Numerically Solving for Spot from an Option Delta

Summary

The document asks whether spot price can be calculated directly from a target option delta when strike, expiry, volatility, rates, and other inputs are known. It discusses Black–Scholes and Bjerksund–Stensland models, which provide option prices and Greeks as functions of the inputs, but says there is no closed-form inverse for spot from delta in this setup.

Instead, the proposed practical approach is numerical root-finding: vary spot until the model delta matches the target. Newton–Raphson, bisection, or Brent’s method are named as possible methods. The explanation attributes the lack of a direct formula to the appearance of the standard normal cumulative distribution function in the delta calculation, whose inverse does not yield a simple closed-form expression. The document gives no implementation details, convergence comparisons, or treatment of choosing a search interval, so those details must be handled in an actual solver.

Key ideas

• The document states that spot cannot be recovered from delta with a closed-form formula under the specified option models.
• The spot can be estimated by numerically finding where model delta matches the target delta.
• Newton–Raphson, bisection, and Brent’s method are suggested as root-finding choices.
• The inversion is difficult because the delta formulas involve the standard normal cumulative distribution function.
• A practical implementation still needs appropriate numerical settings and a suitable search interval.

Tags

# Solve for spot price given delta


# Solve for spot price given delta












I can use Black Scholes or Bjerksund Stensland to solve for delta given spot price, strike, expiration, vol, interest rate, etc. But is there a direct solution to solve for spot price given delta, strike, expiration, vol, etc? I can search for delta by plugging in different values of spot price and seeing what value corresponds most closely to the target delta (iterative / Newton-Raphson method), but curious if it can be calculated directly.

## Answer by Hans-Peter Schrei (score 4, accepted)

https://quant.stackexchange.com/a/74969

There is no closed-form solution for the spot price given delta, strike, expiration, volatility, interest rate, and other parameters. As you mentioned, an iterative method such as the Newton-Raphson method or a search algorithm like bisection or Brent's method can be used to find the spot price that corresponds most closely to the target delta.

The reason for this is that the Black-Scholes and Bjerksund-Stensland models, which do provide closed-form solutions for option pricing and Greeks (like delta), do so in terms of the cumulative distribution function of the standard normal distribution. This function, which is used to compute the probabilities of the option being exercised, is not easily invertible.

As a result, when you try to solve for the spot price given delta, you're essentially trying to invert the cumulative distribution function of the standard normal distribution, which doesn't have a closed-form solution.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.