Part I: Win Probability as a Stochastic Process

A novice’s description of the Bachelier model for option pricing.

A Novice’s Guide to the Simplest Option Pricing Model

Binary European Call Option

Let’s unpack that.

• Option - The right, but not the obligation, to buy (call) or sell (put) an underlying asset at a predetermined “strike” price.
• European - Type of option where the option can only be exercised at its expiration date.
• Binary - Option that is worth either 1 (e.g., underlying asset price is greater than strike at expiration for a call) or 0.

The Bachelier Model

How do you price an option now that may or may not have value at a pre-defined expiration date? A common approach in quantitative finance is to assign a dynamic to the underlying asset price. Given this assumed dynamic, one can either use stochastic calculus or simulate several trajectories of the underlying to determine a price based on this ensemble of possible trajectories.

The Bachelier Model is the simplest such model.

• Ignore the opportunity cost of risk-free return on the option’s price (“premium”).
• Assume T discrete periods until expiration.
• Assume each period is independent.

The threat of arbitrage forces the underlying’s price to be a martingale: that is, in expectation, the price at the next period must be the price at the current period (otherwise, a risk-free profit could be obtained).

By assumption, we’ll fix the volatility of the asset and assume each period’s change in follows a Normal distribution.

Therefore, the change in price each period can be modeled as a random normal with mean 0 and a fixed standard deviation.

Analytical Approach

Assume the change in price of the underlying at each period is normally distributed with mean 0 and standard deviation .

Assume there are T discrete periods until expiration.

A basic result from probability is that the sum of independent normals is itself a normal, where the summation’s mean is the sum of the combined means and the summation’s variance is the sum of the combined variances.

Then the price of the underlying asset at time T is the initial price plus a normally distributed variable.

\[ P_t = P_0 + \sum_{i = 1}^{T}(N(0, \sigma)) \]

Each period’s variance is \[\sigma ^ 2\].

The variance for T periods is \[T(\sigma ^ 2)\].

The standard deviation for T periods is \[\sqrt{T} * \sigma\].

Concrete Example

Assume the current price of an underlying is $100. Assume the underlying’s daily price movements follow a normal distribution with mean 0 and standard deviation 1. Price a binary call option that expires after 25 days and is either (1) worth $1 if the underlying’s price exceeds $110 or (2) expires worthless otherwise.

After 25 days, the change in the underlying’s prices is normally distributed with standard deviation \[\sqrt{25}*1 = 5\].

What’s the probability of a $10 price increase for the underlying? That’s two standard deviations above the mean of 0. Using the usual rules of thumb, about 4% of observations are greater than 2 standard deviations away from the mean (above or below). So a $10 increase should be have about a 2% chance.

Show code

sd <-
  sqrt(25)

pnorm(10, mean = 0, sd = sd, lower.tail = F) %>%
  round(., digits = 3)

[1] 0.023

Therefore, the expected value of this binary call (under this model’s assumptions) is roughly $0.02.

Simulation Approach

Show code

sim_trajectory <- function(price_init = 100, mean = 0, period_sd = 1, periods = 25){
  
  p_0 <- price_init
  
  p_t <- rep(NA_real_, periods)
  
  period_price_changes <- rnorm(n = periods, mean = mean, sd = period_sd)
  
  for(i in 1:periods){
    if(i == 1){
      p_t[i] = p_0 + period_price_changes[i]
    } else {
      p_t[i] = p_t[i -1] + period_price_changes[i]
    }
  }
  
  return(p_t[periods])
}

final_prices <-
  replicate(10000, sim_trajectory())

hist(final_prices)

Show code

sum(final_prices > 110)/length(final_prices)

[1] 0.02