Financial Derivatives: Pricing and Arbitrage Opportunities

Posted on Nov 16, 2024 in Language

Closing Positions and Cash Settlement

To close a short or long position, enter a futures position of equal magnitude but opposite direction. If a position isn’t closed before contract expiry, the futures exchange calculates profit or loss and adjusts the account accordingly. For example, SPI200 futures contracts have a standard size of $A25.

Hedging Contracts

The number of contracts to hedge is calculated as: No of contracts = Beta * (value to be hedged / value of one SPI200)

Capital Asset Pricing Model (CAPM)

Return of portfolio = risk-free rate + beta * (market return – risk-free rate)

Forward Price with Two Exchange Rates

r = local rate and f = foreign rate

Black-Scholes Formula for Call Option Value

Where N(.) represents a standard normal cumulative distribution function, found on Z table or using NORM.S.DIST in Excel.

Put-Call Parity

American vs. European Options

Arbitrage Opportunity for Long Call Option

Steps:

1. Calculate the lower bound.
2. If the option price is too low (less than the lower bound):
   • Enter a long call position at the option price.
   • Short sell shares at the current stock price.
   • Invest the difference (short sale – option price) at the risk-free rate.
   • If S < X, we exercise the call and gain the difference.
   • If S > X, we exercise the call, pay the spot price to close the short position, and receive the invested surplus.
   • Profit in both scenarios.
3. Example: An American-style call option on Suncorp with three months to expiry, a strike price of $4.50, trading at $1.20. Suncorp shares are at $6.00, with a 5% risk-free interest rate.

Immediate exercise yields a $0.30 profit (buy at $4.50, sell at $6, less option price $1.20).

Arbitrage Opportunity for Long Put Option

Steps:

• Calculate the lower bound.

• If the option price is lower than the lower bound:
  • Enter a long put option.
  • Buy shares at the current spot price.
  • Exercise the right to sell at the strike price.
  • Profit = (strike price – spot price) + option price.
• Example: Strike price $7.50, spot price $6.90, option price $0.35, profit = $0.25.

Arbitrage Opportunity for Short Call Option

Steps:

• Calculate the upper bound (stock price).
• If the option premium is higher than the stock price:
  • Short call and receive the premium.
  • Buy stock.
  • Invest the difference.
  • If S < 10, the option expires worthless.
  • If S > 0, the option is exercised. Losses are recouped by selling stock.
• Example: Call option on XYZ with a $10 strike, current price $8.50, premium $9.

Arbitrage Opportunity for Short Put Option

Steps:

• Calculate the upper bound (strike price).
• If the option premium is higher than the strike price:
  • Short put and receive the premium.
  • Invest everything in the bank.
  • If S > strike price, the option is not exercised.

Binomial Tree

S0 is the current strike price at time 0. St (Su) = $100 if price goes up, St (Sd) = $81 if prices go down. C = call option payoff.

Replication Approach

Construct a portfolio of ∆ shares and borrowed money to match the option payoff. Example:

Purchase 1 share and borrow $78.48. Construction cost = (∆ * S) – B = (1 * 90) – 78.48 = 11.52.

Delta Hedging Method

Create a portfolio with ∆ stocks and a short option for a certain payoff. Discount to present value to find the option price. Example:

V1 is the payoff. Vu and Vd should be equal. Delta hedging can also be used with put options.

Payoff can also be found using the Black-Scholes formula.

Calculate Up Movement

Calculate Down Movement

d = 1/u

Risk-Neutral Valuation Method

Find P, the probability of up/down movements if individuals were risk-neutral.

1-P is the downward probability. Calculate the expected option payoff and discount back at the risk-free rate.

Example:

For European options, calculate all possible payoffs and path probabilities, then discount to time 0.

For American options, use the risk-neutral valuation method recursively.

Pascal’s Triangle

Path Dependent Payoff – Lookback Options

Floating lookback call: payoff = max(0, ST – Smin)

Floating lookback put: payoff = max(0, Smax – ST)

Fixed lookback call: payoff = max(0, Smax – X)

Fixed lookback put: payoff = max(0, X – Smin)

Steps:

• List all possible paths and find the max/min price.
• Calculate path probability and multiply by the payoff.
• Sum all potential payoffs.
• Discount to present value.

Path Dependent Payoff: Barrier Options

Like a normal binomial distribution, but we exclude all paths that are both not in the money and dont hit the barrier:

• Exclude all nodes, and the paths to these nodes, which are out of the money

• Exclude all nodes that are in the money, but do not hit the barrier during their path

• Only use paths that are in the money and hit the barrier during their path

• Find these payoffs as normal, multiplying the path probabilities by the payoff at the end, summing all these payoffs then discounting to present value

Remember the different types of barriers:

Up and in example ^

= Up if barrier (H) is above starting share price, = Down if H is below starting share price