Problem 26.5 Easy Free 4 variations

Straddle Delta via Parity

A straddle on underlying $S$ consists of a European call and a European put, both with strike $K$ and expiry $T$ . Assume Black–Scholes dynamics. If the put has $\Delta = -0.31$ , find the $\Delta$ of the straddle.

Work it out, then compare with the answer.

SolutionShow the worked solution

Under Black–Scholes with no dividends, put–call parity holds at every time and for every spot $S$ :

C(S,t) - P(S,t) = S - Ke^{-r(T-t)}.

Differentiating with respect to $S$ gives

\Delta_C - \Delta_P = 1, \qquad\text{so}\qquad \Delta_C = 1 + \Delta_P = 1 - 0.31 = 0.69 .

Delta is linear in the position, because it is the derivative of a sum of prices. The straddle is one call plus one put, so

\Delta_{\text{straddle}} = \Delta_C + \Delta_P = 1 + 2\Delta_P = 1 - 0.62 = 0.38 .

\boxed{\Delta_{\text{straddle}} = 0.38}

Key idea. A straddle is delta-neutral only when $\Delta_P = -\tfrac12$ . Here the put has $|\Delta_P| < \tfrac12$ , so the call dominates and the straddle is net long delta.

Variations

The same idea with a twist. Each solution gives only the changed step.

Variation 26.5a

A European call and a European put on the same underlying share a strike $K$ and expiry $T$ . The call has $\Delta = 0.21$ . Find the $\Delta$ of the put.

Work it out, then compare with the answer.

Solution

Differentiate parity $C - P = S - Ke^{-rT}$ with respect to $S$ to get $\Delta_C - \Delta_P = 1$ . Then

\Delta_P = \Delta_C - 1 = 0.21 - 1 = -0.79 .

\boxed{\Delta_P = -0.79}

Variation 26.5b

A European call and a European put on the same underlying share a strike $K$ and expiry $T$ . The call has $\Gamma = 0.02$ . Find the $\Gamma$ of the put.

Work it out, then compare with the answer.

Solution

Differentiate $\Delta_C - \Delta_P = 1$ once more with respect to $S$ . This gives $\Gamma_C - \Gamma_P = 0$ , so the put’s gamma equals the call’s.

\boxed{\Gamma_P = 0.02}

Variation 26.5c

A straddle on underlying $S$ consists of a European call and a European put, both with strike $K$ and expiry $T$ . Assume Black–Scholes dynamics. If the call has $\Gamma = 0.03$ , find the $\Gamma$ of the straddle.

Work it out, then compare with the answer.

Solution

Parity gives $\Gamma_P = \Gamma_C$ , and gamma adds across positions. Hence

\Gamma_{\text{straddle}} = \Gamma_C + \Gamma_P = 2\Gamma_C = 2(0.03) = 0.06 .

\boxed{\Gamma_{\text{straddle}} = 0.06}

Variation 26.5d

European call options at strike $K=24$ have $\Delta = 0.25$ , and the underlying trades at $S_0 = 21$ . You sell $100$ put options at strike $K=24$ on this underlying. How many units of the underlying should you buy or sell to delta-hedge the position? Enter $-x$ if you should sell $x$ units.

Work it out, then compare with the answer.

Solution

From parity, $\Delta_P = \Delta_C - 1 = 0.25 - 1 = -0.75$ . The spot price $S_0 = 21$ is not needed.

You are short 100 puts, so the delta of the position is

-100 \times (-0.75) = +75 .

To make the portfolio delta-neutral you must hold $-75$ units of the underlying. That is, you sell 75 units.

\boxed{-75 \text{ (sell 75 units)}}

Spotted an error? Email us