In the Black-Scholes' setting, the delta hedge ratio of a European call option is given by $N(d_1)$, which is an increasing function of the underlying equity spot $S_0$. Does this property hold generally, that is, for general cases where the volatility may be stochastic and may depend on the spot $S_0$?

  • $\begingroup$ what if spot goes up but at same time vol goes down? i think it is possible that delta then goes down. Any argument against this line of thinking? $\endgroup$
    – mbison
    Dec 21 '15 at 19:50
  • $\begingroup$ see quant.stackexchange.com/questions/22135/… for a related question that I think will give some counterexamples $\endgroup$
    – Mark Joshi
    Aug 27 '16 at 6:52

For homogenous diffusion models (i.e. models such that the distribution of $\ln(S_t)-\ln(S_0)$ is level-independent, e.g. Black-Scholes, Heston, Bates etc.), this would indeed hold.

To illustrate this, consider an exponential Lévy model for the spot price under the risk-neutral measure $\mathbb{Q}$ $$ S_t = S_0 e^{X_t},\ \forall t \in [0,T] $$

The price of a call option expiring at $T$ and struck at $K$ in that case reads $$ C_0 = C(S_0;K,T) = \Bbb{E}^\Bbb{Q}_0 \left[ \frac{(S_T - K)^+}{B_T} \right] $$ Now computing the delta yields \begin{align} \Delta &= \frac{\partial C_0}{\partial S_0} \\ &= \Bbb{E}^\Bbb{Q}_0 \left[ \frac{1\{ S_T \geq K\}}{B_T} \frac{\partial S_T}{\partial S_0} \right] \\ &= \Bbb{E}^\Bbb{Q}_0 \left[ 1\{ S_T \geq K\} \frac{S_T B_0}{S_0 B_T} \right] \\ &= \Bbb{E}^\Bbb{S}_0 \left[ 1\{ S_T \geq K\} \right] \\ &= \Bbb{S}( S_T \geq K ) \\ &= \Bbb{S} ( X_T \geq \ln(K/S_0) ) \end{align} Such that, all else equal, increasing $S_0$ increases the Delta.

  • $\begingroup$ Its an interesting question. I was wondering if we could not get anything more general by studying the Gamma? $\endgroup$
    – Quantuple
    Aug 26 '16 at 15:47

For the general case, the first question is "what do you mean by delta". Only in local volatility models is it theoretically possible to hedge continuously with the spot to replicate the option payoff. With stochastic volatility or jumps, a spot hedge will not be perfect.

One answer to "what is delta" is the derivative of option price with respect to spot, holding other model parameters fixed. That is not natural, though, since it depends on how you parameterize your model. You can change coordinates, giving what is financially an exactly equivalent model, but delta defined this way will be different.

A common solution is to hold market data fixed. There is some calibration of the model to market data that is treated as a black box. Delta is defined as the change of option price (under the calibrated model) with respect to change of spot where other calibration market data is held fixed.

A third possibility is to define delta as the hedge ratio which produces the minimum variance (according to the model) for option price as spot moves an infintesimal amount. Here when spot and vol moves are correlated, the vol is assumed to move as spot moves according to that correlation.

A fourth possibility specifically in the case of vanilla options is to use the delta under the Black-Scholes model for the inplied volatility of the option in question. So when different options are priced as having different volatilities, as for a stochastic volatility model, the delta used is from the Black-Scholes model with different volatilites for different strikes and maturities.

In the case of the local volatility model, where there is an exact spot hedge, the answer to the question is "yes". The delta is the probability under the asset measure of the call finisihing in the money, and this probability increases monotonically with the spot level.

For other cases, it will depend on which definition of delta. Too many possibilities for me here!


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