# Asteria Step Option

Deribit have been issuing classic European and American options of mainstream crypto assets for years but failed at igniting the market as the same scale as traditional finance. There are three main reasons:

1\. Fragmented liquidity of option market

2\. Professional finance team is necessary

3\. The average crypto traders have difficulties to learn and master option contracts. &#x20;

The first two dilemmas are solved by the Peer-to-Pool model. Firstly, the shared LP capital pool provides the liquidity by having opposite positions available to buyers. Secondly, Asteria build a professional, comprehensive, and stable pricing and hedging system for all automated market makers/liquidity provider&#x73;_._

While only creative product targeted on crypto investor trading pattern would ultimately solve the third problem. Team Asteria designed a user-friendly product specifically for the majority of crypto market players who are in favor of high-leveraged and short-term trading: **Asteria Step Option**.

## **Product Design**

Asteria step option has two options structures I) StepUp (call) and II) StepDown (put). These are designed to match users' two-way demand for bullish and bearish conditions alike. By using a three-level-step structure to facilitate investors' best interests - allowing them to navigate market risk and reward. Asteria Step Option uses the 15-minute option expiration window and if desired realizes the pricing and trading through an automated market maker mechanism. The product is similar to American options in that customers can freely choose to exercise it ten minutes before the expiration of the contract. The payoff structure of StepUp and StepDown options is as follows:

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FgwWD82FOVJ2KL1RJM3oZ%2Fimage.png?alt=media\&token=ac9e5606-476e-4ac5-b530-3faf7fd92399)

**StepUp Option**: During the 15-minute option period, if the price of the underlying asset increases by more than K1, the investor will get the payoff of R1. If the price of the underlying asset increases by more than K2, the investor will get the payoff of R2. If the price of the underlying asset increases by more than K3, the investor gets the payoff of R3.

**StepDown Option**: During the 15-minute option period, if the price of the underlying asset decreases by more than K1, the investor will get the payoff of R1. If the price of the underlying asset decreases by more than K2, the investor will get the payoff of R2. If the price of the underlying asset decreases by more than K3, the investor gets the payoff of R3.

The maximum lifespan of Asteria Step Option contract is 15 minutes, but as the volatility of the underlying asset changes, or according to the volatility of different underlying, the Asteria platform may adjust the length of the option to expiration (the structure of existing options remains unchanged). This streamlines professional option products, reducing the risk of investment and market making for investors.

The automatic quotation system of Asteria Step Option will quote the price of the underlying 5 times at 1 minute interval in the first 5 minutes of the 15 minutes period for investors. At any time, the option buyer would get the best quotation during the 5-minute purchasing period.

Buyers of Asteria Step Option can independently choose their own settlement time according to the price fluctuation of underlying within 10 minutes after the purchasing window is closed, with very high degrees of flexibility - as with American Options. If the position is held until the end of 15 minutes, the contract will be settled at the closing price.

## **Pricing Model**

Asteria Step Option pricing contracts professionally:

Asteria Step Option contract rules: take call option (_StepUp_) for example, is valid for 15 minutes and can be exercised 10 minutes before expiration:

– According to the current underlying asset price<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FnTKh69pGbBoIFWnPVHZM%2Fimage.png?alt=media&#x26;token=1f54ffd0-4941-4148-9f23-d75d2e5def9d" alt="" data-size="line">, three groups of strike prices are set in advance <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FJwbyVu1tnP4jOGVFTOIx%2Fimage.png?alt=media&#x26;token=893e3072-8a84-4c88-81be-966559e6ce54" alt="" data-size="line">.

– When the price of the underlying asset exceeds <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FeWLJFjHDUwS32nst8n8Q%2Fimage.png?alt=media&#x26;token=cff307f6-6813-426d-b510-dbb64e5cd60b" alt="" data-size="line">, if choosing to exercise, the investor can get <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F8MMPZ8JsmzJXmoWrunYa%2Fimage.png?alt=media&#x26;token=fa69b852-57b9-4bef-89e1-e52d38bc7a5a" alt="" data-size="line">, where <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FufcSHbtaAmeMDxl5Y2Au%2Fimage.png?alt=media&#x26;token=e4b2c029-de30-4659-aee1-74da81c5faa2" alt="" data-size="line">.

– When the price of the underlying asset exceeds <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F7GgBaadiavi9GksuUg2w%2Fimage.png?alt=media&#x26;token=e3d20577-8083-4262-9f2a-ef07a6da18de" alt="" data-size="line">, if choosing to exercise, the investor can get <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FlQWYjy8bB0wnnQQRp5oJ%2Fimage.png?alt=media&#x26;token=9b5aaa7c-fe6c-4456-9d87-c76163b8463b" alt="" data-size="line">.

– When the price of the underlying asset exceeds<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FnoS0GhFpjLo4FgRBr4jm%2Fimage.png?alt=media&#x26;token=62b60459-c94d-4d79-aa55-1d95732710ac" alt="" data-size="line"> , if choosing to exercise, the investor can get <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F9LgIgumL5gjTBlQ9Rj4T%2Fimage.png?alt=media&#x26;token=d1df2dff-4c71-4890-ac23-a8552c707e2b" alt="" data-size="line">.

– Each option corresponds to one unit of digital currency (the contract size can be adjusted).

Assuming that the digital currency price<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FTNF2v8LaAKAW62puE2BJ%2Fimage.png?alt=media&#x26;token=8af03572-d6ea-43c5-8b8c-3a6ba4b21143" alt="" data-size="line">under the risk-neutral measure is subject to the geometric Brownian motion of the Black-Scholes-Merton type:

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fro1JaiF62w39sMmxnpdj%2Fimage.png?alt=media\&token=1c28b6ab-36c1-43fb-8ca3-c096dd4c0f58)

Where <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FHeQWtxpzK2bozlCOqVKC%2Fimage.png?alt=media&#x26;token=b0ec6feb-1457-49c1-a84c-fb5cf89e8cf9" alt="" data-size="line"> is the risk-free rate of return per unit time that the market maker can obtain; <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FhSq4YjftAtaQJ86hN7Fo%2Fimage.png?alt=media&#x26;token=d615ad0d-59cb-4c6c-a7fa-f412a289bb50" alt="" data-size="line"> is the unit time volatility of the underlying asset’s rate of return; <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fxa64ZPxh2Aj5TyrprodG%2Fimage.png?alt=media&#x26;token=e260cc38-a043-4495-9c27-10a9677afc86" alt="" data-size="line"> is the Brownian motion under the risk-neutral measure.

According to the Ito lemma, it is known that under the risk-neutral measure, <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FVfbBEBrppai5HJqd9NZq%2Fimage.png?alt=media&#x26;token=f74fc35c-885c-4851-aae3-c7c570ad1bda" alt="" data-size="line">obeys the linear Ito process:

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FeKC7GCLAPtjgSXgqkiXx%2Fimage.png?alt=media\&token=d30f25eb-ed44-4d14-bab7-5a711d66018d)

Where<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FqE4dDDztQbEDEgLvhz24%2Fimage.png?alt=media&#x26;token=31345181-6f98-4d3d-965d-7791e5791fc6" alt="" data-size="original">

Based on the above assumptions, the price <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fwlnrql8yZs3V6rgabB0z%2Fimage.png?alt=media&#x26;token=b3bcb7ed-94b9-4da6-8f1b-76bf674d5a44" alt="" data-size="line"> of any derivative product regarding the crypto currency price <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FqqOVk5wfpYZb45Jc45KW%2Fimage.png?alt=media&#x26;token=f3c36465-798a-40fd-ae24-8c71e3aba918" alt="" data-size="line"> at time <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FtzlrioVFNHmMhB8q30AL%2Fimage.png?alt=media&#x26;token=458173d7-9e95-48d4-acb3-4cac41ea94e6" alt="" data-size="line"> satisfies the following partial differential equation (also known as BSM partial differential equation):

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FcuPtMISzg1JMsCBR5Kb0%2Fimage.png?alt=media\&token=b0d977ad-6d00-49e8-b94b-1901300a28e6)

Here, for easy calculation, <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FzJ3CZec6xdBXFWdIoLAR%2Fimage.png?alt=media&#x26;token=9ea463d9-6624-4eda-983d-f67f3f5b48b7" alt="" data-size="line">is replaced with <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FWtqCjpRwAguIaqa0gp2R%2Fimage.png?alt=media&#x26;token=8d62cb61-bb52-4c6b-8157-d2a3bd6fa364" alt="" data-size="line">.

– For the plain vanilla European call option with strike price K, this partial differential equation satisfies the boundary condition <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fgc6R4NtErQ07KAcfltWs%2Fimage.png?alt=media&#x26;token=440de042-5d41-45e9-a4a0-164c2177832d" alt="" data-size="line">. The solution of this partial differential equation is the Black-Scholes-Merton option pricing formula:

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fes99rQ3M2D9cYh41lHsA%2Fimage.png?alt=media\&token=7093b916-e395-4324-97b4-66593ed963d3)

&#x20;Where ![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FZ1nXXE7CyeX2sBnklflg%2Fimage.png?alt=media\&token=0faa1cd0-000a-478f-813a-b0b847c3d6d4) ，<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FUqTWgXqBXGja1sFFT67l%2Fimage.png?alt=media&#x26;token=4322b1f0-ea36-41ed-9de0-392bd9412622" alt="" data-size="line">，<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fg7ZL6iGn1pB1FpL59nJy%2Fimage.png?alt=media&#x26;token=cffa844d-58f2-40de-af10-5ec47e566d0e" alt="" data-size="line"> is the cumulative distribution function of the standard normal distribution.

– Since the provided options are all American options, in order to provide various products in the future, numerical methods are used to solve the option price (the provided standard contracts have analytical forms, which is conducive to rapid and accurate risk management of mainstream contracts). Commonly used methods include binomial tree option pricing, or more naive Monte Carlo simulation, and finite difference.

For the sake of calculation speed and accuracy, the finite difference method (**Crank-Nicolson**) is used to calculate the option price.

– First, divide the time from 0 to the expiration time _T_ into _N_ equal parts <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FIzrx74Yv1r6zXCzn1ikJ%2Fimage.png?alt=media&#x26;token=1d9927e0-7620-4fb7-bf7b-b68bdc9d597c" alt="" data-size="line">, and the length of each part is recorded as<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FZ7PBDMSHDAQLemcJ1J3g%2Fimage.png?alt=media&#x26;token=b00c140f-9d3c-4fee-9469-38fdc2488b49" alt="" data-size="line">; at the same time, the logarithmic price <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FPgHJIbdZznQcMrdxEoIa%2Fimage.png?alt=media&#x26;token=9924423e-4802-4370-aa3d-a20225709a72" alt="" data-size="line"> of digital currency is divided into _2M_ equal parts, namely <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FzJN8QWeWsXs7NoX9b3rr%2Fimage.png?alt=media&#x26;token=3aa9e604-8e37-47f4-8c0a-26f86412ccb8" alt="" data-size="line">, where <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F0vjjGmn27BFzyC0nPNih%2Fimage.png?alt=media&#x26;token=9d9b92d2-5c73-49fc-b7f4-60fe54b0577c" alt="" data-size="line"> and the length of each part is recorded as <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FzfdNo5EoepQAyDctbHIq%2Fimage.png?alt=media&#x26;token=f7b9dc17-160e-4128-b452-42a0d2bf17c1" alt="" data-size="line">. Here,  <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FQl1AiwnIlyxjSO1mYPII%2Fimage.png?alt=media&#x26;token=b7f3b3c4-393f-4559-93bc-7210d3316ac3" alt="" data-size="line">is used to describe the limit of the logarithmic price, usually considering the logarithmic price of 3 standard deviations, that is, <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FstY8Sm5Il7sr88aMUahQ%2Fimage.png?alt=media&#x26;token=d52ac1f0-a80c-4de4-873c-c561ed7fd711" alt="" data-size="line">.

– Mark the divided time and logarithmic price on the coordinate axis, where the horizontal axis is time and the vertical axis is logarithmic price, thus obtaining a two-dimensional grid. The next step is to calculate the option price for each point on the grid. The price of the option that needs to be quoted corresponds to the point <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FHV6hxsPe9S7u0MkLMh1Z%2Fimage.png?alt=media&#x26;token=2016b53d-4490-44c1-8aa1-73c02642a907" alt="" data-size="line"> on the grid.

– Let the price of the option at point <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FK7Dj7alkQfWE0yYdfzM2%2Fimage.png?alt=media&#x26;token=84e47fb1-89ed-4d60-9c24-2c2ba506177e" alt="" data-size="line"> be<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FOLOtTLGBdISxhomDNspr%2Fimage.png?alt=media&#x26;token=e8aeeddc-97fb-4eb5-9563-943a1ea3e741" alt="" data-size="line"> . Since the BSM partial differential equation holds for any point on the grid, corresponding option price can be calculated starting from the penultimate column of the grid, namely <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F0iM57uwihXrypMgNvpsR%2Fimage.png?alt=media&#x26;token=7d61ed4c-7ac8-4b58-84c7-300740d12dae" alt="" data-size="line">. Take <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FftKu276bxgJBOGCHG2Vo%2Fimage.png?alt=media&#x26;token=a0dec4f5-bc04-4f77-919c-149168d86ec1" alt="" data-size="line"> as an example,

• Use <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FMEHEiSM7lPrvCkO79gy7%2Fimage.png?alt=media&#x26;token=ba96c22d-6ea0-46c0-a592-b134c13b39d9" alt="" data-size="original"> to estimate the option price in the middle of the two moments <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FDPkMsNHnmMSo4RO8QpoA%2Fimage.png?alt=media&#x26;token=92ab9831-c2dd-494a-94eb-ab03b7942102" alt="" data-size="line">; use <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F9NXj05Hx4n68zZqzAClH%2Fimage.png?alt=media&#x26;token=c95c4924-c25d-41b5-bf70-4f68da0495ce" alt="" data-size="original"> to represent the option price in the middle of these two moments, but the corresponding underlying price is the <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FED9IDMpfbdRf1yIEI8B7%2Fimage.png?alt=media&#x26;token=10c04bf9-9c64-49a3-a243-640e8ff4e63a" alt="" data-size="line"> point; use ![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FYzNurUON8tOD5WswflWk%2Fimage.png?alt=media\&token=5a08570a-615e-465c-8909-cb6e7f19fc1f) to indicate the option price in the middle of those two moments, but the corresponding underlying price is the <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F8HKCkphkAtz3ZTNOlevo%2Fimage.png?alt=media&#x26;token=346026e5-6ed3-4179-b57a-8804128082eb" alt="" data-size="line"> point.

– So one can estimate the partial derivative of the option price in the middle of those two moments with respect to time _t_ and the first and second partial derivatives of the logarithmic price _X_.

• ![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fs1Vuel6XWAd49cE3olEl%2Fimage.png?alt=media\&token=53388da8-07a8-4cbd-b253-f09cb1c8780d)

• ![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FYltgZHqK9h7VOAEMnMr3%2Fimage.png?alt=media\&token=8f6b5eeb-bdfd-4c07-a55f-81c0db3a1aa0)

•![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FkWCD1YuipQlA52REs5vf%2Fimage.png?alt=media\&token=742ac838-d1cf-49cc-8937-6bf432b2c361)

– Substituting partial differential equations gives the following equation:

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FKJ6ajXCH2iuCDSM6EXWM%2Fimage.png?alt=media\&token=de5a4ca7-fa8f-4a35-ad00-726f3863d655)

Where:

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FP1cHJDdavXF7XTqD4Xsz%2Fimage.png?alt=media\&token=27ab09ab-e50f-4324-8bb3-6622476d84bd)

– Since the values of <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F6RUTQltIdLastfNIDHa4%2Fimage.png?alt=media&#x26;token=051eb94a-9270-47fe-a09e-4492e0a2325f" alt="" data-size="line">,<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fc4x2fueuEMOdTEiqFfsO%2Fimage.png?alt=media&#x26;token=0635d52a-b10d-4d85-859d-cc030fa5d851" alt="" data-size="line"> and <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FgSlcOSe1Q1hiwhUGkfMw%2Fimage.png?alt=media&#x26;token=464681b9-36f1-4bc5-8745-69089a4e89ed" alt="" data-size="line">are equal to the intrinsic value of the option, all information on the right-hand side of the equation is known. For example, <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FszFnwk3eKzpZopMvUTs2%2Fimage.png?alt=media&#x26;token=10e07055-9bec-4e3c-ab2a-b5bb9769faaa" alt="" data-size="line">at time <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FraXaZGeLBKEiwuX72aNi%2Fimage.png?alt=media&#x26;token=8dbccacf-e95a-41b8-ba64-a05cfae5ece7" alt="" data-size="line">, when <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FH8Onn6hWQkveVxpEDs6Y%2Fimage.png?alt=media&#x26;token=4fa86d2b-73ae-4b74-b89f-14948e84512c" alt="" data-size="line">.

– Therefore, it is only necessary to list the linear equations for each<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FCg8gW68O2llKWaXxkvUH%2Fimage.png?alt=media&#x26;token=b46dbceb-e1bd-4ca0-ade8-62ca70dd5672" alt="" data-size="line"> at time<img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FkSTYuaIlXIcKNMC1bwWJ%2Fimage.png?alt=media&#x26;token=c748be6a-e00c-4b5b-aae4-5e1905c59943" alt="" data-size="line">, and then solve the linear equations to obtain the option price at time <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FieOHrWObWpU6RFGkMvie%2Fimage.png?alt=media&#x26;token=a262790f-f93a-483b-93e2-6a78a41af259" alt="" data-size="line">. Then just repeat the process at time <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FnTzhF9YRgShcYoV5rH7W%2Fimage.png?alt=media&#x26;token=d2e2802c-b9d5-4489-a971-ba44afee78cf" alt="" data-size="line"> and before to get the option price at each node.

– Therefore, at each <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F27ELVdhPYmDVB6hYSaEY%2Fimage.png?alt=media&#x26;token=ecb36a4a-c035-4fac-b9bc-a111a376266f" alt="" data-size="line"> time <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F91A4eYYHhVsCV0VGIBly%2Fimage.png?alt=media&#x26;token=386041d8-5a00-4b16-9584-519e6eb130f9" alt="" data-size="line">, there are the following <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FTDGw7cAvI3YwbuZr0qqY%2Fimage.png?alt=media&#x26;token=ca7c977b-8ab2-4cc9-973c-e6b7cd32f7f9" alt="" data-size="line"> linear equations, satisfying:

![](https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FxBhBshq9zxr77Lb6IQLj%2Fimage.png?alt=media\&token=17ba067d-68a5-4e0e-8d17-1e186b9ab153)

&#x20;Where <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Frxwm6glIFsK2ueycs4lb%2Fimage.png?alt=media&#x26;token=17cc76d3-0b91-4c19-870d-b6ac5c96b95e" alt="" data-size="line">, <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FTqJHHlHmjkBNfyLm8uPX%2Fimage.png?alt=media&#x26;token=5a260e18-7899-427e-bf90-b1e838c86dd2" alt="" data-size="line"> corresponds to the option price with the underlying logarithmic price <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FqJUYJfLDXbIIz7X277NJ%2Fimage.png?alt=media&#x26;token=b944e3d3-6823-47ac-b6c9-7b30804e39b0" alt="" data-size="line">at time <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FHt2XMuQDDh9UuuTvAkrK%2Fimage.png?alt=media&#x26;token=fd947193-d382-4e02-a577-326684500ab3" alt="" data-size="line">, and so on; <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FCR83qw7RxUkfsZkBH4jI%2Fimage.png?alt=media&#x26;token=c671f136-5688-4e00-ba78-69864576c76c" alt="" data-size="line">, and so on; when the underlying price is at <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fo5sqRuMlLaZ0UKMk8Qas%2Fimage.png?alt=media&#x26;token=5bfd60e3-a930-4ee1-a77c-9399e439669f" alt="" data-size="line"> and <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FlxQOG5rIRBwL2JIieGvC%2Fimage.png?alt=media&#x26;token=952343d6-609a-4e27-905c-bdcaa929116e" alt="" data-size="line">, determine <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FrlUqZNkAm4QJpsIDPnJc%2Fimage.png?alt=media&#x26;token=7fdefc39-d240-4fb2-acd3-60fbfb96e953" alt="" data-size="line"> and <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FucGqqWHk321Etb3JORQP%2Fimage.png?alt=media&#x26;token=25f028c7-35d8-44f8-8dde-6bbd0caddec2" alt="" data-size="line"> according to the boundary conditions of different option structures.

– Since the American Step option is provided, after calculating the option price at each time <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FggrQ7JDRnSm0u540rNc9%2Fimage.png?alt=media&#x26;token=e6731710-b547-4175-a28d-4e0eb834e529" alt="" data-size="line">, it is necessary to compare the calculated option price with the intrinsic value of the option at this time to judge whether the investor will choose to exercise the option. Therefore, in the iterative process from <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FVuMB174fjw5FMVuEWbde%2Fimage.png?alt=media&#x26;token=3aa73122-9ed6-4e69-953d-990b7c67dc9a" alt="" data-size="line"> to <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2Fwu5sWOaHTNbO6zpxspCE%2Fimage.png?alt=media&#x26;token=b8071d80-0fd5-4438-bd34-13b6cc90d3a4" alt="" data-size="line">, it is necessary to retain the larger one of the calculated option price and intrinsic value. Note: When investors are not allowed to exercise their rights in the first five minutes of the contract, there is no need to compare option prices and intrinsic values in this range.

– For options contracts, when the underlying logarithmic price is at <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F2wVrn7sqZgq2jDiEolTX%2Fimage.png?alt=media&#x26;token=a0b3fd08-e7f4-4dc4-bf06-e5ac5eca6173" alt="" data-size="line"> or <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FgER5VaIhQNr5JH5lK5Hv%2Fimage.png?alt=media&#x26;token=df02759e-c3aa-46b4-8742-0661cac3f7b1" alt="" data-size="line">, there is <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2F1ESSoB65gEPOl3J3lmew%2Fimage.png?alt=media&#x26;token=10ed5fb4-eaa6-4a08-852a-e5affe1af648" alt="" data-size="original">. Therefore, it can be assumed that <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FrczI8vmqpUrXDPkD5nk1%2Fimage.png?alt=media&#x26;token=da012eb1-144d-4b53-b9d3-dc1b871a46dd" alt="" data-size="line">, and <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FC3FNiFnnSTm6jvQB68F0%2Fimage.png?alt=media&#x26;token=1c0fdf50-2985-4fe3-bef4-95fb889f0166" alt="" data-size="line">, that is, <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FjE6R1slWsa0TSDu0NIKF%2Fimage.png?alt=media&#x26;token=a7203361-4fe7-45e3-a8d2-a216262d5bb4" alt="" data-size="line">, and <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FGBYN1ANcdbxEd4nEe6A7%2Fimage.png?alt=media&#x26;token=bb384df7-f208-444a-a84c-2a458f5c08d4" alt="" data-size="line">. This result can actually help set <img src="https://1317347883-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-M_8X8tg9KcpJeRr0sk6%2Fuploads%2FckCMe4lJL77rO8vWSBME%2Fimage.png?alt=media&#x26;token=1c08ef0c-a46d-480d-bc3b-010b742d9568" alt="" data-size="line"> more efficiently.

