Question

The stock of Company ABC is currently trading at a price of $50. According to analyst forecasts, the share price will be either $45 or $55 at the end of six months. Suppose that the risk-free interest rate is 10% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of $50 using the Delta-hedging method? Verify your result using the Risk-neutral method.

Answer 1

Spot Price = $50 | Strike Price = $ 50 | Up-State Price = $ 55 | Down-state Price = $45 | Risk free rate = 10%

For Delta-hedging value of a put option, we will create a portfolio where we sell Delta number of shares and buy a put option. Delta is the number of shares to buy/sell such that value of the portfolio is equal in both upstate and downstate.

Let f be the price of put option, Using this and spot price, we can find the value of the portfolio.

At t = 0: Selling Delta shares at spot price = - 50 * Delta

Buying a put option = f

Value of portfolio = f - 50 * Delta ------------- Equation (1)

At t = 1: When price rise to $ 55, Value of portfolio = 0 - 55 * Delta (For put option Strike Price should be greater than Stock Price, else it is worthless)

When price goes down to $ 45, Value of portfolio = (50 - 45) - 45 * Delta = 5 - 45*Delta

In Delta-hedging, the value of portfolio at t = 1 should be equal in both up and down cases.

=> - 55 * Delta = 5 - 45 * Delta

=> 10 * Delta = -5

Delta = -0.50

In 6 months, Price = $55, the value of portfolio = -55*(-0.50) = $ 27.5

In 6-months, Price = $45, the value of portfolio = 5 - 45 * (-0.50) = $ 27.5

At Delta of -0.50, there is no uncertaintly in value of our portfolio after 6-months as it creates a risk-free hedge.

Since it is risk-free hedge, we can calculate portfolio's value at t = 0 using the risk-free rate with continuous compounding.

Value of portfolio at (t=0) = Value of portfolio after 6-months * e^-RT

T = 6 months or 0.50

Value of portfolio at (t=0) = 27.5 * e^{-10% * 0.50}

Value of portfolio at (t=0) = $ 26.16

Now that we know value of our portfolio at t = 0, we can equalise it to the portfolio value expression we created at t = 0 or Equation (1).

Value of portfolio at (t=0) = f - 50 * Delta

We know value and Delta, therefore, we can solve the equation for f.

=> 26.16 = f - 50 * (-0.5)

=> f = 26.16 - 25

Value of Put Option using Delta-hedging = $ 1.16

Using Risk-neutral method, we will verify the Value of put option which has been calculated above.

Spot Price = $50 | Strike Price = $ 50 | Up-State Price = $ 55 | Down-state Price = $45 | Risk free rate = 10%

First we will find the risk-neutral probabilities using the below formula. Let probability of Up-state is p and probability of down-state is (1 - p)

At Risk-neutral environment, Forward price at t=0 should equal expected stock price.

=> Spot Price * e^RT = Probability of up * Up-state price + Probability of down * Down-state price

=> 50 * e^{10% * 0.5} = p * 55 + (1 - p) * 45

=> 52.56 = 55p + 45 - 45p

=> 7.56 = 10p

=> p = 0.756

Probability of up = 0.756 | Probability of down = 1 - 0.756 = 0.244

Now we will calculate the payoffs after 6-months.

For put price, Payoff at t = Max(Strike price - Stock price, 0)

At t=1: Payoff in upstate = Max(50 - 55, 0) = 0

Payoff in downstate = Max(50 - 45, 0) = 5

To find the Value of the put option, we will discount the Expected payoff at t=1 to t=0 using the risk-free rate.

Value of the Put option = (Probability of up * Payoff at up-state + Probability of down * Payoff at down-state)* e^-RT

Value of the Put option = (0.756 * 0 + 0.244 * 5) * e^{-10% * 0.5}

Value of the Put option = 1.22 * e^{-10% * 0.5}

Value of the Put option using Risk-neutral method = $1.16

Hence, The Value of Put Option calculated using Delta Hedging method of $1.16 has been verified using Risk-neutral method.