Reconocimiento en el mercado de trabajo y el sistema educativo

Since in typical situations it is not difficult to find a proper form of the call-put parity, we shall usually restrict our attention to the case of a call option. In some circumstances, it will be convenient to explicitly account for the dependence of the option’s price on its strike price K, as well as on the parameters r and σ of the model. For this reason, we shall sometimes write Ct= c(St, T− t, K, r, σ) and Pt= p(St, T − t, K, r, σ) in what follows.

3.2.5 The Black-Scholes PDE

Suppose we are given a Borel-measurable function g : R → R. Then we have the following re-sult, which generalizes Theorem 3.2.1. Let us observe that the problem of attainability of any P^∗-integrable European contingent claim can be resolved by invoking the predictable representa-tion property (completeness of the multidimensional Black-Scholes model is examined later in this chapter.

Corollary 3.2.3 Let g : R → R be a Borel-measurable function, such that the random variable X = g(S_T) is integrable under P^∗. Then the arbitrage price in MBS of the claim X which settles at time T is given by the equality π_t(X) = v(S_t, t), where the function v : R₊× [0, T ] → R solves the Black-Scholes partial differential equation

∂v

∂t +1

2σ²s²∂²v

∂s² + rs∂v

∂s− rv = 0, ∀ (s, t) ∈ (0, ∞) × (0, T ), (3.68) subject to the terminal condition v(s, T ) = g(s).

3.2.6 Sensitivity Analysis

We will now examine the basic features of trading portfolios involving options. Let us start by introducing the terminology widely used in relation to option contracts. We say that at a given instant t before or at expiry, a call option is in-the-money and out-of-the-money if St > K and St < K, respectively. Similarly, a put option is said to be in-the-money and out-of-the-money at time t when St < K and St> K, respectively. Finally, when St= K both options are said to be at-the-money. The intrinsic values of a call and a put options are defined by the formulae

I_t^C= (St− K)⁺, I_t^P = (K− St)⁺, (3.69) respectively, and the time values equal

J_t^C= Ct− (St− K)⁺, J_t^P = Pt− (K − St)⁺, (3.70) for t∈ [0, T ]. It is thus evident that an option is in-the-money if and only if its intrinsic value is strictly positive. A short position in a call option is referred to as a covered call if the writer of the option hedges his or her risk exposure by holding the underlying stock; in the opposite case, the position is known as a naked call. When an investor who holds a stock also purchases a put option on this stock as a protection against stock price decline, the position is referred to as a protective put. While writing covered calls truncates, roughly speaking, the right-hand side of the return distribution and simultaneously shifts it to the right, buying protective puts truncates the left-hand side of the return distribution and at the same time shifts the distribution to the left. The last effect is due to the fact that the cost of a put increases the initial investment of a

portfolio. Note that the traditional mean-variance analysis pioneered in Markowitz (1952)⁶ is not an appropriate performance measure for portfolios containing options because of the skewness that may be introduced into portfolio returns. For more details on the effectiveness of option portfolio management, the interested reader may consult Leland (1980), Bookstaber (1981), and Bookstaber and Clarke (1984, 1985).

To measure quantitatively the influence of an option’s position on a given portfolio of financial assets, we will now examine the dependence of its price on the fluctuations of the current stock price, time to expiry, strike price, and other relevant parameters. For a fixed expiry date T and arbitrary t≤ T, we denote by τ the time to option expiry – that is, we put τ = T − t. We write p(St, τ, K, r, σ) and c(St, τ, K, r, σ) to denote the price of a call and a put option, respectively. The functions c and p are thus given by the formulae

c(s, τ, K, r, σ) = sN (d₁)− Ke^−rτN (d₂) (3.71) and

p(s, τ, K, r, σ) = Ke^−rτN (−d2)− sN(−d1), (3.72) where

d_1,2= d_1,2(s, τ, K, σ, r) = ln(s/K) + (r±¹₂σ²)τ σ√

τ .

Recall that at any time t∈ [0, T ], the replicating portfolio of a call option involves αtshares of stock and β_tunits of borrowed funds, where

α_t= c_s(S_t, τ ) = N

d₁(S_t, τ )

, β_t= c(S_t, τ )− αtS_t. (3.73) The strictly positive number αt, which determines the number of shares in the replicating portfolio, is commonly referred to as the hedge ratio or, briefly, the delta of the option.

It is not hard to verify by straightforward calculations that c_s = N (d₁) = δ > 0,

c_ss = n(d₁) sσ√

τ = γ > 0, cτ = sσ

2√

τ n(d1) + Kre^−rτN (d2) = θ > 0, c_σ = s√

τ n(d₁) = λ > 0, cr = τ Ke^−rτN (d2) = ρ > 0, c_K = −e^−rτN (d₂) < 0,

where n stands for the standard Gaussian probability density function – that is n(x) = 1

√2πe^−x2/2, ∀ x ∈ R.

Similarly, in the case of a put option we get

p_s = N (d₁)− 1 = −N(−d1) = δ < 0, p_ss = n(d1)

sσ√

τ = γ > 0, pτ = sσ

2√

τn(d1) + Kre^−rτ(N (d2)− 1) = θ, p_σ = s√

τ n(d₁) = λ > 0,

pr = τ Ke^−rτ(N (d2)− 1) = ρ < 0, pK = e^−rτ(1− N(d2)) > 0.

6The more recent literature include Markowitz (1987),Huang and Litzenberger (1988),and Elton and Gruber (1995).

3.2. THE BLACK-SCHOLES OPTION VALUATION FORMULA 61 Consequently, the delta of a long position in a put option is a strictly negative number (equivalently, the price of a put option is a strictly decreasing function of a stock price). Generally speaking, the price of a put moves in the same direction as a short position in the asset. In particular, in order to hedge a written put option, an investor needs to short a certain number of shares of the underlying stock. Another useful coefficient which measures the relative change of an option’s price as the stock price moves is the elasticity. For any date t≤ T, the elasticity of a call option is given by the equality

η^c_t= cs(St, τ )St/Ct= N

d1(St, τ ) St/Ct, and for a put option it equals

η_t^p= ps(St, τ )St/Pt=−N

−d1(St, τ ) St/Pt.

Let us check that the elasticity of a call option price is always greater than 1. Indeed, for every t∈ [0, T ], we have

η_t^c= 1 + e^−rτKC_t⁻¹N

d₂(S_t, τ )

> 1.

This implies also that C_t− cs(S_t, τ )S_t< 0, so that the replicating portfolio of a call option always involves the borrowing of funds. Similarly, the elasticity of a put option satisfies

η_t^p= 1− Ke^−rτP_t⁻¹N

−d2(St, τ )

< 1.

This in turn implies that P_t− Stp_s(S_t, τ ) > 0(this inequality is obvious anyway) and thus the replicating portfolio of a short put option generates funds which are invested in risk-free bonds.

These properties of replicating portfolios have special consequences when the assumption that the borrowing and lending rates coincide is relaxed. It is instructive to determine the dynamics of the option price C. Using Itˆo’s formula, one finds easily that under the martingale measure P^∗we have

dC_t= rC_tdt + σC_tη_t^cdW_t^∗.

This shows that the appreciation rate of the option price in a neutral economy equals the risk-free rate r; however, the volatility coefficient equals ση^c_t, so that, in contrast to the stock price volatility, the volatility of the option price follows a stochastic process.

The position delta is obtained by multiplying the face value⁷ of the option position by its delta.

Clearly, the position delta of a long call option (or a short put option) is positive; on the contrary, the position delta of a short call option (and of a long put option) is a negative number. The position delta of a portfolio is obtained by summing up the position deltas of its components. In this context, let us make the trivial observation that the position delta of a long stock equals 1, and that of a short stock is−1. It should be stressed that the option’s (or option portfolio’s) position delta measures only the market exposure at the current price levels of underlying assets. More precisely, it gives the first order approximation of the change in option price, which is sufficiently accurate only for a small move in the underlying asset price. To measure the change in the option delta as the underlying asset price moves, one should use the second derivative with respect to s of the option’s price – that is, the option’s gamma. The gamma effect means that position deltas also move as asset prices fluctuate, so that predictions of revaluation profit and loss based on position deltas are not sufficiently accurate, except for small moves. It is easily seen that bought options have positive gammas, while sold options have negative gammas. A portfolio’s gamma is the weighted sum of its options’ gammas, and the resulting gamma is determined by the dominant options in the portfolio. In this regard, options close to the money with a short time to expiry have a dominant influence on the portfolio’s gamma. Generally speaking, a portfolio with a positive gamma is more attractive than a negative gamma portfolio. Recall that by theta we have denoted the derivative of the option price with respect to time to expiry. Generally, a portfolio dominated by bought options will have a negative theta, meaning that the portfolio will lose value as time passes (other variables

7The face value equals the number of underlying assets,e.g.,the face value of an option on a lot of 100 shares of stock equals 100.

held constant). In contrast, short options generally have positive thetas. The derivative of the option price with respect to volatility is known as the vega of an option. A positive vega position will result in profits from increases in volatility; similarly, a negative vega means a strategy will profit from falling volatility.

Example 3.2.1 Consider a call option on a stock S, with strike price $30and with 3 months to expiry. Suppose, in addition, that the current stock price equals $31, the stock price volatility is σ = 10% per annum, and the risk-free interest rate is r = 5% per annum with continuous compounding.

We may assume, without loss of generality, that t = 0and T = 0.25. Using (3.43), we obtain (approximately) d₁(S₀, T ) = 0.93, and thus d₂(S₀, T ) = d₁(S₀, T )−σ√

T = 0.88. Consequently, using formula (3.42) and the following values of the standard Gaussian probability distribution function:

N (0.93) = 0.8238 and N (0.88) = 0.8106, we find that (approximately) C₀ = 1.52, φ¹₀ = 0.82 and φ²₀ =−23.9. This means that to hedge a short position in the call option, which was sold at the arbitrage price C0 = $1.52, an investor needs to purchase at time 0the number δ = 0.82 shares of stock (this transaction requires an additional borrowing of 23.9 units of cash). The elasticity at time 0of the call option price with respect to the stock price equals

η^c₀= N

d₁(S₀, T ) S₀ C0

= 16.72.

Suppose that the stock price rises immediately from $31 to $31.2, yielding a return rate of 0.65% flat.

Then the option price will move by approximately 16.5 cents from $1.52 to $1.685, giving a return rate of 10.86% flat. Roughly speaking, the option has nearly 17 times the return rate of the stock;

of course, this also means that it will drop 17 times as fast. If an investor’s portfolio involves 5 long call options (each on a round lot of 100 shares of stock), the position delta equals 500× 0.82 = 410, so that it is the same as for a portfolio involving 410shares of the underlying stock. Let us now assume that an option is a put. The price of a put option at time 0equals (alternatively, P0 can be found from the put-call parity (3.66))

P₀= 30 e^−0.05/4N (−0.88) − 31 N(−0.93) = 0.15.

The hedge ratio corresponding to a short position in the put option equals approximately δ =−0.18 (since N (−0.93) = 0.18), therefore to hedge the exposure, using the Black-Scholes recipe, an investor needs to short 0.18 shares of stock for one put option. The proceeds from the option and share-selling transactions, which amount to $5.73, should be invested in risk-free bonds. Notice that the elasticity of the put option is several times larger than the elasticity of the call option. If the stock price rises immediately from $31 to $31.2, the price of the put option will drop to less than 12 cents.