Black-Scholes Model

Study guides: Exam FAM, Exam ALTAM, Exam INV-201

The Black-Scholes Model prices a European option on a non-dividend-paying stock by assuming the stock price follows a geometric Brownian motion with constant volatility $\sigma$ and a constant continuously compounded risk-free rate $r$ ; the price is the risk-neutral expected present value of the option's payoff. The same formula values the guarantees embedded in equity-linked life insurance and annuity contracts.

$c = S_0\,N(d_1) - K e^{-rT} N(d_2)$

$p = K e^{-rT} N(-d_2) - S_0\,N(-d_1)$

$d_1 = \frac{\ln(S_0/K) + \left(r + \tfrac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}$

$d_2 = d_1 - \sigma\sqrt{T}$

$S_0$ is today's stock price, $K$ the strike, $T$ the time to expiry in years and $N(\cdot)$ the standard normal CDF. Under the risk-neutral measure $\ln S_T$ is normal with mean $\ln S_0 + (r - \sigma^2/2)T$ and variance $\sigma^2 T$ , so $S_T$ is lognormal; $N(d_2)$ is the risk-neutral probability that a call finishes in the money
Delta (units of stock in the replicating portfolio): $N(d_1)$ for the call, $-N(-d_1)$ for the put. Valuing at time $t$ replaces $T$ by $T - t$
Call and put satisfy put-call parity, $c - p = S_0 - Ke^{-rT}$ , which is a quick check on any pair of prices
Embedded guarantees (AMLCR): a guaranteed minimum maturity benefit $G$ at time $T$ on a fund $F_T = F_0\,(S_T/S_0)(1-m)^T$ is a put on the fund. Its value at issue is ${}_Tp_x$ times a Black-Scholes put with $F_0(1-m)^T$ in place of $S_0$ and $G$ in place of $K$ , where $m$ is the annual management charge

Pricing a Put and a Call on the Same Stock A non-dividend-paying stock trades at $S_0 = 100$ . The risk-free force of interest is $r = 0.04$ and $\sigma = 0.25$ . Find the Black-Scholes prices and deltas of one-year European call and put options with strike $K = 105$ .
Answer
 $\begin{align*} d_1 &= \frac{\ln(100/105) + (0.04 + 0.03125)(1)}{0.25} \\ &= \frac{-0.04879 + 0.07125}{0.25} \\ &= 0.08984 \end{align*}$ 
So $d_2 = 0.08984 - 0.25 = -0.16016$ , $N(d_1) = 0.53579$ , $N(d_2) = 0.43638$ , and $Ke^{-rT} = 105e^{-0.04} = 100.8829$ .
 $\begin{align*} c &= 100(0.53579) - 100.8829(0.43638) \\ &= 53.579 - 44.023 \\ &= 9.556 \end{align*}$ 
 $\begin{align*} p &= 100.8829(0.56362) - 100(0.46421) \\ &= 56.860 - 46.421 \\ &= 10.439 \end{align*}$ 
Parity check: $c - p = -0.883 = 100 - 100.883$ . The call's delta is $0.536$ and the put's is $-0.464$ : a writer of the put hedges by shorting $0.464$ shares per option.

Cost of a Guaranteed Minimum Maturity Benefit A single premium of $100{,}000$ from a life aged 55 is invested in an equity fund that tracks a stock index. A management charge of $m = 0.25\%$ is deducted from the fund at the start of each year. At the end of 10 years, if the policyholder is alive, the insurer tops the fund up to the premium (a GMMB with $G = 100{,}000$ ). Given ${}_{10}p_{55} = 0.92$ , $r = 0.04$ and $\sigma = 0.20$ , find the value of the guarantee at issue.
Answer The guarantee pays $(G - F_{10})^+$ where $F_{10} = 100{,}000\,(S_{10}/S_0)(0.9975)^{10}$ , so it is a put on a fund worth $F_0(1-m)^{10} = 97{,}527.94$ in today's money, with strike $100{,}000$ .
 $\begin{align*} d_1 &= \frac{\ln(97{,}527.94/100{,}000) + (0.04 + 0.02)(10)}{0.20\sqrt{10}} \\ &= \frac{-0.02503 + 0.6}{0.63246} \\ &= 0.90911 \end{align*}$ 
$d_2 = 0.27665$ , $N(-d_1) = 0.18165$ , $N(-d_2) = 0.39102$ , and $Ge^{-10r} = 67{,}032.00$ .
 $\begin{align*} \text{put} &= 67{,}032.00(0.39102) - 97{,}527.94(0.18165) \\ &= 26{,}211.16 - 17{,}715.68 \\ &= 8{,}495.47 \end{align*}$ 
 $\begin{align*} \text{Value} &= {}_{10}p_{55} \times 8{,}495.47 \\ &= 0.92 \times 8{,}495.47 \\ &= 7{,}815.84 \end{align*}$ 
The guarantee costs about $7.8\%$ of the premium, which the insurer recovers from the management charges or a separate risk charge.

How it appears on the exam

FAM asks for the price of a European call or put on a non-dividend-paying stock, and its delta, from $S_0$ , $K$ , $r$ , $\sigma$ and $T$ ; a companion part often checks or uses put-call parity.

ALTAM asks for the value of a GMMB or GMDB "using the Black-Scholes model": the put is written on the fund net of management charges, then multiplied by a survival probability (GMMB) or summed over the year of death (GMDB).

ALTAM follows on with a delta hedge: the units of stock and the bond holding in the replicating portfolio at issue and after one step, and the hedging error from rebalancing only at discrete times.

Exam traps

Using an annual effective rate for $r$ : Black-Scholes needs the continuously compounded rate, so $i = 5\%$ becomes $r = \ln 1.05 = 0.04879$ .

Writing the GMMB put on $F_0$ instead of $F_0(1-m)^T$ . Leaving out the charges understates the guarantee's cost.

Forgetting the survival probability on an embedded option: the option value is the market price, but the insurer only pays if the life is alive at $T$ (GMMB) or dies before $T$ (GMDB).

Sign slips in the put: it is $N(-d_2)$ and $N(-d_1)$ , and the put's delta is negative. A positive put delta means the hedge has been set up in the wrong direction.