Black-Scholes Model

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The Black-Scholes model was originally developed for pricing European stock options, but it has been extended to other asset classes (e.g., foreign exchange, equity indices, and commodity derivatives). In these scenarios, due to the different characteristics of the underlying assets (e.g., dividends, holding costs, foreign exchange rate differences), the Black-Scholes formula requires adjustments. Below, we explain the variations and adjustments of the Black-Scholes model for foreign exchange options, equity options (e.g., stocks or indices), and commodity options.

1. Basic Framework of the Black-Scholes Model

The core formula of the Black-Scholes model provides a closed-form solution for pricing European call and put options. Its basic form is:

1.1 European Call Option Pricing Formula

$C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$

1.2 European Put Option Pricing Formula

$P = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)$

1.3 Parameter Definitions

$S_0$ : Current price of the underlying asset;
$K$ : Strike price of the option;
$T$ : Time to maturity (in years);
$r$ : Risk-free interest rate (annualized);
$q$ : Yield of the underlying asset (e.g., dividend yield or holding cost);
$\sigma$ : Volatility of the underlying asset;
$N(x)$ : Cumulative distribution function of the standard normal distribution;
$d_1$ and $d_2$ are defined as:
$d_1 = \frac{\ln(S_0 / K) + (r - q + \frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}.$

Core Idea

$S_0 e^{-qT}$ : Present value of the underlying asset, adjusted for holding costs (e.g., dividends or foreign exchange rate differences).
$K e^{-rT}$ : Present value of the strike price.
$q$ : Represents dividend yield (for stocks), foreign interest rate differential (for foreign exchange), or storage costs (for commodities).

2. Black-Scholes Model for Foreign Exchange Options

In foreign exchange markets, the underlying asset is a currency pair (e.g., EUR/USD). Foreign exchange options involve two interest rates: Domestic Rate ( $r_d$ ) and Foreign Rate ( $r_f$ ). Thus, the Black-Scholes formula for foreign exchange options requires adjustments.

2.1 European Foreign Exchange Option Formula

Call Option Pricing Formula:
$C = S_0 e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2)$
Put Option Pricing Formula:
$P = K e^{-r_d T} N(-d_2) - S_0 e^{-r_f T} N(-d_1)$

2.2 Parameter Definitions

$r_d$ : Risk-free interest rate of the domestic currency;
$r_f$ : Risk-free interest rate of the foreign currency;
Other parameters ( $S_0, K, T, \sigma$ ) are the same as in the standard formula;
$d_1$ and $d_2$ are defined as:
$d_1 = \frac{\ln(S_0 / K) + (r_d - r_f + \frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}.$

2.3 Adjustment Logic

Yield ( $q$ ) Replaced by Foreign Rate ( $r_f$ ):
- Holding foreign currency $S_0$ generates a yield of $r_f$ (similar to dividend yield for stocks).
Discount Factor:
- The strike price $K$ is discounted using the domestic rate $r_d$ .

2.4 Explanation

The price of a foreign exchange option is essentially an option on a forward contract, influenced by two risk-free rates:
- $r_f$ : Yield from holding foreign currency;
- $r_d$ : Discount factor for the strike price.

3. Black-Scholes Model for Equity Options (Dividend-Paying Stocks or Indices)

Stocks or equity indices may generate dividend yields, which significantly impact option prices. In the Black-Scholes formula, this is adjusted by introducing a dividend yield ( $q$ ).

3.1 European Equity Option Formula

Call Option Pricing Formula:
$C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)$
Put Option Pricing Formula:
$P = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)$

3.2 Parameter Definitions

$q$ : Dividend yield (or index yield), representing the yield from holding the underlying asset;
Other parameters ( $S_0, K, T, \sigma, r$ ) are the same as in the standard formula;
$d_1$ and $d_2$ are defined as:
$d_1 = \frac{\ln(S_0 / K) + (r - q + \frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}.$

3.3 Adjustment Logic

Dividend Yield ( $q$ ):
- Investors holding stocks or indices receive periodic dividends, reducing the present value of the stock. Thus, the discount factor $e^{-qT}$ is introduced.
Discount Factor:
- $S_0 e^{-qT}$ represents the present value of the underlying asset, adjusted for future dividends.

3.4 Explanation

The introduction of the dividend yield $q$ allows the formula to adapt to pricing options on dividend-paying assets. For non-dividend-paying stocks or indices, set $q = 0$ , and the formula reverts to the standard Black-Scholes formula.

4. Black-Scholes Model for Commodity Options

Commodity options require consideration of two factors: storage costs ( $c$ ) and convenience yield ( $y$ ). These jointly affect the spot price and futures price of the commodity.

4.1 European Commodity Option Formula

Call Option Pricing Formula:
$C = S_0 e^{-(q + c - y)T} N(d_1) - K e^{-rT} N(d_2)$
Put Option Pricing Formula:
$P = K e^{-rT} N(-d_2) - S_0 e^{-(q + c - y)T} N(-d_1)$

4.2 Parameter Definitions

$q$ : Holding yield (e.g., dividend yield; if no dividends, $q = 0$ );
$c$ : Storage cost, representing the cost of holding the commodity (e.g., warehousing fees);
$y$ : Convenience yield, representing the benefit of holding the commodity (e.g., ensuring stable production);
Other parameters ( $S_0, K, T, \sigma, r$ ) are the same as in the standard formula;
$d_1$ and $d_2$ are defined as:
$d_1 = \frac{\ln(S_0 / K) + (r - q - c + y + \frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}.$

4.3 Adjustment Logic

Storage Cost ( $c$ ):
- Holding commodities incurs storage costs, so $c$ is added to adjust the present value of the underlying asset.
Convenience Yield ( $y$ ):
- Holding commodities may provide convenience yield (e.g., ensuring stable supply for production), which reduces holding costs.
Net Holding Cost:
- The adjusted holding cost is $q + c - y$ , affecting the present value of the commodity.

4.4 Explanation

Commodity option prices are significantly influenced by net holding costs. For different commodities (e.g., energy, precious metals, agricultural products), the magnitude of storage costs and convenience yield varies, requiring calibration based on specific market conditions.

5. Summary: Variations of the Black-Scholes Formula

Asset Class	Holding Yield ( $q$ )	Adjustment Factors	Formula Characteristics
Stocks/Equities	Dividend Yield ( $q$ )	Dividend yield reduces present value: $e^{-qT}$	Dividends lower the present value of the underlying asset, affecting option prices.
Foreign Exchange	Foreign Interest Rate ( $r_f$ )	Interest rate differential: $r_d - r_f$	Foreign interest rate acts as a yield, domestic interest rate as a discount factor.
Commodities	$q + c - y$ (Net Holding Cost)	Storage cost ( $c$ ), convenience yield ( $y$ )	Storage costs increase holding costs, convenience yield reduces holding costs, affecting the present value of the commodity.

The core idea of the Black-Scholes model remains unchanged, but by adjusting the holding yield (e.g., dividend yield, foreign interest rate, net holding cost) and discount factors, it can adapt to the pricing needs of different asset classes such as foreign exchange, equities, and commodities. These adjustments make the Black-Scholes formula more versatile and widely applicable across financial markets.

Below is a Python implementation of a simple European currency option:

import math

def bs_call(s, x, t, rd, rf, sigma):
    """
    Calculate the price of a European currency call option.

    Args:
        s: Spot price
        x: Strike price
        t: Time to maturity
        rd: Domestic risk-free rate
        rf: Foreign risk-free rate
        sigma: Volatility

    Returns:
        Option price
    """

    d1 = (math.log(s / x) + (rd - rf + sigma ** 2 / 2) * t) / (sigma * math.sqrt(t))
    d2 = d1 - sigma * math.sqrt(t)
    return s * math.exp(-rf * t) * math.norm(d1) - x * math.exp(-rd * t) * math.norm(d2)


def bs_put(s, x, t, rd, rf, sigma):
    """
    Calculate the price of a European currency put option.

    Args:
        s: Spot price
        x: Strike price
        t: Time to maturity
        rd: Domestic risk-free rate
        rf: Foreign risk-free rate
        sigma: Volatility

    Returns:
        Option price
    """

    return bs_call(s, x, t, rd, rf, sigma) - s + x

Example: Consider a six-month European dollar call/euro put option. The USD/EUR exchange rate is 1.56, the strike price is 1.6, the European risk-free rate is 8% per annum, the US foreign risk-free rate is 6% per annum, and the volatility is 12% per annum. $S=1.56,X=1.6,T=0.5,r=0.06,r_f=0.08,\sigma=0.12.$

d_{1}=\frac{\ln(1.56/1.6)+(0.06-0.08+0.12^{2}/2)0.5}{0.12\sqrt{0.5}}=-0.3738

d_{2}=d_{1}-0.12\sqrt{0.5}=-0.4587

N(d_{1})=N(-0.3738)=0.3543

N(d_{2})=N(-0.4587)=0.3232

c=1.56e^{-0.08\times0.5}N(d_1)-1.6e^{-0.06\times0.5}N(d_2)=0.0291

Thus, the option premium is $0.0291 per euro. Alternatively, the premium can be quoted at a rate of 0.0120 euros per dollar (0.0291/1.56), or as 1.8654% of the spot price in euros. Therefore, if the notional amount of the option is 100 million euros, the total option premium is 1,865,384.62 euros, or 1,865,384.62×1.56=2,910,000,000 dollars.

s = 1.56
x = 1.6
t = 0.5
rd = 0.06
rf = 0.08
sigma = 0.12

c = bs_call(s, x, t, rd, rf, sigma)
print(c)

Put-Call Parity for European Options

Higgins (1902) and Nelson (1904) described the put-call parity relationship, which gives the value of a put option and a call option with the same strike price. If the parity relationship does not hold, arbitrage opportunities exist. This relationship is based on several assumptions, such as the ability to short the underlying asset easily, no bid-ask spreads, and no transaction costs. However, it does not rely on any assumptions about the distribution of the underlying security's price.

Foreign Exchange Options

c=p+Se^{-r_{f}}T-Xe^{-rT}\quad p=c-Se^{-r_{f}}T+Xe^{-rT}

Forward Price

Foreign exchange forward and spot transactions are common in the foreign exchange market. Forward transactions involve the exchange of foreign currency at a predetermined price and date, while spot transactions involve immediate exchange at the current market price.

For forward transactions, there are two common interest rate calculation methods: annual interest rate and continuous compounding. Their formulas are as follows:

Annual Interest Rate Method:
In the annual interest rate method, the forward exchange rate can be calculated using the spot exchange rate, domestic interest rate ( $r_d$ ), and foreign interest rate ( $r_f$ ). Assuming the forward exchange rate is $F$ , the spot exchange rate is $S$ , the domestic interest rate is $r_d$ , the foreign interest rate is $r_f$ , and the forward delivery time is $T$ years, the formula for the forward exchange rate is:

F=S\times(1+r_d)^T\:/\:(1+r_f)^T

where $^{\wedge}$ denotes exponentiation.

Continuous Compounding Method:
In the continuous compounding method, the formula for the forward exchange rate is slightly different. Assuming the forward exchange rate is $F$ , the spot exchange rate is $S$ , the domestic interest rate is $r_d$ , the foreign interest rate is $r_f$ , and the forward delivery time is $T$ years, the formula for the forward exchange rate is:

F=S\times e^{(r_d-r_f)\times T}

where $e$ is the base of the natural logarithm.