

tσ( τ ) 2 dτ is used to replace σ.

How to obtain σ( t ) given the implied volatility measured at time tof a European option expiring at time t. Now

σimp ( t , t ) =

1

t − t t

tσ( τ ) 2 dτ

so that

t

tσ( τ ) 2 dτ = σ2

imp ( t , t )( t − t ) .

Differentiate with respect to t, we obtain

σ( t ) = σimp ( t , t ) 2 + 2( t − t ) σimp ( t , t )∂σ imp ( t , t )

∂t.

54



Practically, we do not have a continuous differentiable implied volatil-ity function σimp ( t , t ), but rather implied volatilities are available atdiscrete instants ti . Suppose we assume σ( t ) to be piecewise con-stant over ( t i−1 , t i ), then

( t i − t ) σ2imp ( t , t i) −( t i−1 − t ) σ2

imp ( t , t i−1 )

= ti

t i−1 σ2

( τ ) dτ = σ2

( t )( t i − t i−1 ) , t i−1 < t < t i ,

σ( t ) = ( t i − t ) σ2imp ( t , t i−1 ) −( t i−1 − t ) σ2

imp ( t , t i−1 )

ti −

ti−

1, t i−1 < t < t i .

55

Quanto-prewashing techniques



1. Consider two assets whose dynamics follow the lognormal pro-cesses

df f

= µf dt + σf dZ f

dgg

= µg dt + σg dZ g.

By the Ito Lemma

d( fg )fg

= ( µf + µg + ρσ f σg) dt + σ dZ

where σ2 = σ2f + σ2

g + 2 ρσ f σg;

d( f /g )f /g

= ( µf −µg −ρσ f σg + σ2g ) dt + σ dZ

where

σ2 = σ2

f + σ2g −2ρσ f σg.

56

Proof



Proof

d( fg ) = f dg + g df + ρσf σgfgdt

arising from dfdg and

observing dZ f dZ g = ρ dt

d( fg )fg = dgg + df f + ρσ f σg dt

= ( µf + µg + ρσ f σg) dt + σf dZ f + σg dZ g. (1)

Observe that the sum of two Brownian processes remains to beBrownian. Recall the formula

VAR( X + Y ) = VAR( X ) + VAR( Y ) + 2COV( X, Y ) .

Hence, the sum of σf dZ f + σg dZ g can be expressed as σ dZ , where

σ2 = σ2f + σ2

g + 2 ρσ f σg.

57



2. S = foreign asset price in foreign currencyF = exchange rate

= domestic currency price of one unit of foreign currencyS = F S = foreign asset price in domestic currency

q = dividend yield of the asset .

Under the domestic risk neutral measure Qd, the risk neutraldrift rate of

S and

F are

δdS = rd −q and δd

F = rd −r f .

Under the foreign Qf , the risk neutral drift rate for S and 1 /F

areδf

S = r f −q and δf 1 /F = r f −r d.

59



We obtain

δdS = δd

S

−δd

F

−ρσF σS

= ( r d −q) −( r d −r f ) −ρσF σS = ( r f −q) −ρσF σS .

Comparing with δf S = r f

−q and δd

S , there is an extra term

−ρσF σS .

The risk neutral drift rate of the asset is changed by the amount

−ρσF σS when the risk neutral measure is changed from the foreigncurrency world to the domestic currency world.

61



Foreign exchange options

Under the domestic risk neutral measure Qd, the exchange rateprocess follows

dF F = ( r d −r f ) dt + σF dZ F .

Suppose the terminal payoff is max( F T −X d, 0), then the price of the exchange rate call option is

V ( F, τ ) = F e−r f τ N ( d1 ) −X de−r dτ N ( d2 ) ,

where

d1 = ln F X d + r d −r f +

σ2F

2 τ σF √ τ

, d2 = d1 −σF √ τ .

65



Under the domestic risk neutral measure Qd

dS = δd dt + σ dZ



S = δd

S dt + σS dZ S

dF F

= δdF dt + σF dZ F

S = F S = asset price in domestic currency

dS S

= δdS dt + σS dZ S .

where Z S , Z F , and Z S are all Qd-Brownian processes.

By Ito’s lemma,

δdS = δd

S + δdF + ρSF σS σF

σ2S = σ2

S + σ2F + 2 ρSF σS σF .

Under the risk neutral measures,

δdS = rd −q, δd

F = rd −r f , δf S = r f −q,

δdS = δd

S

−δd

F

−ρSF σS σF = r f

−q

−ρSF σS σF = δf

S

−ρSF σS σF .

67



1. Dene

c1 ( S,F,τ ) /F =

c1 ( S, τ ) so that

c1 ( S, 0) = max( S −X f , 0) .

This call option behaves like the usual vanilla call option in theforeign currency world so that

c1 ( S, 0) = Se−qτ N ( d(1)

1 ) −X f e−r f τ N ( d(1)2 )

where

d(1)1 =

ln S X f

+ δf S +

σ2S

2 τ

σS √ τ

, d(1)2 = d(1)

1 −σS √ τ .

68



2. c2 ( S,F, 0) = max( S T −X d, 0)

c2 ( S,F,τ ) = S e−qτ N ( d(2)1 )

−X de−r dτ N ( d(2)

2 )

where

d(2)1 = ln

S X d + δ

dS +

σ2S 2 τ

σS √ τ , d(2)

2 = d(2)1 −σS √ τ ,

σ2S = σ2

S + σ2F + 2 ρSF σS σF .

69

3. For c3 ( S, 0) = F0 max( ST X f , 0), the payoff is denominated in



3. For c3 ( S, 0) F 0 max( S T −X f , 0), the payoff is denominated in

the domestic currency world, so the risk neutral drift rate is of the stock price δd

S . The call price is

c3 ( S, τ ) = F 0 e−r dτ [Se δdS τ N ( d(3)

1 ) −X f N ( d(3)2 )]

where

d(3)1 =

ln S X f

+ δdS +

σ2S

2 τ

σS √ τ

, d(3)2 = d(3)

1 −σS √ τ .

• The price formula does not depend on the exchange rate F sincethe exchange rate has been chosen to be the xed value F 0 .

• The currency exposure is reected through the dependence onσF and correlation coefficient ρSF .

70



Digital quanto option relating 3 currency worlds

F S \U = SGD currency price of one unit of USD currency

F H \S = HKD currency price of one unit of SGD currency

• Digital quanto option payoff: pay one HKD if F S \U is abovesome strike level K .

• We may interest F S \U as the price process of a tradeable assetin SGD. The dynamics is governed by

dF S

\U

F S \U = ( r SGD −r USD ) dt + σF S \U dZ S F S \U .

71



Exchange options

Exchange asset 2 for asset 1 so that the terminal payoff is



Exchange asset 2 for asset 1 so that the terminal payoff is

V ( S 1 , S 2 , 0) = max( S 1 −S 2 , 0) .

AssumedS 1S 1 = δS 1 dt + σ1 dZ 1 , δS 1 = r −q1 ,

dS 2S 2

= δS 2 dt + σ2 dZ 2 , δS 2 = r −q2 .

Dene S = S 1S 2

, σ2S = σ2

1 −2 ρ12 σ1 σ2 + σ22 . Write

V ( S 1 , S 2 , 0)S 2

= maxS 1S 2 −1 , 0 .

V ( S 1 , S 2 , τ ) = e−rτ S 1 eδ

S 1τ

N ( d1 ) −S 2 eδ

S 2τ

N ( d2 ) , where

d1 =ln S 1

S 2+ δS 1 −δS 2 + σ

2S

2 τ

σS √ τ , d2 = d1 −σS √ τ .

73



We then have d Z 2 = dZ 2 −σ2 dt where d Z 2 is under Q . In a similarmanner, we obtain

d

Z 1 = dZ 1 −ρ12 σ2 dt.

Putting everything together,

dS S

= ( δS 1 −δS 2 ) dt + ( σ1 d

Z 1 −σ2 d

Z 2 )

andσ1 d Z 1 −σ2 d Z 2 = σ d Z

where

σ2 = σ21 + σ22 −2 ρ12 σ1 σ2 .

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Use of the exchange option formula to price quanto options

Consider

C 2 ( S,F, 0) = F max( S −X, 0) = max( S −XF, 0)C 4 ( S,F, 0) = S max( F −X, 0) = max( S −XS, 0)

where S = F S . Both can be considered as exchange options.

Though an exchange option appears to be a two-state option, itcan be reduced to an one-state pricing model when the similarity

variable S = S 1

S 2is dened. Similarly, the two-state quanto options

can be reduced to one-state pricing models.

76

quanto lecture note

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