Using Put Options in Volatility Trades
6.5 THE EQUIVALENCE OF PUT AND CALL OPTIONS
The fact that the profit or loss to a long volatility strategy is the same whether one uses puts or calls is no coincidence. Puts and calls are very similar instruments and it is in fact possible to turn one option into another via a very simple trade. In order to see how this is done
consider two fund managers, A and B, both with negative views on the direction of a particular stock price. The stock price at the start of the period under consideration is $110 and there is a one-year put and call option with an exercise price of $100. The put and call are priced at $2.50 and $12.50 respectively. The put is out-of-the-money and the call is in-the-money. Manager A takes the simple route and buys one put option paying up $250. Manager B takes a more complicated route and buys the call (paying up $1,250) and simultaneously shorts 100 units of stock at $110.
We now consider the situation of both managers if the stock price falls all the way down to $70 by the option expiry date. Manager A sells his put at the intrinsic price of $30 and receives $3,000 thus making a profit of 3,000 - 250 = $2,750. The call option that manager B owns obviously expires worthless, and so he loses all his $1,250 investment. However, manager B is also short 100 units of stock and he buys this back at $70 making 100 X (110 - 70) = $4,000 profit. Manager B has thus made an overall profit of 4,000 - 1,250 = $2,750 which is exactly the same as that made by manager A. Table 6.4 and Figure 6.5 show the profits of both fund managers for various final stock prices. The net profit to both managers is identical everywhere and so the combination of call option and short stock must be identical to being long a put option. We say that the portfolio of long a call option and short stock is in fact a synthetic put option. We can write this as an equation:
Long put = Long call + Short stock
or if we replace the + short stock term with the more meaningful long stock term we have:
Long put = Long call - Long stock
So by running a short stock position alongside a long call position one has essentially turned a call into a put. It is easy to see how this works graphically. The profit profile of the call option is kinked at $100. Above $100 the call option gives the holder the exposure to 100 units of long stock. If the portfolio is at the same time short 100 units ol stock these two stock exposures cancel and the resultant combination has zero exposure. Below $100 the call option has no stock exposure but the portfolio is still short 100 stock units and so the resultant combination is net short 100 units. This is exactly the same expiring
(6.1)
(6.2)
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w 2,000 - ^s^.
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i 1.000 - -s.
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70 80 90 100 110 120 13C
Stock price ($)
Figure 6.5 Long put versus long call + short stock
stock exposure of a long put option. Initially, in some exchange traded options markets, only call options were available. Participants wanting to take negative positions synthesised put options by buying calls and simultaneously shorting stock.
I he example in Table 6.4 was chosen for simplicity. At $110 the put ciiid call were conveniently priced at the rounded figures of $2.50 and S I 2.50 respectively.
It is possible to show that the equivalence of a put
and call option holds true whatever the original stock price and at all times, not just on expiry. If this were not the case then a risk-free arbitrage profit could be obtained irrespective of volatility or price direction. This means that the price of a put and call option with the same terms, must be inextricably linked at all times.
The price of one must be a simple function of the other and in the absence of interest rates and dividends is given by the expressions
Put price = Call price + Exercise price - Stock price (6.3) or
Call price = Put price - Exercise price + Stock price (6.4) In the above example, if $12.50 is the correct price for the call option then the put price should be 12.50 + 100 - 110 = $2.50. To illustrate what can happen if this were not the case consider again the initial situation of the two fund managers above. Say manager A buys the put at $2.50 and at the same time manager B buys the call at $13.50 (not at the correct price of $12.50) and shorts 100 stock at $110.
It is easy to show that whatever the final expiring stock price, the profit to manager B is always $100 less than manager A and this is because manager B paid $100 too much to synthesise the put. Either the put is too cheap at $2.50 or the call is too expensive at $13.50. If this situation existed in reality, a third individual, Mr C, could come into the market and create a risk-free arbitrage profit as follows. Mr C would buy one put as $2.50, sell short one call at $13.50 and buy 100 units of stock at $110 and wait until expiry. On expiry he would unwind the positions with a $100 risk-free profit whatever the stock price. Table 6.5 gives the details for various expiring stock prices.
If the final stock price is above $100 the loss on the short call and long put is completely cancelled by the profit on the long stock. Below $100, the loss on the long stock is completely cancelled by the profit on the long put and short call.
Whatever happens, a profit of $100 results because one option was either $1 too expensive or the other was $1 too cheap. If the market price of the call is correct at
$13.50 then the put price must be = 13.50 +100-110 = $3.50. Alternatively if the put is correctly priced at $2.50 then the call must be 2.50 - 100 +110 = $12.50.
Just as one can generate a put from a call and stock, it is also possible to generate a call from a put and stock. To generate a call we need
to be able to add two instruments together that on expiry will (1) have a kink at
$100, (2) have zero stock exposure below $100 and (3) have 100 units long exposure above $100. The put has minus 100 units exposure below $100 and so these could be neutralised by adding 100 long stock units. This would also have the effect of introducing 100 long units above $100. So a call can be synthesised by adding 100 units of stock to a put or
Long call = Long put + Long stock
Readers familiar with basic algebra will note that expression (6.5) is a simple rearrangement of expression (6.2).
So in reality we only need one type of option since we can always generate one type from the other. And the same goes for synthesising short option positions. It is left as an exercise for the reader to show the following:
Short call = Short put - Long stock Short put = Short call 4- Long stock
One can take the subject of synthesising instruments one more stage and consider generating a synthetic stock position from put and call options. A portfolio long a put option and short the same call option is essentially short the underlying. This is demonstrated clearly by looking at the expiring option prices given in Figure 6.6. The upper panel shows the exposures of the separate components and the lower panel the exposure of a portfolio short 100 units of stock—they are identical. Above $100, the put expires worthless but the short call gives exposure of short 100 units. Below $100, the call is worthless but the long put gives exposure of short 100 units. So at all prices the combination of long put and short call is equivalent to being short 100 stock units. It is possible to show that this will be so at all stock prices and at all times, not just expiry. The identity can be expressed algebraically as
Short stock = Long put + Short call
Similarly it is easy to show that a long stock position can be synthesised by a short put and a long call. i.e.
Long stock = Short put + Long call
(6.5)
(6.6) (6.7)
(6.8)
(6.9)
Figure 6.6 Synthesising a short stock position
So we see that stock, calls and puts are interchangeable. Any two can he used to generate the third. This link between the two types of options and the underlying stock means that the prices of all three must stay in line according to the expressions given above. If one instrument becomes too expensive relative to the other two. it is easy 10 svnthesise an equivalent long position cheaply, short the expensive iL;ii instrument and run to expiry. Whatever the final stock price, a • isk lice arbitrage profit will result.
In all of the above it should be emphasised that when turning one option position into another or when synthesising stock from options, the ratio of stock to option is always one to one or more precisely 100 stock to one option exercisable into 100 shares. We have made no mention of delta. Put and call prices are linked via the zero arbitrage expressions (6.3) and (6.4). It should be no surprise that the deltas of the two options are also linked. Recall expression (6.2) again which states that at all times and at all stock prices, a long put position behaves identically to one long of a call option and short stock. Deltas are simply rates of changes of option prices with respect to the underlying stock price and so it should be clear that the rate of change of the put price will be identical to the rate of change of the price of the portfolio long of a call and short stock or
Long put = Long call - long stock
Rate of change of (long put) = Rate of change of (long call) - rate of change of (long stock)
The rate of change of a stock position with respect to the stock price is one and so Delta of put = Delta of call - 1 (6.10) So in the long volatility example outlined above when the delta of the one year call is 0.50 the delta of the corresponding put is 0.50 - 1.00 = -0.50. At the higher stock price when the call delta is 0.66 that of the put is 0.66 - 1.00 = -0.34, etc.
We can take this argument to one final stage and derive the gamma of a put from the gamma of a call. Recall that the gamma of an option is the rate of change of delta and so we can apply this to expression (6.10) as follows:
Rate of change of (delta of put) = Rate of change of (delta of call) - Rate of change of (1)
The rate of change of 1 is of course zero so
Rate of change of (delta of put) = Rate of change of (delta of call) or
Gamma of put = Gamma of call (6.11) This is where we started. The gammas of puts and calls with the same terms are identical and this is why volatility traders view the two
instruments as the same. All of the above connections involving the prices, deltas and gammas of puts in terms of calls stems from the fact that a risk-free profit could be obtained if the links were broken.