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Please determine the payoff for the following long position in a knock-in put option, based on the price path summary below: Starting price of underlying (at inception): $106.1 Maximum traded price of underlying: $112.65 Minimum traded price of underlying: $96.78 Terminal price of underlying (at maturity): $103.19 Strike rate, K: $102.77 Barrier, H: $110.24 Premium, p: $4.1 (Assume continuous price observations)

Question

Please determine the payoff for the following long position in a knock-in put option, based on the price path summary below:

Starting price of underlying (at inception): $106.1

Maximum traded price of underlying: $112.65

Minimum traded price of underlying: $96.78

Terminal price of underlying (at maturity): $103.19

Strike rate, K: $102.77

Barrier, H: $110.24

Premium, p: $4.1

(Assume continuous price observations)

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Solution

The payoff for a knock-in put option depends on whether the price of the underlying asset hits the barrier level during the life of the option. If the price hits the barrier, the option is "knocked in," or activated, and the payoff is the maximum of zero or the difference between the strike price and the terminal price of the underlying asset. If the price does not hit the barrier, the option is not activated and the payoff is zero.

In this case, the maximum traded price of the underlying asset is 112.65,whichisabovethebarrierlevelof112.65, which is above the barrier level of 110.24. Therefore, the option would be knocked in.

The payoff would then be the maximum of zero or the difference between the strike price (102.77)andtheterminalpriceoftheunderlyingasset(102.77) and the terminal price of the underlying asset (103.19). Since 102.77islessthan102.77 is less than 103.19, the payoff would be $0 because the holder of the put option would not exercise the option (as they would not sell for less than the current price).

However, we must also account for the premium paid for the option (4.1).So,thenetpayoffwouldbe4.1). So, the net payoff would be 0 - 4.1=4.1 = -4.1.

So, the payoff for this long position in a knock-in put option would be -4.1.Thismeanstheholderoftheoptionwouldhaveanetlossof4.1. This means the holder of the option would have a net loss of 4.1.

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