Andreas Park PRO
Professor of Finance at UofT
Trading Case: OP1
The quick and dirty on OP1
New symbols?
What's notable about the security?
Calls and Puts on DM for different strike prices
standard European call options; specs for underlying as per case
What do you have to do?
identify no-arbitrage conditions and devise trading strategy to exploit them
Anything special?
- what's the fair value?
- use your knowledge on options pricing to identify no-arbitrage conditions
Some background on options
Payoffs at maturity
strike price X
strike price X
call option for strike price X
put option for strike price X
call = right to buy a certain quantity at a pre-determined price
put = right to sell a certain quantity at a pre-determined price
Common questions
- Why do we have them?
- Who cares - it's just a cash flow stream
- Why use them?
- Lots of reasons: leverage speculation, insurance, etc.
The Black-Scholes Formula
Price of a European call:
\[C=S\cdot N(d_{1})-X\cdot e^{-rT}\cdot N(d_{2})\]
\(C=\)Current call option value
\(S=\)Current stock price
\(N(d)\)= probability that a random draw from a normal distribution will be less than \(d\),
\(X=\)Exercise price
\(r=\)Risk-free interest rate (annualized, continuously compounded with the same maturity as the option)
\(T =\)time to maturity of the option in years
\(\sigma = \)Standard deviation of the stock.
where
\(d_{1} = \frac{ \ln\left( \frac{S}{X}\right) + T\left(r + \frac{ \sigma^2}{2}\right)}{ \sigma \sqrt{T}}, \)
\(d_{2} = d_{1} - \sigma \sqrt{T}.\)
Black-Scholes is a Model, not "the truth"
Black-Scholes "hugs" the curve of the payoff at maturity and gets closer has \(t\to\) maturity
A Common Application
Portfolio insurance
-
when things go bad, you want your investment loss limited,
-
but you also want to benefit from the upside
slightly (but crucially) imprecise because the plot ignores the cost for the option \(\to\) insurance isn't free!
\(S\)
secure \(X\)
benefit from upside
+
- you bought a stock at \(S\)
- value in \(t=3\) months for price \(S_t\)
\(S_t=\) stock price
- Now add a put with strike \(X\)
- Its payoff:
\(S\)
An Alternative way to create the same payoff!
\(S\)
secure \(X\)
benefit from upside
+
- you bought a call with strike \(S\)
\(S\)
- Suppose you also bought riskless bonds with combined value of \(X\)
\(X\)
Compare costs of the two portfolios "today"
Cost of strategy 1: stock + put = \(S+P\)
Cost of strategy 2: bond + call = \(PV(X)+C\)
BUT: they provide the same payoff profile at maturity
\(\Rightarrow\) The Law of One Price applies
stock + put = bond + call
\(\Leftrightarrow\) \(S+P = PV(X)+C\)
And that's what we call put-call-parity
Please note: this doesn't tell you the right price for a call, it tells you the relationship between put and call prices.
The quick and dirty on OP1-B
New symbols?
What's notable about the security?
- Calls and Puts on DM for different strike prices
- In OP1, you cannot trade DM directly, but now you can!
nothing new
What do you have to do?
the previous version of the case did not allow you to trade based on a standard concept from options pricing \(\ldots\) but now develop an arbitrage strategy that allows you to benefit of deviations in put-call parity
MGT435: Options Case OP1
By Andreas Park
