Topic 2 – No-arbitrage basics

2.1 Standard equity-style parity

European call and put on the same underlying, with:

• current spot price: $S_0$
• strike: $K$
• maturity: $T$
• risk-free rate (continuously compounded): $r$
• zero dividends

Put–call parity:

$$C_0 - P_0 = S_0 - K e^{-rT}$$

Equivalently:

$$C_0 = P_0 + S_0 - K e^{-rT}$$ $$P_0 = C_0 - S_0 + K e^{-rT}$$

Replication idea:

Consider two portfolios:

1. Portfolio A: long call $+C$ and long discount bond that pays $K$ at $T$. Current cost: $C_0 + K e^{-rT}$.

2. Portfolio B: long put $+P$ and long stock $+S$. Current cost: $P_0 + S_0$.

Check their payoff at maturity:

• If $S_T > K$:
  • A: call is in the money $\Rightarrow S_T - K$, plus $K$ from bond ⇒ $S_T$.
  • B: put expires worthless, stock is worth $S_T$ ⇒ $S_T$.
• If $S_T \le K$:
  • A: call worthless, bond pays $K$ ⇒ $K$.
  • B: stock worth $S_T$, put worth $K - S_T$ ⇒ $S_T + (K - S_T) = K$.

So payoffs are identical in all states. No-arbitrage implies their initial prices must be equal, which gives the formula above.

2.2 Forward-style parity (useful in commodities)

Equity-style spot + bond is often inconvenient in power & commodities where:

• forwards/futures are the primary liquid object,
• “spot” is ill-defined or only a settlement index.

Define:

• Forward price for delivery at $T$: $F_{0,T}$
• Discount factor: $D(0,T)$ (e.g. $e^{-rT}$)

For options on forwards (common in power/gas):

$$C_0 - P_0 = D(0,T) \cdot (F_{0,T} - K)$$

This is the same parity, just expressed in terms of the forward instead of spot.

Interview line:

“In commodities, I tend to write put–call parity as $C - P = D(0,T)(F - K)$. It’s more natural because forwards are the primary quotes and carry/seasonality are already baked into $F$.”

3.1 Call option bounds

Call premium $C_0$ must satisfy:

$$\max(0, S_0 - K e^{-rT}) \;\le\; C_0 \;\le\; S_0$$

• Upper bound $C_0 \le S_0$: A call is never worth more than the stock itself; with the stock you can do at least as well as the call in all states. If $C_0 > S_0$, short call, buy stock → non-negative payoff and upfront profit.

• Lower bound $C_0 \ge \max(0, S_0 - K e^{-rT})$: Start from parity $C_0 = P_0 + S_0 - K e^{-rT}$. Since $P_0 \ge 0$,

$$C_0 \ge S_0 - K e^{-rT}.$$

Also, a call can never be negative ⇒ $C_0 \ge 0$. Combine them: $C_0 \ge \max(0, S_0 - K e^{-rT})$.

3.2 Put option bounds

Similarly, put premium $P_0$ must satisfy:

$$\max(0, K e^{-rT} - S_0) \;\le\; P_0 \;\le\; K e^{-rT}$$

• Upper bound $P_0 \le K e^{-rT}$: The put can never pay more than $K$, so its present value can’t exceed the PV of $K$.

• Lower bound from parity and $C_0 \ge 0$:

$$P_0 = C_0 - S_0 + K e^{-rT} \ge -S_0 + K e^{-rT}$$

and also $P_0 \ge 0$. Combine them.

4.1 Monotonicity in strike

For two strikes $K_1 < K_2$, same maturity $T$:

• Calls: $C(K_1) \ge C(K_2)$
• Puts: $P(K_1) \le P(K_2)$

Pathwise for calls:

$$(S_T - K_1)^+ \ge (S_T - K_2)^+ \quad \text{for all } S_T$$

So call with lower strike dominates higher-strike call ⇒ must be more expensive.

For puts:

$$(K_1 - S_T)^+ \le (K_2 - S_T)^+ \quad \text{for all } S_T$$

So higher-strike put dominates ⇒ must be more expensive.

If your call surface increases with strike, something is wrong: either quotes are off, or you are misreading them.

4.2 Monotonicity in underlying price

If spot increases, calls should be worth more and puts less.

Formally, for fixed $K, T$:

• $C(S)$ is increasing in $S$,
• $P(S)$ is decreasing in $S$.

Pathwise dominance:

• If you add $\Delta S > 0$ to the underlying level in every state, the call payoff never decreases (and typically increases); the put payoff never increases.

In models, this is reflected by:

• call delta $\partial C / \partial S \in (0, 1)$
• put delta $\partial P / \partial S \in (-1, 0)$

4.3 Monotonicity in maturity (caveat: rates, dividends)

Under constant non-negative rates and no dividends:

• $C(K, T)$ is non-decreasing in $T$: more time ⇒ more optionality.
• $P(K, T)$ is non-increasing in $T$: with more time, the PV of strike shrinks and chance of ending ITM is nuanced; but the no-arb result (in frictionless, non-negative rate environment) gives non-increasing behaviour.

In real markets:

• Term structure of rates and dividends (or cost-of-carry, convenience yield) can bend this a bit.
• But gross violations (“1-year call cheaper than 1-month call at same strike in a stable market”) are a red flag.

1. Cast of characters

Think of every option as a little contract between two people:

• Underlying: the thing whose price moves (stock, power forward, gas, etc.).
• Strike $K$: the “deal price” written in the contract.
• Expiry $T$: the last day the contract matters.
• Buyer (long option):
  • Pays premium today.
  • Has a right, not an obligation.
  • Worst case: loses the premium.
• Seller / writer (short option):
  • Receives premium today.
  • Has an obligation if the buyer exercises.
  • Best case: keeps the premium; worst case: can lose a lot.

The option price / premium is just:

“How much the buyer pays today to shift some risk to the seller.”

2. Call option – “right to buy later”

Plain English:

• A call is the right to buy the underlying at strike $K$ on/at expiry.
• The buyer of a call likes up moves in the underlying.
• The seller of a call is giving away upside in exchange for the premium.

Payoff at expiry (call buyer):

• If final price $S_T \le K$: right is useless → payoff = 0.
• If $S_T > K$: exercise → payoff = $S_T - K$.

Graph in words:

• Draw horizontal axis = final price $S_T$.
• Vertical axis = payoff.
• Below $K$: flat at zero.
• Above $K$: straight line with slope +1 starting from zero at $S_T = K$.

So:

• Long call payoff:
  • Bottom is floored at zero.
  • Upside is unlimited.
• Long call profit = payoff − premium.
  • Shifts the whole graph down by the premium.

How a pro describes it:

“A long call is a cheap way to be long convexity: you pay a fixed premium, your downside is capped, your upside is open-ended.”

Seller side:

• Short call payoff = −(long call payoff).
• Short call profit = premium − payoff.
  • You earn premium if market is quiet or down.
  • You bleed if market rallies through the strike.

“Short call = I’ve sold upside. I collect premium; I’m short convexity and short gamma.”

3. Put option – “right to sell later”

Plain English:

• A put is the right to sell the underlying at strike $K$ on/at expiry.
• The buyer of a put likes down moves (it’s protection against drops).
• The seller of a put is insuring someone against a drop.

Payoff at expiry (put buyer):

• If $S_T \ge K$: you wouldn’t sell at K, so payoff = 0.
• If $S_T < K$: you exercise → payoff = $K - S_T$.

Graph:

• Horizontal axis = $S_T$.
• Vertical axis = payoff.
• Above $K$: flat at zero.
• Below $K$: straight line going up as price goes down (slope −1).

Again:

• Long put payoff:
  • Upside is limited (max is K if S_T→0).
  • Downside is floored at zero.
• Long put profit = payoff − premium.

Typical practitioner line:

“A long put is downside insurance: I pay a premium so I know my minimum selling price.”

Seller:

• Short put payoff = −(long put payoff).
• Profit = premium − payoff.

“Short put = I’m selling downside insurance. I earn premium most of the time, but I eat the crash.”

4. Buyer vs seller: who wants what?

5. What does the option price “mean”?

The premium today is roughly:

“Discounted expected payoff under a risk-neutral world + risk/technical adjustments.”

Intuitively, it:

• Increases with:
  • Time to expiry (more time for things to happen),
  • Volatility (bigger possible moves),
  • Interest rates (for calls via carry),
  • Moneyness (ITM has intrinsic value).
• Decreases if:
  • Deep out of the money,
  • Almost at expiry with nothing happening.

Practitioner sentence:

“High premium = the market expects big moves or is scared of the tails.”

6. One simple picture (mentally)

If you visualise only two graphs, you’re 80% there:

1. Long call P&L at expiry (vs $S_T$):
   • Flat negative (−premium) until $S_T$ reaches about $K + \text{premium}$,
   • Then an upward line with slope +1.
2. Long put P&L at expiry:
   • Flat negative (−premium) above about $K - \text{premium}$,
   • Then line with slope −1 as price falls.

Everything else (delta, gamma, parity) is just different ways of talking about those two shapes.

7. How to sound relaxed when talking about options

You can summarise like this in an interview:

• “A call is a right to buy; a put is a right to sell. The buyer pays a premium and gets convex exposure; the seller receives the premium and is short convexity. At expiry, a long call pays $(S_T - K)^+$, a long put pays $(K - S_T)^+$. The option price is what the market charges today to transfer that upside or downside risk.”

If you want, next I can give you one tiny code cell that plots the four basic payoff shapes (long/short call/put) so you can see them and memorise the picture.