Question

Assume no arbitrage unless otherwise noted.

A stock is currently priced at $39.00. The risk free rate is 4.9% per annum with continuous compounding. Every 6 months, its price will either go up by 17% or down by 19%. Consider a European put with strike $42.00 expiring in 12 months.

(a) Using the binomial tree model, compute the price of a European put option at the initial node, the two intermediate nodes, and the three terminal nodes.

Enter the following solutions as dollar values, including dollar symbols ($), to two decimal places.

Note: The solutions to later parts of this problem may require more precision than solutions to earlier parts. Therefore, you need to record your solutions to a higher precision than required for use in subsequent computations in order to avoid rounding errors later on.

Top terminal node:
Middle terminal node:
Lower terminal node:
Upper intermediate node:
Lower intermediate node:
Initial node:

(b) Estimate the Δ of the put at the two intermediate nodes and the initial node using the solutions to part (a)

Enter the following solutions to three decimal places.

Upper intermediate node:
Lower intermediate node:
Initial node:

(c) Estimate the Γ of the put at the initial node using the solutions to part (b)

Enter your solution to three decimal places.

(d) You own 1 put today. Use part (b) to determine how much stock you should buy to be Δ-neutral (that is, to ensure that the portfolio value does not change regardless of where the stock goes to at the intermediate time step 6 months from today).

Enter your solution to three decimal places. Note that a negative answer means selling stocks

Buy ___ shares

(e) Suppose the stock goes up at the intermediate time step. How should you change your position in order to remain Δ-neutral? In other words, how many stocks should you add to your holding from part (d)? This is an example of dynamic hedging.

Enter your solution to three decimal places. Note that a negative answer means selling stocks

Buy___ shares

Answer 1

r = 4.9% = 0.049; u = 1 + 17% = 1.17; d = 1 - 19% = 0.81; S0 = 39; K = 42; t = time per period = 6 months = 0.5 year; T = time to maturity = 12 months = 1 year

Risk neutral probability of up movement, p = (e^rt - d) / (u - d) = (e^{0.049 x 0.5} - 0.81) / (1.17 - 0.81) = 0.5967

Before we get in to the question, let's have a look at the stock tree.

Node	Formula	Value
S0		39.00
Su = u x S0	1.17 x 39	45.63
Sd = d x S0	0.81 x 39	31.59
Suu = u x Su	1.17 x 45.63	53.39
Sud = d x Su	0.81 x 45.63	36.96
Sdd = d x Sd	0.81 x 31.59	25.59

We are now ready to get into the questions:

(a) Using the binomial tree model, compute the price of a European put option at the initial node, the two intermediate nodes, and the three terminal nodes.

Top terminal node: Puu = max (K - Suu, 0) = max (42 - 53.39, 0) = 0.00
Middle terminal node: Pud = max (K - Sud, 0) = max (42 - 36.96, 0) =  5.04
Lower terminal node: Pdd = max (K - Sdd, 0) = max (42 - 25.59, 0) = 16.41
Upper intermediate node: Pu = [p x Puu + (1 - p) x Pud] x e^-rt = [0.5967 x 0 + (1 - 0.5967) x 5.04] x e^{0.049 x 0.5} = 2.08
Lower intermediate node: Pd = [p x Pud + (1 - p) x Pdd] x e^-rt = [0.5967 x 5.04 + (1 - 0.5967) x 16.41] x e^{0.049 x 0.5} = 9.87
Initial node: P0 = [p x Pu + (1 - p) x Pd] x e^-rt = [0.5967 x 2.08 + (1 - 0.5967) x 9.87] x e^{0.049 x 0.5} = 5.35

(b) Estimate the Δ of the put at the two intermediate nodes and the initial node using the solutions to part (a)

Enter the following solutions to three decimal places.

Upper intermediate node: Δu = (Puu - Pud) / (Suu - Sud) = (0 - 5.04) / (53.39 - 36.96) = -0.307
Lower intermediate node: Δd = (Pud - Pdd) / (Sud - Sdd) = (5.04 - 16.41) / (36.96 - 25.59) = -1.000
Initial node: Δ = (Pu - Pd) / (Su - Sd) = (2.08 - 9.87) / (45.63 - 31.59) = -0.554

(c) Estimate the Γ of the put at the initial node using the solutions to part (b)

Γ = ( Δu - Δd) / (Su - Sd) = (-0.307 - (-1.000)) / (45.63 - 31.59) =  0.049

Enter your solution to three decimal places.

(d) You own 1 put today. Use part (b) to determine how much stock you should buy to be Δ-neutral (that is, to ensure that the portfolio value does not change regardless of where the stock goes to at the intermediate time step 6 months from today).

You should buy  -Δ = 0.554 number of stocks

Enter your solution to three decimal places. Note that a negative answer means selling stocks

Buy 0.554 shares

(e) Suppose the stock goes up at the intermediate time step. How should you change your position in order to remain Δ-neutral? In other words, how many stocks should you add to your holding from part (d)? This is an example of dynamic hedging.

You should buy  -Δu = 0.307 number of stocks

Enter your solution to three decimal places. Note that a negative answer means selling stocks

Buy 0.307 shares