STOCHASTIC METHODS IN FINANCE 2021–22 STAT0013
Exercises 10: Black-Scholes formula. Risk-Neutral pricing. SDEs.
The Black-Scholes formula for the price of a European call option under the standard assumptions, with strike price K and time to expiry T , is
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S0N(d1) − Ke−rT N(d2),
where N(·) denotes the cumulative distribution function of a standard Nor-
log(S0 ) + r + σ2 T d1= K √ 2
log(S0 ) + r − σ2 T √
d2= K √ 2 =d1−σT σT
The Black-Scholes formula for the price of a European put option with strike price K and time to expiry T , is
Ke−rT N(−d2) − S0N(−d1), where the notation is the same as above.
1. A financial institution sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stock. We assume that the stock price is S0 = 49, the strike price is K = 50, the risk-free inter- est rate is r = 5% per annum, the stock price volatility is σ = 20% per annum and the time to maturity is 20 weeks. Assume that the assumptions of the Black-Scholes model hold.
(a) Find the value of the European call option on the 100,000 shares.
(b) Find the value of a European put option with same strike price and expiration date on the 100,000 shares.
(c) Verify that the put-call parity holds in this case.
2. Assume that the current stock price is £110, the riskless interest rate is 3% per annum and the volatility is 7% per annum. Let S2 be the stock price after two years. What is the value of a financial product that pays S2 − 120 if S2 > 120, 100 − S2 if S2 < 100 and zero otherwise? Assume that all Black-Scholes assumptions hold (including no dividends).
3. For each of the following European-style derivatives with the given payoffs (in £), draw the payoff diagram at maturity and find the price of the derivative. In each case assume that the Black-Scholes assumptions hold, that all derivatives have maturity dates 1 year from now and that the underlying asset price process follows the stochastic differential equation
dSt = μStdt + σStdBt,
with parameters μ = 0.1 and σ = 0.4. Assume also that the underlying
asset is currently priced at £4 and that the risk-free rate is 0%.
(a) Apayoffof0ifST ≤4,andapayoffofST −4ifST ≥4.
(b) Apayoffof0.2ifST ≤4,apayoffofST −4if4≤ST ≤5,anda payoffof1ifST ≥5.
4. A stock price St follows the usual model dSt = μStdt + σStdBt with expected return μ = 0.16 and a volatility σ = 0.35. The current price is £38.
(a) What is the probability that a European call option on the stock with an exercise price of £40 and a maturity date in six months
will be exercised? Using the Black- Scholes formula, find the price of the call option if the risk-free interest rate is μ.
(b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised? Find the price of the put option, using put-call parity.
5. Assume that {Bt}t≥0 is a standard Brownian motion. Let Xt = B2t − Bt
Is Xt a Brownian motion? Explain briefly your answer.
6. Assume that {Bt}t≥0 is a standard Brownian motion.
(a) Let Xt = 2Bt/4. Show that Xt is a Brownian motion. Explain briefly your answer.
(b) Compute for 0 < s < t the covariance: cov(B3t − tB2t, Bs)
(c) Recall that sinh(x) = ex−e−x , x ∈ R. For t > s, compute: 2
E [e−t/2 sinh(Bt )|Bs ]
CIR model for a security with price Xt is determined via the
following SDE:
dXt = r(θ − Xt)dt + σXtdBt
(a) What is the interpretation of the parameters r, θ, σ?
(b) Find a function f so that the process Yt = f(Xt) satisfies an SDE with unit diffusion coefficient.
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