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# Components of the BSM Model

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

#### European call

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

#### European Put

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

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

## Interpretation of the BSM Model

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 bond component.

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:

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

## Call Options

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.

## Put options

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.

### Example: Interpreting BSM Model Components

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:

$$\text{Replicating strategy cost}=-0.2237\times48+0.3686\times38.05=3.29$$

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: 1.$9.57
2. $11.94 3.$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;

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