###### Standard III (C) – Suitability

1. When Members and Candidates are in an advisory relationship with a client,... **Read More**

The BSM model for pricing options on a non-dividend-paying stock is given by:

$$c_{0}=S_{0}N(d_{1})-e^{-rT}KN(d_{2})$$

$$p_{o}=e^{-rT}KN(-d_{2})-S_{0}N(-d_{1})$$

Where:

$$d_{1}=\frac{ln\bigg(\frac{S}{K}\bigg)+\bigg(r+\frac{1}{2}\sigma^{2}\bigg)T}{\sigma\sqrt{T}}$$

$$d_{2}=d_{1}-\sigma\sqrt{T}$$

\(N(x)=\) standard normal cumulative distribution function

\(N(–x) = 1 – N(x)\)

BSM Model has the following variables:

\(T=\) time to option expiration

\(r=\) continuously compounded risk-free rate

\(S_{0}=\) current share price

\(K=\) exercise price

\(\sigma=\) annual volatility of asset returns

The BSM model can be interpreted as the present value of the expected option payoff at expiration. It can be expressed as:

$$c_{0}=PV(S_{0}e^{rT}N(d_{1})-KN(d_{2}))$$

$$p_{0}=PV(KN(-d_{2})-S_{0}e^{rT}N(-d_{i}))$$

Where the present value factor in this case is \(e^{-rT}\)

Alternatively, the BSM model can be described as having two components, a ** stock component,** and a

The stock component for call options is \(S_{0}N(d_{1})\) while the bond component is \(e^{-rT}KN(d_{2})\). Thus, the BSM model call value is the difference between the stock component and the bond component.

The stock component is \(S_{0}N(d_{1})\) and the bond component is \(e^{rT}KN(-d_{2})\) for put options. Thus, the BSM model put value is the bond component minus the stock component.

An option can be thought of as a dynamically managed portfolio of the underlying stock and zero-coupon bonds. The initial cost of this replicating strategy is given as:

$$\text{Replicating strategy cost}=n_{s}S+n_{B}B$$

The equivalent number of underlying shares is \(n_{S}=N(d_{1})>0\). \(n_{S}\) greater than 0 implies that we are buying the stock. On the other hand, the equivalent number of bonds is \(n_{B}=-N(d_{2})<0\). \(n_{B}\) less than 0 implies that we are selling the bond. Selling a bond is the same as borrowing money. Thus, a call option can be viewed as a* leveraged *position in the stock where \(N(d_{1})\) units of shares are purchased using \(e^{-rT}KN(d_{2})\) of borrowed money.

The equivalent number of underlying shares is \(n_{S}=-N(-d_{1})<0\). This can be interpreted as selling the shares of the underlying stock as \(n_{s}<0\). Further, the equivalent number of bonds is \(n_{B}=-N(-d_{2})>0\). The bond is being bought here as \(n_{B}\) is greater than 0. Buying a bond similar to lending money. Therefore, a put can be viewed as buying a bond where this purchase is partially financed by short selling the underlying stock.

Consider the following information relating to call and put options on an underlying stock

\(S_{0}=48\)

\(K=40\)

\(r=2.5\%\) (Continuously compounded)

\(T=2\)

\(\sigma=30\%\)

Current market price of call option = 14

Current market price of put option = 3

The following values have been calculated using the above information

$$PV(K)=40\times e^{-0.025\times2}=38.05$$

$$d_{1}=0.7597$$

$$d_{2}=0.3354$$

$$N(d_{1})=0.7763$$

$$N(d_{2})=0.6314$$

We can determine the replicating strategy cost and arbitrage profits on both options as follows:

According to the no-arbitrage approach to replicating the call option, a trader can:

Purchase \(n_{s}=N(d_{1})=0.7763\) shares of stock by borrowing \(n_{B}=-N(d_{2})=0.6314\) shares of zero-coupon bonds priced at \(B=ke^{-rT}=$38.05\) per bond.

$$\text{Replicating strategy cost}=n_{S}S+n_{B}B$$

$$\text{Replicating strategy cost}=0.7763\times48+(-0.6314\times38.05)=$13.24$$

An arbitrage profit can be realized on the call option by writing a call at the current market price of $14 and purchasing a replicating portfolio for $13.24.

Thus,

$$\text{Arbitrage profit}=$14-$13.24=$0.76$$

For the put option, we have:

$$N(-d_{1})=1-N(d_{1})=1-0.7763=0.2237$$

$$N(-d_{2})=1-0.6314=0.3686$$

The no-arbitrage approach to replicating the put option involves:

Purchasing \(n_{B}=N(-d_{2})=0.3686\) shares of zero-coupon bonds priced at \(40e^{-0.025\times2}=$38.05\) per bond and short-selling \(n_{S}=-N(-d_{1})=-0.2237\) shares of stock resulting in short proceeds of \($48\times0.2237=$10.74\)

Thus, the replicating strategy cost for the put option is:

$$\text{Replicating strategy cost}=-0.2237\times$48+0.3686\times$38.05=$3.29$$

The trader can exploit arbitrage profits by selling the replicating portfolio and purchasing puts for an arbitrage profit of $0.29 per put.

i.e., \(\text{Arbitrage profit}=$3.29-$3=$0.29\)

## Question

Common stock is currently trading at $50. A call option is written on it with an exercise price of $45. Further, the continuously compounded risk-free rate of interest is 4%, the interest rate volatility is 30%, and the time to the option expiry is 2 Years. Using the BSM model, the following components have been calculated:

\(PV(K)=$41.54\)

\(d_{1}=0.6490\)

\(d_{2}=0.2248\)

\(N(d_{1})=0.7418\)

\(N(d_{2})=0.5889\)

\(c_{0}=12.63\)

The value of the replicating portfolio is

closest to:

- $9.57
- $11.94
- $12.63
## Solution

The correct answer is C:The no-arbitrage approach to replicating the call option involves purchasing \(n_{s}=N(d_{1})=\text{0.7418 shares}\) of stock partially financed with \(n_{B}=-N(d_{2})=-0.5889\) shares of zero-coupon bonds priced at \(B=Ke^{rT}=\text{\$41.54 per bond}\).

Cost of replicating portfolio \(=n_{S}S+n_{B}B\)

$$c=0.7418(50)+(-0.5889)41.54=$12.63$$

*Reading 38: Valuation of Contingent Claims*

*LOS 38 (g). interpret the components of the Black–Scholes–Merton model as applied to call options in terms of a leveraged position in the underlying; *