The option value at the expiration is determined by the differential between the spot price at expiration and the exercise price. If the spot price is greater than the exercise price then the call option will have a positive value equal to ST - X. Otherwise, it will have a zero value as it will expire worthless. Similarly, the value of the put option will be equal to X - ST if the exercise price is greater than the spot price at expiration and otherwise zero.
According to one-period binomial model, only two possibilities will be there in the future. Either the price will move to S1+ or it will move to S1- from the current price of S0. We can then easily calculate the option value at the expiration at both the nodes and then multiply those values with the respective upside and downside risk-neutral probabilities and sum those value to get the expected value of the option at the expiry. By discounting that value with the risk-free rate, we can calculate the value of the option today. It is as simple as this.
Up move = u = S1+/S0
Down move = d = S1-/S0
The only things we require for the option valuation is the upside move, downside move, the risk-neutral probabilities, and the risk-free rate. We do not require the actual probabilities of up and down moves as we will see later.
We can hedge short call position with the long position in the underlying. But the hedging won't require one unit of underlying per unit of the call option. Let's assume that we would require h units of the underlying.
Therefore, the value of the combined portfolio today, V0 = hS0- c0
When the price of the underlying moves to S1+ or S1- at expiry, the value of the combined portfolio will be:
V1+ = hS1+ - c1+
or
V1- = hS1- - c1-
But for hedging to work, both of these values should be equal at the expiration. On equating these value, we get
h = (c1+ - c1-)/(S1+ - S1-)
The perfectly hedged portfolio should earn risk-free rate. Therefore, V1+ = V1- = V0
On solving these equations, we get
c0 = [πc1+ + (1-π)c1+]/(1+r)
where
π = Risk-neutral up-probability = (1+r-d)/(u-d)
Similarly, we can the value of put option as
p0 = [πp1+ + (1-π)p1+]/(1+r)
Important points:
- The volatility of the underlying is reflected in the difference between S1+ and S1- and affects c1+ and c1- and is an important factor in determining the value of the option.
- The actual probabilities of up and down moves are not needed to calculate the value of options
- The risk-neutral probabilities of up move (π) and down move (1- π) are also called as synthetic or pseudo probabilities and they produce a weighted average of future call value i.e. expected future value which is discounted by the risk-free rate to get the value of the call option.
Check your concepts:
(58.25) Which of the following is used to discount the expected payoff based on risk-neutral probabilities to get the value of a call option in binomial pricing model?
(a) Risk-free rate plus risk premium
(b) Required return on underlying minus risk-free rate
(c) Risk-free rate
(58.26) Which of the following is directly represented by the difference between the up and down factors in the binomial model?
(a) Risk-neutral probability
(b) Expected value of the stock
(c) Volatility of the underlying
Solutions:
(58.25) Correct Answer is C: The expected payoff based on risk-neutral probabilities is discounted by risk-free rate to get the value of call option in the binomial model.
(58.26) Correct Answer is C: The difference between up and down factors represent the volatility of the underlying in the binomial model.
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