Question

1. A Whatnot pays the difference between the final price and the minimum price of a stock over the period of the asset. For example, if the price of a stock is 200, 220, and 234 in the previous periods (here periods 0, 1, and 2), the minimum price is 200, the final price is 234, and the asset would pay 234-200=34.. If the price path is 200, 220, 200, the minimum price is 200, the final price is 200, and the asset would pay 0.

So given all this, you have a Whatnot that expires in 2 periods where the stock price in period 0 is 60. The stock either moves up with u=1.2, or down with d=1/1.2. The interest rate is 5 percent. What is the value of your asset today in period 0? Please include all work with steps and explanations.

Answer 1

Please see the two period binomal tree for the stock price:

We have , S = 60, u = 1.2, d = 1/1.2, R = 1 + interest rate = 1 + 5% = 1.05

uS = 1.2 x 60 = 72; dS = 1 / 1.2 x 60 = 50, u²S = 1.2² x 60 =  86.40, udS = 1.2 x 1/1.2 x 60 = 60, d²S = 60 / 1.2² = 41.67

Risk neutral probability, p = (R - d)/(u - d) = (1.05 - 1/1.2) / (1.2 - 1/1.2) = 0.590909091

The asset is a call option with minimum stock price over a path as the strike price. Let's denote this call option by C.

At t = 2, when

• Final Stock price is u²S = 86.40, the path of the stock price is: S => uS => u²S. Hence, the minimum price of a stock over the period of the asset = S = 60. Hence, payoff from call option, C_uu = Final stock price - the minimum price of a stock over the period of the asset = 86.40 - 60 = 26.40
• Final Stock price is udS = 60, there are two possible paths:
  • If the path of the stock price is: S => uS => udS. Hence, the minimum price of a stock over the period of the asset = S = 60. Hence, payoff from call option, C_ud = Final stock price - the minimum price of a stock over the period of the asset = 60 - 60 = 0
  • If the path of the stock price is S => dS => udS. Hence, the minimum price of a stock over the period of the asset = dS = 50. Hence, payoff from call option, C_du = Final stock price - the minimum price of a stock over the period of the asset = 60 - 50 = 10
• Final Stock price is d²S = 41.67, the path of the stock price is: S => dS => d²S. Hence, the minimum price of a stock over the period of the asset = d²S = 41.67. Hence, payoff from call option, C_dd = Final stock price - the minimum price of a stock over the period of the asset = 41.67 - 41.67 = 0

Under this situation, the price of the asset today = price of the call option today = C₀

= 1 / R x [p²C_uu + p(1 - p)C_ud+ (1 - p)pC _du+ (1‐p)²C_dd]

= 1 / 1.05 x (0.590909091² x 26.40 + 0.590909091 x (1 - 0.590909091) x 0 + (1 - 0.590909091) x 0.590909091 x 10 + (1 - 0.590909091)² x 0 = 1 / 1.05 x 11.64 =  11.08

Hence, the value of your asset today in period 0 = 11.08