Option portfolio and Black Scholes

Quantitative Risk Management in R

Alexander McNeil

Instructor

European options and Black-Scholes

  • European call option: gives right but not obligation to buy stock for price K at time T
  • European put option: gives right but not obligation to sell stock for price K at T
  • Value at time t < T depends on:
    • Stock price S, time to maturity T-t, interest rate r, annualized volatility $\sigma$ or sigma
    • Pricing by Black-Scholes formula
Quantitative Risk Management in R

Pricing a first call option

K <- 50

T <- 2

t <- 0

S <- 40

r <- 0.005

sigma <- 0.25

Black_Scholes(t, S, r, sigma, K, T, "call")

2.619183

Black_Scholes(t, S, r, sigma*1.2, K, T, "call")

3.677901
  • Price increases with volatility
  • Option above is out-of-the-money
Quantitative Risk Management in R

Implied volatility X needs change

  • Volatility not directly observable
  • Market participants use implied volatility, the value of volatility implied by quoted option price
Quantitative Risk Management in R

The VIX index

plot(VIX)

Quantitative Risk Management in R

Let's practice!

Quantitative Risk Management in R

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