Binomial Model to Black-Scholes Model: A pokemon analogy
Take this journey from Binomial model to Black-Scholes model through a pokemon analogy
Pokemons are pocket monsters. Each pokemon has its own strengths and weaknesses. Also, a pokemon can evolve and its evolved form is even more powerful than its less evolved version. For example, Pikachu (my favorite one) can evolve to become Raichu and its brute strength increases significantly although at the cost of agility.
Let’s consider our own Pikachu- Binomial model. It can price European options by taking into consideration below variables:
- Size of up and down moves: U and D
- Probability of an up move: (exp{rt}-D)/(U-D)
- Probability of a down move: 1-U
Like our Pikachu, this model is highly flexible because it can be used to price both European and American options. It does so by making a tree of prices based on size of up and down down and multiplying them with their corresponding probabilities. Next a corresponding tree of option’s payoffs is formed by calculating the respective payoffs at each node of the tree. As the last step, the final price can be calculated by taking the expected value at terminal nodes and multiplying it with the discount factor to obtain the present value. However, the Binomial model lacks strength because it does not exactly know the size of up and down moves as these parameters are dependent on volatility, so these parameters need to be provided as inputs.
Now, our Pikachu evolves to become Raichu, i.e., Black-Scholes model. Raichu has immense strength because it can price European options very effectively (given that assumptions of the model hold true). It does so by incorporating the volatility in the model itself. Black-Scholes model incorporates volatility as a stochastic component and actually this is the addition they did which led to them winning the Nobel prize formula.
