International Finance

Options Primer: Pricing and Payoffs

Main issues

• What is an option? Calls, puts, and the four basic positions

• Payoff diagrams: visualizing option positions at expiration

• FX options: conventions, intrinsic value, and key price drivers

• Put-call parity: the fundamental link between calls, puts, and forwards

• Binomial option pricing: one period, multi-period, and convergence to Garman-Kohlhagen

This primer provides the pricing background used in Lecture 8 (Nonlinear Exposure and FX Options).

This is a technical primer. Lecture 8 focuses on why firms need options and how to use them strategically. This primer covers the mechanics: what options are, how their payoffs work, and how they are priced using no-arbitrage arguments.

What Is an Option?

Definition

An option gives the holder the right, but not the obligation, to buy or sell an asset at a predetermined price.

• Call option: right to buy the underlying at the strike price \(X\)

• Put option: right to sell the underlying at the strike price \(X\)

Key distinction from forwards:

• A forward obligates both parties. An option gives a choice.

• This asymmetry has value — the buyer pays a premium upfront.

Long vs. short positions

	Long (buyer)	Short (seller/writer)
Pays	Premium upfront	—
Receives	Right to exercise	Premium upfront
Risk	Limited to premium	Short call: potentially unlimited loss. Short put: large but bounded loss, up to \(\approx X\) per unit, net of premium
Motivation	Insurance / speculation	Earn premium income

• The buyer pays for optionality — the right to walk away if the option is not favorable

• The seller is obligated to deliver if the buyer exercises

• Zero-sum game: buyer’s gain = seller’s loss

European vs. American

• European option: can only be exercised at expiration

• American option: can be exercised at any time before expiration

In foreign exchange markets:

• Most traded FX options are European style

• Simplifies pricing: only need to consider the terminal payoff

• American options are more complex (early exercise premium) but rare in FX

Throughout this primer, all options are European.

Payoff Diagrams

The four basic positions

Long call: \(\max(S_T - X,\; 0)\)

• Profits when \(S_T > X\); loss capped at zero payoff

Short call: \(-\max(S_T - X,\; 0)\)

• Mirror image of long call; seller’s obligation

Long put: \(\max(X - S_T,\; 0)\)

• Profits when \(S_T < X\); loss capped at zero payoff

Short put: \(-\max(X - S_T,\; 0)\)

• Mirror image of long put; seller’s obligation

Payoff diagrams

Four panels showing the payoff (not profit) of each position. Notice the kinks at the strike price X. The buyer’s positions have a floor at zero; the seller’s positions have a ceiling at zero. Payoff is not the same as profit — we haven’t subtracted the premium yet.

From payoff to profit

Profit = Payoff \(-\) Premium

• The buyer pays the premium upfront, so profit shifts the payoff curve down by the premium amount

• Break-even for a long call: \(S_T = X + \text{Premium}\)

• Break-even for a long put: \(S_T = X - \text{Premium}\)

• Maximum loss for the buyer: the premium paid (limited downside)

• Maximum loss for the seller: potentially unlimited for a short call; large but bounded (up to \(\approx X\) per unit, net of premium) for a short put

Profit diagrams

The dashed lines show payoff, the solid lines show profit (payoff minus premium). Notice how the break-even points shift away from the strike by the premium amount. The shaded regions show where each position makes or loses money.

FX Options

FX option = right to exchange currencies

An FX option gives the right to exchange one currency for another at a fixed rate.

• A “EUR call / USD put” is the right to buy EUR and sell USD at strike \(X\) (USD/EUR)

• A “EUR put / USD call” is the right to sell EUR and buy USD at strike \(X\)

Convention: An FX option is simultaneously a call on one currency and a put on the other.

Example: A EUR call with \(X = 1.10\) USD/EUR gives the right to buy EUR 1 for USD 1.10.

• Exercise if \(S_T > 1.10\): buy EUR cheaply at 1.10 and sell at market rate \(S_T\)
• Don’t exercise if \(S_T < 1.10\): buy EUR at the cheaper market rate instead

The forward rate and option moneyness

For a European FX option, the payoff at maturity depends on the terminal spot rate \(S_T\):

\[\text{Call: } \max(S_T - X,\, 0) \qquad \text{Put: } \max(X - S_T,\, 0)\]

Today, the no-arbitrage benchmark for maturity \(T\) is the forward rate (CIP):

\[F_{0,T} \;=\; S_0 \times \frac{1 + r}{1 + r^*}\]

Forward moneyness:

• \(X < F_{0,T}\): call is ITM forward; put is OTM forward
• \(X = F_{0,T}\): both are at-the-money forward (ATMF)
• \(X > F_{0,T}\): call is OTM forward; put is ITM forward

Key takeaway. The forward rate is the no-arbitrage anchor for option moneyness. The terminal payoff still depends on \(S_T\), not on \(F\).

At-the-money forward and put-call parity

For European options with the same strike and expiry, put-call parity gives:

\[c - p \;=\; \frac{F_{0,T} - X}{1 + r}.\]

When \(X = F_{0,T}\):

\[c - p \;=\; 0 \quad\Longrightarrow\quad c = p.\]

This is why the at-the-money forward (ATMF) strike is the natural reference: call and put have the same price, and the forward is the no-arbitrage anchor. The full derivation appears in the next section.

The expressions \(\max(F - X, 0)\) and \(\max(X - F, 0)\) are sometimes called “forward intrinsic value” or “forward moneyness.” They are useful classification labels, not exercise values for European options (which can only be exercised at \(T\), against \(S_T\)). Present-value lower bounds on European option prices are \(c \geq \max(F_{0,T} - X, 0)/(1+r)\) and \(p \geq \max(X - F_{0,T}, 0)/(1+r)\), i.e., the forward moneyness discounted to today.

Factors affecting option prices

Five key inputs determine option value (table holds the forward \(F\) fixed when varying \(r\)):

Factor	Call	Put	Intuition
Forward \(F\) \(\uparrow\)	\(\uparrow\)	\(\downarrow\)	Higher expected terminal rate
Strike \(X\) \(\uparrow\)	\(\downarrow\)	\(\uparrow\)	Harder/easier to exercise
Volatility \(\sigma\) \(\uparrow\)	\(\uparrow\)	\(\uparrow\)	More chance of large moves
Time to expiry \(T\) \(\uparrow\)	\(\uparrow\)	\(\uparrow\) (\(^*\))	More time for favorable moves
Domestic rate \(r\) \(\uparrow\) (holding \(F\) fixed)	\(\downarrow\)	\(\downarrow\)	Future payoffs discounted more heavily

\(^*\) Generally true for plain European options; can be affected by rates/carry in edge cases.

Note. If spot \(S_0\) and \(r^*\) are held fixed instead, a higher domestic rate raises the forward \(F\), which tends to raise calls and lower puts. The table separates this forward effect from pure discounting.

Volatility is the most important input — it measures how much the exchange rate might move. Higher volatility \(\Rightarrow\) more “upside” for the option buyer \(\Rightarrow\) higher price.

Put-Call Parity

The fundamental link

For European options with the same strike and expiry:

\[\boxed{c - p = \frac{F - X}{1 + r}}\]

where \(c\) = call price, \(p\) = put price, \(F\) = forward rate, \(X\) = strike, \(r\) = domestic interest rate.

This is a no-arbitrage relationship, not a model — it holds regardless of what pricing model you use.

Put-call parity links calls, puts, and forwards. It is model-free because it relies only on the absence of arbitrage. If it were violated, you could construct a riskless profit by trading the three instruments.

Proof by replication

Consider two portfolios at time 0:

Portfolio A: Long call + invest \(\frac{X}{1+r}\) in domestic bonds

Portfolio B: Long put + long forward at rate \(F\) + invest \(\frac{F}{1+r}\) in domestic bonds

At expiration \(T\):

State	Portfolio A	Portfolio B
\(S_T > X\)	\((S_T - X) + X = S_T\)	\(0 + (S_T - F) + F = S_T\)
\(S_T \leq X\)	\(0 + X = X\)	\((X - S_T) + (S_T - F) + F = X\)

Both give \(\max(S_T, X)\) at \(T\). Same payoff \(\Rightarrow\) same price today:

\[c + \frac{X}{1+r} = p + 0 + \frac{F}{1+r} \quad \Rightarrow \quad c - p = \frac{F - X}{1+r}\]

Visual: long call + short put = forward

Key implication: A forward is just a combination of a call and a put at the same strike.

At strike \(X = F\): the call and put have equal value (\(c = p\)).

Numerical example

Setup: \(S_0 = 1000\), \(r = 5\%\), \(r^* = 4\%\), \(X = 1050\)

Forward rate: \(F = 1000 \times \frac{1.05}{1.04} \approx 1009.6\)

Put-call parity: \(c - p = \frac{F - X}{1 + r} = \frac{1009.6 - 1050}{1.05} = \frac{-40.4}{1.05} = -38.5\)

So \(c = p - 38.5\). The call is cheaper than the put because the strike is above the forward.

• If someone quotes you \(p = 60\), then \(c = 60 - 38.5 = 21.5\)

• If you know the call price, you get the put price for free (and vice versa)

One-Period Binomial Model

Setup

Two possible states at \(T\): the exchange rate goes up to \(S_u\) or down to \(S_d\).

The call has known payoffs: \(c_u = \max(S_u - X, 0)\) and \(c_d = \max(S_d - X, 0)\).

Question: What is the fair price \(c_0\) today?

The replication idea

Key insight: We can replicate the option payoff using two instruments:

1. \(\Delta\) units of a forward contract (costs nothing to enter)

2. \(B\) dollars invested in a domestic bond (earns \(r\))

Replicating portfolio payoffs at \(T\):

• Up state: \(\Delta(S_u - F) + B(1 + r) = c_u\)

• Down state: \(\Delta(S_d - F) + B(1 + r) = c_d\)

Two equations, two unknowns. Solve for \(\Delta\) and \(B\).

Solving for the option price

Subtract down from up:

\[\Delta(S_u - S_d) = c_u - c_d \quad \Rightarrow \quad \boxed{\Delta = \frac{c_u - c_d}{S_u - S_d}}\]

\(\Delta\) is the hedge ratio — how many forwards replicate the option.

From the down-state equation:

\[B = \frac{c_d - \Delta(S_d - F)}{1 + r}\]

Option price = cost of replicating portfolio = \(B\) (forward is free to enter)

\[\boxed{c_0 = B = \frac{c_d - \Delta(S_d - F)}{1 + r}}\]

No probabilities needed!

Notice: we never used the probability of up vs. down.

• The option price comes purely from no-arbitrage — if the option is mispriced relative to the replicating portfolio, there is a riskless profit

• Different investors can disagree about probabilities and still agree on the price

• This is the power of replication: price by matching payoffs, not by forecasting

This is the same logic behind CIP (Lecture 4) and forward pricing:

• If two strategies have the same payoff, they must have the same price

• Otherwise: arbitrage

Risk-neutral pricing

An equivalent and often more convenient approach. Define:

\[\boxed{q = \frac{F - S_d}{S_u - S_d}}\]

Then the option price can be written as:

\[c_0 = \frac{q \cdot c_u + (1 - q) \cdot c_d}{1 + r}\]

• \(q\) is the risk-neutral probability — it makes the expected rate equal to the forward

• Not a real-world probability — it’s the probability that is consistent with no-arbitrage pricing

• Equivalent to the replication approach, but often faster to compute

Numerical example

Setup: \(S_0 = 1000\), \(S_u = 1100\), \(S_d = 950\), \(r = 5\%\), \(r^* = 4\%\), \(X = 1050\)

\(F = 1000 \times \frac{1.05}{1.04} \approx 1009.6\), \(\quad c_u = 50\), \(\quad c_d = 0\)

Delta: \(\Delta = \frac{50 - 0}{1100 - 950} = \frac{50}{150} = \frac{1}{3}\)

Bond: \(B = \frac{0 - \frac{1}{3}(950 - 1009.6)}{1.05} = \frac{0 + 19.87}{1.05} = 18.92\)

Option price: \(c_0 = B = 18.92\)

Check with risk-neutral pricing:

\(q = \frac{1009.6 - 950}{1100 - 950} = \frac{59.6}{150} = 0.397\)

\(c_0 = \frac{0.397 \times 50 + 0.603 \times 0}{1.05} = \frac{19.87}{1.05} = 18.92\) \(\checkmark\)

Multi-Period Binomial Model

From one period to two

The tree recombines: an up move followed by a down move gives the same node as down-then-up. This is critical for computational efficiency — with N periods, we have N+1 terminal nodes instead of 2^N.

Working backwards

Backward induction: Start from the terminal payoffs and work backwards one period at a time.

At each node, apply the one-period formula with the same risk-neutral probability:

\[q = \frac{f_{\Delta t} - d}{u - d} \qquad \text{where } f_{\Delta t} = \frac{1 + r_{\Delta t}}{1 + r^*_{\Delta t}}\]

Step 1: Terminal payoffs — \(c = \max(S_T - X, 0)\) at each final node

Step 2: At each period-1 node: \(c = \frac{q \cdot c_u + (1-q) \cdot c_d}{1 + r_{\Delta t}}\)

Step 3: At the root: same formula using the period-1 values

Each backward step is just the one-period model applied locally.

Two-period example

Using \(u = 1.10\), \(d = 0.95\), \(X = 1050\), \(r_{\Delta t} = 2.5\%\), \(r^*_{\Delta t} = 2.0\%\):

Terminal payoffs:

• \(S_{uu} = 1210\): \(c_{uu} = \max(1210 - 1050, 0) = 160\)

• \(S_{ud} = 1045\): \(c_{ud} = \max(1045 - 1050, 0) = 0\)

• \(S_{dd} = 902.5\): \(c_{dd} = 0\)

Risk-neutral probability: \(q = \frac{1.025/1.02 - 0.95}{1.10 - 0.95} = \frac{0.0549}{0.15} = 0.366\)

Period 1 (up node): \(c_u = \frac{0.366 \times 160 + 0.634 \times 0}{1.025} = \frac{58.6}{1.025} = 57.1\)

Period 1 (down node): \(c_d = \frac{0.366 \times 0 + 0.634 \times 0}{1.025} = 0\)

Period 0: \(c_0 = \frac{0.366 \times 57.1 + 0.634 \times 0}{1.025} = \frac{20.9}{1.025} = 20.4\)

From 2 to N periods

The two-period model extends naturally:

• Split the time horizon \(T\) into \(N\) equal periods of length \(\Delta t = T/N\)

• Each period has up factor \(u\) and down factor \(d\)

• Tree has \(N+1\) terminal nodes (recombining), backward induction from the end

Key question: How to choose \(u\) and \(d\)?

Cox-Ross-Rubinstein (CRR) calibration:

\[\boxed{u = e^{\sigma\sqrt{\Delta t}}, \qquad d = \frac{1}{u} = e^{-\sigma\sqrt{\Delta t}}}\]

• \(\sigma\) = annualized volatility of the exchange rate

• As \(N\) increases, \(\Delta t \to 0\) and the binomial model produces finer and finer approximations

• The tree structure ensures \(d = 1/u\) so the tree recombines

Convergence to Black-Scholes

Binomial \(\to\) Garman-Kohlhagen

The oscillating blue line is the binomial price as a function of the number of periods N. It converges rapidly to the GK price (red dashed line). The oscillation between odd and even N is typical of binomial models. By N=50 or so the price is very close to GK.

The Garman-Kohlhagen formula

As \(N \to \infty\), the CRR binomial price converges to:

\[\boxed{c = e^{-rT}\Big[F \cdot N(d_1) - X \cdot N(d_2)\Big]}\]

where:

\[d_1 = \frac{\ln(F/X) + \frac{1}{2}\sigma^2 T}{\sigma \sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}\]

and \(F = S_0 \cdot e^{(r - r^*)T}\) is the forward rate (continuous compounding).

• This is the Garman-Kohlhagen (1983) formula — the FX version of Black-Scholes

• \(N(\cdot)\) is the standard normal CDF

• Put price: use put-call parity, or \(p = e^{-rT}\left[X\, N(-d_2) - F\, N(-d_1)\right]\)

Compounding convention. With continuous compounding (as in GK), put-call parity reads \(c - p = e^{-rT}(F - X)\). The discrete one-period form \(c - p = (F - X)/(1+r)\) used earlier in the primer is the same identity under simple-compounding notation; do not mix the two on the same slide.

Interpreting the GK formula

The GK formula has a clean financial interpretation:

• \(N(d_2) \approx\) risk-neutral probability that the call expires in the money

• \(N(d_1)\) is the forward delta of the call. The spot delta is \(e^{-r^* T} N(d_1)\).

• \(e^{-rT}\) = discounting to present value

Inputs: \(S_0\), \(X\), \(\sigma\), \(r\), \(r^*\), \(T\) — all observable except \(\sigma\).

In practice:

• \(\sigma\) is either estimated from historical data or implied from market prices

• Implied volatility is the \(\sigma\) that makes GK match the observed option price

• This is exactly the concept explored in Lecture 8 (volatility smile, risk premium)

Summary

Concept	Key takeaway
Options	Right, not obligation — buyer pays premium for asymmetric payoff
Payoffs	Kink at strike; four basic positions (long/short \(\times\) call/put)
FX conventions	EUR call = USD put; forward rate anchors ATMF and put-call parity
Put-call parity	\(c - p = (F-X)/(1+r)\): model-free, links calls, puts, forwards
Binomial model	Replication + no-arbitrage \(\Rightarrow\) unique price without probabilities
Risk-neutral pricing	Probability \(q\) that makes expected return = forward; equivalent to replication
Multi-period	Backward induction; CRR calibration: \(u = e^{\sigma\sqrt{\Delta t}}\)
Convergence	Binomial \(\to\) Garman-Kohlhagen as \(N \to \infty\)

Next: Lecture 8 applies these tools to corporate hedging — option strategies, nonlinear exposure, and the information content of implied volatility.