thank you very much for trying to answer this question, and I hope it will be helpful to everyone in my situation.
I am preparing for an interview, and I've come across these three questions on the internet for the position I'm applying for:
1. Give the price of an ATMF call, and deduce the ATM call price.
2. Using the call-put parity, provide the Put price.
3. From the put price, deduce the price of an autocall at one of the observation dates (autocall trigger at 100%).
I don't think I need some help for the first two questions, but just to be sure : $\text(Call_{ATMF}) = 0.4 * S * \sigma * \sqrt(T-t)$ and then the ATM call is just (by the $\delta = 0.5$ when ATM): $$\text(Call_{ATM})=\text(Call_{ATMF})+(F-S)*0.5$$ $$\text(Call_{ATM}) \approx \text(Call_{ATMF}) + Sr(T-t)*0.5 $$ Then for the ATM Put we just have (with K = S) with the C-P Parity: $$ \text(Put_{ATM}) = Se^{-r(T-t)} - S + C \approx C - Sr(T-t) $$
Now with these prices, how can I deduce the price of an autocallable with a 100% trigger at an observation date ? Firstly, is the question clear? What type of autocallable? Just the ATM Put with an early redemption at 100%? It would just be $0.5 \times \text{Spot}$ then, as the ATM digital price providing 100% of spot is 50% of the spot...
Or maybe the question is clear for someone familiar with autocallable products and all the prices above are quite useful for pricing an autocallable (e.g. Pheonix or Athena...) ?
Anyway, thank you in advance for your help and I hope this topic will help others !
