2: Pricing of an Option

Mini Chapter Two

Pricing of an Option

• Most pricing models calculate the price of an option using some form of hedging such that the portfolio of the option along with the hedges neutralise market price risks at all times. In theory this requires rebalancing the hedges continuously through the tenor of the option the cost of which is the premium of the option. Cost of hedging therefore is primarily a function of:

Strike/forward price of the underlying – the distance between the strike and current market price suggests the ‘moneyness’ of the option. More out of the money or less in the money the strike is versus the current forward, lower is the option price.

Current price of the underlying – as an extension of the above, price movement of the underlying dynamically changes the moneyness of the option by way of its distance from the fixed strike. Price of the option would correlate with the moneyness of the option in the same manner as for the strike price.

Risk free Interest Rate for the tenor of the option – It should naturally follow that this risk free option strategy (dynamically hedged at all times) would then have a return equal to the risk free rate and hence the option price is a function of the level of risk free interest rate. From the formulae below one can see that the value of a call option increases as risk free rate goes up and vice versa for put options.

• Volatility of the underlying’s price – higher the volatility of the underlying, higher the probability of a larger payoff. In the forward space it is the implied/expected volatility that we account for while pricing the option.

Tenor of the option – a longer tenor offers more time for the underlying’s volatility to increase the moneyness and hence the price of an option.

• Price of an option can also be thought of as denoting the probability weighted payoff of the option at maturity (discounted for its Present Value).

As an example, to price a strike S (for an asset that takes discrete values), 3m Call option where Probability of S_i = P(S_i), S_t = price at which cumulative probability equals 1.

Expected Payoff of 100 Strike 3m Call = {P(S _i) <100)*0} + ⅀ _{i = 101 to t} {P(S_i)*(S_i-100)}

t in the equation above is the count on the observations that cover the entire probability distribution.

• It would be intuitive now to think of the option price to be composed of intrinsic (distance between the strike and current forward price) and extrinsic value (time value of the option). As we approach option expiry almost all its value becomes intrinsic. For instance a call option would have a positive intrinsic value (is in the money) if the expected forward price at expiry is greater than the strike and is zero for all values at or below the strike. Moneyness of an option as mentioned above refers to its intrinsic value.

Graph 2 – Long Call Option price across Option expiries

Source: Pandemonium.

Graph 3 – Long Put Option price across Option expiries

Graph 4 – Short Call Option price across Option expiries

Graph 5 – Short Put Option price across Option expiries