Sukrit Mittal
Franklin Templeton Investments
Before tackling continuous-time models and multiple assets, we need to understand:
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.
We work with only two points in time:
This is deliberately simplistic. Real markets trade continuously, but the two-period model:
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.
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.
Example: A bond with \(A(0) = 100\) and annual interest rate \(r = 5\%\) will be worth exactly \(A(T) = 105\) in one year.
\[ K_S = \frac{S(T) - S(0)}{S(0)}, \qquad K_A = \frac{A(T) - A(0)}{A(0)} = r_F \]
Consider an investor holding:
The portfolio value at time \(t\) is
\[ V(t) = x S(t) + y A(t) \]
The pair \((x, y)\) is called a portfolio. Here, \(x\) and \(y\) are called positions.
Between \(t = 0\) and \(t = T\), your wealth changes by:
\[ V(T) - V(0) = x\big(S(T) - S(0)\big) + y\big(A(T) - A(0)\big) \]
The return on the portfolio is:
\[ K_V = \frac{V(T) - V(0)}{V(0)} = \frac{x\big(S(T) - S(0)\big) + y\big(A(T) - A(0)\big)}{xS(0) + yA(0)} \]
This return can be rewritten in terms of asset returns. Dividing numerator and denominator appropriately:
\[ K_V = w_S K_S + w_A K_A \]
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 \(w_S + w_A = 1\).
These assumptions are idealizations, but they approximate reality in liquid, developed markets.
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.
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.
These assumptions are idealizations, but they approximate reality in liquid, developed markets.
Investors can always meet their obligations. No bankruptcy or counterparty risk.
No fees, taxes, or bid-ask spreads. Every trade executes at the quoted price.
Reality check: These two assumptions are clearly false but simplify the analysis. More realistic models incorporate these frictions.
Short positions require borrowing assets (for stocks) or money (for bonds).
Can you profit if a stock price goes down? Yes—by short selling.
Steps:
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 \(10 \times 100 = 1000\).
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.
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:
Classic example:
Key features:
If such an opportunity existed and was known, rational investors would:
But this would immediately drive prices back into equilibrium (buying in Delhi raises prices there, selling in Mumbai lowers prices there).
We assume that markets do not allow arbitrage opportunities.
Real-world timing: In modern electronic markets, arbitrage opportunities vanish in milliseconds, not hours or days.
But these are exceptions. For liquid, developed markets, the no-arbitrage assumption is excellent.
There is no portfolio \((x, y)\) such that:
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.
The future stock price can take only two values:
\[ S(T) = \begin{cases} S_u(T) & \text{with probability } p \text{ (up state)}\\ S_d(T) & \text{with probability } 1-p \text{ (down state)} \end{cases} \]
with \(0 < p < 1\) and \(S_d(T) < S_u(T)\).
This is the simplest model of uncertainty:
Despite its simplicity, this model:
Suppose:
\[ K_S(T)= \begin{cases} 25\% & \text{with probability } p \text{ (up state)}\\ 5\% & \text{with probability } 1-p \text{ (down state)}\end{cases} \]
If \(p = 0.6\) (60% chance of up state):
\[ E(K_S) = 0.6 \times 25\% + 0.4 \times 5\% = 15\% + 2\% = 17\% \]
The risk premium is \(17\% - 10\% = 7\%\)—the extra expected return for bearing risk.
To avoid arbitrage, the following condition must hold:
\[ \frac{S_d(T)}{S(0)} < \frac{A(T)}{A(0)} < \frac{S_u(T)}{S(0)} \]
Interpretation: The risk-free growth factor \((1+r_F) = \frac{A(T)}{A(0)}\) must lie strictly between the down and up growth factors of the stock.
Equivalently, in terms of returns:
\[ K_d < r_F < K_u \]
where \(K_d = \frac{S_d(T) - S(0)}{S(0)}\) and \(K_u = \frac{S_u(T) - S(0)}{S(0)}\).
Intuition:
Either case creates arbitrage. We'll prove this formally in the next slides.
Suppose \(\frac{A(T)}{A(0)} \leq \frac{S_d(T)}{S(0)}\) (equivalently, \(r_F \leq K_d\)).
This means the stock beats the bond even in the worst case. That's too good to be true!
At \(t=0\):
Initial cost: \(V(0) = 1 \cdot S(0) - \frac{S(0)}{A(0)} \cdot A(0) = 0\) âś“
Portfolio: \(x=1\), \(y = -\frac{S(0)}{A(0)}\)
\[ V(T) = S(T) - \frac{S(0)}{A(0)} A(T) \]
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!
Suppose \(\frac{A(T)}{A(0)} \geq \frac{S_u(T)}{S(0)}\) (equivalently, \(r_F \geq K_u\)).
This means the bond beats the stock even in the best case. Why would anyone hold the stock?
At \(t=0\):
Initial cost: \(V(0) = -1 \cdot S(0) + \frac{S(0)}{A(0)} \cdot A(0) = 0\) âś“
Portfolio: \(x=-1\), \(y = \frac{S(0)}{A(0)}\)
You must buy back the stock to return it. Your portfolio is worth:
\[ V(T) = -S(T) + \frac{S(0)}{A(0)} A(T) \]
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!
How do we measure the risk and expected return of a portfolio? This section introduces the key statistical tools.
Let:
\[ S(T)=\begin{cases} 100 & \text{with probability } 0.8 \\ 60 & \text{with probability } 0.2 \end{cases} \]
Stock returns: \[ K_S = \begin{cases} \frac{100-80}{80} = 25\% & \text{with probability } 0.8 \\ \frac{60-80}{80} = -25\% & \text{with probability } 0.2 \end{cases} \]
Suppose you invest \(10,000\) with: \(x = 50\) shares; \(y = 60\) bonds.
Initial value: \(V(0) = 50 \times 80 + 60 \times 100 = 4000 + 6000 = 10{,}000\) âś“
Weights: 40% in stocks, 60% in bonds
\[ V(T) = 50 \times S(T) + 60 \times 110 = 50 \times S(T) + 6600 \]
\[ V(T) = \begin{cases} 50 \times 100 + 6600 = 11{,}600 & \text{if stock goes up} \\ 50 \times 60 + 6600 = 9{,}600 & \text{if stock goes down} \end{cases} \]
Corresponding returns:
\[ K_V = \begin{cases} \frac{11600-10000}{10000} = 16\% & \text{with probability } 0.8 \\ \frac{9600-10000}{10000} = -4\% & \text{with probability } 0.2 \end{cases} \]
Expected return (mean):
\[ E(K_V) = 16\% \times 0.8 + (-4\%) \times 0.2 = 12.8\% - 0.8\% = 12\% \]
Risk (standard deviation):
First, compute the variance:
\[ \text{Var}(K_V) = E[(K_V - E(K_V))^2] = (16\% - 12\%)^2 \times 0.8 + (-4\%-12\%)^2 \times 0.2 \]
\[ = (4\%)^2 \times 0.8 + (-16\%)^2 \times 0.2 = 0.0016 \times 0.8 + 0.0256 \times 0.2 \]
\[ = 0.00128 + 0.00512 = 0.0064 \]
Standard deviation:
\[ \sigma_V = \sqrt{0.0064} = 0.08 = 8\% \]
All bonds:
All stocks:
Mixed portfolio (60-40): \(E(K_V) = 12\%\), \(\sigma_V = 8\%\)
Observation: The mixed portfolio has intermediate risk and return, demonstrating diversification.
Design a portfolio with initial wealth of \(10{,}000\), split 50:50 between stocks and bonds (by value).
Compute:
Use the same asset prices as above:
\[ S(T) = \begin{cases} 100 & \text{with probability } 0.8 \\ 60 & \text{with probability } 0.2 \end{cases} \]
Initial wealth: 10,000, split 50-50.
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%).
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).
Long forward position: Agrees to buy the asset at \(t=T\) for price \(F\)
Short forward position: Agrees to sell the asset at \(t=T\) for price \(F\)
Note these are symmetric: one party's gain is the other's loss (zero-sum).
What should \(F\) be? If \(F\) is set wrong, one party gets a guaranteed advantage—that's arbitrage!
No-arbitrage forward price:
\[ F = S(0) \cdot \frac{A(T)}{A(0)} = S(0) \cdot (1+r_F) \]
Intuition: The forward price is the current stock price "grown" at the risk-free rate.
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 \(r_F\) (costs \(0\) today, owe \(S(0)(1+r_F)\) at time \(T\))
Both strategies deliver one stock at \(T\) and cost zero today. By no-arbitrage, they must cost the same at \(T\):
\[ F = S(0)(1+r_F) \]
Unlike forward contracts (which are obligations), options give you rights without obligations.
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 \(T\).
You choose whether to exercise:
If exercising would lose money, simply walk away—that's the power of an option.
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: \[ P(T) = \max(X - S(T), 0) = \begin{cases} X - S(T) & \text{if } S(T) < X \\ 0 & \text{if } S(T) \geq X \end{cases} \]
Options have asymmetric payoffs:
This asymmetry has value, so you must pay a premium \(C(0)\) or \(P(0)\) to acquire the option.
Stock at \(S(0) = 100\). Buy a call option with strike \(X = 100\) for premium \(C(0) = 10\).
At maturity:
Maximum loss: 10. Maximum gain: unlimited!
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.
Let us represent portfolios as \((x, y, z)\) where:
Portfolio value:
\[V(t) = xS(t) + yA(t) + zC(t)\]
To price the option, we find \(C(0)\) such that a replicating portfolio exists.
\[ S(0) = 100, \quad S(T) = \begin{cases} 120 & \text{with probability } p \\ 80 & \text{with probability } 1-p \end{cases} \]
Call option with strike \(X = 100\):
\[ C(T) = \max(S(T) - 100, 0) = \begin{cases} 20 & \text{(up state)} \\ 0 & \text{(down state)} \end{cases} \]
Risk-free asset: \(A(0) = 100\), \(A(T) = 110\) (so \(r = 10\%\)).
We seek a portfolio \((x, y)\) such that:
\[ xS(T) + yA(T) = C(T) \quad \text{in both states} \]
This gives us a system of two equations (one for each state):
\[ \begin{cases} x \cdot 120 + y \cdot 110 = 20 & \text{(up state)} \\ x \cdot 80 + y \cdot 110 = 0 & \text{(down state)} \end{cases} \]
Step 1: Subtract the second equation from the first:
\[ x(120 - 80) = 20 - 0 \implies 40x = 20 \implies x = \frac{1}{2} \]
Step 2: Substitute \(x = 1/2\) into the second equation:
\[ \frac{1}{2} \cdot 80 + y \cdot 110 = 0 \implies 40 + 110y = 0 \implies y = -\frac{40}{110} = -\frac{4}{11} \]
The replicating portfolio is:
Net cost today:
\[ C(0) = \frac{1}{2} \times 100 - \frac{4}{11} \times 100 = 50 - 36.36 = 13.64 \]
Up state: \(V(T) = \frac{1}{2} \times 120 - \frac{4}{11} \times 110 = 60 - 40 = 20\) âś“
Down state: \(V(T) = \frac{1}{2} \times 80 - \frac{4}{11} \times 110 = 40 - 40 = 0\) âś“
The portfolio perfectly replicates the option payoff, so by no-arbitrage: \(C(0) = 13.64\).
| Feature | Forward | Future | Option |
|---|---|---|---|
| Obligation | Yes | Yes | No (for buyer) |
| Upfront cost | \(0\) | Margin | Premium |
| Loss potential | Unlimited | Unlimited | Limited (premium) for buyer |
| Flexibility | Low | Medium | High |
| Trading venue | OTC | Exchange | Both |
Foreign exchange (FX) markets enable trading between currencies. They are among the largest and most liquid markets in the world.
Example: If \(X(t) = 83\) INR/USD, then 1 USD costs 83 INR.
Domestic risk-free asset: Bank account in domestic currency
Foreign risk-free asset: Bank account in foreign currency
\[ V(t) = y_d A_d(t) + y_f A_f(t) X(t) \]
where:
Key insight: The exchange rate \(X(t)\) is uncertain, making foreign holdings risky from the domestic investor's perspective.
Apply the same binomial framework to exchange rates:
\[ X(T) = \begin{cases} X_u(T) & \text{with probability } p \text{ (foreign currency appreciates)} \\ X_d(T) & \text{with probability } 1-p \text{ (foreign currency depreciates)} \end{cases} \]
By analogy with the stock-bond case, to rule out arbitrage:
\[ \frac{X_d(T)}{X(0)} < \frac{1+r_d}{1+r_f} < \frac{X_u(T)}{X(0)} \]
Economic interpretation:
The ratio \(\frac{1+r_d}{1+r_f}\) is the relative interest rate factor.
This is the foundation of covered interest rate parity, one of the most fundamental relationships in international finance.
If this condition is violated:
Such opportunities are immediately exploited by banks and hedge funds, keeping exchange rates and interest rates in equilibrium.
By no-arbitrage, the forward exchange rate must be:
\[ F = X(0) \cdot \frac{1+r_d}{1+r_f} \]
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:
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.
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:
Options are not just for speculation—they are powerful tools for risk management and designing specific payoff profiles.
Options combine:
Let's explore some concrete strategies.
Scenario: You have \(1000\) to invest. Stock trades at \(S(0) = 100\):
\[ S(T) = \begin{cases} 120 & \text{with probability } 0.5 \\ 80 & \text{with probability } 0.5 \end{cases} \]
Buy 10 shares for \(1000\).
Payoff:
Expected value: \(E(V) = 0.5 \times 1200 + 0.5 \times 800 = 1000\) (0% expected return—fair bet)
Risk (std dev): \(200\)
Assume call options (strike \(X = 100\)) cost \(C(0) = 13.64\) each.
Buy \(\frac{1000}{13.64} \approx 73.3\) options.
Payoff:
Expected value: \(E(V) = 0.5 \times 1466 + 0.5 \times 0 = 733\)
Risk (std dev): \(733\)
| Strategy | Up State | Down State | Expected | Risk |
|---|---|---|---|---|
| Stock | \(1200\) | \(800\) | \(1000\) | \(200\) |
| Options | \(1466\) | \(0\) | \(733\) | \(733\) |
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.
Now let's see how options can reduce risk, not increase it.
Scenario: You own a volatile stock with \(S(0) = 100\):
\[ S(T) = \begin{cases} 160 & \text{with probability } 0.5 \text{ (if up)} \\ 40 & \text{with probability } 0.5 \text{ (if down)} \end{cases} \]
At \(t=0\):
The \(31.81\) premium grows to \(31.81 \times 1.10 = 35\) by time \(T\).
Your portfolio value is:
\[ V(T) = S(T) - C(T) + 35 \]
where \(C(T) = \max(S(T) - 100, 0)\) is the call payoff you owe (since you sold it).
Up state (\(S(T) = 160\)):
Down state (\(S(T) = 40\)):
Let's compare the risk profiles:
Payoffs: \(S(T) \in \{40, 160\}\)
Range: \(40\) to
\(160\)
Risk (spread): \(160 - 40 =
120\)
Expected value: \(E(V) = 0.5
\times 160 + 0.5 \times 40 = 100\)
Payoffs: \(V(T) \in \{75, 135\}\)
Range: \(75\) to
\(135\)
Risk (spread): \(135 - 75 =
60\)
Expected value: \(E(V) = 0.5
\times 135 + 0.5 \times 75 = 105\)
Interpretation: By selling the call, you:
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.
Even the simplest market model—two periods, two assets, two states—reveals profound insights: