Topic 6 Value of a Call Option

Intrinsic value of a call option

S − X (spot price minus exercise price). Zero when S ≤ X, increases 1-for-1 with asset price when S > X.

Time value of an option

The part of an option's price reflecting the probability of expiring in-the-money before maturity. Maximum when S = X. At expiry, time value = 0.

C = TV + IV

A call option's price equals Time Value plus Intrinsic Value. Below X: C = TV only. Above X: C = TV + IV.

Delta (∂C/∂S > 0)

Sensitivity of call price to spot price. E.g. Delta = 0.88 means a £1 rise in S increases C by £0.88.

Gamma (∂²C/∂S² > 0)

Second-order sensitivity of the call to spot price. Captures the non-linearity; needed alongside Delta for accurate price-change estimates.

Theta (∂C/∂T > 0)

Sensitivity of call to time to expiry. Longer maturity = higher call value. E.g. Theta = −13.50: one fewer trading day → −13.50 × 0.004 = −£0.054.

Rho (∂C/∂r > 0)

Sensitivity of call to risk-free rate. Higher r → higher call value (deferred purchase earns more interest). E.g. Rho = 5.40, r rises 1% → +£0.054.

Vega (∂C/∂σ > 0)

Sensitivity of call to volatility. Higher σ raises in-the-money probability → higher C. E.g. Vega = 9.80, σ falls 1% → −£0.098.

Exercise (strike) price and call value

Higher X → lower call value. A higher spot price is needed to be in-the-money, reducing the probability.

Effect of rising spot price on a put

A rise in S reduces put value (puts are in-the-money when S < X).

Put-Call Parity formula

S₀ + P − C = Xe^(−rT), rearranged: P = C − S₀ + Xe^(−rT). European options only.

Put-Call Parity assumptions

Frictionless markets, no-arbitrage, no short-selling restrictions, risk-free rate exists, no dividends during option life.

Why Put-Call Parity doesn't hold for American options

American options allow early exercise, creating extra value. Boundary conditions apply instead of strict parity.

Synthetic security

A combination of puts, calls, stocks and/or bonds that replicates another security's payoff. E.g. short call + long put + long share → risk-free payoff of X.

Synthetic stock futures

Long call + short put (same underlying, expiry, strike) replicates the payoff of a futures contract on that stock.

European vs American option

European: exercised only at expiry. American: can be exercised at any point up to expiry (early exercise). American ≥ European in value.

Why options must have a positive premium

By no-arbitrage: the long position can only gain, so a premium must be paid to compensate the short (writer).

In-the-money call option

When S > X: spot price exceeds exercise price, giving positive intrinsic value.

Out-of-the-money call option

When S < X: no intrinsic value, only time value (if before expiry).

Vega-hedging example

Options with Vegas 10 and 5: long 1 of the first + short 2 of the second → net Vega = 0, portfolio insensitive to volatility.

Effect of volatility on puts and calls

Higher volatility increases both call and put values; loss is always capped at the premium, so upside from wider price swings dominates.

Present value of exercise price (continuous compounding)

PV(X) = Xe^(−rT). Discounts the strike price to today using the risk-free rate.

Break-even for a long call

S = X + Premium. The spot must exceed the strike by at least the premium paid.

Bid vs ask price

Bid: highest price a buyer will pay. Ask: lowest a seller accepts. Buyers pay the ask; sellers receive the bid.