A pricing system is offline, but you still have these figures for an option on a stock: - theta $\Theta = -1.9$ - delta $\Delta = 0.3$ - gamma $\Gamma = 0.03$ - risk-free rate $r = 0.03$ - variance $\sigma^2 = 0.2$ - stock price $S = 25$ Assuming Black-Scholes, what is the option worth, to two decimal places?

Option value recovered from its Greeks

Question

A pricing system is offline, but you still have these figures for an option on a stock:

theta $\Theta = -1.9$
delta $\Delta = 0.3$
gamma $\Gamma = 0.03$
risk-free rate $r = 0.03$
variance $\sigma^2 = 0.2$
stock price $S = 25$

Assuming Black-Scholes, what is the option worth, to two decimal places?

Short answer

$6.67$ . Any Black-Scholes price satisfies $\Theta + \tfrac12\sigma^2S^2\Gamma + rS\Delta = rV$ . Here $-1.9 + 1.875 + 0.225 = 0.2$ , so $V = 0.2/0.03 \approx 6.67$ . The trick fails when $r=0$ , since $V$ then drops out of the equation.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free