Question

spot price: 66

strike price 68

risk-free interest rate is 6% per annum with continuous compounding, please undertake option valuations and answer related questions according to following instructions: Binomial trees: Additionally, assume that over each of the next two four-month periods, the share price is expected to go up by 11% or down by 10%.

1. Use a two-step binomial tree to calculate the value of an eight-month European call option using risk-neutral valuation.
2. Use a two-step binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation.

Answer 1

Spot price = 66 | Strike price = 68 | Up-state factor = 11% | Down-state factor = 10% | Total time period = 8 months

Number of steps = 2 of 4 months each | Risk-free rate = 6%

We can first calculate Upstate and Downstate prices in period 1 and 2 of 4 months months using their factors.

t = 1: Price (u) = 66 * (1 + 11%) = 73.26 | Price (d) = 66 * (1 - 10%) = 59.40

t = 2: Price (uu) = 73.26 * (1+11%) = 81.32 | Price (ud) = 73.26 * (1-10%) = 65.93 |

Price (du) = 59.40*(1+11%) = 65.93 | Price (dd) = 59.40 * (1-10%) = 53.46

Let probability of going up be p and probability of going down be (1 - p)

p = (Spot Price * e^RT - Down-state Price) / (Up-state Price - Down-state Price)

p = (66 * e^6%*1/3 - 59.40) / (73.26 - 59.40) = (67.33 - 59.40) / (73.26 - 59.40)

p = 57.24%

(1 - p) = 1 - 57.24% = 42.76%

a) Using the price calculated, we can calculate the payoffs in case of a call. Strike Price = 68

Payoff at Time t for a call option = Max(Current Price - Strike Price, 0)

We will go back in time from t = 2 to t=0

t = 2: Payoff (uu) = Max(Price(uu) - S, 0) = Max(81.32 - 68,0) = 13.32

Payoff(ud) = Payoff(du) = Max(65.93 - 68,0) = 0

Payoff (dd) = Max(53.46 - 68,0) = 0

Now by discounting expected payoffs at t=2 to t=1 using Risk-free rate, we can find the payoff for each state at t=1. Here we treat each step as an individual binomial tree.

t = 1: Payoff(u) = (Probability of up * Payoff (uu) + Probability of down * Payoff (ud)) * e^-RT

Payoff(u) = (57.24%* 13.32 + 42.76% * 0) * e^-6%*1/3 ------------------- 4 months = 4/12 = 1/3

Payoff(u) = 7.62 * e^-6%*1/3

Payoff(u) = 7.47

Payoff(d) = (Probability of up * Payoff (du) + Probability of down * Payoff (dd)) * e^-RT

Payoff (d) = (57.24%* 0 + 42.76% * 0) * e^-6%*1/3

Payoff(d) = 0

Now we can discount the expected payoff at t =1 to t = 0 using the risk-free rate and that would be the value of the call option.

Value of the Option or Payoff at (t=0) = (Probability of up * Payoff (u) + Probability of down * Payoff (d)) * e^-RT

Value of the Option or Payoff at (t=0) = (57.24% * 7.47 + 42.76% * 0) * e^-6%*1/3

Value of the Option or Payoff at (t=0) = 4.28 * e^-6%*1/3

Value of the Call Option = 4.19

Below is how Two-step Binomial tree for Call option would look like:

b) For put option, Payoff at time t = Max(Strike Price - Current Price, 0)

Similarly, we can calculate the Payoffs at each state using the Prices calculated at the beginning.

We will go from t=2 to t=0 and calculate Payoffs at each period to find the value of the put option.

t = 2: Payoff (uu) = Max(Strike Price - Price(uu), 0) = Max(68 - 81.32,0) = 0

Payoff (ud) = Payoff (du) = Max(Strike Price - Price(ud), 0) = Max(68 - 65.93, 0) = 2.07

Payoff(dd) = Max(Strike Price - Price(dd), 0) = Max(68 - 53.46, 0) = 14.54

For t=1, we will discount the expected payoff from t=2 using the risk-free rate.

t = 1: Payoff(u) = (Probability of up * Payoff (uu) + Probability of down * Payoff (ud)) * e^-RT

Payoff (u) = (57.24% * 0 + 42.76% * 2.07) * e^-6%*1/3

Payoff (u) = 42.76% * 2.07 * e^-6%*1/3

Payoff (u) = 0.88 * e^-6%*1/3

Payoff (u) = 0.87

Payoff(d) = (Probability of up * Payoff (du) + Probability of down * Payoff (dd)) * e^-RT

Payoff (d) = (57.24% * 2.07 + 42.76% * 14.54) * e^-6%*1/3

Payoff (d) = 7.40 * e^-6%*1/3

Payoff (d) = 7.25

Now we can discount the expected payoff at t=1 to t=0 using the risk-free rate to find the value of the put option.

Value of the Option or Payoff at (t=0) = (Probability of up * Payoff (u) + Probability of down * Payoff (d)) * e^-RT

Value of the Option or Payoff at (t=0) = (57.24% * 0.87 + 42.76% * 7.25) * e^-6%*1/3

Value of the Option or Payoff at (t=0) = 3.60 * e^-6%*1/3

Value of the Put Option = 3.53

Below is how Two-step Binomial tree for Put option would look like: