ETF5952 · Quantitative Methods for Risk Analysis
Option Pricing and Option Risk
This chapter of Monash University's ETF5952 Quantitative Methods for Risk Analysis is the derivatives capstone: it turns the return-distribution machinery of the earlier weeks into an option price and a hedge. Starting from the call and put payoffs, you price a one-period binomial option two equivalent ways — replication with the hedge ratio (delta) and risk-neutral valuation — then move to lognormal prices and the Black–Scholes call and put, closing with the comparative statics that say which way value moves as S, K, volatility and maturity change. These no-arbitrage skills are examined by hand in the closed-book final and rehearsed in the R workshops.
What this chapter covers
- 01European payoffs at maturity: call C_T = max{S_T − K, 0}, put P_T = max{K − S_T, 0}, floored at zero
- 02One-period binomial, no-arbitrage: two states s_u, s_d and a risk-free rate r_f
- 03Hedge ratio (delta): x = (C_u − C_d) / (s_u − s_d), the shares that make the portfolio risk-free
- 04Risk-neutral probability q = (S₀ e^{r_f} − s_d)/(s_u − s_d) and price c₀ = e^{−r_f}[q C_u + (1−q) C_d]
- 05Replication and risk-neutral valuation give the SAME no-arbitrage price
- 06Lognormal prices: ln(S_{t+τ}/S_t) ~ N(τμ, τσ²), with the mean correction E[S] = S_t e^{τμ + ½τσ²}
- 07Black–Scholes call c_t = S_t N(d_1) − K e^{−r_f τ} N(d_2), with d_1, d_2 = d_1 − σ√τ
- 08Black–Scholes put p_t = K e^{−r_f τ} N(−d_2) − S_t N(−d_1) and put–call parity
- 09Comparative statics: call ↑ in S, put ↑ in K, BOTH ↑ in volatility σ; maturity τ generally ↑
One-period binomial call priced by hedge ratio and risk-neutral valuation
- +1Terminal payoffs: C_u = max{120 − 105, 0} = 15; C_d = max{90 − 105, 0} = 0.
- +1Hedge ratio (delta): x = (C_u − C_d)/(s_u − s_d) = (15 − 0)/(120 − 90) = 15/30 = 0.5 shares per call.
- +1Risk-neutral probability: q = (S₀ e^{r_f} − s_d)/(s_u − s_d) = (100 × e^{0.03} − 90)/30 = (103.0455 − 90)/30 = 0.4348, so 1 − q = 0.5652.
- +1Risk-neutral price: c₀ = e^{−0.03}[0.4348 × 15 + 0.5652 × 0] = 0.97045 × 6.5227 = 6.33.
- +1Replication cross-check: short 1 call, long 0.5 shares. Payoff up 0.5 × 120 − 15 = 45; down 0.5 × 90 − 0 = 45 — risk-free 45. Today 0.5 × 100 − c₀ = 45 × e^{−0.03} = 43.670.
- +1Solve: 50 − c₀ = 43.670 ⇒ c₀ = 6.33 — identical to the risk-neutral price, confirming no arbitrage.
Key terms
- European call / put
- A call gives the right to buy the stock at strike K at maturity, paying C_T = max{S_T − K, 0}; a put gives the right to sell, paying P_T = max{K − S_T, 0}. Both payoffs are floored at zero, so the buyer's loss is capped at the premium.
- Strike price (K)
- The fixed price at which the option holder may buy (call) or sell (put) the underlying stock at maturity. A call is in the money when S_T > K; a put when S_T < K.
- Hedge ratio (delta)
- The number of shares x = (C_u − C_d)/(s_u − s_d) that, held against a written option, makes the one-period portfolio risk-free (same payoff in both states). In Black–Scholes it equals N(d_1).
- Risk-neutral probability (q)
- The weight q = (S₀ e^{r_f} − s_d)/(s_u − s_d) that makes the discounted stock a fair game. It is not the real probability; under it every asset is discounted at the risk-free rate.
- Risk-neutral valuation
- Pricing an option as the discounted risk-neutral expectation of its payoff: c₀ = e^{−r_f}[q C_u + (1 − q) C_d]. It gives the same no-arbitrage price as building a hedge by replication.
- Lognormal price
- When log returns are i.i.d. normal, ln(S_{t+τ}/S_t) ~ N(τμ, τσ²) and the price S_{t+τ} is lognormal, with expected value E[S_{t+τ}] = S_t e^{τμ + ½τσ²} — note the ½τσ² mean correction.
- Black–Scholes formula
- The continuous-time European option price under lognormal stocks: call c_t = S_t N(d_1) − K e^{−r_f τ} N(d_2), put p_t = K e^{−r_f τ} N(−d_2) − S_t N(−d_1), with d_1 = [ln(S_t/K) + (r_f + ½σ²)τ]/(σ√τ) and d_2 = d_1 − σ√τ.
- Comparative statics
- How an option's value responds to its inputs: a call rises in S and falls in K (a put the reverse), while BOTH calls and puts rise in volatility σ, because the zero floor caps the downside and dispersion helps the open upside.
Option Pricing and Option Risk FAQ
Why do the hedge-ratio (replication) and risk-neutral methods give the same option price?
Because both are the same no-arbitrage argument. Replication builds a portfolio of the stock and a bond that reproduces the option's two possible payoffs; that portfolio must cost what the option costs, or a free lunch exists. Rearranging that price algebraically gives exactly the discounted risk-neutral expectation c₀ = e^{−r_f}[q C_u + (1 − q) C_d]. So the two answers are identical by construction — computing both is a built-in check on your arithmetic.
Why does higher volatility raise the value of BOTH a call and a put?
An option's payoff is floored at zero — the holder can never owe more than the premium. Higher volatility widens the distribution of the terminal stock price, which helps the uncapped side of the payoff (a very high S_T for a call, a very low S_T for a put) while the zero floor caps the other side. More dispersion can only help, so both call and put values rise in σ. This asymmetry is the single most tested comparative-static in the chapter.
Can AI help me with option pricing and option risk in ETF5952?
Yes — Sia can explain the method step by step: how to write the call and put payoffs, how to find the hedge ratio and the risk-neutral probability q, how to price a one-period binomial option two ways and cross-check them, how to build d_1 and d_2 for Black–Scholes, and how to read the comparative-statics signs. It works through practice problems with you and checks your formula and arithmetic, but it does not do graded assessments for you or promise a particular mark — always follow Monash's assessment and academic-integrity rules and confirm details on Moodle.
Exam move
Anchor everything on no-arbitrage: two portfolios with the same future payoff must cost the same today. Learn the two payoffs cold (call max{S_T − K, 0}, put max{K − S_T, 0}), then drill the one-period binomial both ways on the same numbers — hedge ratio x = (C_u − C_d)/(s_u − s_d) and risk-neutral price c₀ = e^{−r_f}[q C_u + (1 − q) C_d] with q = (S₀ e^{r_f} − s_d)/(s_u − s_d) — because agreement between the two is a free self-check. For Black–Scholes, practise building d_1 and d_2 = d_1 − σ√τ carefully (the ½σ² term and the σ√τ subtraction are where marks leak), price the call, get the put from N(−d) or from put–call parity c − p = S − K e^{−r_f τ}, and always run parity as a check. Know the comparative-statics signs — call up in S, put up in K, both up in volatility σ — and state a direction plus one reason. The final exam is worth 40%, is closed book and centrally scheduled in the Monash Semester-1 period (~June 2027 — confirm the exact date and length on Moodle); because the duration is not published in the unit materials, budget your time in proportion to the marks and bank the definitional marks before grinding arithmetic. Assignment 2 is set in the style of the final, so rehearse there.
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