How to use middleware, asyncData, fetch in Nuxt.js v2 with typescript and Option API

I'm working hard to introduce typescript to Nuxt.js with Option API.

In Current Code, we use middleware, asyncData and fetch methods. After I switched lang to ts (<script lang="ts">), typescript says error for the above methods.

like this (about middleware)

 Object literal may only specify known properties, and 'middleware' does not exist in type 'ComponentOptions<Vue, DefaultData<Vue>, DefaultMethods<Vue>, DefaultComputed, PropsDefinition<DefaultProps>, DefaultProps>'.

Then, I googled this problem and found this solution, but it does not work maybe because this solution is about Nuxt with Class API...

summary of my .vue file:

<script lang="ts">
import Vue from 'vue'

export default Vue.extend({
  async middleware({store, params, from}) {
    // Error is occuring here...
    // Object literal may only specify known properties, and 'middleware' does not exist in type 'ComponentOptions<Vue, DefaultData<Vue>, DefaultMethods<Vue>, DefaultComputed, PropsDefinition<DefaultProps>, DefaultProps>'.
  },
  asyncData({params}) {
    ~~~
  },
});
</script>

tsconfig.json:

{
  "compilerOptions": {
    "target": "ES2018",
    "module": "ESNext",
    "moduleResolution": "Node",
    "lib": [
      "ESNext",
      "ESNext.AsyncIterable",
      "DOM"
    ],
    "esModuleInterop": true,
    "allowJs": true,
    "sourceMap": true,
    "strict": true,
    "noEmit": true,
    "baseUrl": ".",
    "paths": {
      "~/*": [
        "./*"
      ],
      "@/*": [
        "./*"
      ]
    },
    "types": [
      "@types/node",
      "@nuxt/types"
    ]
  },
  "exclude": [
    "node_modules"
  ]
}

Is there any way to use those methods in Nuxt with Option API ?