Financial Mathematics
Swaraj Patil
Summer’25 SoS
—
DHRUVAM
INTRODUCTION
I started this project with an aim
to improve my knowledge of
finance and find the intersection
with mathematics. I wanted to
explore what actually the financial
terms we hear in everyday life
actually mean and how they are
deeply woven into the whole
structure of finance alongside the
mathematics supporting them. I
took the course Financial Markets
offered by Yale University under
the professor Robert Shilling
online as an attempt to learn
general notions of financial
markets and how they function in
real world.
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1. Introduction to Financial Markets
Financial markets are platforms where buyers and sellers engage in
the trade of financial assets such as stocks, bonds, derivatives, and
currencies. They play a critical role in the allocation of resources
and the facilitation of capital formation.
2. Bonds: The Backbone of Fixed Income
What is a Bond?
A bond is a financial instrument representing a loan from an
investor to a borrower, typically corporate or governmental. The
borrower agrees to pay back the principal (face value) on a fixed
maturity date and often makes periodic interest payments known as
coupons.
Types of Bonds
• Zero-Coupon Bonds: Sold at a discount, no intermediate
payments, pays full face value at maturity.
• Coupon Bonds: Periodic interest payments and repayment of
principal at maturity.
• Government Bonds: Issued by sovereigns, often seen as riskfree.
• Corporate Bonds: Issued by companies; higher risk and return.
Risks Involved
• Credit Risk: Issuer defaults on payment.
• Inflation Risk: Reduced purchasing power.
• Liquidity Risk: Difficulty selling the bond without affecting its
price.
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Current Bond Market Conditions
As of 2025, global bond markets are facing pressures from rising
interest rates, resulting in significant mark-to-market losses. Longterm U.S. Treasury yields have surged, causing price drops across
fixed-income portfolios.
3. The 2008 Financial Crisis: Lessons in Risk
The global financial crisis of 2008 was triggered by the collapse of
mortgage-backed securities and excessive risk-taking by financial
institutions. This event caused a severe liquidity crunch, leading to
the failure of major institutions and a collapse in global markets.
Bonds, especially those tied to subprime mortgages, lost massive
value, highlighting the need for robust risk management and
transparency.
4. Stocks:
Equity Ownership and Market Participation
Stocks represent ownership in a company. Investors earn returns
through dividends and capital gains. Compared to bonds, stocks are
riskier but offer higher potential returns.
Key Concepts
• Dividends: Profit distribution.
• Volatility: Stock prices fluctuate based on performance,
sentiment, and macroeconomic factors.
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5. Derivatives: Managing Financial Exposure
Derivatives derive their value from underlying assets such as stocks,
bonds, or interest rates. They are used for hedging, speculation, and
arbitrage.
Common Types of Derivatives
• Options: Right, not obligation, to buy (call) or sell (put).
o American Options: Exercise anytime before expiration.
o European Options: Exercise only at expiration.
• Futures and Forwards: Agreement to buy/sell at future date
and price.
• Swaps: Exchange of cash flows, often fixed vs floating interest
rates.
Why Use Derivatives?
• Risk management
• Speculation
• Exploiting arbitrage opportunities
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6. Arbitrage and How Institutions and Retail
Investors Make Money
What is Arbitrage?
Arbitrage involves exploiting price differences in different markets
to earn risk-free profits. It is a cornerstone of institutional trading.
Types of Arbitrage
• Convertible Arbitrage: Bond vs stock mispricing.
• Interest Rate Arbitrage: Mismatch in bond yields.
• ETF Arbitrage: Difference between ETF price and net asset
value.
Institutional Advantages
• Better data access
• Algorithmic trading
• Lower transaction costs due to scale
Retail Investing
Retail investors use platforms like Zerodha, Robinhood, or Upstox.
Though lacking institutional resources, they can still profit through
informed investing, mutual funds, ETFs, or SIPs (Systematic
Investment Plans).
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7. Interest Rates: The Time Value of Money
Simple vs Compound Interest
• Simple Interest: Interest on principal only.
• Compound Interest: Interest on both principal and
accumulated interest.
8. Present and Future Value of Cash Flows
Present Value (PV)
Future Value (FV)
Where:
• : Future value or cash flow
• : Interest rate per period
• : Time periods
Used for evaluating investments, comparing loans, and assessing
projects.
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9. The Role of Exchanges and OTC Markets
Exchanges
Physical or electronic venues where securities are listed (e.g., NSE,
NYSE).
Over-the-Counter (OTC)
Decentralized markets where trading is direct between parties,
often for derivatives and bonds.
Market Makers
Ensure liquidity by quoting both buy and sell prices continuously.
10. Market Indexes
Indexes summarize the performance of a group of assets:
• S&P 500: U.S. large-cap stocks
• Nifty 50: Indian benchmark equity index
• Dow Jones: U.S. blue-chip companies
Used to measure market trends and as benchmarks for portfolio
performance.
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11. Financial Institutions vs Retail Investors
Financial Institutions
• Mutual funds, hedge funds, banks
• Use leverage and complex models
• Engage in arbitrage and hedging
Retail Investors
• Individual market participants
• Rely on basic research, platforms, and SIPs
• Influenced by market sentiment and social media
Appendix and References
• Investopedia.com
• NSE India (nseindia.com)
• BSE India
• Khan Academy Finance Lectures
• “Options, Futures, and Other Derivatives” by John C. Hull
• U.S. Treasury Reports
• SEBI publications
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Options
A financial option is a contract that gives the buyer the right, but
not the obligation, to buy or sell an underlying asset at a
predetermined price on or before a specified date.
•
•
Call Option: Gives the holder the right to buy an asset. A call
option is profitable if the asset's price (S) rises above the strike
price (K). The payoff at expiration is max(ST−K,0).
Put Option: Gives the holder the right to sell an asset. A put
option is profitable if the asset's price (S) falls below the strike
price (K). The payoff at expiration is max(K−ST,0).
Key Terms:
• Underlying Asset (S): The financial instrument (e.g., stock,
index, commodity) on which the option is based.
• Strike Price (K): The predetermined price at which the asset
can be bought or sold.
• Expiration Date (T): The date after which the option is void.
• Premium: The price of the option contract itself.
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The Black-Scholes-Merton Model
The Black-Scholes-Merton (BSM) model provides a theoretical
estimate of the price of European-style options. It's derived from
the Black-Scholes Partial Differential Equation (PDE),
which describes how the price of an option evolves over time.
The PDE is:
𝜕𝑉 1 2 2 𝜕 2 𝑉
𝜕𝑉
+ 𝜎 𝑆
+
𝑟𝑆
− 𝑟𝑉 = 0
𝜕𝑡 2
𝜕𝑆 2
𝜕𝑆
Where:
• V is the option price as a function of asset price S and time t.
• r is the continuously compounded risk-free interest rate.
• σ is the volatility of the underlying asset's returns.
The Greeks
The "Greeks" are partial derivatives of the option pricing formula.
They measure the sensitivity of an option's price to changes in its
underlying parameters.
Delta (Δ)
Delta measures the rate of change of the option price with respect to
a $1 unit change in the underlying asset's price. It represents the
option's price sensitivity to the underlying's price movements.
𝜕𝑉
Δ=
𝜕𝑆
•
•
•
For a Call Option: Δ𝐶 = 𝑁(𝑑1 )
For a Put Option: ΔP=N(d1)−1
Range: Call delta is [0,1]; Put delta is [−1,0].
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Gamma (Γ)
Gamma measures the rate of change of Delta with respect to a $1
unit change in the underlying asset's price. It shows how much the
delta will change as the underlying moves.
𝜕2 𝑉
𝜕Δ
ΔΓ = 𝜕𝑆 2 = 𝜕𝑆
•
𝑁 ′ (𝑑1 )
For both Calls and Puts: Γ = 𝑆 𝜎 𝑇−𝑡 Where N′(x) is the
𝑡
√
Probability Density Function (PDF) of the standard normal
′(
distribution:𝑁 𝑥 ) =
1
√2𝜋
𝑒
−
𝑥2
2
Theta (Θ)
Theta measures the rate of change of the option price with respect
to the passage of time. It's commonly known as "time decay." It's
usually negative because an option loses value as its expiration date
approaches.
𝜕𝑉
Θ=−
𝜕𝑡
•
•
𝑆 𝑁 ′ (𝑑 )𝜎
1
For a Call Option: Θ𝐶 = − 𝑡2 𝑇−𝑡
− 𝑟𝐾𝑒 −𝑟(𝑇−𝑡) 𝑁(𝑑2 )
√
For a Put Option: Θ𝑃 = −
𝑆𝑡 𝑁′ (𝑑1 )𝜎
2√𝑇−𝑡
+ 𝑟𝐾𝑒 −𝑟(𝑇−𝑡) 𝑁(−𝑑2 )
Vega (ν)
Vega measures the rate of change of the option price with respect to
a 1% change in the volatility of the underlying asset.
𝜕𝑉
𝜈=
𝜕𝜎
′
• For both Calls and Puts: 𝜈 = 𝑆𝑡 √𝑇 − 𝑡𝑁 (𝑑1 )
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Rho (ρ)
Rho measures the rate of change of the option price with respect to
a 1% change in the risk-free interest rate.
𝜕𝑉
𝜌=
𝜕𝑟
•
•
For a Call Option: 𝜌𝐶 = 𝐾(𝑇 − 𝑡)𝑒 −𝑟(𝑇−𝑡) 𝑁(𝑑2 )
For a Put Option: 𝜌𝑃 = −𝐾(𝑇 − 𝑡)𝑒 −𝑟(𝑇−𝑡) 𝑁(−𝑑2 )
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Some terms in markets
1. Open Interest (OI)
Open Interest is a measure used in futures and options markets. It
represents the total number of outstanding derivative contracts that
have not been settled or closed. For every buyer of a contract, there
must be a seller; one long position and one short position together
create one unit of open interest.
What it indicates:
• Market Activity & Money Flow: Increasing OI indicates new
money flowing into the market, suggesting a strengthening
trend. Decreasing OI means positions are being closed, which
can signal a weakening trend.
• Confirmation: High OI is often seen as confirmation of a price
trend's strength.
Key Point: Volume measures the number of contracts traded on a
given day, while Open Interest measures the number of contracts
still active at the end of the day.
2. Volatility (σ)
What it is: Volatility measures the magnitude of an asset's price
fluctuations over a certain period. It is a statistical measure of the
dispersion of returns for a given security or market index. In
simpler terms, it tells you how much the price is swinging up and
down.
What it indicates:
• Risk: Higher volatility means the price of an asset can change
dramatically over a short time period in either direction,
indicating higher risk.
• Option Pricing: Volatility is a critical input in options pricing
models (like the Black-Scholes model). Higher volatility leads
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to higher option premiums because there's a greater chance the
option will end up being profitable.
o Historical Volatility (HV): Calculated from past price
movements.
o Implied Volatility (IV): Derived from current option
prices, representing the market's forecast of future
volatility.
3. Liquidity
What it is: Liquidity describes the degree to which an asset can be
quickly bought or sold in the market at a price reflecting its current
value.
What it indicates:
• Ease of Trading: High liquidity means there are many buyers
and sellers, so trades can be executed easily and quickly
without significantly affecting the price.
• Low Transaction Costs: Liquid markets typically have a tight
bid-ask spread (the difference between the highest price a
buyer is willing to pay and the lowest price a seller is willing to
accept), resulting in lower transaction costs.
• Market Health: Stocks or options with high liquidity are
generally considered less risky to trade.
4. Volume
What it is: Volume is the total number of shares (for stocks) or
contracts (for options/futures) traded during a specific time period
(e.g., a day). Each transaction contributes to the total volume count.
What it indicates:
• Market Interest: High volume shows high interest in a security
at its current price.
• Trend Strength: A price move (up or down) accompanied by
high volume is considered more significant and powerful than
a move on low volume. It suggests strong conviction behind the
price action.
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•
Confirmation: Traders look for volume to confirm trends and
chart patterns. A breakout from a pattern on high volume is
more likely to be sustained.
5. Moving Averages (MA)
What it is: A Moving Average is a widely used technical analysis
indicator that helps smooth out price action by creating a single,
flowing line representing the average price over a specific period.
What it indicates:
• Trend Direction: The slope of the moving average line indicates
the direction of the trend. An upward-sloping MA suggests an
uptrend, while a downward-sloping MA suggests a downtrend.
• Support and Resistance: In an uptrend, a shorter-term moving
average can act as a dynamic support level. In a downtrend, it
can act as dynamic resistance.
• Crossover Signals: When a shorter-term MA (e.g., 50-day)
crosses above a longer-term MA (e.g., 200-day), it is often
interpreted as a bullish "Golden Cross." The opposite is a
bearish "Death Cross."
Common Types:
• Simple Moving Average (SMA): The simple average of a
security's price over a defined number of periods.
• Exponential Moving Average (EMA): Gives more weight to
recent prices, making it more responsive to new information.
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6. Futures
A futures contract is a legal agreement to buy or sell a specific
asset (like a commodity or a financial instrument) at a
predetermined price on a specified future date. Unlike an option, a
futures contract is an obligation. The buyer must buy the asset,
and the seller must sell the asset, unless the position is closed
before the expiration date. Futures are widely used by traders for
speculation and by corporations for hedging against price
fluctuations.
7. ETFs (Exchange-Traded Funds)
An Exchange-Traded Fund (ETF) is a type of investment fund
that is traded on a stock exchange, just like an individual stock. An
ETF holds a basket of assets, such as stocks, bonds, or commodities.
For example, a Nifty 50 ETF would hold all 50 stocks in the Nifty 50
index. They offer the diversification of a mutual fund combined
with the trading flexibility of a stock, as they can be bought and
sold throughout the trading day at market-determined prices.
8. Index
A market index is a statistical measure that represents the
performance of a specific group of assets, providing a snapshot of a
particular market or sector. It's created by selecting a representative
portfolio of securities and is used as a benchmark to track market
trends and evaluate the performance of investments.
• Key Examples:
o Nifty 50: Represents the weighted average of the 50
largest and most liquid Indian companies listed on the
National Stock Exchange (NSE).
o S&P 500: Tracks the performance of 500 of the largest
U.S. publicly traded companies.
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Calculating Investment Returns
Measuring the performance of your investments is crucial for
understanding your financial progress. While a simple percentage
gain seems straightforward, different methods are required to
accurately account for the time period and the timing of cash flows.
1. Absolute Return
The most basic measure of return is the absolute return, which
tells you the total percentage gain or loss on an investment,
irrespective of the time period.
Mathematical Formula
Final Value − Initial Value
Absolute Return = (
) × 100\%
Initial Value
Example: If you invest ₹50,000 and its value grows to ₹65,000,
your absolute return is:
(50,00065,000−50,000)×100%=(50,00015,000)×100%=30%
Limitation: This metric is useful but incomplete, as a 30% return
over one year is far better than a 30% return over ten years. It
ignores the time factor.
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2. Compound Annual Growth Rate (CAGR)
The (CAGR) provides a smoothed-out, annualized rate of return. It
tells you the constant yearly rate at which your investment would
have grown if it had compounded at the same rate each year. It's the
standard for comparing investments that involve a single lump-sum
investment over a period.
Mathematical Derivation
The formula for CAGR is derived from the standard formula for
compound interest:
Final Value = Initial Value × (1 + Rate)𝑛
To find the rate (CAGR), we rearrange the equation to solve for it:
1. Divide both sides by the Initial Value:
Final Value
= (1 + CAGR)𝑛
Initial Value
2. Take the n-th root of both sides (or raise to the power of 1/n):
1
𝑛
Final Value
(
) = 1 + CAGR
Initial Value
3. Subtract 1 to isolate CAGR. This gives us the final formula:
1
𝑛
Final Value
CAGR = (
) −1
Initial Value
Where:
Final Value: The ending value of the investment.
o Initial Value: The beginning value of the investment.
o n: The total number of years the investment was held.
Example: You invest ₹50,000, and after 4 years, it grows to
₹90,000.
CAGR=(50,00090,000)41−1=(1.8)41−1≈1.158−1=0.158
o
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Your CAGR is 15.8%. This means your investment grew at an
average rate of 15.8% per year for four years.
Limitation: CAGR is only accurate for a single investment with no
additional deposits or withdrawals in between. If you have multiple
cash flows, you must use XIRR.
3. Extended Internal Rate of Return (XIRR)
The Extended Internal Rate of Return (XIRR) is the most
powerful and accurate method for calculating returns when you
have multiple cash flows (deposits or withdrawals) at
irregular intervals. This is typical for systematic investment
plans (SIPs), recurring deposits, or any portfolio where you add or
remove money over time.
Mathematical Concept
Mathematically, XIRR is the discount rate that makes the Net
Present Value (NPV) of all cash flows (both inflows and
outflows) equal to zero.
The formula at its core is:
𝑛
𝐶𝐹𝑖
NPV = ∑
=0
(𝑑𝑖 −𝑑0 )
𝑖=0 (1 + Rate) 365
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Let's break down this complex formula:
• Rate: This is the XIRR that we are trying to find. It's the single
annualized rate that explains the performance of the entire
series of transactions.
• 𝐶𝐹𝑖 : The cash flow for the i-th transaction.
o Outflows (investments, purchases) are represented by
negative numbers.
o Inflows (withdrawals, sales, or the final market value) are
represented by positive numbers.
• 𝑑𝑖 : The date of the i-th cash flow.
• 𝑑0 : The date of the very first cash flow (the starting point).
• (𝑑𝑖 − 𝑑0 )/365: This is the crucial part. It calculates the time
period in years between each cash flow and the starting date,
accurately handling irregular intervals.
How it Works: The equation cannot be solved directly for the
"Rate." It is found using an iterative process (trial and error), where
a financial calculator or software like Excel (=XIRR() function) tries
different discount rates until it finds the one that makes the sum of
all the discounted cash flows equal to zero.
Example
Imagine the following transactions for a mutual fund SIP:
| Date
| Transaction
| Cash Flow (CFi)
| 01-Jan-2023 | Initial Investment | -₹10,000
| 01-Jul-2023 | Additional SIP
| -₹5,000
| 01-Jan-2024 | Additional SIP
| -₹5,000
| 01-Jan-2025 | Final Market Value | +₹23,500
To calculate the XIRR, the software would solve the following
equation for "Rate":
−10,000
−5,000
−5,000
23,500
0=
+
+
+
0
181
730
365
(1 + Rate)365 (1 + Rate)365 (1 + Rate)365 (1 + Rate)365
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Solving this equation (using Excel's =XIRR function with the values
and dates) gives an XIRR of approximately 8.45%. This is the true
annualized return on your investment, accurately accounting for the
size and timing of each contribution.
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Interest Rates
Computation of Interest Rates
In pure discount bonds the interest rate is implicit in the difference
between the purchase price and the amount that the holder of the
bond will receive at maturity. In general, consider a pure discount
bond that sells today at a price
P(0) and matures with a nominal payment of P(T). We will say that
the bond's interest rate is the value r that solves:
𝑃(0)(1 + 𝑟) = 𝑃(𝑇)
or
𝑟=
𝑃(𝑇) − 𝑃(0)
𝑃(0)
Simple versus Compound Interest; Annualized Rates
Traditionally, interest rates have been classified as simple or
compound, depending on whether interest is paid on the interest
received or not. An important issue with compound interest is the
frequency of the interest payments: a bank can pay interest on the
balance of the account every month, or every week, or some other
time period. Both the rate and the periodicity of the payments will
affect the final value of the investment. If rQ is the quarterly simple
rate, one dollar today is worth 1 + 4𝑟𝜌 dollars after a year. If rQ is
the quarterly rate and interest is compounded quarterly, one dollar
today would increase to
1+rQ dollars three months from now, (1+rQ)(1+rQ) dollars six
4
months from now, and (1 + 𝑟𝑄 ) dollars after a year.
- 23 -
Continuous Interest
The continuously compounded interest rate is the rate paid at
infinite frequency. That is, we say that the bank pays a continuous
rate rC when the effective annual rate r is obtained from:
𝑟𝐶 𝑛
1 + 𝑟 = lim (1 + ) = 𝑒 𝑟𝐶
𝑛→∞
𝑛
If the continuous rate rC is paid over a period t different from the
year, the value of one dollar invested today will be:
𝑒 𝑟𝑐⋅𝑡
Present Value
DEFINITION: Present value of a future payment is that amount
which, when invested today at a given interest rate, would result in
the given value of the future payment.
We see that the present value V(0) is obtained as:
𝑉
𝑉 (0) =
1+𝑟
where V is the value of the future payment and r is the interest rate
per period in question. We say that the future payoff amount V is
discounted, since the present value is less than the future payment
V. The factor 𝑑 = 1/(1 + 𝑟)by which we multiply the future payoff
value is called the discount factor.
In the case of continuous compounding with a continuous rate rC
and a period of t years, the discount factor is:𝑑 = 𝑒 −𝑟𝐶 𝑡
- 24 -
Present and Future Values of Cash Flows
The previous discussion is easily extended to a stream of several
payments, a cash flow. Consider a time interval [0, T] consisting of
m equally spaced periods, with interest rate r per period. Suppose
that we receive a payment P0 at time zero and invest it in the bank;
in addition, we receive P1 after the first period and invest it in the
bank; we receive P2 after the second period and invest it in the
bank, and so on, until the final time T, at the end of the last time
interval, at which moment we receive Pm. The values Pi can also be
negative, which means a cost (instead of investing it, we have to
borrow the amount Pi from the bank, at the same interest rate). The
future value of the cash flow is (compounded every period):
𝑃(𝑇) = 𝑃0 (1 + 𝑟)𝑚 + 𝑃1 (1 + 𝑟)𝑚−1 +⋅⋅⋅ +𝑃𝑚
Similarly, we now extend the notion of the present value to cash
flows of several payments. Given a cash flow of payments with
values V0, ..., Vm, where Vi is the payment at the end of the ith
period, the present value of the cash flow is (compounded every
period):
𝑉1
𝑉𝑚
𝑉 (0) = 𝑉0 +
+⋅⋅⋅ +
(1 + 𝑟)
(1 + 𝑟)𝑚
It can be checked that the present value V(0) and the future value
V(T) of a cash flow are related by:
𝑉 (𝑇)
𝑉 (0) =
(1 + 𝑟)𝑚
In order to compute the present values in various examples, it is
convenient to recall the following summation formula for a
geometric sequence:
1
1
1
1
1
+
+⋅⋅⋅ +
= (1 −
)
(1 + 𝑟) (1 + 𝑟)2
(1 + 𝑟)𝑚 𝑟
(1 + 𝑟)𝑚
This expression means that the cash flow of equal payments of P
dollars after each of m periods has the present value:
- 25 -
𝑃
1
(1 −
)
(1 + 𝑟)𝑚
𝑟
where r is the interest rate per period. The payments of P dollars are
sometimes called annuities. Equivalently, inverting the equation, in
order to pay off a loan of
V(0) dollars in equal installments of P dollars at the end of each
period, for m periods, the amount P is set equal to:
𝑟(1 + 𝑟)𝑚 𝑉 (0)
𝑃=
(1 + 𝑟)𝑚 − 1
We say that the loan is amortized over m periods.
𝑉 (0) =
- 26 -