Incorporating Carry Benefits into the BSM Model
Just like the BSM model for stocks, the BSM applied to currencies can be viewed as having two components—a foreign exchange component and a bond component.
For call options, the foreign exchange component is
$$\displaystyle Se^{-r^fT}N(d_1)$$,
and the bond component is
$$\displaystyle e^{-rT}XN(d_2)$$.
The BSM call model applied to currencies is simply the foreign exchange component minus the bond component.
For put options, the foreign exchange component is
$$\displaystyle Se^{-R^fT}N(-d_1)$$ ,
and the bond component is
$$\displaystyle e^{-rT}XN(-d_2)$$.
The BSM put model is the bond component minus the foreign exchange component.
Suppose you were initiating a call or put option position and using the BSM model to evaluate the cost. While there are some securities that the BSM model will work on naturally, others will require a modification of the model to reduce the carrying cost.
Why's that?
That's it!
Any of these are correct because each cash flow will impact the BSM model. For dividends, foreign interest, or coupon payments, the cash flow received is called a carry benefit, while carry costs, such as commodity storage, can be treated as negative carry costs.
For the continuous BSM model, these carry benefits are treated as a continuous yield as $$\gamma$$.
The carry-benefit adjusted BSM model is
$$\displaystyle c=Se^{-\gamma T}N(d_1) - e^{-rT}XN(d_2)$$.
For a put, it's
$$\displaystyle p= e^{-rT}XN(-d_2) - Se^{-\gamma T}N(-d_1)$$
where
$$\displaystyle d_1 = \frac{ln(\frac{S}{X}) + (r - \gamma + \frac{\sigma ^2}{2})T}{\sigma \sqrt{T}}$$
$$\displaystyle d_2 = d_1 - \sigma \sqrt{T}$$.
The value of the put option can also be found by using the put-call parity ratio.
$$\displaystyle p+ Se^{-\gamma T} = c + e^{-rT}X$$
To summarize:
[[summary]]
Given these equations, you can see that the carry-adjusted BSM model is still the present value of the expected option payoff at expiration. And the carry benefits offset any cost of borrowing.
How does that impact the value of a call option?
Not quite.
You'd think it would increase the value, but there shouldn't be any arbitrage opportunities.
No.
A cash flow definitely impacts the value.
That's it!
The carry benefits lower the value of a call option because of the no-arbitrage principle. If the underlying pays a dividend, it's assumed that the value of the underlying will decrease, so the expected future value will also decrease.
But for a put option, an increase in carry benefits will increase the value of the put because the carry benefits offset any interest paid.
Since carry benefits impact _d_1 and _d_2, that also impacts the probability of being in the money on a long call. Do you think the probability is higher or lower?
Recall that the BSM model has two components—a stock and a bond. For the carry benefit's adjusted BSM, it's the same two components.
For call options, the stock component is
$$\displaystyle Se^{-\gamma T}N(d_1)$$,
and the bond component is
$$\displaystyle e^{-rT}XN(d_2)$$.
For a put option, the stock component is
$$\displaystyle Se^{-\gamma T}N(-d_1)$$,
and the bond component is
$$\displaystyle e^{-rT}XN(-d_2)$$.
Specifically for stock options, γ = δ, where δ is the continuously compounded dividend yield.
The dividend-yield BSM model can still be viewed as a dynamically managed portfolio of the stock and zero-coupon bonds.
For calls, the equivalent number of units of stock is now
$$\displaystyle n_S = e^{-\delta T}N(d_1) > 0$$,
and the number of bonds is
$$\displaystyle n_B = -N(d_2)< 0$$.
For puts, the equivalent number of units of stock is
$$\displaystyle n_S = -e^{- \delta T}N(-d_1)$$,
and bonds is
$$\displaystyle n_B = N(-d_2) > 0$$.
So for the BSM model, buying the underlying stock replicates the call option. But how do you think a dividend impacts the number of shares that need to be purchased?
Bingo!
No.
The value of the option is lower.
Since the underlying price is assumed to be reduced by a dividend payment, the probability of it reaching in-the-money status is also reduced. So as the carry benefit increases, the probability of being in the money on a call option decreases.
Exactly!
The dividend received would lower the number of shares to be purchased for the replicating strategy. Remember that dividends reduce the value of the option, so there are fewer shares that need to be purchased.
It's key to remember that dividends influence the dynamically managed portfolio by lowering the number of shares to buy for calls and lowering the number of shares to short for puts. Higher dividends will also lower the number of bonds to short for calls and lower the number of bonds to buy for puts.
That's not it.
The dividend is a cash flow received by the stockholder.
Not quite.
Since there's a cash flow, it impacts the value of the option and the number of shares to be purchased.
But how do carry benefits impact other option types? For foreign exchange options, γ = _r__f_, which is the continuously compounded foreign risk-free interest rate. The underlying instrument is the foreign exchange spot rate, and the volatility is captured by the log return of the spot exchange rate. There's also a key detail to remember: the underlying and the exercise price must be quoted in the same currency unit.
The carry benefit is the interest rate in the foreign currency because it could be invested in the foreign country's risk-free instrument.
Dividends are paid
Foreign interest is paid
Coupon payments are paid
It lowers it
It increases it
It has no impact on it
Lower
Higher
It lowers the number of shares
It increases the number of shares
There's no impact on the number of shares
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