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We derive a Put Option price associated with selling strategy of the underlying security in a random interest rate environment. This extends Put Option pricing under linear investment strategy from the Black-Scholes setting to Hull-White stochastic interest rate model. As an application, Call Option price for the linear investment strategy in the Hull-White model is established. Our results address recent emergence of developing dynamic investment strategies for the purpose of reducing the investor risk exposure associated with European-type options.

A steady growth of financial derivatives market over the past decades led to various generalizations of the classical Black-Scholes model. Notably, a pioneering work of Wang et al. ( [

The gist of considering dynamic investment strategies, such as presented here, is two-fold. Firstly, unlike in the classical Black-Scholes model where the investor buys options and has no position in the underlying stock throughout the option time horizon, the dynamic investment strategy requires the investor to continuously trade the stock, whereby lowering the investor risk which is manifested by the lower option price. Secondly, the interest rates are no longer constant and are assumed stochastic.

This paper is concerned with Put Option hedging by linear investment strategy under the Hull-White stochastic interest rates model. European Put Option with the linear investment strategy triggers stock selling whenever the stock price falls below the strike price and stays in the range [ ( 1 − α ) K , K ] . Following [

Q ( S ) = { β S ≤ ( 1 − α ) K ( 1 − β ) α K [ S − ( 1 − α ) K ] + β ( 1 − α ) K ≤ S ≤ K 1 S ≥ K (1.1)

where

S is stock price.

Q ( S ) is the stock investment proportion, which is equal to the value of the stock investment divided by A, where A is the entire investment amount.

K is strike price of the option.

α is the investment strategy index, indicating the stock investment occurs during the period in which the stock price drops from K to ( 1 − α ) K .

β is the minimum value of the stock investment proportion.

It was found in [_{T} based on the linear investment with parameters α , β , strike price K reads as follows:

V T = { 0 S T ≥ K 1 − β α ( 2 α − 1 2 K + ( 1 − α ) S T − S T 2 2 K ) ( 1 − α ) K ≤ S T ≤ K ( K − β S T ) − ( 1 − β ) K ( 2 − α ) 2 S T ≤ ( 1 − α ) K (1.2)

where S_{T} is the terminal stock price.

We will use the above formula for the stock price that satisfies a stochastic differential equation (SDE) with drift depending on the random interest rate, whose SDE follows the Hull-White model.

The evolution of the stock price S t satisfies the following SDE

d S t = μ t S t d t + σ 1 S t d W 1 , t (2.1)

with mean return μ t , constant volatility σ 1 and a standard Brownian motion W 1 , t . The stock price dynamic under the risk-neutral measure is then as follows

d S t = r t S t d t + σ 1 S t d W 1 , t (2.2)

where r t is the interest rate.

By Ito formula the stock price at time T can be expressed as

S T = S 0 e ∫ 0 T ( r s − σ 1 2 2 ) d s + ∫ 0 T σ 1 d W 1 , t (2.3)

where S_{0} is the initial stock price.

Wang et al. [

d r t = ( θ ( t ) − a r t ) d t + σ 2 d W 2 , t (2.4)

with a and σ 2 constants and W 2 , t standard Brownian motion independent from W 1 , t .

Remark 2.1. Special case of Hull-White model, θ ( t ) = a b , becomes the Vasicek model.

In general, Calin [

θ ( t ) = ∂ t f ( 0 , t ) + a f ( 0 , t ) + σ 2 2 a ( 1 − e − 2 a t ) (2.5)

where f ( 0 , t ) is the yield curve determined by the bond price.

The solution to (2.4) reads

r t = r 0 e − a t + e − a t ∫ 0 t θ ( s ) e a s d s + σ 2 e − a t ∫ 0 t e a s d W 2 , s . (2.6)

Note that the first two terms are deterministic and the last is a Wiener integral, thus the process r t is normally distributed, with mean and variance

E [ r t ] = r 0 e − a t + e − a t ∫ 0 t θ ( s ) e a s d s V a r [ r t ] = σ 2 2 2 a ( 1 − e − 2 a t ) (2.7)

Integrating (2.6) yields

∫ 0 t r s d s = r 0 ( 1 − e − a t ) a + ∫ 0 t e − a s ∫ 0 s θ ( u ) e a u d u d s + σ 2 ∫ 0 t e − a s ∫ 0 s e a u d W 2 , u (2.8)

when the interest rates are stochastic, the bond price is calculated by conditional expectation

P ( t , T ) = E [ e − ∫ t T r s d s | F t ] (2.9)

where F t denotes the information available in the market at time t.

Lemma 2.1. The Hull-White zero-coupon bond price is as follows

P ( 0 , T ) = e r 0 ( e − a T − 1 ) a − ∫ 0 T e − a s ∫ 0 s θ ( u ) e a u d u d s + σ 2 2 2 a 2 [ T + 1 − e − 2 a T 2 a − 2 a ( 1 − e − a T ) ] (2.10)

Proof: By (2.9)

P ( 0 , T ) = E [ e − ∫ 0 T r s d s | F 0 ] = e r 0 ( 1 − e − a T ) a e − ∫ 0 T e − a s ∫ 0 s θ ( u ) e a u d u d s E [ e − σ 2 ∫ 0 T e − a s ∫ 0 s e a u d W 2 , u d s | F 0 ] (2.11)

The proof will be carried out in several steps.

Step 1: Set X T = ∫ 0 T e − a s ∫ 0 s e a u d W 2 , u d s , then

E [ X T ] = ∫ o T e − a s E [ ∫ 0 s e a u d W 2 , u ] d s = 0 (2.12)

since ∫ 0 s e a u d W 2 , u is gaussian with mean 0 and variance e 2 a s − 1 2 a .

Step 2:

By product rule

d ( X s ∫ 0 s e a u d W 2 , u ) = ∫ 0 s e a u d W 2 , u d X s + X s d ∫ 0 s e a u d W 2 , u + d X s d ∫ 0 s e a u d W 2 , u ︸ 0 = e − a s ( ∫ 0 s e a u d W 2 , u ) 2 d s + X s e a s d W 2 , s (2.13)

Integrating the above gives

X T ∫ 0 T e a s d W 2 , s = ∫ 0 T e − a s ( ∫ 0 s e a u d W 2 , u ) 2 d s + ∫ 0 T X s e a s d W 2 , s (2.14)

By taking the expectation and using the fact that the Wiener integral has zero mean, we obtain

E [ X T ∫ 0 T e a s d W 2 , s ] = ∫ 0 T e − a s E [ ( ∫ 0 T e a s d W 2 , s ) 2 ] d s = ∫ 0 T e − a s ( e 2 a s − 1 2 a ) d s = 1 a 2 ( e a T + e − a T 2 − 1 ) (2.15)

Step 3: Applying Ito lemma:

d ( X T 2 ) = 2 X T d X T + ( d X T ) 2 = 2 X T e − a T ∫ 0 T e a s d W 2 , s d t (2.16)

then integrating and applying step 2, yields

E [ X T 2 ] = 2 ∫ 0 T e − a s E [ X s ∫ 0 s e a u d W 2 , u ] d s = 2 a 2 ∫ 0 T ( 1 + e − 2 a s 2 − e − a s ) d s = 1 a 2 [ T + 1 2 a ( 1 − e − 2 a T ) + 2 a ( e − a T − 1 ) ] (2.17)

Step 4: Using a stochastic variant of Fubini’s theorem we interchange the Riemann and the Wiener integrals as follows

X T = ∫ 0 T e − a s ∫ 0 s e a u d W 2 , u d s = ∫ 0 s e a u ∫ 0 T e − a s d s d W 2 , u = ∫ 0 T e − a s d s ∫ 0 s e a u d W 2 , u = 1 a ( 1 − e − a T ) ∫ 0 s e a u d W 2 , u (2.18)

which implies that X_{T} is normally distributed with mean 0 and variance E [ X T 2 ] computed in step 3. Therefore

E [ e − σ 2 ∫ 0 T e − a s ∫ 0 s e a u d W 2 , u d s ] = E [ e σ 2 X T ] = e ( σ 2 2 2 ) V a r ( X T ) = e σ 2 2 2 a 2 [ T + 1 − e − 2 a T 2 a − 2 a ( 1 − e − a T ) ] (2.19)

which gives rise to the formula (zero coupon bond price).

The stochastic model for the bond price under Hull-White model is as follows [

d P ( t , T ) = r t P ( t , T ) d t + ν ( t , T ) P ( t , T ) d W t ≡ r t P ( t , T ) d t − σ 2 a ( 1 − e − a ( T − t ) ) P ( t , T ) d W t (3.1)

In order to simplify the calculation of option value under the stochastic interest rate, we use the technique of changing the measure and numeraire. Following general considerations in Brigo and Mercurio [

Proposition 2.3.1. [

d B t = ( ... ) d t + σ t B d W t d P t = ( ... ) d t + σ t P d W t (3.2)

Then the drift of process X under numeraire P is

μ t P ( X t ) = μ t B ( X t ) − σ t ( X t ) ( σ t B B t − σ t P P t ) (3.3)

and

d W t P = d W t + ( σ t B B t − σ t P P t ) d t . (3.4)

Note. By the Proposition for money market account d B t = r t B t d t and zero-coupon bond d P ( t , T ) = r t P ( t , T ) d t − σ a ( 1 − e − a ( T − t ) ) P ( t , T ) d W 2 , t , r t for Hull-White model under T-forward measure Q T satisfies the following SDE

d r t = ( θ ( t ) − a r t − σ 2 2 a ( 1 − e − a ( T − t ) ) ) d t + σ d W t T (3.5)

where

d W t T = d W t + σ a ( 1 − e − a ( T − t ) ) d t . (3.6)

Now that we obtained the evolution of Hull-White under T-forward measure, we solve (3.5) via multiplying by the integrating factor e a t to get

d ( r t e a t ) = e a t θ ( t ) d t − e a t σ 2 2 a ( 1 − e − a ( T − t ) ) d t + e a t σ 2 d W 2 , t T . (3.7)

Integrating from 0 and t yields

r t = r 0 e − a t − σ 2 2 a [ ( e a t − 1 ) e − a T ( 2 e a T − e a t − 1 ) 2 a ] e − a t + e − a t ∫ 0 t θ ( s ) e a s d s + σ 2 ∫ 0 t e − a ( t − s ) d W 2 , s T (3.8)

By integrating over [0, T] we obtain

∫ 0 T r t d t = r 0 1 − e − a T a − σ 2 2 a [ e − 2 a T ( ( 2 a T − 3 ) e 2 a T + 4 e a T − 1 ) 2 a 2 ] + ∫ 0 T e − a t ∫ 0 t θ ( s ) e a s d s d t + σ 2 ∫ 0 T ∫ 0 t e − a ( t − s ) d W 2 , u T d t . (3.9)

The Put Option price is expressed as a product of the expectation of V_{T} under the T-forward measure and the price of zero-coupon bond. Notice that by (2.3) stock price S_{T} is lognormally distributed and we denote its probability density by f ( s ) for S T = s .

Theorem 4.1. (Put Option Price). The Put Option price at time 0 under the Hull-White interest rate is given by

P T = P ( 0 , T ) ( ( K − ( 1 − β ) K ( 2 − α ) 2 ) N [ d 1 ] − β e μ T + 1 2 σ T 2 N [ d 2 ] ) − P ( 0 , T ) ( 1 − β α 2 α − 1 2 K [ N ( d 3 ) − N ( d 1 ) ] ) − P ( 0 , T ) ( 1 − β α ( 1 − α ) e μ T + 1 2 σ T 2 [ N ( d 4 ) − N ( d 2 ) ] ) + P ( 0 , T ) ( β − 1 2 α K e μ T + 1 2 σ T 2 [ N ( d 4 ) − N ( d 2 ) ] ) (4.1)

where

P ( 0 , T ) = e r 0 ( e − a T − 1 ) a − ∫ 0 T e − a s ∫ 0 s θ ( u ) e a u d u d s + σ 2 2 2 a 2 [ T + 1 − e − 2 a T 2 a − 2 a ( 1 − e − a T ) ]

d 1 = ln ( 1 − α ) K − μ T σ T d 2 = ln ( 1 − α ) K − μ T − σ T 2 σ T d 3 = ln K − μ T σ T d 4 = ln K − μ T − σ T 2 σ T μ T = ln S 0 − σ 1 2 2 T + r 0 1 − e − a T a − σ 2 2 a [ e − 2 a T ( ( 2 a T − 3 ) e 2 a T + 4 e a T − 1 ) 2 a 2 ] + ∫ 0 T e − a t ∫ 0 t θ ( s ) e a s d s d t σ T 2 = σ 1 2 T + σ 2 a 2 [ T − 2 1 − e − a T a + 1 − e − 2 a T 2 a ]

Proof.

E T [ V T ] = ∫ 0 ( 1 − α ) K [ ( K − β S T ) − ( 1 − β ) K ( 2 − α ) 2 ] f ( S T ) d S T + ∫ ( 1 − α ) K K 1 − β α [ 2 α − 1 2 K + ( 1 − α ) S T − S T 2 2 K ] f ( S T ) d S T = ∫ 0 ( 1 − α ) K [ ( K − β S T ) − ( 1 − β ) K ( 2 − α ) 2 ] f ( S T ) d S T + ∫ ( 1 − α ) K K 1 − β α [ 2 α − 1 2 K + ( 1 − α ) S T ] f ( S T ) d S T − ∫ ( 1 − α ) K K [ 1 − β α S T 2 2 K ] f ( S T ) d S T (4.2)

We split evaluating E T [ V T ] into integrals I 1 , I 2 and I 3 as follows

I 1 = ∫ 0 ( 1 − α ) K [ ( K − β S T ) − ( 1 − β ) K ( 2 − α ) 2 ] f ( S T ) d S T I 2 = ∫ ( 1 − α ) K K 1 − β α [ 2 α − 1 2 K + ( 1 − α ) S T ] f ( S T ) d S T I 3 = − ∫ ( 1 − α ) K K [ 1 − β α S T 2 2 K ] f ( S T ) d S T (4.3)

Set y = ln S T , then f ( e y ) e y is the probability density function of ln S T , and the mean and variance under the T-forward measure can be expressed form (2.3) as follows

μ T = E T [ ln S T ] = E T [ ln S 0 + ∫ 0 T ( r t − σ 1 2 2 ) d t + ∫ 0 T σ 1 d W 1 , t T ] = ln S 0 − σ 1 2 2 T + E T [ ∫ 0 T r t d t ] + E T [ ∫ 0 T σ 1 d W 1 , t T ] ︸ 0 = ln S 0 − σ 1 2 2 T + r 0 1 − e − a T a − σ 2 2 a [ e − 2 a T ( ( 2 a T − 3 ) e 2 a T + 4 e a T − 1 ) 2 a 2 ] + ∫ 0 T e − a t ∫ 0 t θ ( s ) e a s d s d t (4.4)

σ T 2 = V a r T [ ln S T ] = V a r T [ ln S 0 + ∫ 0 T ( r t − σ 1 2 2 ) d t + ∫ 0 T σ 1 d W 1 , t T ] = V a r T [ ln S 0 ] + V a r T [ ∫ 0 T ( r t − σ 1 2 2 ) d t ] + V a r T [ ∫ 0 T σ 1 d W 1 , t T ] = V a r T [ ∫ 0 T σ 1 d W 1 , t T ] + V a r T [ ∫ 0 T σ 2 e − a t ∫ 0 t e a s d W 2 , s T d t ] = ∫ 0 T σ 1 2 d t + V a r T [ σ 2 ∫ 0 T ∫ 0 t e − a ( t − s ) d W 2 , s T d t ] = σ 1 2 T + σ 2 a 2 [ T − 2 1 − e − a T a + 1 − e − 2 a T 2 a ] . (4.5)

Moreover, we have

f ( e y ) e y = 1 2 π σ T e − 1 2 ( y − μ T ) 2 σ T 2 (4.6)

and therefore

I 1 = ∫ 0 ( 1 − α ) K ( ( K − β S T ) − ( 1 − β ) K ( 2 − α ) 2 ) f ( S T ) d S T = ∫ 0 ( 1 − α ) K ( K − ( 1 − β ) K ( 2 − α ) 2 ) f ( S T ) d S T − β ∫ 0 ( 1 − α ) K S T f ( S T ) d S T = ∫ − ∞ ln ( 1 − α ) K ( K − ( 1 − β ) K ( 2 − α ) 2 ) f ( e y ) e y d y − β ∫ − ∞ ln ( 1 − α ) e y f ( e y ) e y d y = ( K − ( 1 − β ) K ( 2 − α ) 2 ) 1 2 π σ T ∫ − ∞ ln ( 1 − α ) K e − 1 2 ( y − μ T ) 2 σ T 2 d y − β 1 2 π σ T ∫ − ∞ ln ( 1 − α ) K e y e − 1 2 ( y − μ T ) 2 σ T 2 d y

Setting z = y − μ T σ T gives

I 1 = ( K − ( 1 − β ) K ( 2 − α ) 2 ) 1 2 π ∫ − ∞ ln ( 1 − α ) K − μ T σ T e − 1 2 z 2 d z − β 1 2 π ∫ − ∞ ln ( 1 − α ) K − μ T σ T e μ T + z σ T e − 1 2 z 2 d z = ( K − ( 1 − β ) K ( 2 − α ) 2 ) 1 2 π ∫ − ∞ ln ( 1 − α ) K − μ T σ T e − 1 2 z 2 d z − β 1 2 π ∫ − ∞ ln ( 1 − α ) K − μ T σ T e μ T + 1 2 σ T 2 e − 1 2 ( z − σ T 2 ) d z = ( K − ( 1 − β ) K ( 2 − α ) 2 ) N [ ln ( 1 − α ) K − μ T σ T ] − β e μ T + 1 2 σ T 2 N [ ln ( 1 − α ) K − μ T − σ T 2 σ T ] (4.7)

where N ( x ) is the standard normal cumulative distribution function.

Similarly,

I 2 = ∫ ( 1 − α ) K K 1 − β α ( 2 α − 1 2 K + ( 1 − α ) S T ) f ( S T ) d S T = ∫ ( 1 − α ) K K ( 1 − β α 2 α − 1 2 K ) f ( S T ) d S T + 1 − β α ( 1 − α ) ∫ ( 1 − α ) K K S T f ( S T ) d S T = 1 − β α 2 α − 1 2 K ∫ ln ( 1 − α ) K ln K f ( e y ) e y d y + 1 − β α ( 1 − α ) ∫ ln ( 1 − α ) K ln K e y f ( e y ) e y d y = 1 − β α 2 α − 1 2 K 1 2 π σ T ∫ ln ( 1 − α ) K ln K e − 1 2 ( y − μ T ) 2 σ T 2 d y + 1 − β α ( 1 − α ) 1 2 π σ T ∫ ln ( 1 − α ) K ln K e y e − 1 2 ( y − μ T ) 2 σ T 2 d y

= 1 − β α 2 α − 1 2 K 1 2 π ∫ ln ( 1 − α ) K − μ T σ T ln K − μ T σ T e − 1 2 z 2 d z + 1 − β α ( 1 − α ) 1 2 π ∫ ln ( 1 − α ) K − μ T σ T ln K − μ T σ T e μ T + z σ T e − 1 2 z 2 d z = 1 − β α 2 α − 1 2 K [ N ( ln K − μ T σ T ) − N ( ln ( 1 − α ) K − μ T σ T ) ] + 1 − β α ( 1 − α ) e μ T + 1 2 σ T 2 [ N ( ln K − μ T − σ T 2 σ T ) − N ( ln ( 1 − α ) K − μ T − σ T 2 σ T ) ] (4.8)

and

I 3 = − ∫ ( 1 − α ) K K [ 1 − β α S T 2 2 K ] f ( S T ) d S T = β − 1 2 α K ∫ ( 1 − α ) K K S T 2 f ( S T ) d S T = β − 1 2 α K ∫ ln ( 1 − α ) K ln K e 2 y f ( e y ) d y = β − 1 2 α K ∫ ln ( 1 − α ) K ln K e y e y f ( e y ) d y = β − 1 2 α K 1 2 π σ T ∫ ln ( 1 − α ) K ln K e y e − 1 2 ( y − μ T 2 ) σ T 2 d y = β − 1 2 α K 1 2 π ∫ ln ( 1 − α ) K − μ T σ T ln K − μ T σ T e μ T + z σ T e − 1 2 z 2 d z = β − 1 2 α K e μ T + 1 2 σ T 2 [ N ( ln K − μ T − σ T 2 σ T ) − N ( ln ( 1 − α ) K − μ T − σ T 2 σ T ) ] (4.9)

Corollary. 4.1. (Put Option price for Vasicek Model). Observe that in the special case we recover the Put Option price for the Vasicek model when θ ( t ) = a b .

European Call Option under the linear investment strategy triggers stock buying whenever the stock price exceeds the strike price. The investment fraction is defined by:

Q ( S ) = { 0 S ≤ K β α K ( S − K ) K ≤ S ≤ ( 1 + α ) K β S ≥ ( 1 + α ) K

where

S is stock price.

Q ( S ) is the stock investment proportion, which is equal to the value of the stock investment divided by A, where A is the entire investment amount.

K is strike price of the option.

α is the investment strategy index, indicating the stock investment occurs during the period in which the stock price increases fromK to ( 1 + α ) K .

β is the maximum value of the stock investment proportion.

Zhang et al. [

Theorem. 5.1. The Call Option price with the linear investment strategy at time 0 for the Hull-White model is given by

C T = P ( 0 , T ) ( 1 + β α ( 1 − μ T + ln K − σ T 2 ) ) e μ T + 1 2 σ T 2 [ N ( d 1 ) − N ( d 2 ) ] − P ( 0 , T ) ( 1 + β α ) K [ N ( d 3 ) − N ( d 4 ) ] − P ( 0 , T ) β α σ T 2 π e μ T + 1 2 σ T 2 ( e − d 2 2 2 − e − d 1 2 2 ) + P ( 0 , T ) ( 1 − β α ln ( 1 + α ) ) e μ T + 1 2 σ T 2 N ( − d 1 ) + P ( 0 , T ) K ( β − 1 ) N ( − d 3 )

with P ( 0 , T ) , d 1 , d 2 , d 3 , d 4 , μ T and σ T 2 defined below

P ( 0 , T ) = e r 0 ( e − a T − 1 ) a − ∫ 0 T e − a s ∫ 0 s θ ( u ) e a u d u d s + σ 2 2 2 a 2 [ T + 1 − e − 2 a T 2 a − 2 a ( 1 − e − a T ) ] d 1 = ln ( 1 + α ) K − μ T − σ T 2 σ T d 2 = ln K − μ T − σ T 2 σ T d 3 = ln ( 1 + α ) K − μ T σ T d 4 = ln K − μ T σ T μ T = ln S 0 − σ 1 2 2 T + r 0 1 − e − a T a − σ 2 2 a [ e − 2 a T ( ( 2 a T − 3 ) e 2 a T + 4 e a T − 1 ) 2 a 2 ] + ∫ 0 T e − a t ∫ 0 t θ ( s ) e a s d s d t σ T 2 = σ 1 2 T + σ 2 a 2 [ T − 2 1 − e − a T a + 1 − e − 2 a T 2 a ]

Proof. The formula for C_{T} has been derived in Zhang et al. [_{T} in Hull-White model is analogous to that of Vasicek model we omit the proof of the formula C_{T}.

Corollary. 5.1. (Call Option price for Vasicek Model). Observe that in the special case we recover the Call Option price for the Vasicek model when θ ( t ) = a b .

We obtained the closed form of the Put and Call Option price for the linear investment strategy under the Hull-White stochastic interest rates. In particular, a protective put option can serve as an insurance policy against losses for the stock holder. Since the option price associated with trading of the underlying security is based on continuous stock trading (impossible to implement!), a feasible discrete variant is in order. Recently Li et al. [

The authors declare no conflicts of interest regarding the publication of this paper.

Korzeniowski, A. and Ghorbani, N. (2021) Put Options with Linear Investment for Hull-White Interest Rates. Journal of Mathematical Finance, 11, 152-162. https://doi.org/10.4236/jmf.2021.111007