Maths IA · Statistics and probability · topic 53 of 97
Pricing an option: from coin flips to Black-Scholes · IB Maths IA idea: guide, sample graph and step-by-step plan
coin flips can price a bet!An option is a bet on a share price, and yet its fair price does not depend on how likely you think the price is to rise. You're going to find out why that is true, and build the price from coin flips.
- ✓ you like binomial probability and want to see it used in finance
- ✓ you enjoy seeing a discrete model turn into a continuous one
- ✓ you are happy to explain one new idea in full (risk-neutral pricing)
The big idea
Split the time to expiry into steps. At each step the share price goes up by a factor u or down by a factor d. Work backwards from the payoffs and the price today is a discounted expected value, but under a special 'risk-neutral' probability q, not the real one. Over n steps the number of up-moves is binomial, so the price is a sum of binomial terms. As n grows the price settles down to the Black-Scholes value, which uses the normal distribution. The tree is mechanical. The marks come from your reasoning: why q is what it is, how fast the price converges, and how far real option prices stray from the model.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Price a call option on a share that can only go up or down once. Build a portfolio of shares and cash that copies the option's payoff.
→ A pair of linear equations whose solution gives the price. Show why the price does not use the real probability of an up-move.Extend to 2, 3, then n steps and price by working back through the tree.
→ A sum of binomial probabilities times payoffs. Write it as one formula and check it against your backward calculation.Plot the tree price against n and compare it with the Black-Scholes price.
→ A sequence that wobbles around a limit. Explain the wobble, which depends on whether the strike sits on a node.Price a put option on the same tree and check that call minus put equals share price minus the discounted strike.
→ An identity tested on your own numbers. Say what it means and what breaks if it fails.Take quoted option prices for one share and find the volatility that makes the model match one of them.
→ Solving an equation for the unknown volatility. Compare the volatilities you find across strikes.Vary the spread of the share price and plot how the option price responds.
→ A function to differentiate or compare numerically. Say what happens to the price when volatility rises, and why.Your Maths IA research question: run the check
which option? which inputs?Write your own question first. Then make sure it ticks every box:
- Names the option: call or put, strike, time to expiry, and the share it is written on.
- States your inputs: starting price, interest rate and volatility, and where each comes from.
- Says what you compare the tree with: the Black-Scholes formula, real quoted prices, or both.
- Says how you will measure convergence: for example the error against the number of steps.
- Shows that you can explain risk-neutral pricing in your own words, not only compute with it.
The model
One step of the tree: the price can move up or down, the option pays something in each case, and the price today is a discounted weighted average. Draw one step for your own option.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
What an option is, why its price matters, and what you expect the tree price to do as steps increase.
The tree, the payoffs, a replicating portfolio and the price, with every symbol defined.
Where q comes from, why it differs from the real probability, and a check that the price is fair.
Backward pricing for 2 and 3 steps, then a binomial sum for n steps, tested against your spreadsheet.
Tree price against n, the Black-Scholes value as a limit, and an explanation of the size and the wobble of the error.
Quoted option prices beside model prices, and the volatility needed to match each.
Back to your aim: how many steps are enough and how close the model gets to the market.
Constant volatility, no costs, shares that never jump. Say which assumption matters most.
Where every price, rate and formula came from.
What your Maths IA graph could look like
show the error shrinkingA made-up one-step model to show the form of a strong graph, not the answer: the share can rise 20% or fall 10%, strike 100, interest 5% per period. Yours will look different. example only
- The option's price curve lies above the payoff line. That gap is the value of waiting.
- Axis labels give what is plotted: share price today and option value, in the same currency.
- Show the payoff line too, so the reader sees what the tree adds.
- Plot values from your own tree as points: agreement with the formula is your evidence it is right.
- Another graph to consider: tree price against the number of steps, with the Black-Scholes value as a line.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
How fast does the tree price settle?
Compare prices for different numbers of steps with the formula's value. Work it out for your own option, and explain why the prices wobble rather than approach smoothly.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what a call option pays. Before any calculation, guess whether the price should rise or fall when the share's volatility increases.
- Choose the optionFix the share price, strike, time to expiry, interest rate and volatility, and say where each number comes from.
- Build a one-step treeFind the payoffs and copy them with a portfolio of shares and cash. Solve for the price and for q.
- Trial run (1 lesson)Price a 2-step tree by hand and in a spreadsheet. Check that both agree, then check q lies between 0 and 1.
- Write the n-step priceExpress the price as a sum of binomial probabilities times payoffs, discounted. Test it against your backward calculation.
- Study convergencePlot the price against n up to a few hundred steps. Compare it with the Black-Scholes value and describe the error.
- Meet real pricesTake quoted option prices and compare them with your model. Find the volatility that matches each, and look at the pattern.
- Go further, with a reasonPrice a put and check call-put parity, or study how the price changes with volatility. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means in the real setting and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Probability tree and expected value – Price an option as the expected payoff over the tree, discounted, with branch probabilities and .
- Binomial distribution – Write the number of up-moves as and the price as .
- Exponentials and compound growth – Set and discount by per step.
- Normal distribution – Use the Black-Scholes formula, which contains the normal cumulative distribution function, as the limit of the tree.
- Convergence and error – Plot and measure the difference between tree price and limit as increases.
Key equations
Syllabus links
- 1.4Compound interest and depreciation
- 1.9Binomial theorem
- 2.9Exponential and logarithmic functions
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 4.9Normal distribution
- 1.4Compound interest and depreciation
- 4.7Discrete random variables and expected value
- 4.8Binomial distribution
- 4.9Normal distribution
- 4.15Combinations of normal variables; central limit theoremHL
Read up first
Data and sources to start from
Use made-up inputs for the model, then compare with a few quoted option prices and a rate from a public source.
- Cboe Global Markets: delayed quotes for listed options
- Yahoo Finance: option chains and historical share prices
References to cite
- Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229-263.
- Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
- Hull, J. C. (2021). Options, futures, and other derivatives (11th ed.). Pearson.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
q is not the real probability- Using the real chance of the share rising instead of q. The price does not depend on it, and saying why is a core part of the work.
- Applying u and d without checking that d < 1 + r < u. If that fails, the model allows a risk-free profit.
- Using the annual rate on each step. The rate per step depends on the step length.
- Calling a few steps 'converged' without showing the error against n.
- Quoting the Black-Scholes formula without saying what its inputs mean.
- Treating a market price that disagrees with the model as the model being wrong, without looking at volatility.
More Maths IA ideas like this
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Two shares or one? Diversification and portfolio risk●●○
Lotteries: is a ticket ever worth buying?●●○
Goals and tries: are they random?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.