Calculate the value today (t=0) of a two-year European put-option written on one gram of gold, with an exercise price of DKK 260.

Steps to Solve

1. Set Up the Binomial Tree for Gold Prices

   • The initial price of gold is ( S_0 = 250 ) DKK.
   • The expected increase is 12%, so the up factor ( u = 1.12 ) and the down factor ( d = 0.88 ).
   • The tree for two years looks like:
     • At ( t = 0 ): ( S_0 = 250 )
     • At ( t = 1 ):
       • Up: ( S_u = S_0 \times u = 250 \times 1.12 = 280 )
       • Down: ( S_d = S_0 \times d = 250 \times 0.88 = 220 )
     • At ( t = 2 ):
       • Up-Up: ( S_{uu} = S_u \times u = 280 \times 1.12 = 313.6 )
       • Up-Down: ( S_{ud} = S_u \times d = 280 \times 0.88 = 246.4 )
       • Down-Up: ( S_{du} = S_d \times u = 220 \times 1.12 = 246.4 )
       • Down-Down: ( S_{dd} = S_d \times d = 220 \times 0.88 = 193.6 )

2. Calculate Risk-Neutral Probabilities

   • Use the formula for risk-neutral probabilities ( p ) and ( 1-p ):
     • ( p = \frac{e^{rt} - d}{u - d} )
   • For ( r = 0.0075 ) (0.75% nominal risk-free rate) and ( t = 1 ):
     • ( u = 1.12 )
     • ( d = 0.88 )
   • Calculate ( p ): $$ p = \frac{e^{0.0075} - 0.88}{1.12 - 0.88} $$

3. Calculate Value of European Put Option at Each Node

   • For a put option with exercise price ( K = 260 ):
     • At ( t = 2 ):
       • ( V_{uu} = \max(260 - S_{uu}, 0) = \max(260 - 313.6, 0) = 0 )
       • ( V_{ud} = \max(260 - S_{ud}, 0) = \max(260 - 246.4, 0) = 13.6 )
       • ( V_{du} = \max(260 - S_{du}, 0) = 13.6 ) (same as ( V_{ud} ))
       • ( V_{dd} = \max(260 - S_{dd}, 0) = \max(260 - 193.6, 0) = 66.4 )

4. Discount Back to Present Value

   • Use the risk-neutral probabilities to calculate expected values at ( t = 1 ):
     • For ( t = 1 ):
       • ( V_{u} = e^{-rt}(p \cdot V_{uu} + (1 - p) \cdot V_{ud}) )
       • ( V_{d} = e^{-rt}(p \cdot V_{du} + (1 - p) \cdot V_{dd}) )

5. Calculate Present Value Today

   • Use the values from ( t = 1 ) to find the present value at ( t = 0 ):
     • ( V_0 = e^{-rt}(p \cdot V_u + (1 - p) \cdot V_d) )