Why Higher Interest Rates Raise a Call Option’s Delta

# why Delta increases as interest rate increases


# why Delta increases as interest rate increases












I just would like to know why $\Delta$ increases as $r$ increases.

I would like an intuitive answer, without model (I can compute my greeks myself).

Thanks

## Answer by Quantuple (score 7, accepted)

https://quant.stackexchange.com/a/29585

[Mathematically]

Risk-neutral pricing means that \begin{align} C_0(K,T) &= \mathbb {E}_0\left[\frac{1}{B_T} (S_T - K)^+\right] \\ &= \mathbb {E}_0\left[\left(\frac {S_T}{B_T} - \frac {K}{B_T}\right)^+\right] \end{align}

Now simply notice that the dynamics of $$\tilde{S}_t := \frac {S_t}{B_t},\ \forall t \geq 0$$ is independent of $r$ (see the very definition of the risk neutral measure associated to the numéraire $B_t$) while the present value of the strike $K/B_T$ decreases as $r$ increases.

This is a "model-free" result in the sense that it does not depend on working modelling assumptions (i.e. no specific (jump)-diffusion model).

[Intuitively]

Increasing $r$ will cause the forward price $F (0,T)$ to increase (model-free cash & carry replication argument), which in turn means that the undiscounted call price, $\mathbb {E}[(S_T-K)^+] $, will increase because the forward price represents the expectation of the stock price $S_T $ under the risk-neutral measure (hence increasing forward means shifting the pdf towards the right).

In parallel however, the discount factor (measuring the present value of future cashflows) will decrease as $r$ increases.

Everything else equal, it is straightforward to see that the forward price and the discount factor will move in the exact same relative proportions... but in opposite directions thereby compensating each other's effect.

The game changer is the fact that the present value of the strike price as seen of today will decrease regardless, hence causing the call price to increase.

## Answer by Mats Lind (score 4)

https://quant.stackexchange.com/a/29596

Just to strengthen the intuition in the perfect answer above: With r going very high (and hence F), all prices on cash instruments are expected to gain fast with time (to compensate for the carry) and the call-strike is expected to be deep[er] in the money; hence with a delta close[r] to one.