Numerical Simulation of Exchange Option with Finite Liquidity: Controlled Variate Model
NNUMERICAL SIMULATION OF EXCHANGE OPTION WITH FINITELIQUIDITY: CONTROLLED VARIATE MODEL ∗ KEVIN S. ZHANG † AND
TRAIAN A. PIRVU ‡‡ Abstract.
In this paper we develop numerical pricing methodologies for European style ExchangeOptions written on a pair of correlated assets, in a market with finite liquidity. In contrast to thestandard multi-asset Black-Scholes framework, trading in our market model has a direct impact on theasset’s price. The price impact is incorporated into the dynamics of the first asset through a specifictrading strategy, as in large trader liquidity model. Two-dimensional Milstein scheme is implementedto simulate the pair of assets prices. The option value is numerically estimated by Monte Carlo withthe Margrabe option as controlled variate. Time complexity of these numerical schemes are included.Finally, we provide a deep learning framework to implement this model effectively in a productionenvironment.
Key words.
Exchange Option, FX, price impact, XVA, illiquid market, Monte Carlo, deeplearning
AMS subject classifications.
1. Introduction.
The Black-Scholes (BS) model was truly a breakthrough forpricing single asset options. It assumes participants operate in a perfectly liquid,friction-less and complete market. In practice, one or more of these assumptions areviolated. When the liquidity restriction is relaxed, trading will impact the price of theunderlying assets. Wilmott (2000) [28] was one of the pioneers of these price impactmodels. He considered price impacts depending upon different trading strategies suchas buy and hold, limit order and portfolio optimization. To account for price impact,Liu and Yong (2005) [18] included an additional term in the asset price stochasticdifferential equation (SDE). This inclusion indirectly adds a valuation adjustment tothe price of the option. Such an adjustment stems from a lack of liquidity, and maybe classified as liquidity valuation adjustment (LVA). Various non-linear BS-like partialdifferential equations (PDE), capturing the resulting price impact from trading havebeen studied [12, 6, 1, 4]. All these models share the similarity of being single-assetLVA models.
Exchange Options provide the utility of exchanging one asset for another. Underthe BS assumption for binary asset markets, Margrabe (1978) [19] derived a closed formsolution for the price of Exchange Options. The Exchange Option plays an essentialrole in currency markets. The
Foreign Exchange (FX)
Option is an Exchange Optionwhere the assets are currencies. A common concern is raised when one considers theinteraction between liquid and illiquid currencies. A trader might ask, “How reliable isthe price of a 3-month European style USD/UAH (Ukrainian Hryvnia, an infrequentlytraded currency) FX Option?”. In this work, we are interested in these type of scenar-ios. Recent studies on Exchange Options, such as [3, 2, 27, 13], exhibit deviation fromthe assumptions of BS. The aforementioned studies predominately involve stochasticvolatility models. Similar to Exchange Options, studies on Spread Option pricing havebeen conducted in the presence of full or partial price impact [25, 22].In this paper, we consider a binary-asset market with a single illiquid asset. Under ∗ Preprint.
Funding:
This work was funded by NSERC grant 5-36700. † Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada. ‡ Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada.1
This manuscript is for review purposes only. a r X i v : . [ q -f i n . P R ] J un KEVIN S. ZHANG, TRAIAN A. PIRVU this consideration, we construct a price impact model, called the finite liquidity marketmodel (FLMM). The model is a system of SDEs, one for each asset. The liquid asset isunchanged, the illiquid is modified to incorporate the resulting price impact from trad-ing. Existence and uniqueness conditions on the SDES are established for the FLMM(see section 7). By replicating a portfolio, We derive the partial differential equation(PDE) characterization of option prices. Further, we consider a market consisting ofmarket makers, who trade by Delta Hedging. We utilize the Milstein method andsimulate the FLMM SDEs as inspired by [10, 14]. The Margrabe Exchange Option isused as the control variate for our Monte Carlo (MC) pricing of the option. Motivatedby [7, 5], we apply deep feed-forward network to our MC pricing engine and achievesaccurate high speed pricing.The remainder of the content written in this paper is organized in the followingsections. Section 2 discusses the model framework. In Section 3, we analyze the priceimpact effect when majority of the market participants implement Delta Hedging. InSection 4, we apply Milstein’s method to simulate the path-wise price and sensitivity.Subsequently, we deploy control variate MC for estimation. Section 5 contains themethodology of
Deeply Learning Derivative for Exchange Option with price impact.In Section 6, we make some concluding statements for the readers. The last Section isan Appendix containing the proofs of our results.
2. Model Framework.
In this section we describe the dynamics of FLMM. Thereis a filtered probability space (cid:0) Ω , P , F ( t ) (cid:1) that satisfies the usual conditions. There aretwo risky assets whose prices are assumed to be a two-dimensional correlated Iˆto process S ( t ) = (cid:0) S ( t ) , S ( t ) (cid:1) . There is also a risk-free asset D ( t ) . The uncertainty in this modelis driven by a two-dimensional independent Brownian Motion W ( t ) = (cid:0) W ( t ) , W ( t ) (cid:1) .The system of SDEs which captures the asset price dynamics can be illustrated asfollows: dS ( t ) S ( t ) = µ ( t ) dt + σ dW ( t ) + λ (cid:0) t, S ( t ) , S ( t ) (cid:1) df (cid:0) t, S ( t ) , S ( t ) (cid:1) ,dS ( t ) S ( t ) = µ ( t ) dt + σ ρdW ( t ) + σ (cid:112) − ρ dW ( t ) , (2.1) dD ( t ) D ( t ) = − rdt, where µ i ( t ), σ i , ρ are the drift process, volatility and correlation of each Iˆto Processrespectively. The novelty here is the term λ (cid:0) t, S ( t ) , S ( t ) (cid:1) df (cid:0) t, S ( t ) , S ( t ) (cid:1) , and itrepresents the price impact λ ( t, s , s ) from a trading strategy f ( t, s , s ) . We willassumed the price impact is always non-negative, that is λ ( t, s , s ) ≥ . Let us pointout the two-dimensional market model used by Margrabe (1978) [19] is a special caseof this model when λ ( t, s , s ) = 0.We plan to obtain a canonical SDE of Asset 1 , and this will allow for a betterunderstanding of the model’s dynamics. In order to achieve this, we first apply Itˆo’sTheorem to compute the following differential df (cid:0) t, S ( t ) , S ( t ) (cid:1) . Then, we isolate the dS ( t ) terms, and compute the following quadratic/cross-variation terms: dS ( t ) dS ( t ), dS ( t ) dS ( t ) and dS ( t ) dS ( t ). By doing so we arrive at: dS ( t ) = ¯ µ (cid:0) S ( t ) (cid:1) dt + ¯ σ (cid:0) S ( t ) (cid:1) dW ( t ) + ¯ σ (cid:0) S ( t ) (cid:1) dW ( t ) , (2.2) This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION µ ( t, s , s ) = 11 − λf s (cid:16) µ s + λf t + s µ λf s + f s s ( ρσ σ s s + σ s λf s )1 − λf s + f s s ( σ s + σ s λ f s + 2 ρσ σ s s λf s )2 (cid:0) − λf s (cid:1) + σ s f s s (cid:17) , ¯ σ ( t, s , s ) = σ s − λf s , ¯ σ ( t, s , s ) = σ s λf s − λf s . With the model dynamics in hand, we can determine the requirements for the SDEdriving S to have a unique solution. In classical literature on SDE such as Oksendal(1992) [21], there are classical theorems for the existence and uniqueness of differentkinds (strong, weak) solutions. The following theorem provides sufficient conditionsfor the existence and uniqueness of FLMM SDEs: Theorem
Finite Liquidity Existence and Uniqueness Theorem I ). Under the assumptions (1) to (6) of (7.1), the SDE of S in (2.2) has a unique strongsolution.Proof. Please refer to the Appendix Section 7.1.The replicating portfolio argument is fundamental to the derivations of BS equa-tion. The replication argument in Chapter 4.5 of Shreve (2004) [26] can be modified toreplicate the option within FLMM framework. The portfolio used for replication willhave two assets and one cash account. The resulting equation will be a linear BS-likePDE of the parabolic family: rV = V t + rs V s + rs V s + V s s − λf s (cid:0) ρσ σ s s + λf s σ s (cid:1) + V s s − λf s ) (cid:0) σ s + λ f s σ s + 2 λf s ρσ σ s s (cid:1) + 12 V s s σ s ,V ( T, s , s ) = h ( s , s ) , with 0 < s , s < ∞ , 0 ≤ t ≤ T , where h ( s , s ) is a general payoff function. Existence results in Chapter 4 of Friedman(1975) [8] yield a unique classical solution for this PDE, granted 1 − λf s satisfiescondition (3) of Theorem 2.1.Feynman-Kac formula allows that the solutions for this PDE to be representedas a conditional expectations. As a by product of Feynman-Kac, we will discover aninduced risk-neutral measure (cid:101) P . Under this measure, we have the pricing formula: V (cid:0) t, s , s (cid:1) = (cid:101) E t,s ,s [ e − r ( T − t ) V (cid:0) T, S ( T ) , S ( T ) (cid:1) ] . (2.3)
3. Analysis of Replication of Exchange Option by Delta Hedging asPrice Impact.
In this section, we show that FLMM has a unique strong solutionfor a specific choice of price impact λ (cid:0) t, S ( t ) (cid:1) df (cid:0) t, S ( t ) , S ( t ) (cid:1) . There have beennumerous studies in the past focused on price impacts from trading. For example, Liuand Yong (2005) [18] studied a price impact model for single asset options. Pirvu etal. (2014) [25] also studied a price impact model for spread option. In this paper, weadopt the following price impact function:¯ λ (cid:0) t, s (cid:1) = (cid:40) (cid:15) (cid:0) − e − β ( T − t ) (cid:1) if S ≤ s ≤ S , , (3.1) This manuscript is for review purposes only.
KEVIN S. ZHANG, TRAIAN A. PIRVU where S and S represents a trading floor and cap of the asset respectively. Thiscause the trading price impact to be truncated within the floor and cap. As for theother parameters, (cid:15) is the price impact per share, and β is a decaying constant. It isimportant to emphasize that ¯ λ ( t, s ) will be employed for numerical approximation.The theoretical λ ( t, s ) should be a function with bounded derivative, that is obtainedthrough standard mollifying ¯ λ ( t, s ).Delta hedging is a strategy adopted by many big financial institutions to reducetheir option portfolio’s exposure against movements in the underlying assets. In thispaper, we assume majority of the market participants implement Delta hedging withthe Delta of the impact-less Exchange Option. Therefore, we choose the trading strat-egy function to be ∆ ( t ) of Margrabe’s option, that is f (cid:0) t, s , s (cid:1) = ∆ ( t ). The closedform expression for ∆ can be found in the Appendix Section (7.3).As a result, the drift and diffusion functions in (2.2) have the following dynamics: (cid:101) µ ( t, s , s ) = 11 − λ Γ (cid:16) µ s + λChm + µ s λ Γ + Spd ( ρσ σ s s + σ s λ Γ )1 − λ Γ + Spd ( σ s + σ s λ Γ + 2 ρσ σ s s λ Γ )2(1 − λ Γ ) + σ s Spd (cid:17) , (cid:101) σ ( t, s , s ) = σ s − λ Γ , (cid:101) σ ( t, s , s ) = σ s λ Γ − λ Γ . Here
Chm , Γ and
Spd are higher order Greeks of Magrabe’s option derived fromMargrabe’s formula. All the Greek formulas are given in the Appendix section 7.3.
Theorem
Existence and Uniqueness of Finite Liquidity MarketModel SDE II ). The SDE of S with drift and diffusion function of (7.3) has a unique strong solu-tion.Proof. Please refer to Appendix 7.1 for the proof.
4. Numerical Simulations.
In this section, our first objective is to simulate theFLMM assets by applying the Milstein Algorithm. Once we have the asset processes,we can use the results in our control variate MC estimator to price the Exchange Optionwith price impact. As a naming convention for our analysis, we refer to the number ofpoints M used to generate the stochastic assets as “path dimension”. The amount ofasset paths N used in the MC estimator will be referred to as “space dimension”. Compared with the more well knownEuler-Maruyama, Milstein is a second-order pathwise method for approximating SDEsolutions. It was created by Milshtein G. N. (1975) [20], this method retains the secondorder terms from Iˆto Taylor expansion. For a 2-dimensional SDE system satisfied by
This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION X ( t ) = (cid:0) X ( t ) , X ( t ) (cid:1) , a second-order approximation of the solution is: X ( t ) ≈ X ( t ) + (cid:90) tt µ (cid:0) X ( u ) (cid:1) du + (cid:90) tt σ (cid:0) X ( u ) (cid:1) dW ( u ) + (cid:90) tt σ (cid:0) X ( u ) (cid:1) dW ( u )+ 12 (cid:88) j,k,l =1 ∂σ j ∂x l σ lk (cid:0) X ( t ) (cid:1)(cid:0) ∆ W j ( t )∆ W k ( t ) + ρ jk ( t − t ) − A jk ( t , t ) (cid:1) ,X ( t ) ≈ X ( t ) + (cid:90) tt µ (cid:0) X ( u ) (cid:1) du + (cid:90) tt σ (cid:0) X ( u ) (cid:1) dW ( u ) + (cid:90) tt σ (cid:0) X ( u ) (cid:1) dW ( u )+ 12 (cid:88) j,k,l =1 ∂σ j ∂x l σ lk (cid:0) X ( t ) (cid:1)(cid:0) ∆ W j ( t )∆ W k ( t ) + ρ jk ( t − t ) − A jk ( t , t ) (cid:1) , According to Giles (2018) [10], the term A ij ( t , t ) is the L´evy Area between two the twodriving Brownian motions. It’s behavior is captured by following stochastic integral: A ij ( t , t ) = (cid:90) tt (cid:0) ∆ W i ( u ) dW j ( u ) − ∆ W j ( u ) dW i ( u ) (cid:1) . (4.1)Since we are only interested in pricing and hedging, it is advantageous to workunder the risk-neutral measure. FLMM in (2.2) with the updated drift and diffusionfunctions of (3) has the following dynamics under (cid:101) P : dS ( t ) = rS ( t ) dt + (cid:101) σ (cid:0) S ( t ) (cid:1) d (cid:102) W ( t ) + (cid:101) σ (cid:0) S ( t ) (cid:1) d (cid:102) W ( t ) ,dS ( t ) = rS ( t ) dt + (cid:101) σ ( t ) d (cid:102) W ( t ) + (cid:101) σ ( t ) d (cid:102) W ( t ) , (4.2) dD ( t ) D ( t ) = − rdt, for simplicity, we set: (cid:101) σ ( t ) = σ s ρ, (cid:101) σ ( t ) = σ s (cid:112) − ρ . The Milstein approximation for (4.2) can be set up by following these procedures:1. Partition [ t, T ] into M equivalent intervals of length ∆ t = T − tM .2. Set the initial values as S (0) = s and S (0) = s .3. Sample { ∆ W ( j ) , ∆ W ( j ) } j =1 , ,...M , where each { ∆ W ( j ) , ∆ W ( j ) } ∼ N ( , ∆ tI ).4. Generate L´evy Areas A ij (0 , ∆ t ).5. Recursively define: S ( m + 1) = S ( m ) + rS ( m )∆ t + (cid:88) i =1 (cid:101) σ i (cid:0) S ( m ) (cid:1) ∆ W i ( m + 1) + 12 (cid:88) i,j,k =1 ∂ (cid:101) σ i ∂s k × (cid:101) σ kj (cid:0) S ( m ) (cid:1)(cid:0) ∆ W i ( m + 1)∆ W j ( m + 1) − ( i = j ) ∆ t − A ij (cid:1) ,S ( m + 1) = S ( m ) + rS ( m )∆ t + (cid:88) i =1 (cid:101) σ i (cid:0) S ( m ) (cid:1) ∆ W i ( m + 1) + 12 (cid:88) i,j,k =1 ∂ (cid:101) σ i ∂s k × (cid:101) σ kj (cid:0) S ( m ) (cid:1)(cid:0) ∆ W i ( m + 1)∆ W j ( m + 1) − ( i = j ) ∆ t − A ij (cid:1) . This manuscript is for review purposes only.
KEVIN S. ZHANG, TRAIAN A. PIRVU
There are many techniques to approximate the L´evy Area, one of the simplest isto generate the stochastic integral (4.1) piece by piece. In this paper, we adopted analgorithm which closely resembles the method found in Scheicher (2007) [24]. Accordingto Scheicher, this algorithm for L´evy Area has complexity cost of O ( K ), where K isthe number of partition of the time interval ∆ t . Algorithm 4.1
L´evy AreaDefine sub-partition length ∆ t := ∆ tK Generate z , z ∼ N K ( , I K ).Generate lower triangular matrix of 1s T , set R := ∆ tT Generate lower and upper diagonal matrices of 1s L and U .Set B := R z and B =: R z A = b T ( U − L ) b return A We may redefine a matrix recursion version of the Milstein Scheme. Consider thefollowing evolutionary dynamic of S ( t ): S ( m + 1) = B ( m ) S ( m ) + 12 b ( m ) . (4.3)The matrix B ( m ) consists of the first order approximation and the vector b ( m ) is thesecond order approximation. For our SDE system (3), B ( m ) and b ( m ) can be definedas follows: B ( m ) = (cid:20) r ∆ t + (cid:101) σ (cid:0) S ( m ) (cid:1) ∆ W ( m + 1) (cid:101) σ (cid:0) S ( m ) (cid:1) ∆ W ( m + 1) (cid:101) σ (cid:0) S ( m ) (cid:1) ∆ W ( m + 1) 1 + r ∆ t + (cid:101) σ (cid:0) S ( m ) (cid:1) ∆ W ( m + 1) (cid:21) , b ( m ) = (cid:20) W T ( m + 1) J Σ W ( m + 1) − tr ( J Σ) − T ( J Σ ◦ A ) T ( m + 1) J Σ W ( m + 1) − tr ( J Σ) − T ( J Σ ◦ A ) (cid:21) . Here J i is the Jacobi matrix of the i -th asset’s diffusion functions at the m -th step.Matrix Σ encapsulates diffusion functions of all assets, also at m -th step. They are ofthe form: J i = (cid:34) ∂ (cid:101) σ i ∂s ∂ (cid:101) σ i ∂s ∂ (cid:101) σ i ∂s ∂ (cid:101) σ i ∂s (cid:35) , Σ = (cid:20)(cid:101) σ (cid:101) σ (cid:101) σ (cid:101) σ (cid:21) . A is the matrix of L´evy Areas at step m , it has the form: A = (cid:20) A A (cid:21) , notice A is an off diagonal matrix, this is because the stochastic integral (4.1) is 0 when i = j .It is mentioned in Higham (2015) [14] that Milstein scheme has complexity of O ( M ) compared to O ( M ) of Euler-Maruyama. This is important because Milsteinscheme will carry a steeper computation time increase as M increases. The model withoutliquidity impact is a special case of FLMM. One would naturally assume there exists ahigh inherited correlation of option prices produced by the two models. It would make
This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION S and S cv represents FLMM and GBM asset prices respectively. Algorithm 4.2
Milstein Control Variate PathInitialize Values S ( t ) = S cv ( t ) = s Define ∆ t =: T − tM for m = 0 to M − do∆W ( m ) = (cid:0) ∆ w ( m ) , ∆ w ( m ) (cid:1) ∼ N (0 , ∆ tI )Set B ( m ), b ( m ), J i , Σ and A S ( m + 1) = B ( m ) S ( m ) + b ( m ) S cv ( m + 1) = B cv ( m ) S cv ( m ) + b cv ( m ) end forreturn S ( M ), S cv ( M )By generating { S ( i ) ( M ), S ( i ) cv ( M ) } i =1 , ,...N , we can define the control variate MCestimator of FLMM Exchange Option as follows: V = e − r ( T − t ) N N (cid:88) i =1 (cid:16)(cid:0) S ( i )1 ( M ) − S ( i )2 ( M ) (cid:1) + + c (cid:0) S ( i ) cv, ( M ) − S ( i ) cv, ( M ) (cid:1) + (cid:17) (4.4) − cV Margrabe , here V Margrabe is the price of Magrabe option given by Margrabe’s formula in a modelwithout liquidity impact. The term c is the optimization constant. In this case, thevariance of our MC estimator is minimized when ˆ c = − Cov ( V FLMM ,V Margrabe ) V ar ( V Margrabe ) . Managing the Greeks is an essential part of trading. Todetermine the Deltas of FLMM Exchange Option, we will adopt the adjoint method ofGiles and Glasserman (2006) [9]. This method first requires the Greeks to be generatedpathwise, then a MC can be applied to estimate the actual value. The adjoint methodis advantageous because these pathwise Greeks can be generated simultaneously withthe assets. Suppose interchangeability exists between the differential operator andexpectation, then the j -th Delta of FLMM Exchange Option is:∆ j ( t ) = ∂∂S j ( t ) (cid:101) E t,s ,s (cid:104) e − r ( T − t ) V (cid:0) S ( T ) (cid:1)(cid:105) = e − r ( T − t ) (cid:101) E t,s ,s (cid:104) ∂∂S j ( t ) V (cid:0) S ( T ) (cid:1)(cid:105) . By relaxing certain regularity conditions outlined in Glasserman (2004) [11], we mayrewrite it as: ∂∂S j ( t ) V (cid:0) S ( T ) (cid:1) = (cid:88) i =1 ∂V∂S i ( T ) ∂S i ( T ) ∂S j ( t ) . During implementation, ∂V∂S i ( T ) can be approximated through algorithmic differ-entiation. While the ∂S i ( T ) ∂S j ( t ) term is obtained from taking the path-wise derivative ofMilstein scheme (5). Set ∆ ij ( t ) = ∂S i ( T ) ∂S j ( t ) , we obtain an approximating scheme for This manuscript is for review purposes only.
KEVIN S. ZHANG, TRAIAN A. PIRVU ∆ ij ( m ) as follows:∆ ij ( m + 1) = ∆ ij ( m ) + r ∆ ij ( m )∆ t + (cid:88) k,l =1 ∂ (cid:101) σ ik ∂s l ∆ lj ( m )∆ W k ( m + 1)+ 12 (cid:88) k,l,p,q =1 ∆ qj ( m ) (cid:16) ∂ (cid:101) σ ik ∂s p ∂s q (cid:101) σ pj (cid:0) S ( m ) (cid:1) + ∂ (cid:101) σ ik ∂s p ∂ (cid:101) σ pl ∂s q (cid:17) , where m = 0 , , ...M −
1. If we define a matrix D ( m ) as: D ij ( m ) = δ ij ( m ) + r ∆ t + (cid:88) k =1 ∂ (cid:101) σ ik ∂s j ∆ W k ( m + 1)+ 12 (cid:88) k,l,p =1 (cid:16) ∂ (cid:101) σ ik ∂s p ∂s j (cid:101) σ pj (cid:0) S ( m ) (cid:1) + ∂ (cid:101) σ ik ∂s p ∂ (cid:101) σ pl ∂s j (cid:17) , then the evolution of ∆ can be redefined using matrix recursion as follows: ∆ ( m + 1) = D ( m ) ∆ ( m ) , where ∆ ( t ) = I . Similar to estimating the option price, we a can use the Delta fromthe Magrabe option as a multivariate control variate. We adopt the method presentedby Rubinstein and Marcus (1985) [23] and set up the estimator for Delta: ∆ = e − r ( T − t ) N N (cid:88) i =1 (cid:16) ∆ ( i ) ( M ) + C ∆ ( i ) cv ( M ) (cid:17) − C ∆ Margrabe . (4.5)The variance of ∆ is minimized when ˆ C = Σ ∆∆ cv Σ − ∆ cv ∆ cv . We implement our MC engine with alternatingspace and path parameter for the purpose of determining the effect on a 99% Gaussianconfidence interval (CI). For consistency, we fix a set of option parameters: s = 60, s = 80, T = 0 . t = 0, σ = 0 . σ = 0 . ρ = 0 . r = 0 .
05. We also fix theprice impact function parameters to: (cid:15) = 0 .
04 and β = 100. The numerical results arepresented below: Table 1: Space (N) Dimension MC Results N M V CI of V CI Length CPU Time
100 100 1 . . , . . . . , . . k
100 1 . . , . . k
100 1 . . , . . m
100 1.00139 [1 . , . . This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION
N M V CI of V CI Length CPU Time . . , . . . . , . . . . , . . . . , . . . . , . . This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU
Table 3: Analysis of Liquidity Premium N = 100000, M = 100 s FLMM Margrabe Excess Price s = 10 10 0 . . . . . . . . . s = 20 20 1 . . . . . . . . . s = 30 30 2 . . . .
504 10 .
499 0 . . . . s = 100 100 9 . . . (cid:15) increases, we observe a higher liquiditypremium. This effect is illustrated in the figures below.Fig. 2: Liquidity Value AdjustmentIt only appears natural to be also interested in the liquidity adjustment for Delta.Using the Margrabe Delta as a reference, one would expect that the strictly greaterprice of our illiquid asset 1 would cause ∆ to be greater and ∆ to be less. Empirically,we observe an excess effect in ∆ , but we also observed an excess effect in ∆ . Weillustrate this surprising result in the figures below. This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION
5. Deep Learning Method.
Artificial neural network have powerful predictivecapabilities, one of the first versions are the FFN. This network is structured as asequence of layers, with various numbers of neurons embedded in each layer. We shalluse N to denote the number of layers, and n i to denote the number of neurons in the i -th layer. In a fully connected FFN, each neuron in the current layer has a connectionwith each neuron in the subsequent layer. The strength of these connections are knownas weights , we denote the weights connected to the j -th neuron in the i -th layer as w [ i ] j .Each neuron also carries a unique bias term b [ i ] j , this term has a similar effect as theregression intercept. The final component of a neuron is the activation function f ( z ),similar to linking functions of non-linear regression, its purpose is to add non-linearity.In this study, we used these types of activation functions:Table 4: Activation FunctionsType Activation FunctionReLU f ( z ) = max( z, f ( z ) = log(1 + e z )The operation of a neuron can be expressed as: z [ i ] j = w [ i ] j h [ i − + b [ i ] j ,h [ i − j = f ( z [ i ] j ) . We also provide a computation graph on the j -th neuron in the i -th layer: This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU h [ i − h [ i − h [ i − h [ i − n w [ i ] i w [ i ] i w [ i ] i w [ i ] in i f (cid:0) z [ i ] j (cid:1) h [ i ] j This process is repeated for every single neuron, which allows us to transversethrough the network and arrive at the output layer h [ N ] = ˆ y (For the purpose of optionpricing, we have a single output h [ N ] , but in general h [ N ] is a vector). This entireprocess is often referred to as forward propagation . The figure below describes theFFN architecture deployed to price Exchange Option under FLMM: Input Layer Hidden Layer 1 Hidden Layer 2 Hidden Layer 3 Hidden Layer 4 Output Neuron s s σ σ rρτ h [1]1 h [1]2 h [1]3 h [1] n h [2]1 h [2]2 h [2]3 h [2] n h [3]1 h [3]2 h [3]3 h [3] n h [4]1 h [4]2 h [4]3 h [4] n V The loss function measures the goodness of fit. We use mean squared error (MSE)as the loss function, which is commonly used in regression analysis. We will use MSEto evaluate the result of the forward propagation. This evaluation is preformed forevery B input, B is known as the batch size . Our loss function is formulated as: L ( ˆy , y ) = B (cid:88) k =1 (ˆ y k − y k ) . Minimization of the loss function follows the steepest descent idea, so one has to com-pute gradient fields with respect to the weights and biases. This is often accomplishedthrough algorithmic differentiation referenced as back propagation . Then, the weightsand biases are updated in the direction of the gradient field, in hope of discovering a“good enough” local minimum. The common choice of methodology for optimization
This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION batch gradient descent method. This method is demonstrated as: w [ i ] , ( new ) j = w [ i ] , ( old ) j − α ∂ L ∂ w [ i ] , ( old ) j , b [ i ] , ( new ) j = b [ i ] , ( old ) j − α ∂ L ∂ b [ i ] , ( old ) j , for j = 1 , , ...n i , and i = 1 , , ...N. In the above expression, α is the learning rate .One batch of forward propagation combined with one instance of back propagationis considered as one iteration of batch training. An epoch encompasses a series of batchtraining that exhausts the entire data set. Normally, the training is either repeated fora fixed number of epochs, or stopped early when the loss function ceases to decreasefurther.The central theorem in neural networks is the universal approximation theorem .This theorem highlights the approximation power of FFNs. Hornik (1989) [15] es-tablished the fact that deep FFNs are universal approximators, in other words, anyfunction can be accurately approximated by some deep FFN. Since option prices aresmooth solutions of PDEs, then it should be feasible to predict these solutions withFFNs. Option pricing can often be computation-ally expensive. Ferguson and Green (2018) [7] demonstrated the power of FFN, andachieved a much faster speed than traditional MC engines when pricing baskets. How-ever, the initial costs comes from generation of option inputs, as well as, estimating thecorresponding option values through MC engines. Furthermore, training and calibrat-ing the FFN takes tedious effort as well. Nevertheless, these “costs” are reasonable tolarge financial institutions, and at least in theory, will integrate well with their oper-ations. This is largely because both the data generation and network training can bedone offline, when the markets are closed. In addition, the input space can be restrictedto reflect a set of likely market scenarios.To build a FFN pricer for our FLMM Exchange Option, we will use Algorithm(4.4) as the underlying MC engine. Our estimator has 7 parameters (cid:0) x = ( s , s , r, ρ, σ , σ , τ ) (cid:1) , a set of these parameters count as 1 sample input. It isimportant to emphasize the particular distribution used to generate the inputs, theseshould be unique for each option. Indeed, some factors to be considered when choosingthe distributions are: • The physical meaning of each underlying parameter. • The payoff function itself should be considered because it is pointless to gen-erate excessive of out-of-money MC paths.Generating the inputs in judicious ways will not only help the loss to convergefaster, but will also help the FFN to approximate a meaningful solution. In our case,we adopted an even spilt between 2 data generation schemes. The first method allowsus to sample unbiasly from the entire input space. The second method will allow us tosample more realistic input parameters, as well as, capture more in-the-money payoutpaths.
This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU
Table 5: Data Generation SchemesParameter Method 1 Method 2 s s ∼ U (0 , s ∼
50 exp( X ), X ∼ N (0 . , . s s ∼ U (0 , s ∼
50 exp( X − X ), X ∼ N (0 . , . σ σ ∼ U (0 , . σ ∼ U (0 , . σ σ ∼ U (0 , . σ ∼ U (0 , . r r ∼ U (0 , . r ∼ U (0 , . ρ ρ ∼ U (1 , − ρ ∼ X − . X ∼ β (5 , τ τ ∼ U (0 , τ ∼ U (0 , Deeply Learning Derivative method can be synthesized bythe following programming architectural graph:
Data GenerationMethod 1Method 2 MC EngineMilsteinControl VariateDeep Feed Forward NetworkTrainingCross-ValidationTestingExport to Production
The FFN contains 4 fully connected deep layerswith 300 ReLu neurons per layer. The output layer contain a single SoftPlus Neuronto ensure the prediction would be positive definite. We generated 1 million inputs, anduses a relatively inaccurate MC engine ( N =100, M =100) to construct the training set.The logic is it has been shown in practice a well-trained deep FFN has the ability toremove the inaccuracy of weak MC estimators. We trained the FNN with mini-batchsize of 1024, and updated the gradient with ADAM optimizer (2015) [17]. We performedvalidation with samples created from a highly accurate MC engine ( N =100k, M =100),at a 100 / This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION L and L simultaneously. Another important observation is that the MAEerror is more consistent than MSE. This implies the smaller errors matched up moreconsistently between training and validation set. Overall, we can conclude there is nosignificant over-fitting.In the testing phase, we generated 1000 highly accurate samples with MC enginespecification ( N =100k, M =100). We test our trained network and came to the follow-ing testing results:Moving on to analyzing the testing set, we observe a strong linear relationshipbetween the predicted value and true value. This is an indication our net performsextremely well in predicting option prices. In the graph above, we observe relativelyfew misclassification points (error that are more than 3 standard deviation away fromthe mean). Furthermore, we observes approximate normality in the residual histogram.The slight leptokurtic shape could hint hyper-parameter tuning might yield betterresults. However due to the close resemblances to normality, the source of error shouldbe relatively homogeneous.We will use option parameters s = 60, s = 80, σ = 0 . σ = 0 .
2, and r = 0 . This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU
Table 6: FLMM Exchange Option Prediction ResultsN=1m, M=100 τ = 0 . τ = 1 τ = 2 Computation Time ρ = 0 . . . . . . . ρ = 0 . . . . . . . ρ = 0 . . . . . . . ρ = 0 . . . . . . . ρ = 0 . . . . . . .
6. Conclusions.
In this paper, we explored the effects of liquidity on pricingExchange Options in a binary-asset market which we refer to as FLMM. In this market,trading only affected the price of one (the iliquid) asset. Subsequently, we establishedthe existence and uniqueness of a strong solution for the SDEs driving the asset priceswithin FLMM. By the standard replication argument we obtained a two-dimensionalBS-like PDE, which characterized the options prices. We simulated asset prices byMilstein algorithm and developed a fast-converging MC estimator with the Margrabeoption as its control variate. Finally, we deployed deep learning and further improvedthe pricing speed.Conforming to our hypothesis, we observed the same transaction cost “super-replication” effect as described by Liu and Yong [18]. This paper may serve as acautionary note for FX traders who regularly deal with option on iliquid currencies.Option issuers may also adopt this model as a LVA model for any type of ExchangeOptions.
7. Appendix.
This section will include some of the formulas and proofs left outfrom the main body.
In this section, (cid:107) · (cid:107) and ||| · ||| represents the supremum norms: (cid:107) f (cid:107) = sup ( s ,s ) ∈D | f ( t, s , s ) | , where D = (cid:0) R + (cid:1) , ||| f ||| = sup ( t,s ,s ) ∈D | f ( t, s , s ) | , where D = [0 , T ] × (cid:0) R + (cid:1) , The following combination of conditions (1) − (6) will guarantee existence and unique-ness of a strong solution for S .(1) (cid:107) λ ( s f s s + s f s s + f s + s f s + s f s s + s f s s + s f s s ) (cid:107) < ∞ . (2) (cid:107) (cid:0) λ s + λ s (cid:1)(cid:0) s f s + s f s + s f s (cid:1) (cid:107) < ∞ . (3) ||| − λf s ||| > δ , for some δ > . This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION (cid:107) ( λ + λ s + λ s )( f t + f s + f s + f s s + f s s + f s s + f s s s + f s s s ) (cid:107) < ∞ . (5) (cid:107) λ ( f ts + f ts + f s s s + f s s s ) + λ s f s s s + λ s f s s s (cid:107) < ∞ . (6) (cid:107) s f s s + s f s s + s f s s + s f s s + s f s s s + s f s s s + s s f s s s + s s f s s s + s f s s s + s f s s s (cid:107) < ∞ . Proof.
Recall that the SDE of S is of the form: dS ( t ) = ¯ µ (cid:0) S ( t ) (cid:1) dt + ¯ σ (cid:0) S ( t ) (cid:1) dW ( t ) + ¯ σ (cid:0) S ( t ) (cid:1) dW ( t ) , where¯ µ ( t, s , s ) = 11 − λf s (cid:16) µ s + λf t + s µ λf s + f s s ( ρσ σ s s + σ s λf s )1 − λf s + f s s ( σ s + σ s λ f s + 2 ρσ σ s s λf s )2 (cid:0) − λf s (cid:1) + σ s f s s (cid:17) , ¯ σ ( t, s , s ) = σ s − λf s , ¯ σ ( t, s , s ) = σ s λf s − λf s . Following the classical existence uniqueness result for SDEs, we have to show thefunctions ¯ µ ( t, s , s ), ¯ σ ( t, s , s ) and ¯ σ ( t, s , s ) are uniformly Lipschitz continuouswith respect to (cid:107) · (cid:107) . Thus, it is sufficient to check the boundedness of their respectivepartial derivatives. Computing the derivatives, we have: (cid:2) ¯ σ (cid:3) s = σ (cid:0) − λf s + s ( λ s f s + λf s s )(1 − λf s ) (cid:1) , (cid:2) ¯ σ (cid:3) s = σ s λ s f s + λf s s (1 − λf s ) , (cid:2) ¯ σ (cid:3) s = σ s (cid:0) ( λ s f s + λf s s )1 − λf s + λf s ( λf s s + λ s f s )(1 − λf s ) (cid:1) , (cid:2) ¯ σ (cid:3) s = σ (cid:0) λf s + s ( λ s f s + λf s s )1 − λf s + λ s f s ( λ s f s + λf s s )(1 − λf s ) (cid:1) . We can clearly see the boundedness requirement for (cid:2) ¯ σ (cid:3) s , (cid:2) ¯ σ (cid:3) s , (cid:2) ¯ σ (cid:3) s and (cid:2) ¯ σ (cid:3) s can be condensed into: (cid:107) λ ( s f s s + s f s s + f s + s f s + s f s s + s f s s + s f s s ) (cid:107) < ∞ , (7.1) (cid:107) (cid:0) λ s + λ s (cid:1)(cid:0) s f s + s f s + s f s (cid:1) (cid:107) < ∞ . (7.2)Furthermore, we will require the denominator terms in the partial derivatives above tosatisfy: ||| − λf s ||| > δ , for some δ > . (7.3) This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU
The partial derivatives (cid:2) ¯ µ s (cid:3) s and (cid:2) ¯ µ s (cid:3) s are: (cid:2) ¯ µ s (cid:3) s = µ (cid:0) − λf s + s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + λf ts + λ s f t − λf s + λf t ( λ s f s + λf s s )(1 − λf s ) + µ s (cid:0) λf s s + λ s f s − λf s + λf s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + 12 σ (cid:0) s f s s s + 2 s f s s (1 − λf s ) + 3 s f s s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + 12 σ s (cid:0) λ f s f s s s + 2 λf s f s s ( λ s f s + λf s s )(1 − λf s ) + 3 λ f s f s s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + ρσ σ s (cid:0) s λf s f s s s + s f s s ( λ s f s + λf s s ) + λf s s f s (1 − λf s ) + 3 s λf s f s s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + ρσ σ s (cid:0) f s s + s f s s s (1 − λf s ) + 2 s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + σ s (cid:0) λf s f s s s + f s s ( λ s f s + λf s s )(1 − λf s ) + λf s f s s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + 12 σ s (cid:0) f s s s − λf s + f s s ( λ s f s + λf s s )(1 − λf s ) (cid:1) . (cid:2) ¯ µ s (cid:3) s = µ s λ s f s + λf s s (1 − λf s ) + λ s f t + λf ts − λf s + λf t ( λ s f s + λf s s )(1 − λf s ) + µ (cid:0) λf s + s ( λ s f s + λf s s )1 − λf s + s λf s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + 12 σ s (cid:0) f s s s (1 − λf s ) + 3 f s s ( λ s f s + λf s s )(1 − λf s ) (cid:1) + 12 σ (cid:16) λ f s s f s + s (cid:0) λ + f s s s f s + f s s (2 λλ s f s + sλ f s f s s ) (cid:1) (1 − λf s ) +3 s λ f s s f s ( λ s f s + λf s s )(1 − λf s ) (cid:17) + ρσ σ s (cid:16) λf s f s + s (cid:0) λ s f s f s + λ ( f s s f s + 2 f s f s f s s ) (cid:1) (1 − λf s ) +3 s λf s f s s ( λ s f s + λf s s )(1 − λf s ) (cid:17) + ρσ σ s (cid:0) f s s + s f s s s (1 − λf s ) + s f s s ( λ s f s + λf s s (1 − λf s ) (cid:1) + σ (cid:16) s λf s f s s + s (cid:0) λ s f s f s s + λ ( f s s f s s + f s f s s s ) (cid:1) (1 − λf s ) + s λf s f s s ( λ s f s + λf s s )(1 − λf s ) (cid:17) + 12 σ (cid:0) s f s s + s f s s s − λf s + s f s s ( λ s f s + λf s s )(1 − λf s ) (cid:1) . This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION µ ( t, s , s ) will be bonded when ||| − λf s ||| > δ and: (cid:107) ( λ + λ s + λ s )( f t + f s + f s + f s s + f s s + f s s + f s s s (7.4) + f s s s ) (cid:107) < ∞ , (cid:107) λ ( f ts + f ts + f s s s + f s s s ) + λ s f s s s + λ s f s s s (cid:107) < ∞ , (7.5) (cid:107) s f s s + s f s s + s f s s + s f s s + s f s s s + s f s s s + s s f s s s (7.6) + s s f s s s + s f s s s + s f s s s (cid:107) < ∞ , The combination of requirements (7.1), (7.2), (7.3), (7.4), (7.5), (7.6) will guarantee s ¯ µ ( t, s , s ), s ¯ σ ( t, s , s ) and s ¯ σ ( t, s , s ) are uniformly Lipschitz continuous in (cid:0) R + (cid:1) . By Itˆo’s Existence and Uniqueness Theorem Itˆo [16] (1979), the SDE for S will have a unique strong solution. To show theSDE have a unique strong solution, it is sufficient to show that the conditions (1) − (6) inAppendix Section 7.1 are satisfied for the particular choice of λ ( t, s ) and f ( t, s , s ) =∆ ( t ). • Condition (1): (cid:107) λ ( s Spd + s Spd + Γ + s Γ + s Spd + s Spd + s Spd ) (cid:107) = (cid:107) λ (cid:16) N (cid:48) ( d + ) σs √ τ (cid:0) d + σs √ τ + 1 (cid:1) + s d + N (cid:48) ( d + ) σ τ s s + 1 σ √ τ s N (cid:48) ( d + ) s + s N (cid:48) ( d + ) σ √ τ s + s N (cid:48) ( d + ) σs √ τ (cid:0) d + σs √ τ + 1 (cid:1) + 2 d + N (cid:48) ( d + ) σ τ s + 2 d − s N (cid:48) ( d + ) σ τ s (cid:17) (cid:107) < ∞ . Proof.
Notice there is a common term of the form N (cid:48) ( d + ) s n . These terms appearsnaturally in higher order Greeks. Consider any real number n , we have: N (cid:48) ( d + ) s n = 1 s n √ π exp (cid:110) − (cid:16) log( s s ) + σ τσ √ τ (cid:17) (cid:111) = 1 s n √ π e (cid:110) − log2( s s (cid:0) σ τ − log( s (cid:1) + (cid:0) σ τ − log( s (cid:1) σ τ (cid:111) e − n log( s ) = 1 √ π exp (cid:110) − log ( s ) + o (cid:0) log( s ) (cid:1) σ τ (cid:111) , which approaches to 0 as s approaches to zero, and approaches to 0 as well as s approaches ∞ . Since n was arbitrary, then all of the functions in Condition(1) are bounded in s . With a similar method involving the common term N (cid:48) ( d + ) s n , we can also show that all of the terms in Condition (1) are boundedin s . We can ultimately conclude that the entire function of Condition (1) isbounded in ( s , s ). • Condition (2): (cid:107) λ s (cid:0) s Γ + s Γ + s Γ (cid:1) (cid:107) < ∞ . This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU
Proof.
Same proof as Condition (1). • Condition (3): ||| − λ Γ ||| > δ , for some δ > . Proof.
This condition already holds in the s , s dimension. For t we havelim t → T ¯ λ ( t, s ) = 0 and lim t → T Γ ( t ) = ∞ for at the money options. Since¯ λ ( t, s ) approach to 0 at a greater rate, then lim t → T ¯ λ ( t, s )Γ ( t ) = 0 . In fact,this ensures the ¯ λ ( t, s )Γ ( t ) term stays small, which ultimately guaranteesthe existence of δ . There is a more detailed explanation in Pirvu et al (2014)[25]. • Condition (4): (cid:107) ( λ + λ s )( Chm + Γ + Γ + Spd + Spd + Spd + Acc + Acc ) (cid:107) < ∞ . Proof.
Same proof as Condition (1). • Condition (5): (cid:107) λ ( Col + Col + Acc + Acc ) + λ s Acc + λ s Acc (cid:107) < ∞ . Proof.
Same proof as Condition (1). • Condition (6): (cid:107) s Spd + s Spd + s Spd + s Spd + s Acc + s Acc + s s Acc + s s Acc + s Acc + s Acc (cid:107) < ∞ . Proof.
Same proof as Condition (1).Since we have shown Condition (1) to (5) in the Appendix Section 7.1 holds for ourprice impact trading strategy λ (cid:0) t, S ( t ) (cid:1) df (cid:0) t, S ( t ) , S ( t ) (cid:1) . We can conclude the SDEs S (3) has a strong solution. Margrabe (1978) [19] derivedthe following closed form price for Exchange Option. V ( t, s , s ) = (cid:101) E (cid:2) e − rτ (cid:0) S ( T ) − S ( T ) (cid:1) + | F ( t ) (cid:3) = s N ( d + ) − s N ( d − ) , (7.7) where d ± = log( s s ) ± σ τσ √ τ , and σ = σ + σ − σ σ ρ. We can derive the Exchange Option Greeks by differentiating formula (7.7). The firstorder Greeks are well known, they are available in papers such as Alos and Thorsten(2017) [3]. ∆ ( t ) = N ( d + ) ∆ ( t ) = − N ( d − ) . Θ( t ) = σs N (cid:48) ( d + )2 √ τ = − σs N (cid:48) ( d − ) . √ τ . This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION ( t ) = ∂ ∆ ( t ) ∂s = N (cid:48) ( d + ) ∂d + ∂s = N (cid:48) ( d + ) σs √ τ . Γ ( t ) = ∂ ∆ ( t ) ∂s = − N (cid:48) ( d − ) ∂d − ∂s = N (cid:48) ( d − ) σs √ τ . Γ ( t ) = Γ ( t ) = ∂ ∆ ( t ) ∂s = N (cid:48) ( d + ) ∂d + ∂s = − σ √ τ N (cid:48) ( d + ) s = − σ √ τ N (cid:48) ( d − + σ √ τ ) s = − σ √ τ s √ π exp (cid:110) − d − − d − σ √ τ − σ τ (cid:111) = − σ √ τ s √ π exp (cid:110) − d − − log (cid:0) s s (cid:1)(cid:111) = − N (cid:48) ( d − ) σs √ τ .Charm ( t ) = ∂ ∆ ( t ) ∂τ = N (cid:48) ( d + ) ∂d + ∂τ = N (cid:48) ( d + ) (cid:16) − log (cid:0) s s (cid:1) στ + σ √ τ (cid:17) .Charm ( t ) = ∂ ∆ ( t ) ∂τ = − N (cid:48) ( d − ) ∂d − ∂τ = N (cid:48) ( d − ) (cid:16) log (cid:0) s s (cid:1) στ + σ √ τ (cid:17) .Speed ( t ) = ∂ Γ ( t ) ∂s = 1 σ √ τ N (cid:48)(cid:48) ( d + ) ∂d + ∂s − N (cid:48) ( d + ) s = 1 σ √ τ − d + N (cid:48) ( d + ) σs √ τ − N (cid:48) ( d + ) s = − Γ s (cid:0) d + σs √ τ + 1 (cid:1) .Speed ( t ) = ∂ Γ ( t ) ∂s = 1 σ √ τ N (cid:48)(cid:48) ( d − ) ∂d − ∂s − N (cid:48) ( d − ) s = 1 σ √ τ − d − N (cid:48) ( d − ) σs √ τ − N (cid:48) ( d − ) s = − Γ s (cid:0) d − σs √ τ + 1 (cid:1) .Speed ( t ) = ∂ Γ ( t ) ∂s = 1 σ √ τ N (cid:48)(cid:48) ( d + ) ∂d + ∂s s = − d + N (cid:48) ( d + ) σ τ s s = − d + Γ σs .Speed ( t ) = ∂ Γ ( t ) ∂s = 1 σ √ τ N (cid:48)(cid:48) ( d − ) ∂d − ∂s s = − d − N (cid:48) ( d − ) σ τ s s = − d − Γ σs .Colour ( t ) = ∂ Γ ( t ) ∂τ = 1 σs (cid:16) − τ N (cid:48) ( d + ) − τ N (cid:48) ( d + ) d + ∂d + ∂τ (cid:17) = N (cid:48) ( d + )2 στ s (cid:110) − d + (cid:16) log( s s ) 1 σ √ τ − σ √ τ (cid:17)(cid:111) = − Γ σ τ (cid:16) σ τ + 4 σ τ − ( s s ) (cid:17) ,Colour ( t ) = ∂ Γ ( t ) ∂τ = 1 σs (cid:16) − τ N (cid:48) ( d − ) − τ N (cid:48) ( d − ) d + ∂d − ∂τ (cid:17) = − Γ σ τ (cid:16) σ τ + 4 σ τ − ( s s ) (cid:17) ,Colour ( t ) = Colour ( t ) = ∂ Γ ( t ) ∂τ = 1 σs (cid:16) τ N (cid:48) ( d + ) + 1 τ N (cid:48) ( d + ) d + ∂d + ∂τ (cid:17) = − Γ σ τ (cid:16) σ τ + 4 σ τ − ( s s ) (cid:17) = − Γ σ τ (cid:16) σ τ + 4 σ τ − ( s s ) (cid:17) . This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU
Acceleration ( t )= ∂Speed ( t ) ∂s = − (cid:16)(cid:0) ∂∂s Γ s (cid:1)(cid:0) d + σs √ τ + 1 (cid:1) + Γ s σ √ τ (cid:0) ∂∂s d + s (cid:1)(cid:17) = − (cid:16) Speed s − Γ s (cid:0) d + σs √ τ + 1 (cid:1) + Γ s σ √ τ (cid:0) σ √ τ − d + s (cid:1)(cid:17) = − σ √ τ s (cid:16) d + (cid:0) d + σ √ τ s + 1 (cid:1) + (cid:0) σ √ τ − d + (cid:1)(cid:17) = − σ s τ (cid:0) d s + 1 (cid:1) ,Acceleration ( t )= ∂Speed ( t ) ∂s = − (cid:16) s (cid:0) ∂∂s Γ (cid:1)(cid:0) d + σs √ τ + 1 (cid:1) + Γ s σ √ τ s (cid:0) ∂∂s d + (cid:1)(cid:17) = − (cid:16) Speed s (cid:0) d + σ √ τ s + 1 (cid:1) − Γ s σ √ τ s σ √ τ s (cid:17) = 2Γ σs s √ τ (cid:0) d σ √ τ + d + + 1 σs √ τ (cid:1) ,Acceleration ( t )= ∂Speed ( t ) ∂s = 2 σ √ τ s (cid:0) ∂∂s d + (cid:1) Γ + d + (cid:0) ∂∂s Γ (cid:1) = 2Γ σ s s τ (cid:0) d + d − − (cid:1) ,Acceleration ( t )= ∂Speed ( t ) ∂s = − (cid:16) s (cid:0) ∂∂s Γ (cid:1)(cid:0) d − σs √ τ + 1 (cid:1) + Γ s σ √ τ s (cid:0) ∂∂s d − (cid:1)(cid:17) = − (cid:16) Speed s (cid:0) d − σ √ τ s + 1 (cid:1) + Γ s σ √ τ s σ √ τ s (cid:17) = 2Γ σs s √ τ (cid:0) d − σs √ τ + d − − σs √ τ (cid:1) ,Acceleration ( t )= ∂Speed ( t ) ∂s = − (cid:16)(cid:0) ∂∂s Γ s (cid:1)(cid:0) d − σs √ τ + 1 (cid:1) + Γ s σ √ τ (cid:0) ∂∂s d − s (cid:1)(cid:17) = − σ s τ (cid:0) d − s + 1 (cid:1) . Acknowledgments.
The authors are grateful to the anonymous referee for acareful checking of the details and for helpful comments that improved this paper.
REFERENCES[1]
D. Ahmadian, O. F. Rouz, K. Ivaz, and A. Safdari-Vaighani , Robust numerical algorithmto the european option with illiquid markets
E. Als and M. Coulon , On the optimal choice of strike conventions in exchange option pricing ,EconPapers, (2018), https://arxiv.org/abs/1807.05396.
This manuscript is for review purposes only.
UMERICAL ANALYSIS OF EXCHANGE OPTION [3] E. Als and T. Rheinlnder , Pricing and hedging margrabe options with stochastic volatilities ,EconPapers, (2017), https://EconPapers.repec.org/RePEc:upf:upfgen:1475.[4]
A. J. Arenas, G. Gonzalez-Parra, and B. M. Caraballo , A nonstandard finite differ-ence scheme for a nonlinear black-scholes equation
R. Culkin and S. R. Das , Machine learning in finance: The case of deep learning for optionpricing , Journal of Investment Management, (2017).[6]
M. Dyshaev and V. Fedorov , The sensitivities (greeks) for some models of option pricingwith market illiquidity , Mathematical notes of NEFU, 26 (2019), https://doi.org/10.25587/SVFU.2019.102.31514.[7]
R. Ferguson and A. D. Green , Applying deep learning to derivatives valuation , SSRN Elec-tronic Journal, (2018), http://dx.doi.org/10.2139/ssrn.3244821.[8]
A. Friedman , Stochastic Differential Equations and Applications , Academic Press, 1st edi-tion ed., 1975.[9]
M. Giles and P. Glasserman , Smoking adjoints: fast evaluation of greeks in monte carlocalculations , Risk Journals, (2005).[10]
M. B. Giles and L. Szpruch , Multilevel monte carlo methods for applications in finance , High-Performance Computing in Finance, (2018), pp. 197–247, https://arxiv.org/pdf/1212.1377.pdf.[11]
P. Glasserman , Monte Carlo methods in financial engineering , Springer, 2004.[12]
K. J. Glover, P. W. Duck, and D. P. Newton , On nonlinear models of markets with fi-nite liquidity: Some cautionary notes , SIAM Journal on Applied Mathematics, 70 (2010),pp. 3252–3271, https://doi.org/10.1137/080736119.[13]
D. Hainaut , Calendar spread exchange options pricing with gaussian random fields , Risks, 6(2018), p. 77, https://doi.org/10.3390/risks6030077.[14]
D. J. Higham , An introduction to multilevel monte carlo for option valuation , International Jour-nal of Computer Mathematics, 92 (2015), pp. 2347–2360, https://doi.org/10.1080/00207160.2015.1077236.[15]
K. Hornik, M. Stinchcombe, and H. White , Multilayer feedforward networks are universalapproximators
I. Itˆo , On the existence and uniqueness of solutions of stochastic integral equations of thevolterra type , Kodai Math, 2 (1979), pp. 158–170, https://doi.org/https://doi.org/10.2996/kmj/1138036013.[17]
D. Kingma and J. Ba , Adam: A method for stochastic optimization , International Conferenceon Learning Representations, (2014).[18]
H. Liu and J. Yong , Option pricing with an illiquid underlying asset market , Journal of Eco-nomic Dynamics & Control, 29 (2005), pp. 2125–2156, https://doi.org/http://apps.olin.wustl.edu/faculty/liuh/Papers/Liu Yong.pd.[19]
W. Margrabe , The value of an option to exchange one asset for another , Journal of Finance,33 (1978), pp. 177–186, https://doi.org/https://doi.org/10.2307/2326358.[20]
G. N. Milshtein , Approximate integration of stochastic differential equations , Theory of Proba-bility & Its Applications., 19 (1975), pp. 557–000, https://doi.org/https://doi.org/10.1137/1119062.[21]
B. Oksendal , Stochastic Differential Equations (3rd Ed.): An Introduction with Applications ,Springer-Verlag, Berlin, Heidelberg, 1992.[22]
T. Pirvu and A. Yazdanian , Numerical analysis for spread option pricing model in illiquidunderlying asset market: Full feedback model , Applied Mathematics & Information Sciences,10 (2015), pp. 1271–1281, https://doi.org/10.18576/amis/100406.[23]
R. Y. Rubinstein and R. Marcus , Efficiency of multivariate control variates in monte carlosimulation , Operations Research, 33 (1985), pp. 661–677, https://doi.org/10.1287/opre.33.3.661, https://doi.org/10.1287/opre.33.3.661.[24]
K. Scheicher , Complexity and effective dimension of discrete lvy areas , Journal of Complexity,23 (2007), pp. 152–168, https://doi.org/doi:10.1016/j.jco.2006.12.006.[25]
A. Shidfar, K. Paryab, A. Yazdanian, and T. A. Pirvu , Numerical analysis for spreadoption pricing model of markets with finite liquidity: first-order feedback model , Interna-tional Journal of Computer Mathematics, 91 (2014), pp. 2603–2620, https://doi.org/https://doi.org/10.1080/00207160.2014.887274.[26]
S. E. Shreve , Stochastic calculus for finance II, Continuous-time models , Springer, New York,NY; Heidelberg, 2004.[27]
S. ul Islam and I. Ahmad , A comparative analysis of local meshless formulation for multi-asset
This manuscript is for review purposes only. KEVIN S. ZHANG, TRAIAN A. PIRVU option models
P. Wilmott and P. J. Schnbucher , The feedback effect of hedging in illiquid markets ,SIAM Journal on Applied Mathematics, 61 (2000), pp. 232–272, https://doi.org/10.1137/S0036139996308534.,SIAM Journal on Applied Mathematics, 61 (2000), pp. 232–272, https://doi.org/10.1137/S0036139996308534.