From a model to a hedge
An option is worth what it costs to hedge it. This lab walks through the whole chain on SPY prices. First estimate a geometric Brownian motion from the daily returns. The volatility σ is the standard deviation of the returns scaled to a year. The drift of the price is μ=r̄·252+σ²/2: the extra σ²/2 comes from Itô's lemma, because the log price grows more slowly than the price itself. Then simulate paths from a matrix of normal draws, once with the exact scheme and once with the Euler scheme. The same draws feed both, so you can compare them path by path.
Next, price a call two ways. The Black-Scholes formula is the exact average under the risk-neutral measure. Monte Carlo reaches it by simulating with the drift r, averaging the payoff and discounting. It is never exact: it is off by about its standard error. Finally, hedge. Sell the call, buy Δ shares and keep the rest in cash. At each rebalancing date, let the cash grow at the rate r, recompute Δ with the time that is left and trade to it. The error at maturity is what the hedge leaves behind.
- Exact and Euler: the same draws give two paths. The gap between them is the discretisation error of the last topic.
- Price = cost of the hedge: with continuous trading the hedge is perfect, with few rebalances an error of order 1/√n is left.
- The real error is a volatility error: rebalancing more often cannot fix a wrong σ.
Lab · what's graded
- Estimate the model10 pts
- Exact paths15 pts
- Euler paths10 pts
- The Black-Scholes call15 pts
- The delta5 pts
- Monte Carlo price10 pts
- The hedge25 pts
- The hedging study10 pts