The pricing and hedging of financial derivatives have become one of the hot research issues in mathematical finance today. In the case of non-risk neutrality, this article uses the martingale method and probability measurement method to study the pricing method and hedging strategy of financial derivatives. This paper also further studies the hedging strategy of financial derivatives in the incomplete market based on the BSM model and converts the solution of this problem into solving a vector on the Hilbert space to its closure. The problem of space projection is to use projection theory to decompose financial derivatives under a given martingale measure. In the imperfect market, the vertical projection theory is used to obtain the approximate pricing method and hedging strategy of financial derivatives in which the underlying asset follows the martingale process; the projection theory is further expanded, and the pricing problem of financial derivatives under the mixed-asset portfolio is obtained. Approximate pricing of financial derivatives; in the discrete state, the hedging investment strategy of financial derivatives H in the imperfect market is found through the method of variance approximation.

#### Keywords

- B-S-M model
- nonlinear differential equation
- financial market
- financial derivatives

#### MSC 2010

- 62J12

Financial derivatives are derived from basic financial assets and are a very important part of global financial innovation in the 1970s and 1980s. Financial derivatives have the functions of arbitrage, risk transfer and price discovery. Financial derivatives can be divided into four types of contracts: forwards, futures, options and swaps according to their own product forms [1, 2]. If divided according to the relationship between the price of financial derivatives and the price of its underlying assets, financial derivatives can be divided into two categories: linear derivatives and non-linear derivatives: linear derivatives include forward contracts and futures contracts With swap contracts, there is a linear relationship between the price of such derivatives and the price of the underlying asset; nonlinear derivatives include options, structured derivative securities, and exotic derivative securities, and their prices are very close to the price of the underlying asset [3, 4]. Complex nonlinear relationship. Because the structure of linear derivatives is relatively simple, this article will only briefly explain it. This article mainly studies the pricing and hedging of nonlinear derivatives, which mainly refers to the pricing and hedging of options.

Futures contracts and forward contracts have similar properties. The main difference between the two is that futures contracts are customised and issued by exchanges. They are standardised contracts and are generally settled centrally by settlement companies and have a unique settlement system; forward contracts are generally traded off-exchange by both parties to the transaction. Therefore, in many literature research records at home and abroad, it is believed that futures and forwards can use the same pricing model. When the forward and futures contracts of the same underlying asset have the same expiry date, their contract prices are also very similar [5, 6]. Any financial product pricing model must have certain prerequisites, while the forward and futures contract pricing models include the following four basic assumptions: (1) There are no transaction costs, and the number of transactions can be subdivided indefinitely; (2) The tax rate for all trading profits is the same; (3) All traders borrow or lend funds at the same risk-free interest rate; (4) There is no risk-free arbitrage opportunity in the market.

The pricing includes the following three situations: The pricing formula of a forward contract in which the underlying asset does not pay income: ^{r}^{(T−t)}. The pricing formula of a forward contract for which a security pays a known cash return: ^{r}^{(T−t)}. The pricing formula of a forward contract for which a security pays a known dividend rate: ^{(r−q)(T−t)}. Swap can be regarded as a combination of a series of forward contracts. Therefore, the pricing of the swap can be obtained through the combination of a series of forward contract prices, and the pricing method of the swap contract can be derived from the pricing of the forward contract.

We assume that the price process {_{t}_{t}_{t}

Assuming that Eqs. (1) and (2) are satisfied and the underlying risk asset does not pay dividends within the validity period, then for European option _{T}_{T}_{T}_{T}_{T}_{T}_{+}, the price at time _{T}_{T}_{+}, the price at time _{T}_{T}_{+}, where
_{2} − _{1}, let ^{v}^{v}dV

Get:
_{2} can be derived in the same way. Let the variable substitute:

Get:
_{2} − _{1},

Assuming that the price process of the risk underlying asset stock {_{t}_{t}_{t}

Assuming that {_{t}_{t}

The price of the European call option _{T}_{T}_{+} at _{T}_{T}_{+}, the price at time

The trading strategy {_{T}_{T}_{t}^{*} measurement:

And because
^{*}, _{u}

In an incomplete market, even if there is no arbitrage opportunity in the price system because there are multiple equivalent martingale measures, any financial derivative product may correspond to multiple pricing. For the pricing of financial derivatives to be unique, other pricing methods must be introduced on the condition of the principle of no-arbitrage pricing, resulting in a variety of different approximate pricing methods. Suppose there are _{t}_{t∈(0,T}) is the price process of risky assets, and ^{0} ≡ 1 is risk-free assets. These are all defined in the probability space (Ω, _{t}_{t∈[0,T}], and _{t}^{0}, _{t}_{t}_{t∈[0,T]}, where _{t}_{t}_{t∈[0,T]}, and _{t}_{t}_{t∈[0,T]}. For any time _{t}_{t}_{t}_{t}_{t}

From the definition of self-financing transaction, we know that when the cost process C of the hedging strategy is constant, this transaction strategy can be called self-financing. In an incomplete market, for the unreachable financial derivative asset H, there is no self-financing strategy (_{T}_{T}_{T}_{T}_{T}_{T}_{T}_{T}_{T}_{T}

The investment strategy is called a self-financing strategy, and its value process _{t}_{t∈[0,T]} can be decomposed into the sum of a constant and a random integral about _{t}

If there is no arbitrage opportunity in a given price system, then _{e} ≠ ∅.

The requirement of _{T}_{T}^{2}(Ω, _{t}^{2}(Ω, _{t}^{2}(Ω, _{t}^{2}(Ω, _{t}^{2}(Ω, _{t}_{T}^{2}(Ω, _{t}_{T}_{0}, _{0} is a risk-free asset, its price is always equal to 1, and _{e}, The financial derivative ^{Q}^{Q,H}) = 0 is the self-financing strategy.
_{e} is ^{P,H}^{Q}_{T}^{Q,H}_{T}^{P,H}_{e}, this situation needs to be transformed into situation 1 to solve the problem. The steps are as follows: First, the original price system needs to be converted and measured, so that the transformed price system is a martingale measure. The situation can be used when the above conditions are met. 1 method to answer [6].

When _{T}_{0}, _{e}. Assuming that
_{T}_{T}^{S0},^{S}

Then

In the discussion in the first two sections of this chapter, the approximate pricing of financial derivatives

Assuming that _{1}, _{2},···_{n}_{i}_{i}X_{i}_{i}X_{i}_{1}, _{2}, _{n}

In the discrete case, all financial derivative investments _{i}_{j}

When the underlying asset of the financial derivative H is one of the risky assets (the first type), then for the call option:

Put options:

From the calculations of

The paper introduces the pricing methods and hedging strategies of financial derivatives in the context of non-risk-neutral significance. Through the distribution of the option price process, using the equivalent martingale measure and the actual probability measure, without paying dividends, the generalised European option pricing formula is derived, and the parity between European call options and put options is also obtained. Relationship; then expand the pricing method of financial derivatives that do not pay dividends to financial derivatives that pay continuous dividends. Using Ito's formula again, the specific hedging strategies of European call options and put options are obtained under the conditions of paying dividends and not paying dividends. These results also include traditional European option pricing formulas and hedging strategies in the sense of risk-neutral pricing. The derivation of these conclusions and the choice of strategies have important practical economic significance, especially when the price of the underlying asset fluctuates drastically, and the expected return rate of the underlying asset differs greatly from the risk-free interest rate in the market, which has a wider application economy. value. These results are of great help to the further study of option pricing and hedging issues in the future.

We start by constructing subjective evaluation indicators and objective evaluation indicators based on game theory to balance personal empowerment and objectiveness. Then, the weight set is assigned to determine the complete optimal weight result. Finally, in response to the above theoretical results, suggestions for improving college students’ creative ability are proposed: creating a relaxed learning environment for practicing innovative personality development, encouraging students to understand cutting-edge technologies in science and technology, condensing scientific research issues, and proposing innovative ideas and opinions; building a multi-level open development platform, provide a suitable environment for students to practice hands-on; strengthen the ability of teachers of innovation and entrepreneurship, and escort the cultivation and development of students’ creative ability. In addition, attention should be paid to cultivating students’ innovative ideological awareness and practical ability, strengthening the accumulation of knowledge and ability, and enhancing the transformation of results to implement each training link.

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