Skip to content

Quant

Black-Scholes

  • Originally to valuate European call options
  • American equivalents: Bjerksund-Stendland model, binomial, trinomial models

  • Uses 5 Factors:

    1. Volatility
    2. Price of underlying asset
    3. Strike price
    4. Time to expiration
    5. Risk free interest rate

  • Black-Scholes Asumptions

    • Price follows a random walk approximately Geometric brownian motion with constant drift and volatility (i.e. log(variance) is constant)
    • No dividends over life of option
    • Movements are random, market is random
    • No transaction costs
    • RFR and volatility are constant (not a strong assumption for volatility, since that is influenced by supply/demand)
    • Returns are log normal
    • Option is European (can only be exercised at expiration)

Greeks

  1. Delta

    First derivative with respect to price. Rate of change of equilibrium price (aka BS price) with respect to asset price.

  2. Gamma

    Second derivative with respect to price.

  3. Theta

    First derivative with respect to time-to-maturity. Rate of change of equilibrium price with respect to time-to-maturity.

  4. Vega

    Rate of change of equilibrium price with respect to asset volatility.

  5. Rho

    Rate of change of equilibrium price with respect to RF interest rate.

Pay Off Diagrams

  • Plot of Underlying Price vs. P&L
  • 3 Key Points:
    1. Maximum Loss
    2. Maximum Gain
    3. Break-even Point

Call Options

  • Break-even: K + P (where K is strike price and P is cost of option)
  • 5 reasons to buy a call option
    1. Bet on upside move with minimal cost (lot of a exposure for little cost)
    2. Unlimited Upside
    3. Limited Downside: Can only lose what you paid for the option
    4. Increase in Volatility: Option is priced based on its volatility, so all we need is an increase in volatility to increase the value of our option
    5. Hedge Short Position: Unlimited upside offsets risk of short as shorts have unlimited downside

Call-Spread

  • Max Value: difference in strike prices. \(v_{max} = K_2 - K_1\)
    • Where \(K_2=Sold\) and \(K_1=bought\) strike prices
  • Max Loss: \(\text{max_loss} = \text{max_value} -P_{cs}\)
    • Max value - Price of call-spread

Put-Call Parity

  • Represents an arbitrage opportunity
  • \(\text{call_price}+\text{present_value_discounted} = \text{put_price} + \text{spot_price}\)
    • (where present value is discounted from the value at RFR)

Questions

  • What factors in production could cause a backtested strategy to perform different than expected?
    • Slippage, transaction costs, systemic risk, outside events that cannot be modeled such as state of global economy/climate/legislation/etc