Fundamental theorem of derivatives trading by Jesper Andreasen

Assume zero rates and zero dividends.

Assume standard frictionless markets.

Assume the underlying stock evolves continuously.

There exists two stochastic processes \(\mu, \sigma\) such that

\[\frac{dS(t)}{S(t)} = \mu(t)dt + \sigma(t)dW(t)\]

under \(\text{real measure } P\)

let V be the value of an option book S priced on a model with constant volatility

\[\bar{\sigma}\]

Assume the book is delta hedged, we dynamically trade the stock to keep

\[V_S = 0\]

Theorem 1:

Technical:

The value of option book evolves according:

\[dV(t) = \frac{1}{2} (\sigma(t)^2 - \bar{\sigma}^2)S(t)^2V_{SS}(t)dt\]

Proof:

as interest rate is zero, and \(V_S\) is kept constant then

\[dV = 0\] \[V_SdS = 0\]

then we have

\[dV = V_tdt + V_SdS + \frac{1}{2}V_{SS}(dS)^2 = V_tdt + \frac{1}{2}V_{SS} \sigma^2 S^2dt\]

and finally we have

\[dV = V_tdt + \frac{1}{2}V_{ss}(dS)^2 = 0\]

Looking at the equation above, we see that

  • If we are gamma long, \(V_{SS} > 0\) and the realized volatility is higher, we will make profit
  • The option trader’s job is really about balancing realized against implied volatility
    • realized volatility > implied volatility : go long gamma
    • realized volatility < implied volatility: go short gamma
  • In Black-Scholes context, we can see theorem 1 as an investigation of the self financing condition

Application:

  • Black-Scholes implied volatility has to satisfy:
\[E^Q[\frac{1}{2} \int_{0}^{T} (\sigma(u)^2 - \bar{\sigma}_{impled})S(u)^2V_{SS}(u)du] = 0\]
  • Implied volatility is a weighted average of Q expected volatility
  • Consider an option seller that delta hedges his short option position with short gamma:
\[V_{SS} < 0\] \[V(T) - V(0) = \frac{1}{2} \int_{0}^{T}(\sigma(t)^2 - \bar{\sigma}_{impled})S(t)^2V_{SS}(t)dt\]
  • His total profit will be positive only if most of the time, the below holds:
\[\bar{\sigma}_{implied} > \sigma(t)\]
  • Assuming the option trader is risk averse, so to short gamma, the implied volatility when writing a short position, we expect that:
\[\text{implied volatility} > \text{historical volatility}\]
  • We can understand the gap between implied volatility and historical volatility as premium to underwrite the option, not an arbitrage opportunity