Journal Details
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Journal
eISSN
2444-8656
First Published
01 Jan 2016
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2 times per year
Languages
English
Open Access

# Nonlinear Differential Equations in Preventing Financial Risks

###### Accepted: 18 Apr 2022
Journal Details
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

In recent years, the default risks faced by financial institutions have been increasing, and the corresponding capital requirements have also been increasing. The financial market is full of risks. There are numerous examples of losses or company failures caused by poor risk management in the market. The Association of Interbank Money Market Dealers issued guidelines on credit risk mitigation tools in October 2010. This marks the official launch of the Chinese version of the CDS credit derivatives market. This favorably promotes the release of credit risk. The pricing of credit default swaps is a kind of financial credit derivatives. This allows the credit risk of risky assets to be separated from asset ownership. The market-oriented arrangement for transferring credit risk gives risky assets liquidity characteristics [1]. It has played an important role in bank credit portfolio management and the financial risk control of listed companies. Some scholars have studied the pricing of corporate bonds under the reduced model where the interest rate obeys the Vasicek model, and the default intensity is a linear function of the interest rate. Some scholars focus on the random recovery rate model. They treat the recovery rate as an exponential function of default strength. We established a pricing model and used the technique of solving partial differential equations to give an analytical solution to corporate bonds. This way, the impact of recovery parameters and default strength parameters on credit spreads is analyzed [2]. The above documents assume that the company's asset value Vt meets the standard Brownian motion-driven stochastic differential equation in a given probability space. However, the standard Brownian motion cannot describe the fractal characteristics of financial markets, such as self-similarity and long memory. Fractional Brownian motion has become the easiest way to compensate for the shortcomings of the Brownian motion model because fractional Brownian motion is a Gaussian process.

Its properties mainly include additive invariance, self-similarity, thick tail, discontinuity, and long-term correlation. This makes the fractional Brownian motion neither a Markov process nor a semimartingale. This has brought huge difficulties to avoid arbitrage opportunities in the financial market under fractional Brownian motion [3]. Some scholars have proposed using the fractional Brownian jump-diffusion model to avoid the existence of arbitrage, and the results are still not ideal.

Therefore, this article attempts to study the capital valuation adjustment of the value of the derivatives transaction contract. We use the semi-copy method to design a reasonable hedging portfolio strategy [4]. The partial differential equation of valuation adjustment is derived based on the partial differential equation of the value of the derivative transaction contract and the risk-free value of the contract [5]. At the same time, we apply the Feynman-Kac formula to derive a valuation adjustment expression closer to the market from separating the capital valuation adjustment. As mentioned earlier, the calculation of the capital valuation adjustment further revised the price of the derivative contract value.

Capital hedging portfolio strategy

We use the replication method to establish a reasonable hedging portfolio strategy. This enables hedging of derivatives [6]. Using this method to hedge against the counterparty's default risk has become an important means of bank risk management. At the same time, banks are also using scientific and reasonable calculation methods and appropriate models to measure the default risk of trading contracts. This will strengthen the management of capital valuation adjustments and reduce the cost of capital. We set the price of the underlying asset stock as a Brownian motion dS = μSdt + σSdW. In this way, a hedging portfolio strategy is established. $Π=δS+a1P1+a2P2+aCPC+βS+βC−X1$ \Pi = \delta S + {a_1}{P_1} + {a_2}{P_2} + {a_C}{P_C} + {\beta _S} + {\beta _C} - {X^1}

This model assumes that the random process followed by the price of the underlying asset stock is geometric Brownian motion. The model does not consider the jump behavior of stock prices [7]. This article sets the price process of the underlying asset as a jump-diffusion process in the subsequent model asset assumptions. The purpose is to enable the model to cover this risk factor. At the same time, we have added stock options to the hedging portfolio. The purpose is to hedge against the volatility risk of stock prices.

Assume that the value of the transaction contract between the bank and its counterparty is $V^=V^(t,S,JB,JC)$ \hat V = \hat V\left({t,S,{J_B},{J_C}} \right) . The value of the corresponding transaction contract without default risk is V. Where S represents the stock price that is not affected by the bank or counterparty default. JB and JC respectively represent the indicative function of the bank and counterparty default. JB and JC are independent of each other [8]. There is a deviation between the price of any derivative transaction contract and its true value. We set it to U, then we have $V^=V+U$ \hat V = V + U

The deviation U here means total valuation adjustment, including credit valuation adjustment (CVA), financing valuation adjustment (FVA), margin valuation adjustment (MVA), collateral (COLVA), and capital valuation adjustment (KVA). U = CVA + FVA + MVA + COLVA + KVA. The valuation adjustment is the difference between the value of the derivative contract containing credit risk, financing cost, margin, collateral and capital cost, and the value of the contract without default risk.

When the bank or counterparty defaults, the liquidation or claim of the trading contract position is usually determined by the Mark-to-Market M (t, S) of derivatives [9]. Let M (t, S) = V (t, S) determine the value at market value. When the bank defaults, there are: $V^(t,S,1,0)=gB(V,X)$ \hat V(t,S,1,0) = {g_B}(V,X)

When the counterparty defaults, there are: $V^(t,S,0,1)=gC(V,X)$ \hat V(t,S,0,1) = {g_C}(V,X)

gB (V, X) = [V (S, t) − X (t)]+ + RB [V (S, t) − X (t)] + X (t) in formula (3).

gC (V, X) = RC[V (S, t) − X (t)]+ +[V (S, t) − X (t)] + X (t) in formula (4).

Where RB, RC is the recovery rate of the contract value calculated by market value and the difference between the collateral.

When the value of the transaction contract between the bank and its counterparty is $V^=V^(t,S,JB,JC)$ \hat V = \hat V\left({t,S,{J_B},{J_C}} \right) , it needs to hedge the default risk of the transaction contract [10]. According to the semi-copy method, the specific form of a hedging portfolio strategy Π that satisfies $V^+Π=0$ \hat V + \Pi = 0 is as follows: $Π=δSS+aC(S,t)+a1P1+a2P2+aCPC+ϕK(t)+X(t)+βc+βs+βx+βk$ \Pi = {\delta _S}S + aC\left({S,t} \right) + {a_1}{P_1} + {a_2}{P_2} + {a_C}{P_C} + \phi K(t) + X(t) + {\beta _c} + {\beta _s} + {\beta _x} + {\beta _k}

Any hedging portfolio strategy should include at least different assets such as the subject matter, bonds issued by both parties to the transaction, collateral, capital, and cash accounts. Therefore, the risk should include the stock of δS unit price of S as the basic underlying asset. At the same time, increase the stock option item C(S, t) of the short position, namely a < 0. The purpose is to reduce the risk of volatility in stock prices [11]. The combination strategy also includes bonds issued by the bank with a1 unit price of P1 and a2 unit price of P2. At the same time, it also covers the aC unit price PC bonds issued by the counterparty. When the counterparty defaults, it will cause bank losses. The bank can “borrow” bond PC to hedge against losses caused by the counterparty's default. So take a short bond position PC . aC < 0. In addition, the strategy also includes the capital cost K (t) of ϕ units. One unit of collateral X (t) and repurchase cash accounts with different assets as collateral are included in the cost of different assets. βc, βS, βX and βK are the cash accounts that supply bonds PC, stocks S and stock options C(S, t), collateral, and capital costs, respectively. We explain the dynamic equations of different assets in the strategy. The stock price obeys the jump-diffusion process. This can reflect the actual market situation: $dS=(μ−λv)Sdt+σSdW+(Y−1)SdJ$ dS = \left({\mu - \lambda v} \right)Sdt + \sigma SdW + (Y - 1)SdJ

μ represents the expected return rate of the stock. W represents geometric Brownian motion. σ represents the volatility of the stock. The variable J represents the number of jumps of the underlying stock in [0, T] and it obeys the Poisson jump process with the parameter λ. Y represents the rate of change in the price of the underlying stock. υ = E(Y −1) and Y and dJ are independent of each other. When the stock price obeys the jump-diffusion process, the differential form of the option value C = C(S, t) with the stock as the subject is as follows: $dC(S,t)=(∂C∂t+12σ2S2∂2C∂S2)dt+∂C∂SdS=(∂C∂t+(μ−λv)S∂C∂S+12σ2S2∂2C∂S2)dt+σS∂C∂SdW+[C(YS,t)−C(S,t)]dJ$ \eqalign{& dC(S,t) = \left({{{\partial C} \over {\partial t}} + {1 \over 2}{\sigma ^2}{S^2}{{{\partial ^2}C} \over {\partial {S^2}}}} \right)dt + {{\partial C} \over {\partial S}}dS \cr & = \left({{{\partial C} \over {\partial t}} + \left({\mu - \lambda v} \right)S{{\partial C} \over {\partial S}} + {1 \over 2}{\sigma ^2}{S^2}{{{\partial ^2}C} \over {\partial {S^2}}}} \right)dt + \sigma S{{\partial C} \over {\partial S}}dW + \left[{C\left({YS,t} \right) - C(S,t)} \right]dJ \cr}

The prices issued by the bank are P1 and P2 respectively. The dynamic equations satisfied by bonds with recovery rates of P1 and P2 (P1P2) : $dP1=r1P1dt−(1−R1)P1dJB$ d{P_1} = {r_1}{P_1}dt - (1 - {R_1}){P_1}d{J_B} $dP2=r2P2dt−(1−R2)P2dJB$ d{P_2} = {r_2}{P_2}dt - (1 - {R_2}){P_2}d{J_B}

The basis difference between bonds of different grades is not considered for the time being. We use bonds with different recovery rates to offset the balance sheet impact of derivatives. $r1=r−(1−R1)λB$ {r_1} = r - (1 - {R_1}){\lambda _B} $r2=r−(1−R2)λB$ {r_2} = r - (1 - {R_2}){\lambda _B}

λB is the bank's default rate. r is the risk-free interest rate. Assume that the bond recovery rate of the counterparty is zero and the following dynamic equation is satisfied: $dPc=rcPcdt−PcdJc$ d{P_c} = {r_c}{P_c}dt - {P_c}d{J_c}

The repurchase account βS with stocks as collateral satisfies: $dβS=δ(γS−qS)Sdt$ d{\beta _S} = \delta \left({{\gamma _S} - {q_S}} \right)Sdt

qS represents the determined dividend yield of stock assets. γS = r + Sβ,S, Sβ,S represents the spread that exceeds the risk-free rate of return r. The repurchase account βC with bond Pc as collateral satisfies: $dβC=−aCqCPCdt$ d{\beta _C} = - {a_C}{q_C}{P_C}dt

qc represents the yield of bond Pc. When the borrowing rate of bond Pc is rc, the counterparty's default rate λc is: $λC=rC−qC$ {\lambda _C} = {r_C} - {q_C}

Collateral X (t) and its repurchase account βX satisfy: $dX(t)=−rX(t)dt$ dX(t) = - rX(t)dt $dβX=rXXdt$ d{\beta _X} = {r_X}Xdt $γX=rX−r$ {\gamma _X} = {r_X} - r

Capital K (t) and its repurchase account βK satisfy: $dK(t)=rKdt$ dK(t) = rKdt $dβK=−rKK(t)dt$ d{\beta _K} = - {r_K}K(t)dt $dβK=−rKK(t)dt$ d{\beta _K} = - {r_K}K(t)dt

In addition, the asset constraints of the hedging portfolio strategy are set to: $(δ1+δ2)S+aC(S,t)+βS=0$ \left({{\delta _1} + {\delta _2}} \right)S + aC\left({S,t} \right) + {\beta _S} = 0 $αCPC+βC=0$ {\alpha _C}{P_C} + {\beta _C} = 0

Since the hedging investment strategy Π satisfies $V^+Π=0$ \hat V + \Pi = 0 , the capital constraint of the contract value $V^$ \hat V at this time is: $V^=a1P1+a2P2+X(t)+ϕK(t)+βX+βK=0$ \hat V = {a_1}{P_1} + {a_2}{P_2} + X(t) + \phi K(t) + {\beta _X} + {\beta _K} = 0

Solving Partial Differential Equations

We derive the partial differential equation of the total valuation adjustment from the differential form of the hedging portfolio strategy containing stock options and the cost of capital. Then the Feynman-Kac formula derives the expression of the total valuation adjustment [12]. At the same time, we separate the capital valuation adjustment calculation expression. According to the Ito lemma of the jump-diffusion process, we find that the differential form of the contract value $V^=V^(t,S,JB,JC)$ \hat V = \hat V\left({t,S,{J_B},{J_C}} \right) is: $dV^=∂V^∂tdt+∂V^∂SdS+12σ2S2∂2V^∂S2dt+ΔVBdJB+ΔVCdJC$ d\hat V = {{\partial \hat V} \over {\partial t}}dt + {{\partial \hat V} \over {\partial S}}dS + {1 \over 2}{\sigma ^2}{S^2}{{{\partial ^2}\hat V} \over {\partial {S^2}}}dt + \Delta {V_B}d{J_B} + \Delta {V_C}d{J_C}

$ΔVB=gB−V^$ \Delta {V_B} = {g_B} - \hat V and $ΔVC=gC−V^$ \Delta {V_C} = {g_C} - \hat V are the losses or gains when the bank and the counterparty each default. When the hedging portfolio strategy adopted by the bank is in the following form: $Π=δSS+a1P1+a2P2+acPc+βC+βS+βX+βK+αC(S,t)+X(t)+ϕK(t)$ \Pi = {\delta _S}S + {a_1}{P_1} + {a_2}{P_2} + {a_c}{P_c} + {\beta _C} + {\beta _S} + {\beta _X} + {\beta _K} + \alpha C\left({S,t} \right) + X(t) + \phi K(t)

Over time, the differential form of hedging investment strategy Π is: $dΠ=δSdS+a1dP1+a2dP2+acdPc+dβC+dβS+dβX+dβK+adC(S,t)+dX(t)+ϕdK(t)$ \eqalign{& d\Pi = {\delta _S}dS + {a_1}d{P_1} + {a_2}d{P_2} + {a_c}d{P_c} + d{\beta _C} + \cr & d{\beta _S} + d{\beta _X} + d{\beta _K} + adC(S,t) + dX(t) + \phi dK(t) \cr}

According to the assumptions, we assume δS = δ1 + δ2 and substitute $δ1=−∂V^∂Sδ2=−a∂C∂S$ {\delta _1} = - {{\partial \hat V} \over {\partial S}}{\delta _2} = - a{{\partial C} \over {\partial S}} into the above formula. At the same time, we substitute formula (7) and formula (16), (17), (18), (19), (20), and (21) into formula (27) to sort out. Where P = a1P1 + a2 P2, PD = a1R1P1 + a2 R2 P2 and λC = rCqC. Then there are: $d(V^+Π)=dV^+dΠ$ d\left({\hat V + \Pi} \right) = d\hat V + d\Pi

Reorganize the above formula 1 to get: $dΠ+dV^=εhdJB+[∂tV^+AtV^−(r+λB+λC)V^+λBgB+λCgC−γXX−γKK+r(X+ϕK+βX+βK)−εhλB]dt$ d\Pi + d\hat V = {\varepsilon _h}d{J_B} + \left[\matrix{{\partial _t}\hat V + {A_t}\hat V - \left({r + {\lambda _B} + {\lambda _C}} \right)\hat V + {\lambda _B}{g_B} + {\lambda _C}{g_C} \hfill \cr - {\gamma _X}X - {\gamma _K}K + r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) - {\varepsilon _h}{\lambda _B} \hfill \cr} \right]dt

In the formula $At=12σ2S2∂2∂S2+∂C∂t∂C∂S+12σ2S2∂2C∂S2∂∂S+(γS−qS)S∂∂S$ {A_t} = {1 \over 2}{\sigma ^2}{S^2}{{{\partial ^2}} \over {\partial {S^2}}} + {{\partial C} \over {\partial t}}{{\partial C} \over {\partial S}} + {1 \over 2}{\sigma ^2}{S^2}{{{\partial ^2}C} \over {\partial {S^2}}}{\partial \over {\partial S}} + ({\gamma _S} - {q_S})S{\partial \over {\partial S}} , γXrXr. It can be seen from the above formula that as long as the bank has not defaulted, the risk-free state of the transaction contract can be achieved by hedging the risk through the investment portfolio strategy. When the bank defaults, εh is the bondholder's profit and loss. Assume that the final return of the investment portfolio is H (S) when the transaction is terminated. At the same time, from the formula (28), it can be known that the portfolio strategy Π and the contract value $V^$ \hat V satisfy: $dΠ+dV^=εhdJB+[∂tV^+AtV^−(r+λB+λC)V^+λBgB+λCgC−γXX−γKK+r(X+ϕK+βX+βK)−εhλB]dt$ d\Pi + d\hat V = {\varepsilon _h}d{J_B} + \left[\matrix{{\partial _t}\hat V + {A_t}\hat V - \left({r + {\lambda _B} + {\lambda _C}} \right)\hat V + {\lambda _B}{g_B} + {\lambda _C}{g_C} \hfill \cr - {\gamma _X}X - {\gamma _K}K + r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) - {\varepsilon _h}{\lambda _B} \hfill \cr} \right]dt

Banks take into account the interests of shareholders, so they hope that the hedging strategy in the above formula can be transformed into a form of self-financing. The drift term in the above formula is required to be zero, then the partial differential equation about the transaction contract value $V^$ \hat V is: ${∂tV^+AtV^−(r+λB+λC)V^+λBgB+λCgC+γXX−γKK+r(X+ϕK+βX+βK)−εhλB=0V^(T,S)=H(S)$ \left\{\matrix{{\partial _t}\hat V + {A_t}\hat V - \left({r + {\lambda _B} + {\lambda _C}} \right)\hat V + {\lambda _B}{g_B} + {\lambda _C}{g_C} + {\gamma _X}X \hfill \cr - {\gamma _K}K + r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) - {\varepsilon _h}{\lambda _B} = 0 \hfill \cr \hat V\left({T,S} \right) = H(S) \hfill \cr} \right.

After finishing: ${∂tV^+AtV^−(r+λB+λC)V^=−γXX+γKK−λCgC−γBgB−r(X+ϕK+βX+βK)+λBεhV^(T,S)=H(S)$ \left\{\matrix{{\partial _t}\hat V + {A_t}\hat V - \left({r + {\lambda _B} + {\lambda _C}} \right)\hat V = - {\gamma _X}X + {\gamma _K}K - {\lambda _C}{g_C} - \hfill \cr {\gamma _B}{g_B} - r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) + {\lambda _B}{\varepsilon _h} \hfill \cr \hat V\left({T,S} \right) = H(S) \hfill \cr} \right.

The risk-free value V of the transaction contract calculated by the classic Black-Scholes-Merton model satisfies the standard BlackScholes partial differential equation: ${∂tV+AtV−rV=0V(T,S)=H(S)$ \left\{\matrix{{\partial _t}V + {A_t}V - rV = 0 \hfill \cr V(T,S) = H(S) \hfill \cr} \right.

From equation (2), the total valuation adjustment $U=V^−V$ U = \hat V - V can be obtained, so the partial differential equation that U satisfies is: ${∂tU+AtU−(r+λB+λC)U=−γXX+γKK−λC(gC−V)−λB(gB−V)−r(X+ϕK+βX+βK)+λBεhU(T,S)=0$ \left\{\matrix{{\partial _t}U + {A_t}U - \left({r + {\lambda _B} + {\lambda _C}} \right)U = - {\gamma _X}X + {\gamma _K}K - {\lambda _C}({g_C} - V) \hfill \cr - {\lambda _B}({g_B} - V) - r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) + {\lambda _B}{\varepsilon _h} \hfill \cr U\left({T,S} \right) = 0 \hfill \cr} \right.

According to the Feynman-Kac formula, it can be found that the partial differential equation is expressed as $∂U∂t+μ(x,t)∂U∂x+12σ2(x,t)∂2U∂x2−v(x,t)U=f(x,t)$ {{\partial U} \over {\partial t}} + \mu (x,t){{\partial U} \over {\partial x}} + {1 \over 2}{\sigma ^2}(x,t){{{\partial ^2}U} \over {\partial {x^2}}} - v(x,t)U = f(x,t)

When the boundary condition is U (x, T) = Ψ(x), the solution function of this partial differential equation can be written as the conditional expectation of a random process: $U(x,t)=E[∫tTe−∫tsv(xt)dτf(xs,s)ds+e−∫tTv(xt)dτΨ(xT)|xt=x]$ U(x,t) = E\left[{\int_t^T {{e^{- \int_t^s {v\left({{x_t}} \right)d\tau}}}f\left({{x_s},s} \right)ds + {e^{- \int_t^T {v({x_t})d\tau}}}\Psi \left({{x_T}} \right)|{x_t} = x}} \right]

In the formula, μ, σ, υ, ψ and f (x, t) are known functions. T is the given parameter. x = (xt, t ≥ 0) is the Ito stochastic process determined by dXt = μ(Xt, t)dt +σ(Xt, t)dWt. The partial differential equation of the total valuation adjustment U is: $∂U∂t+AtU−(r+λB+λC)U=−γXX+γKK−λC(gC−V)−λB(gB−V)−r(X+ϕK+βX+βK)+λBεh$ \eqalign{& {{\partial U} \over {\partial t}} + {A_t}U - \left({r + {\lambda _B} + {\lambda _C}} \right)U = \cr & - {\gamma _X}X + {\gamma _K}K - {\lambda _C}\left({{g_C} - V} \right) - {\lambda _B}\left({{g_B} - V} \right) - r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) + {\lambda _B}{\varepsilon _h} \cr}

Boundary condition U (T, S) = 0. According to the Feynman-Kac formula: $U(S,t)=E[∫tTe−∫tsv(S,τ)dτf(Sh,h)ds]$ U\left({S,t} \right) = E\left[{\int_t^T {{e^{- \int_t^s {v\left({S,\tau} \right)d\tau}}}f\left({{S_h},h} \right)ds}} \right]

Where: $v(S,t)=r+λB+λCf(S,t)=−γXX+γKK−λC(gC−V)−λB(gB−V)−r(X+ϕK+βX+βK)+λBεh U(S,t)=E[e−∫ts(r+λB+λC)[−γXX+γKK−λC(gC−V)−r(X+ϕK+βX+βK)+λBεh]dσ]$ \eqalign{& v\left({S,t} \right) = r + {\lambda _B} + {\lambda _C} \cr & f\left({S,t} \right) = - {\gamma _X}X + {\gamma _K}K - {\lambda _C}\left({{g_C} - V} \right) - {\lambda _B}\left({{g_B} - V} \right) - r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) + {\lambda _B}{\varepsilon _h} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,U\left({S,t} \right) = E\left[{{e^{- \int_t^s {\left({r + {\lambda _B} + {\lambda _C}} \right)}}}\left[\matrix{- {\gamma _X}X + {\gamma _K}K - {\lambda _C}\left({{g_C} - V} \right) - \hfill \cr r\left({X + \phi K + {\beta _X} + {\beta _K}} \right) + {\lambda _B}{\varepsilon _h} \hfill \cr} \right]d\sigma} \right] \cr}

Since U (S, t) is the total valuation adjustment, it can be decomposed into: $U=CVA+FVA+MVA+COLVA+KVA$ U = CVA + FVA + MVA + COLVA + KVA

In $CVA=−Et[∫tTλC(u)e−∫ts(r+λB+λC)dτ(Vϕ(u)−gC(u))du]$ CVA = - {E_t}\left[{\int_t^T {{\lambda _C}(u){e^{- \int_t^s {\left({r + {\lambda _B} + {\lambda _C}} \right)d\tau}}}\left({{V_\phi}(u) - {g_C}(u)} \right)du}} \right]

Credit valuation adjustment CVA is the present value of the expected expenses due to the default of the counterparty as estimated by the bank [13]. This reflects the impact of the counterparty's default of derivatives on the value of derivatives. The existence of counterparty credit risk in financial derivatives transactions will bring a certain degree of deviation to the pricing of derivatives contracts. In particular, the price of OTC derivatives trading contracts will deviate from its true value. Therefore, we need to make credit valuation adjustments to the value of derivative contracts. This makes its pricing more reasonable. $FVA=−Et[∫tT(rF(u)−rϕ(u))e−∫ts(r+λB+λC)dτ(Vϕ(u)−ϕ(u))du]$ FVA = - {E_t}\left[{\int_t^T {\left({{r_F}(u) - {r_\phi}(u)} \right){e^{- \int_t^s {\left({r + {\lambda _B} + {\lambda _C}} \right)d\tau}}}\left({{V_\phi}(u) - \phi (u)} \right)du}} \right]

Financing valuation adjustment FVA is the adjustment of the value of financing costs in derivatives transaction contracts [14]. This is a discount on expected financing costs. It reflects the impact of external financing costs on the value of derivatives. $MVA=−Et[∫tT(rF(u)−rψ(u))e−∫ts(r+λB+λC)dτψB(u)du]$ MVA = - {E_t}\left[{\int_t^T {\left({{r_F}(u) - {r_\psi}(u)} \right){e^{- \int_t^s {\left({r + {\lambda _B} + {\lambda _C}} \right)d\tau}}}{\psi _B}(u)du}} \right]

Margin valuation adjustment MVA is the cost of issuing initial margin during the trading cycle. It reflects the financing cost of issuing initial margin for derivatives transactions. Therefore, it can also be called the FVA of the initial margin [15]. Especially when the transaction triggers additional initial reserves, margin valuation adjustments will occur. $COLVA=−Et[∫tT(γX(u)−r(u))e−∫ts(r+λB+λC)dτX(u)+γX(u)βXdu]$ COLVA = - {E_t}\left[{\int_t^T {\left({{\gamma _X}(u) - r(u)} \right){e^{- \int_t^s {\left({r + {\lambda _B} + {\lambda _C}} \right)d\tau}}}X(u) + {\gamma _X}(u){\beta _X}du}} \right]

Collateral valuation adjustment COLVA is the cost incurred by the required collateral in the transaction cycle. $KVA=−Et[∫tT(γX(u)−r(u))e−∫ts(r+λB+λC)dτK(u)−γK(u)βK+λBεK(u)du]$ KVA = - {E_t}\left[{\int_t^T {\left({{\gamma _X}(u) - r(u)} \right){e^{- \int_t^s {\left({r + {\lambda _B} + {\lambda _C}} \right)d\tau}}}K(u) - {\gamma _K}(u){\beta _K} + {\lambda _B}{\varepsilon _K}(u)du}} \right]

The expression of capital valuation adjustment KVA is the discount value of capital, capital cash repurchase accounts, and capital adjustment εK (u). If a bank wants to conduct a transaction, it needs to hold a certain amount of equity capital during the transaction cycle to ensure the smooth progress of the transaction. This will involve the cost of holding equity capital. Banks need additional equity capital when investing in low-risk projects since bank equity holders require a certain return on equity investment. The return on these capitals must be at least higher than the return on equity investment required by the equity holders. The purpose of capital valuation adjustment is to ensure that the return on additional equity capital is not lower than the return on the basic equity investment [16]. The more accurate the calculation of capital valuation adjustments, the more conducive to rationalizing additional capital holdings.

Analysis of Numerical Examples

The price of bonds issued by fund companies depends on many factors. Mainly include asset volatility, bond maturity date, the weight of short-term debt and medium and long-term debt in the default boundary, the debt recovery rate at the time of default, market risk-free interest rate, financing leverage, etc. When drawing up capital structure plans, fund companies usually use debt ratios and debt financing ratios to assess bankruptcy risks. Assume that D1/S0=0.05 and D2/S0=0.1. We use Matlab software to program equation (16) and iteratively solve equation (17). When ρ = 0.15, 0.2, 0.25, 0.3, the influence of the simulated Hurst parameter H on the default probability of short-term bonds is shown in Figure 1 and Figure 2.

It can be seen that the default probability of the short-term debt of the fund company gradually increases with the increase of the Hurst parameter H of the mixed fraction jump-diffusion fraction Brownian motion. When the risk factor becomes larger, the probability of default also rises. This is consistent with empirical evidence.

Conclusion

This article studies the cost of capital by including capital items in the hedging portfolio strategy. First, a hedging portfolio is designed from the perspective of an incomplete hedging strategy. The portfolio includes stocks, bonds with different recovery rates, stock options, capital, and different cash accounts. This article assumes that the stock price of the underlying asset obeys the jump-diffusion process to increase the rationality of asset price changes. This article also considers the risk of fluctuations in the underlying asset price. Based on this, stock options are added to the hedging portfolio to reduce the risk of stock price fluctuations.

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