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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.

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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.

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