Hedge Ratio

Derivatives 3 min readModule 4 of 7Save

What you'll learn

LOS 29.c
Identify an arbitrage opportunity involving options and describe the related arbitrage.

Arbitrage with a One Period Binomial Model

Let’s revisit our original example of a single period binomial model. Recall that the call option has a strike price equal to the stock’s current price of $30, U = 1.333, D = 0.75, and the risk free rate is 7%. We calculated that the probability of an up move is 55% and that of a down move 45%.

If the market price of the one period $30 call option were to be different from the $5.14 value calculated before, there would be an arbitrage opportunity. This arbitrage will involve the call option and shares of the stock. If the option is overpriced in the market, we would sell the option and buy a fractional share of the stock for each option we sold.

If the call option is underpriced in the market, we could purchase the option and short a fractional share of stock for each option purchased.

The fractional share of stock needed in the arbitrage trade (commonly referred to as the hedge ratio or delta), is calculated in the one period model as:

h = \frac{C ^{+} - C ^{-}}{S ^{+} - S ^{-}}

Worked example: Calculating arbitrage profit

In a previous example, a one year call option on a $30 stock (exercise price = $30) was valued at $5.14. We were given that the risk free rate is 7%, and the up move factor and down move factor are 1.333 and 0.75, respectively. If the market price of the call option is $6.50, illustrate how this opportunity can be exploited to earn an arbitrage profit.

Assume we trade 100 call options.

Solution

Given the up and down factors, the stock price would be $40 and $22.50 in the up and down states respectively in one year. The call payoffs would accordingly be $10 and $0 in up and down states, respectively. Because the option is overpriced, we will sell 100 call options and purchase a number of shares of stock determined by the hedge ratio:

h shares to purchase = \frac{$10 - $0}{$40 - $22.50} = 0.5714 shares per option = 100 (0.5714) = 57.14

A portfolio that is long 57.14 shares of stock at $30 per share and short 100 calls at $6.50 per call has a net cost of:

net portfolio cost = 57.14 ($30) - 100 ($6.50) = $1, 064

We will borrow $1,064 at 7%, and then will have to repay $1,064 × (1.07) = $1,138.48 at the end of one year. (In an arbitrage transaction, we assume that we begin with $0.)

The values of this portfolio at maturity if the stock moves up to $40 or down to $22.50 are identical:

portfolio after up move portfolio after down move = 57.14 ($40) - 100 ($10) = $1, 286 = 57.14 ($22.50) - 100 ($0) = $1, 286

Profit on the portfolio at the end of one year in either state after repayment of loan = $1,286 − $1,138.48 = $147.52.

The present value of the arbitrage profit is $147.52 / 1.07 = $137.87.

The previous calculations can be represented by the following:

- h S_{0} + C_{0} = \frac{- h S ^{-} + C ^{-}}{1 + R _{f}}

Therefore, the value of the call (the arbitrage free price) is:

C_{0} = h S_{0} + \frac{- h S ^{-} + C ^{-}}{1 + R _{f}}

Or equivalently,

C_{0} = h S_{0} + \frac{- h S ^{+} + C ^{+}}{1 + R _{f}}

We can value the call option using the first relationship as:

C_{0} = 0.5714 ($30) + \frac{- 0.5714 ( $22.50 ) + $0}{1.07} = $5.12

The small difference is due to rounding.

Similarly, a put option can be valued as:

P_{0} = h S_{0} + \frac{- h S ^{-} + P ^{-}}{1 + R _{f}} = h S_{0} + \frac{- h S ^{+} + P ^{+}}{1 + R _{f}}

And the hedge ratio based on puts is:

h = \frac{P ^{+} - P ^{-}}{S ^{+} - S ^{-}}

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