# Chapter 2 Models and References

Chapter 2

Models and References

This chapter simply describes the basic assumptions and models, Martingale Pricing methods, Girsanolv Theorem, quanto options, reset options, the distribution of the maximum of geometric Brownian Motion and Lookback option.

2.1 Basic Assumptions and Models The assumption in the thesis is the same as which in Black&Scholes(1973), which is so called Perfect Market assumption. The assumptions are as follows: 1. 2. 3. 4. 5. 6.

The dynamics of stock price and exchange risk are both the geometric Brownian Motion. The stocks and exchanges are traded continuously, and can be traded with any kind of units. There are no transaction expenses and taxes, which means there is no transaction cost. It is allowed to short selling the stock and make use of the fund by short selling unlimitedly. There is a risk-free rate. There is no cash dividend of the stock before the derivative matures.

Based on the assumptions above, there must be a risk-neutral probability measure Q, such that the dynamics of the foreign stock price viewed by domestic investor can be illustrated by the formula as follows12: dS = (rf − ρ s , xσ sσ x )dt + σ s dWSQ (t ) S

(2- 1)

In which, S represents the underlying foreign stock price denominated in foreign currency; rf represents the foreign risk-free rate; ρ s , x represents the correlation coefficient between the underlying foreign stock price and exchange rate; 1

If we cancel out the 6th assumption and assume the continuous dividend yield equals to just substitute

2

q , we can

rf − q into rf for Eq(2- 2).

The derivation of the dynamics of the foreign stock price and exchange rate can be referred to Musiela, M. and M. Rutkowski (2004). Martingale Methods in Financial Modeling(2nd), Springer.pp.147~151 3

σ s represents the volatility of the underlying foreign stock price denominated in foreign currency; σ x represents the volatility of exchange rate; finally, dWSQ (t ) represents the foreign stock price’s increment of standard Brownian Motion under the risk-neutral probability measure. Eq(2- 1) shows that under domestic investor’s point of view, the expected average return on the foreign stock denominated in foreign currency equals to the difference between the foreign risk-free rate and the covariance between the foreign stock price and exchange rate. On the other hand, the dynamics of exchange rate can be represented as follows:3 dX = ( r − rf ) dt + σ X dWXQ (t ) X

(2- 2)

In which, X represents the domestic value per unit of foreign currency; r represents the domestic risk-free rate; finally, dWXQ (t ) represents the exchange rate’s increment of standard Brownian Motion under the risk-neutral probability measure. Eq(2- 2) shows that the expected average return on exchange rate equals to the difference between the domestic and foreign risk-free rate. In addition, we have to use the dynamics of the foreign stock price denominated in domestic currency( X t St ) to derive some of the pricing model. Therefore, according to Ito’s lemma, we can derive the dynamics of X t St as follows:

d ( XS ) = rdt + σ s dWSQ (t ) + σ X dWXQ (t ) ( XS )

(2- 3)

= rdt + σ SX dWSXQ (t )

Eq(2- 3) represents that the expected average return on the foreign stock denominated in domestic currency equals to the domestic risk-free rate. However, its volatility is as follows4: 2 = σ S2 + σ X2 + 2 ρ S , X σ Sσ X σ SX

3

The same as 2.

4

Let

(2- 4)

2 = σ S2 + σ X2 + 2 ρ S , X σ Sσ X , and both σ s dWSQ (t ) + σ X dWXQ (t ) σ SX

will be the same distribution. 4

and

σ sX dWSXQ (t )

2.2 Martingale Pricing This section illustrates how to use the Martingale Pricing method (or so called risk-neutral pricing method) to price the derivative under the risk-neutral probability measure Q. We assume that there is a contingent claim maturing on T ; therefore, the final payoff of this contingent claim is χ (T ) . The reasonable price at the initial time, t , is as follows:

χ (t ) = e − r (T −t ) E Q [ χ (T ) | ℑt ], 0 < t < T

(2- 5)

If the contingent claim is a quanto call or put, its final payoff is f ( ST , X T , KT ) , in which ST , and X T respectively represent the foreign stock price and exchange rate at the maturity, T , and KT represents the exercise price. We can derive the quanto call or put price, Ct , or Pt , by Martingale Pricing method as follows:

Ct = e− r (T −t ) E Q [CT | ℑt ] = e− r (T −t ) E Q [ f ( ST , X T , KT ) | ℑt ], 0 < t < T

(2- 6a)

Pt = e − r (T −t ) E Q [ PT | ℑt ] = e − r (T −t ) E Q [ f ( ST , X T , KT ) | ℑt ], 0 < t < T

(2- 6b)

The economic intuition of Eq(2- 6a) and (2-6b)means that the fair price of call or put( Ct or Pt ) must equal to the conditional expected value of the future cash flow continuously discounted by the domestic risk-free rate under the risk-neutral probability measure Q and the information set, ℑt .

2.3 Girsanolv Theorem First of all, we’ll illustrate the one-dimensional Girsanolv Theorem. On the other hand, because the dynamics of quanto option exists two Brownian Motions, we’ll illustrate the two-dimensional Girsanolv Theorem.We illustrate the one-dimensional Girsanolv Theorem as follows5: 5

We just describe the one-dimensional Girsanolv Theorem here. The exact proof can be referred to 5

If E[exp(

1 T 2 β t dt )] < ∞ , and let 2 ∫0

T

ξT = exp( ∫ β t dWt Q − 0

1 T 2 β t dt ) 2 ∫0

(2- 7)

Under the risk-neutral probability measure Q, ξT is a Martingale relative to a natural Brownian filtration. In other words, E Q [ξT | ℑt ] = E R [ξt ], 0 < t < T . Furthermore, the relationship between the new probability measure R and the risk-neutral probability measure Q is as follows:

dR = ξT dQ

(2- 8)

In Eq(2- 8), ξT is called Radon-Nikodyn Derivative. The Brownian Motion under the new probability measure R can be obtained by the transformation of the original Brownian Motion under the risk-neutral probability measure as follows:

(2- 9)

dWt Q = dWt R + β t dt

And, E Q [ξT I A ] = E R [ I A ]

(2- 10)

, if A happens 1 . Eq(2- 10) In which, I A is an indicator function, I A =  0 ,if A does't happen illustrates that although the expected value of ξT I A is more difficult to calculate under the original measure Q, we can transfer it into the new measure R and calculate the expected value of I A under it, which is more easy to compute. The two-dimensional Girsanolv Theorem is described as follows6:

6

Shreve, Steven E., 2003. Stochastic Calculus for Finance II -- Continuous-Time Models (Springer),pp212~214 Be the same as 5. Please refer to Shreve, Steven E., 2003. Stochastic Calculus for Finance II -6

If E[exp(

1 T 2 ( β1t + 2 ρ1,2 β1t β 2t + β 22t )dt ] < ∞ , and let ∫ 0 2

T

T

0

0

ξT = exp( ∫ β1t dW1Qt + ∫ β 2t dW2Qt −

1 T 2 ( β1t + 2 ρ1,2 β1t β 2 t + β 22t )dt ) ∫ 0 2

Similar to the one-dimensional Girsanolv Theorem, ξT =

(2- 11)

dR is the dQ

Radon-Nikodyn Derivative in the two-dimensional Girsanolv Theorem, and the relationship of the Brownian Motion between the new measure R and the original risk-neutral measure Q is as follows:

dW1Qt = dW1tR + ( β1t + ρ1,2 β 2 t )dt  Q R dW2t = dW2t + ( β 2t + ρ1,2 β1t )dt

(2- 12) (2- 13)

2.4 Quanto Options Quanto Options are invented by Reiner(1992). Based on the cash flow at maturity, there are 4 types of quanto options, and we will introduce them briefly in this section.7

2.4. 1

Type I Quanto Option

Let PT be the final payoff of the Type I quanto option and K be the exercise price denominated in foreign currency. We can obtain the final cash flow at maturity as follows:

PTI = X T Max( K − ST , 0)

(2- 14)

This kind of quanto put is liquidated by the foreign currency and transferred into the domestic value by the current exchange rate at the maturity. Therefore, the investor holding this kind of quanto put pays more attention on the foreign stock price risk than the exchange rate risk.

Continuous-Time Models (Springer),pp224~225 7

We just introduce the four kinds of quanto options and their pricing formula in this section. The exact proof can be referred to Reiner, E. (1992). "Quanto Mechanics." Risk March: pp59~63 7

The pricing formula for Type I quanto put is as follows:

Pt I = X t {− St N (− d1I ) + Ke

− r f (T − t )

(2- 15)

N (− d 2I )}

In which,

St σ s2 ln( ) + (rf + )(T − t ) K 2 d1I = σs T −t

(2- 16)

(2- 17)

d 2I = d1I − σ S T − t

2.4. 2

Type II Quanto option

Let PTII be the final payoff of the Type II quanto option and K be the exercise price denominated in domestic currency. We can obtain the final cash flow at maturity as follows: PTII = Max( K − X T ST , 0)

(2- 18)

This kind of quanto put is liquidated directly by the current exchange rate at the maturity. Therefore, the investor holding this kind of quanto put pays attention on the foreign stock price risk and the exchange rate risk at the same time. The pricing formula for Type II quanto put is as follows:

Pt II = − St X t N (− d1II ) + Ke − r (T −t ) N (− d 2II )

(2- 19)

In which,

8

ln(

d1II =

X t St σ2 ) + (r + SX )(T − t ) K 2 σ SX T − t

(2- 20)

d 2II = d1II − σ SX T − t

(2- 21)

σ SX = σ S2 + σ X2 + 2 ρ S , X σ Sσ X

(2- 22)

2.4. 3

Type III Quanto Option

Let PTIII be the final payoff of the Type III quanto option, K be the exercise price denominated in foreign currency and χ be the fixed exchange rate stipulated in advance. We can obtain the final cash flow at maturity as follows: PTIII = χ ⋅ Max( K − ST , 0)

(2- 23)

This kind of quanto put is liquidated by the foreign currency and transferred into the domestic value by the fixed exchange rate stipulated in advance at the maturity. Therefore, the investor holding this kind of quanto put fix the future exchange rate in advance and hedge the foreign stock price risk. The pricing formula for Type III quanto put is as follows:

Pt III = χ{− St e

− ( r − r f +σ SX )(T −t )

N (−d1III ) + Ke − r (T −t ) N (−d 2III )}

(2- 24)

In which,

III 1

d

St σ s2 ln( ) + (rf − ρ S , X σ Sσ X + )(T − t ) K 2 = σs T −t

(2- 25)

(2- 26)

d 2III = d1III − σ S T − t

2.4. 4

Type IV Quanto Option

Let PTIV be the final payoff of the Type IV quanto option and K be the 9

exercise price denominated in exchange rate. We can obtain the final cash flow at maturity as follows: PTIV = ST Max( K − X T , 0)

(2- 27)

This kind of quanto put is an exchange rate put and linked to the foreign stock price. Therefore, the investor holding this kind of quanto put pays more attention to the exchange rate risk than the foreign stock price risk. The pricing formula for Type IV quanto put is as follows:

Pt IV = St {− X t N (− d1IV ) + Ke

− ( r − r f + ρ S , X σ sσ X )(T − t )

N (− d 2IV )}

(2- 28)

In which,

IV 1

d

Xt σ X2 ln( ) + (r − rf + ρ S , X σ sσ X + )(T − t ) K 2 = σX T −t

(2- 29)

(2- 30)

d 2IV = d1IV − σ X T − t

2.5 Multiple-Reset Options At the beginning of this section, we illustrate the characteristic of the reset put and use the single-reset put as an example to show how the exercise price reset. Then, we’ll show the resetting mechanism of the multiple-reset put. The reset put exist the basic characteristics of the general put, that is, if the underlying stock price lower than the exercise price at the maturity, the put is in-the money and its final payoff will be the difference of the exercise price and the underlying stock price. However, at the reset point, if the underlying stock price is out-of-the-mmoney, the new exercise price will be resetted to the current underlying stock price. Let’s use the single-reset put for an example, the formula to represent the resetting process can be written as follows:

10

 K , K > S (t * ) = Max( K , S (t * )) K (t ) =  * *  S (t ) , K < S (t ) *

(2- 31)

In which, K represents the exercise price decided initially( t = 0 ), t * represents the reset point(the time exercise price to be resetted) and T represents the maturity date of the reset put.

Figure 2- 1 The path of the stock price and the single-reset point

Let’s take Figure 2- 1 for example. The stock price of Path 1 at the reset point *

t is higher than the exercise price K , so its exercise price will be resetted. The exercise price at maturity will be S (t * ) and no longer be resetted again. On the other hand, the stock price of Path 2 at the reset point t * is lower than the exercise price K , so its exercise price will not be resetted. Its exercise price is still K at maturity. Let’s go on to illustrate how the exercise price being resetted in the multiple-reset case. We assume that the exercise price will be resetted at n reset points, {t1 , t2 , t3 ,..., tn } . We can rewrite the exercise price at time ti , K (ti ) , as follows:

11

 K (t ) , K (ti −1 ) > S (ti ) K (ti ) =  i −1  S (ti ) , K (ti −1 ) < S (ti ) = Max( K (ti −1 ), S (ti )) = Max( S (ti ), S (ti −1 ), K (ti − 2 )) = ... = Max( S (ti ), S (ti −1 ),..., S (t1 ), K )

(2- 32)

∴ K (tn ) = Max( S (tn ), S (tn −1 ),..., S (t1 ), K )

(2- 33)

We use the 3 reset points as an example, and formula of the exercise price can be rewritten as follows:

K (t3 ) = Max( S (t3 ), S (t2 ), S (t1 ), K )

(2- 34)

Let’s use Figure 2- 2 for example. We can see obviously that at time t1 , the stock price S (t1 ) is lower than the current exercise price K , so it will not be resetted at time t1 , which means the exercise price is still K . At time t2 , we can see that the stock price S (t2 ) is higher than the previous exercise price K , so it will be resetted at that time, which means the exercise price at time t2 is S (t2 ) . Finally, at the last reset point t3 , we also can see obviously that the stock price is higher than the previous exercise price S (t2 ) , so it will be resetted again and the final exercise price will equal to S (t3 ) . The K (t ) in Figure 2- 2 is no other than the path of the resetted exercise price.

Figure 2- 2 The path of the stock price and the three reset points

Since we have known the process of the reset of the exercise price, at maturity, the final payoff of the multiple-reset put can be represented as follows: 12

RPT = Max( K (tn ) − ST , 0), T = tn +1

(2- 35)

The pricing formula of the multiple-reset put is more complicated than the single one just because it implies the multi-variate cumulative normal distribution8. The pricing formula is as follows:

 e − qti Ni (d kA2 , d1A2 ,..., diA−21 ; ∑ A )  ⋅  2    n   − r (T −ti ) A1 A1 RP0 = S0 ∑  e ⋅ N n −i +1 (−di +1 ,..., −d n +1 ; ∑ A1 )   i =1    A3 A3 − N ( − d ,..., − d ; ∑ )     n −i +1 i +1 n +1 A3  Ke− rT ⋅ N n +1 (−d1A4 ,..., −d nA+41 ; ∑ A4 )  +  A5 A5 − qT − S e N ( − d ,..., − d ∑ ; )  1 n +1 n +1 A5   0

(2- 36)

2.6 The Maximum Distribution of Geometric Brownian Motion9 We assume a stochastic process as follows:

dYt = µ dt + σ dWt

(2- 37)

And then, the joint cdf of Yt and M Y ,t ( = MaxYu ) is as follows: 0≤ u ≤ t

FY , M (b, c) = Pr(Yt < b, M Y ,t < c) = N(

2 cµ b − µt b − 2c − µ t 2 ) − e σ N( ) σ t σ t

(2- 38)

From Eq(2- 38), we can obtain the cdf of the maximum M Y ,t as follows:

8

The obvious derivation and definition of the parameters can be referred to Cheng, W.-Y. and S.

Zhang (2000). "The analytics of reset options." Journal of Derivatives.. 9

The obvious derivation can be referred to S.N. Chen(2005), Financial Engineering, 2ed. 13

FM (c) = Pr( M Y ,t < c) = N(

2 cµ c − µt −c − µ t 2 ) − e σ N( ), c ≥ 0 σ t σ t

(2- 39)

And based on Eq(2- 39), we can derive the pdf of the maximum M Y ,t as follows:

∂ FM (c) ∂c 2 cµ ∂  c − µt −c − µ t  2 = N( ) − e σ N( ) ∂c  σ t σ t 

f M (c) =

2 cµ

2 cµ

2µ −c − µ t 1 σ 2 −c − µt c − µt 2 = n( e n( ) − ( 2 )e σ N ( )+ ) σ σ t σ t σ t σ t σ t 1

(2- 40)

Under the risk-neutral probability measure Q, the dynamics of the stock price can be represented as follows:

dSt = rdt + σ dWt Q St

(2- 41)

Let Yt = ln St , and based on Ito’s lemma, we can obtain: σ2

dYt = d ln St = (r − ) dt + σ dWt Q 23 1 424

(2- 42)

µ

Let ST = Max Su , and then, from Eq(2- 38) and (2- 42), we can obtain the joint 0 ≤ u ≤T

cumulative distribution of the stock price and its maximum under the risk-neutral measure Q as follows:

B 2µ 2 P Q ( ST < K , ST < B) = N (− d 2 ) − ( ) σ N (d 3 ) S In which,

14

(2- 43)

S ln( ) + µT σ2 K ,µ = r− d2 = 2 σ T ln( d3 =

(2- 44)

KS ) − µT B2 σ T

(2- 45)

According to the Girsanolv Theorem described in 2.3, we can obtain:

d Y t = d ln S t = ( r −

σ

σ

2

2

dW tQ {

)dt + σ

= dWtR + σ dt

(2- 46)

2

= (r + ) dt + σ dWt 1 42 24 3

R

µ*

Q

E [ ST ⋅ I S

{

T < K ,ST < B

Q

}

] = E [ S0e rT

(r −

Q

σ2 2

)T +σ WTQ

⋅I S

{

σ2

T < K , ST < B

}

]

T +σ WTQ

2 = S0 e ⋅ E [ e14 24 3 ⋅ I{ST