We study the Option pricing with linear investment strategy based on discrete time trading of the underlying security, which unlike the existing continuous trading models, provides a feasible real market implementation. Closed form formulas for Call and Put Option price are established for fixed interest rates and their extensions to stochastic Vasicek and Hull-White interest rates.
Financial derivatives market over the past few decades led to various generalizations of the classical Black-Scholes model. Notably, a seminal work of Wang et al. ( [
A key assumption in the above modeling relies on the fact that the stock trading by the option holder is done in continuous time. To remedy this drawback and accommodate the actual market implementation, Meng Li et al. [
Novelty of dynamic investment strategies, and their discrete time market implementation presented here, is two-fold. Firstly, unlike in the classical Black-Scholes models 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 trade the stock at discrete times associated with the stock price crossing certain levels in the pre-determined price range, whereby lowering the investor’s risk, manifested by the lower option price. Secondly, the interest rates are no longer constant and are assumed stochastic.
In this paper the stock price follows the Geometric Brownian Motion (GBM). We first extend the discrete investment strategy for Call Option [
The remainder of the paper is organized as follows. Section 2 describes the Call Option assumptions and Value function in a discrete dynamic model. Option Value under both fixed interest rate and stochastic interest rate for Vasicek and Hull-White models has been derived. In Section 3, the Put Option assumptions, Value function and price under fixed and stochastic interest rate models are presented. The last section concludes the paper.
The adaptive risk hedging for European Call Option is proposed based on the following assumptions [
· The Call Option holder holds one Option contract and the initial Capital of amount A = Q × K at the beginning of the Option period, where Q is the number of shares of stock for Option contract and K is the Call Option strike price.
· The Call Option holder should buy the underlying stock according to the price changes subject to the discrete trading strategy.
· The potential loss sustained by the Option holder through adaptive trading is not the responsibility of the Option writer.
· There are no transaction costs for buying the underlying stock.
The Call Option contains a capital amount A initially. When the price of the underlying stock goes up to ( K + δ ) , δ ≥ 0 the Option holder spends β 0 A to buy stocks. The parameter β 0 is a constant between 0 and 1, called the initial capital utilization coefficient. The Option holder linearly adjusts the capital utilization to increase the holding while the price continues to increase and hits a series of equally spaced points S n where { S n : S n = K − n Δ , n = 1 , 2 , ⋯ , N } , with Δ a positive constant, the price distance for two consecutive trading actions and N the total number of trades within the capital valid period.
The strategy parameters α (investment index) and β (minimum capital utilization), both positive numbers, illustrate the maximum amount of capital tradable is β A when the price reaches ( 1 + α ) ( K + δ ) . The strategy assumes the Option holder will evenly ( β A N ) distribute the capital over each of the potential trades corresponding to S n , n = 1 , 2 , ⋯ , N by selling β A N S n shares of stock [
It was found in [
V T = { 0 S T < K S T − K K ≤ S T < S 1 S T − β S T K N ∑ n = 1 m 1 K + n Δ + ( β m N − 1 ) K S m ≤ S T < S m + 1 ≤ S N S T − β S T K N ∑ n = 1 N 1 K + n Δ + ( β − 1 ) K S N ≤ S T (2.1)
with the investment parameters α , β , strike price K and the terminal stock price S T where,
α is the investment strategy index, indicating the stock investment occurs during the period in which the stock price increases from K to ( 1 + α ) K .
β is the maximum value of the stock investment proportion.
Furthermore, based on (2.1), the Call Option pricing formula for fixed interest rate and the stock price following the geometric fractional Brownian Motion was presented in [
Assume the dynamic of the stock price S T under the risk-neutral measure follows
d S t = r S t d t + σ 1 S t d W 1 , t , S ( 0 ) = S 0 > 0 , 0 ≤ t ≤ T . (2.2)
By Ito formula, the stock price at the maturity time T can be obtained and reads
S T = S 0 e [ ( r − 1 2 σ 1 2 ) t + σ 1 W 1 , T ] . (2.3)
Based on the value function V T in (2.1), the Call Option price under risk-neutral measure can be evaluated as
C = e − r T E [ V T ] (2.4)
where,
E [ V T ] = ∫ K S 1 ( S T − K ) f ( S T ) d S T + ∫ S m S m + 1 [ ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) S T + ( β m N − 1 ) K ] f ( S T ) d S T + ∫ S N + ∞ [ ( 1 − β K N ∑ n = 1 N 1 K + n Δ ) S T + ( β − 1 ) K ] f ( S T ) d S T (2.5)
with
I 1 = ∫ K S 1 ( S T − K ) f ( S T ) d S T I 2 = ∫ S m S m + 1 [ ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) S T + ( β m N − 1 ) K ] f ( S T ) d S T I 3 = ∫ S N + ∞ [ ( 1 − β K N ∑ n = 1 N 1 K + n Δ ) S T + ( β − 1 ) K ] f ( S T ) d S T (2.6)
Calculating the three integrals I 1 , I 2 and I 3 with a similar method to the derivations in [
μ = E [ ln S T ] = E [ ln S 0 + ( r − 1 2 σ 1 2 ) T + σ 1 W 1 , T ] = ln S 0 + ( r − 1 2 σ 1 2 ) T (2.7)
σ 1 2 = V a r [ ln S 0 + ( r − 1 2 σ 1 2 ) T + σ 1 W 1 , T ] = σ 1 2 T (2.8)
gives the Call Option price.
I 1 = ∫ K S 1 ( S T − K ) f ( S T ) d S T = ∫ ln K ln S 1 ( e y − K ) f ( e y ) e y d y = ∫ ln K ln S 1 ( e y − K ) 1 2 π σ e − 1 2 ( y − μ ) 2 σ 2 d y = ∫ ln K ln S 1 1 2 π σ e y e − 1 2 ( y − μ ) 2 σ 2 d y − K ∫ ln K ln S 1 1 2 π σ e − 1 2 ( y − μ ) 2 σ 2 d y
= ∫ ln K − μ σ ln S 1 − μ σ 1 2 π e μ + σ z e − 1 2 z 2 d z − K ∫ ln K − μ σ ln S 1 − μ σ 1 2 π e − 1 2 z 2 d z = 1 2 π ∫ ln K − μ σ ln S 1 − μ σ e μ + 1 2 σ 2 e − 1 2 ( z − σ ) 2 d z − K 1 2 π ∫ ln K − μ σ ln S 1 − μ σ e − 1 2 z 2 d z = e μ + 1 2 σ 2 [ N ( ln S 1 − μ − σ 2 σ ) − N ( ln K − μ − σ 2 σ ) ] − K [ N ( ln S 1 − μ σ ) − N ( ln K − μ σ ) ] (2.9)
and
I 2 = ∫ S m S m + 1 [ ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) S T + ( β m N − 1 ) K ] f ( S T ) d S T = ∫ S m S m + 1 ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) S T f ( S T ) d S T + ∫ S m S m + 1 ( β m N − 1 ) K f ( S T ) d S T = ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) ∫ S m S m + 1 e y f ( e y ) e y d y + ( β m N − 1 ) K ∫ S m S m + 1 f ( e y ) e y d y = 1 2 π σ ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) ∫ S m S m + 1 e y e − 1 2 ( y − μ ) 2 σ 2 d y + 1 2 π σ ( β m N − 1 ) K ∫ S m S m + 1 e − 1 2 ( y − μ ) 2 σ 2 d y
= 1 2 π ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) ∫ S m − μ σ S m + 1 − μ σ e μ + z σ e − 1 2 z 2 d z + 1 2 π ( β m N − 1 ) K ∫ S m − μ σ S m + 1 − μ σ e − 1 2 z 2 d z = e μ + 1 2 σ 2 ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) [ N ( ln S m + 1 − μ − σ 2 σ ) − N ( ln S m − μ − σ 2 σ ) ] + ( β m N − 1 ) K [ N ( ln S m + 1 − μ σ ) − N ( ln S m − μ σ ) ] (2.10)
Thus,
I 3 = ∫ S N + ∞ [ ( 1 − β K N ∑ n = 1 N 1 K + n Δ ) S T + ( β − 1 ) K ] f ( S T ) d S T = ( 1 − β K N ∑ n = 1 N 1 K + n Δ ) ∫ ln S N + ∞ e y f ( e y ) e y d y + ( β − 1 ) K ∫ ln S N + ∞ f ( e y ) e y d y = ( 1 − β K N ∑ n = 1 N 1 K + n Δ ) 1 2 π σ ∫ ln S N + ∞ e y e − 1 2 ( y − μ ) 2 σ 2 d y + ( β − 1 ) K 1 2 π σ ∫ ln S N + ∞ e − 1 2 ( y − μ ) 2 σ 2 d y = e μ + 1 2 σ 2 ( 1 − β K N ∑ n = 1 N 1 K + n Δ ) [ N ( μ + σ 2 − ln S N σ ) ] + ( β − 1 ) K N ( μ − ln S N σ ) (2.11)
By combining (2.4) to (2.11), the Option Price under fixed interest rate reads
C = e − r T E [ V T ] = e − r T e μ + 1 2 σ 2 [ N ( d 1 ) − N ( d 2 ) ] − K e − r T [ N ( d 3 ) − N ( d 4 ) ] + e − r T e μ + 1 2 σ 2 ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) [ N ( d 5 ) − N ( d 6 ) ] + e − r T ( β m N − 1 ) [ N ( d 7 ) − N ( d 8 ) ] + e − r T e μ + 1 2 σ 2 ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) N ( d 9 ) + e − r T K ( β − 1 ) N ( d 10 ) (2.12)
where,
d 1 = ln S 1 − μ − σ 2 σ d 6 = ln S m − μ − σ 2 σ d 2 = ln K − μ − σ 2 σ d 7 = ln S m + 1 − μ σ d 3 = ln S 1 − μ − σ 2 σ d 8 = ln S m − μ σ d 4 = ln K − μ σ d 9 = μ + σ 2 − ln S N σ d 5 = ln S m + 1 − μ − σ 2 σ d 10 = μ − ln S N σ
and
μ = ln S 0 + ( μ − 1 2 σ 1 2 ) T σ 2 = σ 1 2 T .
Solution to the Vasicek interest rate model [
C = P ( 0 , T ) E T [ V T ] (2.13)
where
S T = S 0 e ∫ 0 T ( r s − 1 2 σ 1 2 ) d s + ∫ 0 T σ 1 d W 1 , t (2.14)
and P ( 0 , T ) is the price of the zero-coupon bond. The mean μ T and variance σ T 2 in the case of Vasicek model were obtained under the T-forward measure by Zhang [
C = P ( 0 , T ) e μ T + 1 2 σ T 2 [ N ( d 1 ) − N ( d 2 ) ] − K P ( 0 , T ) [ N ( d 3 ) − N ( d 4 ) ] + P ( 0 , T ) e μ T + 1 2 σ T 2 ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) [ N ( d 5 ) − N ( d 6 ) ] + P ( 0 , T ) ( β m N − 1 ) [ N ( d 7 ) − N ( d 8 ) ] + P ( 0 , T ) e μ T + 1 2 σ T 2 ( 1 − β K N ∑ n = 1 m 1 K + n Δ ) N ( d 9 ) + P ( 0 , T ) K ( β − 1 ) N ( d 10 ) (2.15)
where
d 1 = ln S 1 − μ T − σ T 2 σ T d 6 = ln S m − μ T − σ T 2 σ T d 2 = ln K − μ T − σ T 2 σ T d 7 = ln S m + 1 − μ T σ T
d 3 = ln S 1 − μ T − σ T 2 σ T d 8 = ln S m − μ T σ T d 4 = ln K − μ T σ T d 9 = μ T + σ 2 − ln S N σ T d 5 = ln S m + 1 − μ T − σ T 2 σ T d 10 = μ T − ln S N σ T
and
μ T = ln S 0 − σ 1 2 2 T − r 0 1 − e − a T a + ( θ a − σ 2 2 a 2 ) [ T − 1 − e − a T a ] + σ 2 2 2 a 1 − e − a T a σ T 2 = σ 2 2 a 2 [ T − 2 1 − e − a T a + 1 − e − 2 a T 2 a ] + σ 1 2 T
with the zero-coupon bond price
P ( 0 , T ) = exp { [ 1 − e − a T a − T ] ( θ a − σ 2 2 2 a 2 ) − σ 2 2 1 − e − a T a 4 a } e − r 0 1 − e − a T a . (2.16)
We derived the Option price under Hull-White interest rates model [
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 2 2 a 2 [ T − 2 1 − e − a T a + 1 − e − 2 a T 2 a ] + σ 1 2 T
and the zero-coupon bond price is
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 ) ] . (2.17)
The Put Option contains a specified capital amount A and holding of Q initially. When the stock price drops from K to ( K − δ ) , δ ≥ 0 the Option holder sells β 0 A proportion of stocks. Parameter β 0 is called the initial capital utilization coefficient which is a constant between 0 and 1. The Option holder linearly adjusts the capital utilization to decrease the holding if the price continues to fall until it reaches ( 1 − α ) ( K − δ ) and the total capital spending reaches β A with β < ⋯ < β 1 < β 0 . The Option holder will only sell when the stock price hits a series of equally spaced points S n , where { S n : S n = K − n Δ , n = 1 , 2 , ⋯ , N } , with Δ a positive constant, the price distance for two consecutive trading actions and N the total number of trades within the capital valid period. The strategy parameters α (investment index) and β (minimum capital utilization), both positive numbers, illustrate the maximum amount of capital tradable is β 0 A when the price reaches ( 1 − α ) ( K − δ ) . The strategy assumes the Option holder will evenly ( β A N ) distribute the capital over each of the potential trades corresponding to S n , n = 1 , 2 , ⋯ , N by selling β A N S n shares of stock .
In the classical European Options, the Option writer will sustain a loss of the amount L as the stock price falls below K
L = Q ( K − S ) = A K ( K − S ) . (3.1)
To adaptive hedging Option, the investor is required to sell a proportion of underlying stock in order to hedge the risk. In a discrete position strategy, the Option holder’s income R based on such transactions throughout the Option valid period can be calculated as,
R = β A N S n ( S n − S ) (3.2)
when the stock price falls from S n to S, with S N ≤ S m + 1 < S < S n .
The Option holder makes a cumulative income R ( S ) as
R ( S ) = ∑ n = 1 m β A N S n ( S n − S ) (3.3)
Thus, the total loss taken by the Option writer can be obtained as
L ( S ) = A K ( K − S ) − ∑ n = 1 m β A N S n ( S n − S ) = A K ( K − S ) − β A N ∑ n = 1 m ( S n − S ) S n = A K ( K − S ) − β A N ∑ n = 1 m S n S n + β A S N ∑ n = 1 m 1 S n = A − A S K − β A m N + β A S N ∑ n = 1 m 1 K − n Δ = ( 1 − β m N ) A − A S K + β A S N ∑ n = 1 m 1 K − n Δ (3.4)
however, for S ≤ S N , the Option writer’s loss L is
L ( S ) = ( 1 − β ) A − A S K + β A S N ∑ n = 1 m 1 K − n Δ . (3.5)
Therefore, the following function represents the Option writer’s Loss based on the stock price S
L ( S ) = { ( 1 − β ) A − A S K + β A S N ∑ n = 1 m 1 K − n Δ S ≤ S N ( 1 − β m N ) A − A S K + β A S N ∑ n = 1 m 1 K − n Δ S N ≤ S m + 1 < S ≤ S m A K ( K − S ) S 1 < S ≤ K 0 S > K (3.6)
Consequently, the intrinsic value function V ( S ) is derived by dividing the loss function by the number of shares of stock within the Option depending on the stock finishing price S T
V ( S T ) = { ( 1 − β ) K − S T + β S T K N ∑ n = 1 m 1 K − n Δ S T ≤ S N ( 1 − β m N ) K − S T + β S T K N ∑ n = 1 m 1 K − n Δ S N ≤ S m + 1 < S T ≤ S m K − S T S 1 < S T ≤ K 0 S T > K (3.7)
Considering the assumptions of section (3.1) and the intrinsic value function V ( S T ) (3.7), the Put Option price under risk-neutral measure is evaluated as
P = e − r T E [ V ( S T ) ]
where,
E [ V ( S T ) ] = ∫ 0 S N [ ( 1 − β ) K − S T + β S T K N ∑ n = 1 m 1 K − n Δ ] f ( S T ) d S T + ∫ S m + 1 S m [ ( 1 − β m N ) K − S T + β S T K N ∑ n = 1 m 1 K − n Δ ] f ( S T ) d S T + ∫ S 1 K ( K − S T ) f ( S T ) d S T (3.8)
calculating the three integrals I 1 , I 2 and I 3 similar to the previous sections and using the probability density function of ln S T , whose mean and variance has been obtained in (2.7) and (2.8) gives the Put Option price.
I 1 = ∫ 0 S N [ ( 1 − β ) K − S T + β S T K N ∑ n = 1 m 1 K − n Δ ] f ( S T ) d S T = ∫ 0 S N ( 1 − β ) K f ( S T ) d S T − ∫ 0 S N ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) S T f ( S T ) d S T = ( 1 − β ) K ∫ − ∞ ln S N f ( e y ) e y d y − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) ∫ − ∞ ln S N e y f ( e y ) e y d y = ( 1 − β ) K 1 2 π σ ∫ − ∞ ln S N e − 1 2 ( y − μ ) 2 σ 2 d y − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) 1 2 π σ ∫ − ∞ ln S N e y e − 1 2 ( y − μ ) 2 σ 2 d y
= ( 1 − β ) K 1 2 π ∫ − ∞ ln S N − μ σ e − 1 2 z 2 d z − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) 1 2 π ∫ − ∞ ln S N − μ σ e μ + z σ e − 1 2 z 2 d z = ( 1 − β ) K 1 2 π ∫ − ∞ ln S N − μ σ e − 1 2 z 2 d z − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) 1 2 π ∫ − ∞ ln S N − μ σ e μ + z σ e − 1 2 ( z − σ ) 2 d z = ( 1 − β ) K N ( ln S N − μ σ ) − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) N ( ln S N − μ − σ 2 σ ) (3.9)
and
I 2 = ∫ S m + 1 S m [ ( 1 − β m N ) K − S T + β S T K N ∑ n = 1 m 1 K − n Δ ] f ( S T ) d S T = ( 1 − β m N ) K ∫ S m + 1 S m f ( e y ) e y d y − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) ∫ S m + 1 S m e y f ( e y ) e y d y = ( 1 − β m N ) K 1 2 π σ ∫ S m + 1 S m e − 1 2 ( y − μ ) 2 σ 2 d y − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) 1 2 π σ ∫ S m + 1 S m e y e − 1 2 ( y − μ ) 2 σ 2 d y
= ( 1 − β m N ) K 1 2 π ∫ S m + 1 − μ σ S m − μ σ e − 1 2 z 2 d z − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) 1 2 π ∫ S m + 1 − μ σ S m − μ σ e μ + 1 2 σ 2 e − 1 2 ( z − σ ) 2 d z = ( 1 − β m N ) K [ N ( ln S m − μ σ ) − N ( ln S m + 1 − μ σ ) ] − ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) [ N ( ln S m − μ − σ 2 σ ) − N ( ln S m + 1 − μ − σ 2 σ ) ] (3.10)
Similarly,
I 3 = ∫ S 1 K ( K − S T ) f ( S T ) d S T = K ∫ ln S 1 ln K f ( e y ) e y d y − ∫ ln S 1 ln K e y f ( e y ) e y d y = K 1 2 π σ ∫ ln S 1 ln K e ( y − μ ) 2 σ 2 d y − 1 2 π σ ∫ ln S 1 ln K e y e ( y − μ ) 2 σ 2 d y = K [ N ( ln K − μ σ ) − N ( ln S 1 − μ σ ) ] − e μ + 1 2 σ 2 [ N ( ln K − μ − σ 2 σ ) − N ( ln S 1 − μ − σ 2 σ ) ] (3.11)
The price of Put Option P with the underlying stock price dynamic and interest rate following the Vasicek model reads
P = P ( 0 , T ) E T [ V T ]
where P ( 0 , T ) is the zero-coupon bond price (2.16) with μ T , σ T 2 the mean and variance under T-forward measure. Thus,
P = P ( 0 , T ) E T [ V T ] = P ( 0 , T ) K ( 1 − β ) N ( d 1 ) − P ( 0 , T ) ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) N ( d 2 ) + P ( 0 , T ) ( 1 − β m N ) K [ N ( d 3 ) − N ( d 4 ) ] − P ( 0 , T ) ( 1 − β K N ∑ n = 1 m 1 K − n Δ ) [ N ( d 5 ) − N ( d 6 ) ] + P ( 0 , T ) [ N ( d 7 ) − N ( d 8 ) ] − P ( 0 , T ) e μ + 1 2 σ 2 [ N ( d 9 ) − N ( d 10 ) ] (3.12)
where,
d 1 = ln S N − μ T σ T d 6 = ln S m + 1 − μ T − σ T 2 σ T d 2 = ln S N − μ T − σ T 2 σ T d 7 = ln K − μ T σ T d 3 = ln S m − μ T σ T d 8 = ln S 1 − μ T σ T d 4 = ln S m + 1 − μ T σ T d 9 = ln K − μ T − σ T 2 σ T d 5 = ln S m − μ T − σ T 2 σ T d 10 = ln S 1 − μ T − σ T 2 σ T
and
μ T = ln S 0 − σ 1 2 2 T − r 0 1 − e − a T a + ( θ a − σ 2 2 a 2 ) [ T − 1 − e − a T a ] + σ 2 2 2 a 1 − e − a T a σ T 2 = σ 2 2 a 2 [ T − 2 1 − e − a T a + 1 − e − 2 a T 2 a ] + σ 1 2 T .
The Put Option price under the extended Vasicek, Hull-White model is expressed by (3.12) where the mean μ T and variance σ T 2 under the T-forward measure are
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 2 2 a 2 [ T − 2 1 − e − a T a + 1 − e − 2 a T 2 a ] + σ 1 2 T
where the zero-coupon bond price is
P ( 0 , T ) = e r 0 ( e − a T − 1 ) a − ∫ 0 T e − a s ∫ 0 s θ ( 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 ) ] .
The findings in this paper show the existence of the closed form solutions for Call and Put Option Pricing under the linear investment strategy and are amenable to actual market implementation. The key benefit (unlike the classical Black-Scholes Model) is that not only such options are more realistic (i.e., stochastic interest rates and discrete time trading strategy), but in addition they command lower pricing (i.e., become more attractive from the investor’s perspective) when compared to Black-Scholes pricing. The future research will focus on extension of the Option pricing model based on dynamic investment strategy to Jump-diffusion processes, which typically fits market data better than the regular diffusion processes.
The authors declare no conflicts of interest regarding the publication of this paper.
Ghorbani, N. and Korzeniowski, A. (2022) Call and Put Option Pricing with Discrete Linear Investment Strategy. Journal of Mathematical Finance, 12, 84-96. https://doi.org/10.4236/jmf.2022.121005