Inferring Black–Scholes Put Delta from a Hedge Position

Summary

The document presents a delta hedging exercise for a writer of European put options under Black–Scholes assumptions. It asks for the option delta expression and its value given the stock price, strike, time to expiry, interest rate, contract quantity, and the number of shares in the hedge. The questioner is unsure how to proceed because volatility is not supplied, despite knowing that the Black–Scholes put delta formula normally depends on volatility through the option’s d-one term.

The setup points to a way to infer delta from the stated hedge rather than first calculate it from the pricing formula: the short option position’s aggregate delta is offset by the stock hedge. Dividing the hedge share count by the number of written contracts therefore reveals the per-option delta magnitude, with the sign determined by whether delta refers to the long put or the written put. The text provides no worked solution or discussion of contract conventions, so the hedge interpretation and units must be made explicit when solving.

Key ideas

• A European put’s Black–Scholes delta is the derivative of its value with respect to the underlying share price.
• The delta of a written option has the opposite sign from the delta of the corresponding long option.
• A delta hedge offsets the aggregate option exposure with an opposite stock position.
• The stated share hedge can be used to infer per option delta even when volatility is absent.
• The problem gives no worked answer, so position signs and contract scaling require care.

Tags

# Black Scholes Model Replicating Strategy Delta Hedged Exam Question


# Black Scholes Model Replicating Strategy Delta Hedged Exam Question












A share is currently priced at 640p. A writer of 100,000 units of a one year European put option with an exercise price of 630p has delta-hedged the option with a portfolio which holds cash and is short 24,830 shares. The continuously compounded risk-free rate of interest is 3% p.a. and no dividends are payable during the life of the option. The assumptions of the Black-Scholes model apply.

Write down an expression for the delta of the option.

Calculate its value in this case.

Please help me out in this question I am unable to solve it. So far I tried differentiating option pricing formulae of black schole with S, price of share that will give me formulae for delta, right but how can we solve this equation without knowing volatility.This is where i am stuck.

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