A Simple Market Model

Sukrit Mittal
Franklin Templeton Investments

Outline

1. Basic notions and assumptions
2. No-arbitrage principle
3. One-step binomial model
4. Risk and return
5. Forward contracts
6. Call and put options
7. Foreign exchange
8. Managing risk with options

Basic Notions and Assumptions

We begin with the simplest possible financial market that still captures uncertainty. This abstraction, while highly simplified, is powerful enough to demonstrate fundamental principles of quantitative finance.

Why Start Simple?

Before tackling continuous-time models and multiple assets, we need to understand:

• How uncertainty affects asset prices
• What prevents arbitrage opportunities
• How derivatives can be priced without predicting the future

The two-period, two-asset model is the “hydrogen atom” of mathematical finance—simple enough to solve exactly, yet rich enough to contain the key insights.

Traded Assets

We assume that two assets are traded:

• One risk-free asset Think of a bank deposit or a government-issued bond. In practice, this could be a treasury bill or a savings account with guaranteed interest. The key property: its future value is known today with certainty.

• One risky asset Typically a stock, but it could also be a commodity, currency, or any uncertain asset. The key property: its future value is random and unknown today.

This dichotomy between certainty and uncertainty is fundamental to all of finance.

Time Structure

We work with only two points in time:

• Today: $t = 0$ (the present, when we make decisions)
• A future date: $t = T$ (e.g., one year from now, when uncertainty resolves)

This is deliberately simplistic. Real markets trade continuously, but the two-period model:

• Eliminates mathematical complications
• Focuses attention on the core logic
• Can be extended to multi-period models later (binomial trees)

Interpretation: Between $t=0$ and $t=T$, no trading occurs. We set up our portfolio at $t=0$, wait, and observe the outcome at $t=T$. This “buy-and-hold” assumption will be relaxed in more advanced models.

Risky and Risk-Free Assets

Risky Asset (Stock)

• Described by the number of shares held
• Current price $S(0)$ is known (observable in the market)
• Future price $S(T)$ is uncertain (depends on future states of the world)

Example: A share of Apple stock costs $S(0) = 175$ today, but its price in one year could be anywhere from $150$ to $200$ depending on earnings, economic conditions, etc.

Risk-Free Asset (Bond)

• Described by money in a bank account or bonds held
• Current price $A(0)$ is known (typically normalized to $1$ or $100$)
• Future price $A(T)$ is known with certainty via the risk-free rate $r$

Example: A bond with $A(0) = 100$ and annual interest rate $r = 5\%$ will be worth exactly $A(T) = 105$ in one year.

Returns

Portfolio

Consider an investor holding:

• $x$ shares of stock (can be fractional)
• $y$ units of the risk-free asset (e.g., dollars in a bank account)

The portfolio value at time $t$ is

The pair $(x, y)$ is called a portfolio. Here, $x$ and $y$ are called positions.

Interpretation

• $V(0)$ is the initial wealth (how much capital you invest)
• $V(T)$ is the final wealth (what you end up with)
• The portfolio $(x, y)$ is chosen at $t=0$ and held until $t=T$

Change in Wealth

Between $t = 0$ and $t = T$, your wealth changes by:

Portfolio Return

The return on the portfolio is:

This return can be rewritten in terms of asset returns. Dividing numerator and denominator appropriately:

where $w_S = \frac{xS(0)}{V(0)}$ and $w_A = \frac{yA(0)}{V(0)}$ are the portfolio weights (fractions of wealth invested in each asset). Note that $w_S + w_A = 1$.

Some Important Market Assumptions

These assumptions are idealizations, but they approximate reality in liquid, developed markets.

Divisibility

Assets can be held in fractional quantities.

Rationale: While you cannot buy 0.5 shares of a single stock in retail markets, institutional investors trading millions of dollars can effectively achieve any fractional exposure through various instruments or by treating portfolios as continuous.

Liquidity

There are no bounds on $x$ and $y$. Assets can be bought or sold in arbitrary quantities at market prices without affecting those prices.

Rationale: This is the “price taker” assumption. In reality, large orders move prices (market impact), but for sufficiently liquid markets and moderate position sizes, this effect is negligible.

Solvency

Investors can always meet their obligations. No bankruptcy or counterparty risk.

No Transaction Costs

No fees, taxes, or bid-ask spreads. Every trade executes at the quoted price.

Reality check: These last two assumptions are clearly false but simplify the analysis. More realistic models incorporate these frictions.

Long and Short Positions

• Long position: $x > 0$ or $y > 0$—holding a positive quantity of an asset. You profit if the price rises.
• Short position: $x < 0$ or $y < 0$—holding a negative quantity. You profit if the price falls.

Short positions require borrowing assets (for stocks) or money (for bonds).

Shorting an Asset

Can you profit if a stock price goes down? Yes—by short selling.

Mechanics of Short Selling

Steps:

1. Borrow shares from a broker (who lends them from their inventory or other clients)
2. Sell them immediately at the current market price $S(0)$
3. Wait for the price to (hopefully) fall
4. Buy back the shares at the new price $S(T)$
5. Return the borrowed shares to the broker

Profit/Loss

• If the price falls ($S(T) < S(0)$): you profit by $S(0) - S(T)$ per share
• If the price rises ($S(T) > S(0)$): you lose $S(T) - S(0)$ per share

Example: Short 100 shares at $50$. If the price drops to $40$, you profit $10 \times 100 = 1000$. If it rises to $60$, you lose $10 \times 100 = 1000$.

Risks of Short Selling

• Unlimited loss potential: A stock price can rise indefinitely, but can only fall to zero
• Margin requirements: Brokers require collateral to ensure you can buy back the shares
• Short squeeze: If many investors short a stock and the price rises, they may all rush to buy back shares, driving the price even higher

Portfolio Representation

Short positions correspond to negative holdings. For example, a portfolio with $x = -10$ means you are short 10 shares.

The value is: \(V(t) = -10 \cdot S(t) + y A(t)\)

At $t=0$, shorting 10 shares gives you $+10 \cdot S(0)$ in cash, which you can invest in bonds or use as collateral.

2. The No-Arbitrage Principle

What Is Arbitrage?

Arbitrage is the possibility of making a risk-free profit with no net investment.

In mathematical terms, arbitrage exists if there is a portfolio $(x, y)$ such that:

1. $V(0) = 0$ (no initial investment)
2. $V(T) \geq 0$ with probability 1 (never lose money)
3. $V(T) > 0$ with positive probability (sometimes make money)

Classic example:

• Buy gold in Delhi at ₹60,000 per 10g
• Simultaneously sell the same gold in Mumbai at ₹60,300 per 10g
• Pocket the difference instantly: ₹300 profit with zero risk

Key features:

• Simultaneous transactions (no time risk)
• No risk (prices are locked in)
• No capital committed (borrow to buy, repay from sale proceeds)

Why This Is “Free Money”

If such an opportunity existed and was known, rational investors would:

1. Borrow unlimited money
2. Execute the arbitrage
3. Scale profits to infinity

But this would immediately drive prices back into equilibrium (buying in Delhi raises prices there, selling in Mumbai lowers prices there).

Why Arbitrage Should Not Exist

We assume that markets do not allow arbitrage opportunities.

Why This Assumption Is Reasonable

• Speed: Arbitrage is exploited by high-frequency trading algorithms in microseconds
• Competition: Thousands of traders and algorithms scan for arbitrage 24/7
• Self-correcting: The act of exploiting arbitrage eliminates the opportunity
  • Buying the cheap asset raises its price
  • Selling the expensive asset lowers its price
  • Prices converge until the arbitrage disappears

Real-world timing: In modern electronic markets, arbitrage opportunities vanish in milliseconds, not hours or days.

Exceptions

• Transaction costs: Small price differences may persist if they’re less than the cost to trade
• Market frictions: Liquidity constraints, margin requirements, or regulatory barriers
• Rare events: During market crashes or flash crashes, temporary arbitrage can appear

But these are exceptions. For liquid, developed markets, the no-arbitrage assumption is excellent.

Mathematical Statement

There is no portfolio $(x, y)$ such that:

• $V(0) = 0$ (zero initial cost)
• $V(T) \ge 0$ with probability 1 (never negative)
• $V(T) > 0$ with positive probability (sometimes positive)

Equivalent statement (contrapositive): If two portfolios have the same payoff at $T$ in all possible states, they must have the same price at $t=0$.

This principle is the foundation of derivative pricing: We price options by replicating their payoffs with portfolios of simpler assets.

3. One-Step Binomial Model

Assumption

The future stock price can take only two values:

with $0 < p < 1$ and $S_d(T) < S_u(T)$.

Why Only Two States?

This is the simplest model of uncertainty:

• One state represents “good news” (stock goes up)
• One state represents “bad news” (stock goes down)
• All information about the future is captured by which state occurs

Despite its simplicity, this model:

• Captures the essence of risk and uncertainty
• Admits no-arbitrage pricing
• Can be extended to multi-period binomial trees (next lectures)
• Converges to continuous-time models (Black-Scholes) as we add more periods

Example

Suppose:

• $S(0) = 100$
• $A(0) = 100$
• Stock return:

• Risk-free return: $K_A=10\%$ (deterministic, same in both states)

Observations

1. The stock has higher average return (assuming $p > 0.5$): $E(K_S) > 10\%$
2. But the stock has risk: sometimes returns only 5%
3. The bond has guaranteed return: always 10%
4. Investors face a risk-return tradeoff

Expected Stock Return

If $p = 0.6$ (60% chance of up state):

The risk premium is $17\% - 10\% = 7\%$—the extra expected return for bearing risk.

No-Arbitrage Constraint

To avoid arbitrage, the following condition must hold:

Interpretation: The risk-free growth factor $(1+r) = \frac{A(T)}{A(0)}$ must lie strictly between the down and up growth factors of the stock.

Equivalently, in terms of returns:

where $K_d = \frac{S_d(T) - S(0)}{S(0)}$ and $K_u = \frac{S_u(T) - S(0)}{S(0)}$.

Why This Must Hold

Intuition:

• If $r \geq K_u$, the bond dominates the stock (higher return with no risk) → everyone sells the stock
• If $r \leq K_d$, the stock dominates the bond (higher return in both states) → everyone sells the bond

Either case creates arbitrage. We’ll prove this formally in the next slides.

Our Example

Check: $\frac{105}{100} = 1.05 < \frac{110}{100} = 1.10 < \frac{125}{100} = 1.25$ ✓

Or: $5\% < 10\% < 25\%$ ✓

The no-arbitrage condition is satisfied.

Case-1: Arbitrage When Bond Return Is Too Low

Suppose $\frac{A(T)}{A(0)} \leq \frac{S^d(T)}{S(0)}$ (equivalently, $r \leq K_d$).

This means the stock beats the bond even in the worst case. That’s too good to be true!

Arbitrage Strategy

At $t=0$:

• Borrow the amount $S(0)$ at the risk-free rate (short bond)
• Buy one share of stock for $S(0)$

Initial cost: $V(0) = 1 \cdot S(0) - \frac{S(0)}{A(0)} \cdot A(0) = 0$ ✓

Portfolio: $x=1$, $y = -\frac{S(0)}{A(0)}$

Payoff at Time $T$

In the up state: \(V(T) = S^u(T) - \frac{S(0)}{A(0)}A(T) > 0 \quad \text{(strictly positive)}\)

In the down state: \(V(T) = S^d(T) - \frac{S(0)}{A(0)}A(T) \geq 0 \quad \text{(non-negative by assumption)}\)

Result: We invested nothing, we never lose money, and we sometimes make money. This is arbitrage!

Numerical Example

Let $S(0) = 100$, $S^d(T) = 110$, $A(0) = 100$, $A(T) = 105$.

Check: $\frac{110}{100} = 1.10 > \frac{105}{100} = 1.05$ (condition violated).

Portfolio: Buy 1 share, borrow 100.

Payoff: $V(T) = S(T) - 105 \in {110-105, 125-105} = {5, 20}$ (always positive!).

Case-2: Arbitrage When Bond Return Is Too High

Suppose $\frac{A(T)}{A(0)} \geq \frac{S^u(T)}{S(0)}$ (equivalently, $r \geq K_u$).

This means the bond beats the stock even in the best case. Why would anyone hold the stock?

Arbitrage Strategy

At $t=0$:

• Sell short one share for $S(0)$ (borrow and sell the stock)
• Invest the proceeds $S(0)$ at the risk-free rate

Initial cost: $V(0) = -1 \cdot S(0) + \frac{S(0)}{A(0)} \cdot A(0) = 0$ ✓

Portfolio: $x=-1$, $y = \frac{S(0)}{A(0)}$

Payoff at Time $T$

You must buy back the stock to return it. Your portfolio is worth:

In the up state: \(V(T) = -S^u(T) + \frac{S(0)}{A(0)}A(T) \geq 0 \quad \text{(non-negative by assumption)}\)

In the down state: \(V(T) = -S^d(T) + \frac{S(0)}{A(0)}A(T) > 0 \quad \text{(strictly positive)}\)

Result: Again, we have arbitrage!

Numerical Example

Let $S(0) = 100$, $S^u(T) = 105$, $A(0) = 100$, $A(T) = 110$.

Check: $\frac{105}{100} = 1.05 < \frac{110}{100} = 1.10$ (condition violated).

Portfolio: Short 1 share, invest 100.

Payoff: $V(T) = -S(T) + 110 \in {-105+110, -95+110} = {5, 15}$ (always positive!).

4. Risk and Return

How do we measure the risk and expected return of a portfolio? This section introduces the key statistical tools.

Example Setup

Let:

• $A(0) = 100$; $A(T) = 110$ (risk-free rate $r = 10\%$)
• $S(0) = 80$;

Stock returns: \(K_S = \begin{cases} \frac{100-80}{80} = 25\% & \text{with probability } 0.8 \\ \frac{60-80}{80} = -25\% & \text{with probability } 0.2 \end{cases}\)

Portfolio Construction

Suppose you invest $10,000$ with: $x = 50$ shares; $y = 60$ bonds.

Initial value: $V(0) = 50 \times 80 + 60 \times 100 = 4000 + 6000 = 10{,}000$ ✓

Weights: 40% in stocks, 60% in bonds

Portfolio Value at $T$

Corresponding returns:

Expected Return and Risk

Expected return (mean):

Risk (standard deviation):

First, compute the variance:

Standard deviation:

Comparison with Pure Strategies

• All bonds:
  • Return: $K_A = 10\%$ (deterministic)
  • Risk: $\sigma_A = 0\%$
• All stocks:
  • Expected return: $E(K_S)=25\% \times 0.8 + (-25\%) \times 0.2 = 20\% - 5\% = 15\%$
  • Variance: $(25\%-15\%)^2 \times 0.8 + (-25\%-15\%)^2 \times 0.2 = 0.01 \times 0.8 + 0.16 \times 0.2 = 0.04$
  • Risk: $\sigma_S = \sqrt{0.04} = 20\%$

Mixed portfolio (60-40): $E(K_V) = 12\%$, $\sigma_V = 8\%$

Observation: The mixed portfolio has intermediate risk and return, demonstrating diversification.

Exercise

Design a portfolio with initial wealth of $10{,}000$, split 50:50 between stocks and bonds (by value).

Compute:

1. The portfolio $(x, y)$
2. Final portfolio value $V(T)$ in each state
3. Expected return $E(K_V)$
4. Risk (standard deviation) $\sigma_V$

Use the same asset prices as above:

• $A(0)=100$ and $A(T)=110$ dollars
• $S(0)=80$ and

Solution Sketch

Initial wealth: 10,000, split 50-50.

• Stock investment: 5,000 → $x = 5000/80 = 62.5$ shares
• Bond investment: 5,000 → $y = 5000/100 = 50$ bonds

Final value: \(V(T) = 62.5 \times S(T) + 50 \times 110 = 62.5 \times S(T) + 5500\)

Up state: $V(T) = 6250 + 5500 = 11{,}750$ → Return = 17.5%

Down state: $V(T) = 3750 + 5500 = 9{,}250$ → Return = -7.5%

Expected return: $E(K_V) = 0.8 \times 17.5\% + 0.2 \times (-7.5\%) = 14\% - 1.5\% = 12.5\%$

Variance: $(17.5\%-12.5\%)^2 \times 0.8 + (-7.5\%-12.5\%)^2 \times 0.2 = 25 \times 0.8 + 400 \times 0.2 = 20 + 80 = 100$

Risk: $\sigma_V = \sqrt{100} = 10\%$

Insight: More stocks → higher expected return (12.5% vs 12%) but also higher risk (10% vs 8%).

5. Forward Contracts

A forward contract is an agreement to buy or sell a risky asset at a specified future time (the delivery date) for a price $F$ fixed today (the forward price).

Key Features

• Obligation (not an option): Both parties must complete the transaction
• No upfront payment: The contract itself costs nothing to enter
• Settlement at maturity: Exchange happens at $t=T$

Terminology

• Long forward position: Agrees to buy the asset at $t=T$ for price $F$
  • Profits if $S(T) > F$ (bought cheap)
  • Loses if $S(T) < F$ (overpaid)
• Short forward position: Agrees to sell the asset at $t=T$ for price $F$
  • Profits if $S(T) < F$ (sold high)
  • Loses if $S(T) > F$ (sold too cheap)

Payoff at Maturity

• Long forward payoff: $S(T) - F$
• Short forward payoff: $F - S(T)$

Note these are symmetric: one party’s gain is the other’s loss (zero-sum).

Pricing the Forward

What should $F$ be? If $F$ is set wrong, one party gets a guaranteed advantage—that’s arbitrage!

No-arbitrage forward price:

Intuition: The forward price is the current stock price “grown” at the risk-free rate.

Derivation

Consider two strategies to own the stock at time $T$:

Strategy 1: Enter a long forward contract (costs $0$ today, pay $F$ at time $T$)

Strategy 2: Buy stock today for $S(0)$, financed by borrowing $S(0)$ at rate $r$ (costs $0$ today, owe $S(0)(1+r)$ at time $T$)

Both strategies deliver one stock at $T$ and cost zero today. By no-arbitrage, they must cost the same at $T$:

6. Call and Put Options

Unlike forward contracts (which are obligations), options give you rights without obligations.

Definitions

Call Option: The right (but not obligation) to buy an asset at a predetermined price $X$ (the strike price) at time $T$.

Put Option: The right (but not obligation) to sell an asset at strike price $X$ at time $T$.

Why “Option”?

You choose whether to exercise:

• Call: Exercise if $S(T) > X$ (buy cheap, sell at market price)
• Put: Exercise if $S(T) < X$ (buy at market price, sell high)

If exercising would lose money, simply walk away—that’s the power of an option.

Payoff at Maturity

• Call option payoff: \(C(T) = \max(S(T) - X, 0) = \begin{cases} S(T) - X & \text{if } S(T) > X \\ 0 & \text{if } S(T) \leq X \end{cases}\)

• Put option payoff: \(P(T) = \max(X - S(T), 0) = \begin{cases} X - S(T) & \text{if } S(T) < X \\ 0 & \text{if } S(T) \geq X \end{cases}\)

Key Observation

Options have asymmetric payoffs:

• Upside: unlimited (for calls) or up to $X$ (for puts)
• Downside: limited to zero (you just don’t exercise)

This asymmetry has value, so you must pay a premium $C(0)$ or $P(0)$ to acquire the option.

Example

Stock at $S(0) = 100$. Buy a call option with strike $X = 100$ for premium $C(0) = 10$.

At maturity:

• If $S(T) = 120$: Exercise, profit = $120 - 100 - 10 = 10$
• If $S(T) = 80$: Don’t exercise, loss = $- 10$ (just the premium)

Maximum loss: 10. Maximum gain: unlimited!

Options Pricing: The Replication Method

Goal: Find the fair price $C(0)$ for a call option.

Key idea: Construct a portfolio of stocks and bonds that replicates the option’s payoff in every possible state. By no-arbitrage, the portfolio and the option must have the same price.

Extended Portfolio Representation

Let us represent portfolios as $(x, y, z)$ where:

• $x$ = shares of stock
• $y$ = units of bond
• $z$ = options held

Portfolio value:

To price the option, we find $C(0)$ such that a replicating portfolio exists.

Example

Call option with strike $X = 100$:

Risk-free asset: $A(0) = 100$, $A(T) = 110$ (so $r = 10\%$).

Replicating the Option

We seek a portfolio $(x, y)$ such that:

This gives us a system of two equations (one for each state):

Solving for $(x, y)$

Step 1: Subtract the second equation from the first:

Step 2: Substitute $x = 1/2$ into the second equation:

Interpretation

The replicating portfolio is:

• Buy $1/2$ share of stock (costs $\frac{1}{2} \times 100 = 50$)
• Short $4/11$ bonds (borrow $\frac{4}{11} \times 100 = 36.36$)

Net cost today:

Verification

Up state: $V(T) = \frac{1}{2} \times 120 - \frac{4}{11} \times 110 = 60 - 40 = 20$ ✓

Down state: $V(T) = \frac{1}{2} \times 80 - \frac{4}{11} \times 110 = 40 - 40 = 0$ ✓

The portfolio perfectly replicates the option payoff, so by no-arbitrage: $C(0) = 13.64$.

7. Foreign Exchange

Foreign exchange (FX) markets enable trading between currencies. They are among the largest and most liquid markets in the world.

Setting

• Domestic currency: Your home currency (e.g., USD, INR)
• Foreign currency: Another currency (e.g., EUR, GBP)
• Exchange rate $X(t)$: Price of one unit of foreign currency in domestic currency

Example: If $X(t) = 83$ INR/USD, then 1 USD costs 83 INR.

Assets in FX Markets

1. Domestic risk-free asset: Bank account in domestic currency
   • Grows at domestic rate: $A_d(T) = A_d(0)(1 + r_d)$
2. Foreign risk-free asset: Bank account in foreign currency
   • Grows at foreign rate: $A_f(T) = A_f(0)(1 + r_f)$
   • Value in domestic currency: $A_f(t) \cdot X(t)$

Portfolio Representation

where:

• $y_d$ = units of domestic currency held
• $y_f$ = units of foreign currency held
• $X(t)$ converts foreign holdings to domestic value

Key insight: The exchange rate $X(t)$ is uncertain, making foreign holdings risky from the domestic investor’s perspective.

Binomial Model for Exchange Rate

Apply the same binomial framework to exchange rates:

• Exchange rate uncertainty:

No-Arbitrage Condition (Interest Rate Parity)

By analogy with the stock-bond case, to rule out arbitrage:

Economic interpretation:

The ratio $\frac{1+r_d}{1+r_f}$ is the relative interest rate factor.

• If $r_d > r_f$: Domestic rates are higher, so domestic currency should depreciate (foreign currency appreciates) on average
• The expected exchange rate growth should compensate for the interest rate differential

This is the foundation of covered interest rate parity, one of the most fundamental relationships in international finance.

Intuition

If this condition is violated:

• Borrow in the low-rate currency
• Invest in the high-rate currency
• Lock in the exchange rate
• Profit risk-free (carry trade arbitrage)

Such opportunities are immediately exploited by banks and hedge funds, keeping exchange rates and interest rates in equilibrium.

FX Forward and Option

FX Forward Rate

By no-arbitrage, the forward exchange rate must be:

Interpretation: If domestic rates are higher ($r_d > r_f$), the foreign currency must trade at a forward premium ($F > X(0)$) to prevent arbitrage.

Example:

• Spot: $X(0) = 80$ INR/USD
• Indian rate: $r_d = 7\%$
• US rate: $r_f = 3\%$
• Forward: $F = 80 \times \frac{1.07}{1.03} = 80 \times 1.0388 \approx 83.11$ INR/USD

Currency Call Option

A currency call option gives the right to buy foreign currency at strike $K$ (in domestic currency).

Payoff: $C(T) = \max(X(T) - K, 0)$

Use case: An Indian company expects to pay 1M in one year. Buy a call option with strike $K = 85$ INR/USD to cap the cost.

• If INR weakens ($X(T) = 90$): Exercise, pay only 85 INR/USD (plus premium)
• If INR strengthens ($X(T) = 80$): Don’t exercise, buy USD at spot

Replication

Just as with stock options, currency options can be priced by replicating their payoff using portfolios of domestic and foreign bonds.

The same algebra applies:

1. Set up two equations (one for each state)
2. Solve for the replicating portfolio $(y_d, y_f)$
3. Compute the option price as the portfolio cost

8. Managing Risk with Options

Options are not just for speculation—they are powerful tools for risk management and designing specific payoff profiles.

Three Key Uses

1. Hedging downside risk (insurance)
2. Enhancing yield (selling options for premium income)
3. Constructing tailored exposures (combining options with underlying assets)

Options combine:

• Probabilistic thinking (understanding states and probabilities)
• No-arbitrage logic (replication and fair pricing)
• Strategic flexibility (choosing when to exercise)

Let’s explore some concrete strategies.

Example 1: Speculative Use of Options

Scenario: You have 1000 to invest. Stock trades at $S(0) = 100$: \(S(T) = \begin{cases} 120 & \text{with probability } 0.5 \\ 80 & \text{with probability } 0.5 \end{cases}\)

Strategy A: Buy Stock Directly

Buy 10 shares for 1000.

Payoff:

• Up state: $V(T) = 10 \times 120 = 1200$ (20% gain)
• Down state: $V(T) = 10 \times 80 = 800$ (20% loss)

Expected value: $E(V) = 0.5 \times 1200 + 0.5 \times 800 = 1000$ (0% expected return—fair bet)

Risk (std dev): $200$

Strategy B: Buy Call Options

Assume call options (strike $X=100$) cost $C(0) = 13.64$ each.

Buy $\frac{1000}{13.64} \approx 73.3$ options.

Payoff:

• Up state: $V(T) = 73.3 \times (120-100) = 73.3 \times 20 = 1466$
• Down state: $V(T) = 73.3 \times 0 = 0$

Expected value: $E(V) = 0.5 \times 1466 + 0.5 \times 0 = 733$

Risk (std dev): $733$

Comparison

Observation: Options provide leverage—higher potential upside (1466 vs 1200) but also higher risk (can lose everything). The expected return is actually negative due to the premium you pay.

Purchasing options is riskier and more speculative than buying the stock directly.

Example 2: Covered Call (Risk Reduction)

Now let’s see how options can reduce risk, not increase it.

Scenario: You own a volatile stock with $S(0) = 100$: \(S(T) = \begin{cases} 160 & \text{with probability } 0.5 \text{ (if up)} \\ 40 & \text{with probability } 0.5 \text{ (if down)} \end{cases}\)

Strategy: Covered Call

At $t=0$:

1. Own 1 share of stock (worth 100)
2. Sell 1 call option (strike $X=100$) for premium $C(0) = 31.81$
3. Invest the premium at 10% risk-free rate

The 31.81 premium grows to $31.81 \times 1.10 = 35$ by time $T$.

Payoff at $T$

Your portfolio value is:

where $C(T) = \max(S(T) - 100, 0)$ is the call payoff you owe (since you sold it).

Up state ($S(T) = 160$):

• Stock value: 160
• Call obligation: $-(160-100) = - 60$
• Premium growth: 35
• Total: $160 - 60 + 35 = 135$

Down state ($S(T) = 40$):

• Stock value: 40
• Call obligation: 0 (option expires worthless)
• Premium growth: 35
• Total: $40 - 0 + 35 = 75$

Covered Call: Risk Analysis

Let’s compare the risk profiles:

Holding Stock Only (No Option)

Payoffs: $S(T) \in {40, 160}$

Range: 40 to 160
Risk (spread): $160 - 40 = 120$
Expected value: $E(V) = 0.5 \times 160 + 0.5 \times 40 = 100$

Covered Call Strategy

Payoffs: $V(T) \in {75, 135}$

Range: 75 to 135
Risk (spread): $135 - 75 = 60$
Expected value: $E(V) = 0.5 \times 135 + 0.5 \times 75 = 105$

Key Insights

1. Risk reduced by 50%: The spread decreased from 120 to 60
2. Downside protection: Worst case improved from 40 to 75 (the premium provides a cushion)
3. Capped upside: You give up gains above 135 (trade-off for receiving premium)
4. Expected value increased: From 100 to 105 (you collect premium income)

Interpretation: By selling the call, you:

• Give up unlimited upside potential
• In exchange for immediate premium income
• Which cushions against losses

This is a risk-reducing strategy, popular among investors who own stocks and want to generate income while limiting volatility.

Real-world use: Portfolio managers use covered calls to enhance yields during sideways markets.

Takeaway

Even the simplest market model—two periods, two assets, two states—reveals profound insights:

Key Principles Established

1. No-arbitrage pricing: Assets must be priced consistently to prevent risk-free profits
2. Replication: Derivatives can be priced by constructing portfolios that mimic their payoffs
3. Risk-return tradeoff: Higher expected returns come with higher risk
4. Diversification: Combining assets can reduce risk

Mathematical Tools Introduced

• Portfolio representation: $V(t) = xS(t) + yA(t)$
• Expected value and variance for measuring risk
• Systems of linear equations for replication
• No-arbitrage constraints

Derivative Insights

• Forwards lock in future prices
• Options provide asymmetric payoffs (limited downside, unlimited upside)
• Options have value due to this asymmetry
• Strategies combining options and stocks can either increase or decrease risk