Topic 5 – Black-Scholes Model + Put-Call parity
Introduction
Options and options trading have existed for hundreds of years. It is only recently that analytical frameworks have been developed to value options mathematically. The Black-Scholes-Merton model was published in 1973. It gives a theoretical estimate for the price of European options, regardless of the risk of the security and its expected return. It utilises the same risk-neutral argument as the Binomial Model of options pricing.
It led to a boom in options trading globally and is still widely used. They received a Nobel prize for economics in 1997.
Learning outcomes
By the end of this topic, you should be able to:
• Understand the pricing of options using the Black-Scholes formula
Expected return and volatility
The expected return, μ, required by investors on a stock depends on the riskiness of the stock. The higher the risk, the higher the expected return. Fortunately for valuing stock options we don’t have to concern ourselves with μ, because of the no-arbitrage argument: setting up of a riskless portfolio with a position in the derivative and a position in the stock. In the absence of arbitrage opportunities, the return from the portfolio must be the risk-free interest rate.
• The expected value of the stock price is S0eT
• The expected return on the stock is – not
• is the expected return in a very short time, t, expressed with a compounding frequency of t
• −/2 is the expected return in a long period of time expressed with continuous compounding (or, to a good approximation, with a compounding frequency of t)
Mutual fund returns (see Hull p348)
• Suppose that returns in successive years are 15%, 20%, 30%, −20% and 25% (ann. comp.)
• The arithmetic mean of the returns is 14%
• The returned that would actually be earned over the five years (the geometric mean) is 12.4% (ann. comp.)
• The arithmetic mean of 14% is analogous to
• The geometric mean of 12.4% is analogous to −/2
Volatility
• The volatility is the standard deviation of the continuously compounded rate of return in 1 year
• The standard deviation of the return in a short time period time t is approximately
Example:
If a stock price is $50 and its volatility is 30% per year what is the standard deviation of the price change in one week?
Answer:
30 * (1/52) ^1/2 = 4.16% so a 1 standard deviation move in the stock price in one week is 50 *0.0416 = 2.08
Uncertainty about a future stock price, as measured by its standard deviation, increases with the square root of time.
Concepts underlying Black-Scholes model:
• The option price and the stock price depend on the same underlying source of uncertainty
• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty
• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate
• This leads to the Black-Scholes-Merton equation for valuing European options
Basic derivation of pricing formula
Black-Scholes formula for options
• Where N() is the cumulative normal distribution function
• K is the option strike
• σ is the annualised lognormal volatility
Watch: Khan Academy video
https://www.khanacademy.org/economics-finance-domain/core-finance/derivative-securities/black-scholes/v/introduction-to-the-black-scholes-formula
• Intuitively from formula for Calls, as S0 becomes very large: call price tends to
S0 – Ke-rT and put price tends to zero
• As S0 becomes very small call price tends to zero and put price tends to Ke-rT – S0
Risk-Neutral valuation
• The variable μ does not appear in the Black-Scholes equation
• The equation is independent of all variables affected by risk preference
• The solution to the equation is therefore the same in a risk-free world as it is in the real world
• This leads to the principle of risk-neutral valuation
Example using Black-Scholes
S = 100, K = 105, σ = 10%, r = 5%
What is the value of a 1-year call option?
[Values for N (x) function can be looked up in the tables at the end of Hull book]
Source: Hull
7.3 Implied vs historical volatility Source Hull p 363
The one parameter in the Black-Scholes model that cannot be directly observed is the volatility of the stock price. It can be estimated from a history of the stock price, which is known as historical volatility. But this is backward looking and for trading purposes it is more useful to have a view of future volatility. In practice traders use implied volatility, which is implied by the option prices observed in the market.
The parameters : S₀, K, r and T can readily be observed in the market. σ has to be estimated through either an iterative approach from the value of C₀ in the Black=-Scholes equation, or when the option prices can be observed in the market, then the equation gives a level of σ.
Implied volatilities are used to monitor the market’s opinion about the volatility of a particular stock.
They are forward looking. Traders will trade and quote prices in volatility % terms, rather than its price. This is convenient because implied volatility is less variable than the option price.
Example of Put-Call Parity
The price of a non-dividend paying stock is $19 and the price of a three-month European call option on the stock with a strike price of $20 is $1. The risk-free rate is 4% per annum. What is the price of a three-month European put option with a strike price of $20?
In this case, , , , , and .
From put–call parity
or
so that the European put price is $1.80.